(theory And Applications Of Transport In Porous Media 32) Viliam Novák, Hana Hlaváčiková - Applied Soil Hydrology-springer International Publishing (2019).pdf

Theory and Applications of Transport in Porous Media

Viliam Novák Hana Hlaváčiková

Applied Soil Hydrology

Theory and Applications of Transport in Porous Media Volume 32

Series editor S. Majid Hassanizadeh, Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands Founding series editor Jacob Bear

More information about this series at http://www.springer.com/series/6612

Viliam Novák Hana Hlaváčiková •

Applied Soil Hydrology

123

Viliam Novák Institute of Hydrology Slovak Academy of Sciences Bratislava, Slovakia

Hana Hlaváčiková Institute of Hydrology Slovak Academy of Sciences Bratislava, Slovakia

ISSN 0924-6118 ISSN 2213-6940 (electronic) Theory and Applications of Transport in Porous Media ISBN 978-3-030-01805-4 ISBN 978-3-030-01806-1 (eBook) https://doi.org/10.1007/978-3-030-01806-1 Library of Congress Control Number: 2018957062 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The water that we are drinking nowadays is the very same as was drunk by dinosaurs millions of years ago. Even the quantity of water on Earth is approximately the same; it is continuously recycled and cleaned by evaporation as a part of the so-called hydrologic cycle powered by the energy of the Sun. The number of inhabitants on Earth has increased about 40 times since the Roman era; the demand for freshwater has grown even more than proportionally since then to cover mankind’s material and hygienic needs. Precipitation is the basic source of freshwater. Rainwater is transformed by the Earth’s ‘skin’, represented by a relatively thin (up to 2 m) layer of soil which divides into the surface and subsurface bodies of water; these are the sources of water for communal and industrial use. The majority of freshwater is consumed simultaneously by plant-canopy transpiration and biomass production. Therefore, soil can be denoted as the basic hydrological interface, transforming rainwater to soil and groundwater, as well as to surface run-off, and thus creating the structure of the land’s water balance. Soil hydrology is the science focused on understanding the role of soil in the Earth’s water cycle and expressing its function in the water cycle in quantitative and qualitative terms. The aim of this book is to present and quantify the role of soil in the hydrologic cycle. It is our intention to do this as simply as possible with minimum mathematics, but without neglecting the complexity of the processes. The book is focused on the physical interpretation of soil-water transport processes, as a part of the soil– plant–atmosphere continuum (SPAC). The basic phenomena and characteristics of soil hydrology as soil-water potential, soil-water retention, soil hydraulic conductivity, and methods for their measurement and calculation are described in detail, to be used as input data for simulation models of soil-water movement. This book describes the basic topics of soil hydrology. However, in addition we presented also results of actual research such as water movement in water-repellent soils, water transport and retention of stony soils, evapotranspiration, and basic principles of soil-water-flow modelling. Illustrations are mostly derived from the results of our original research. v

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This book is intended to serve as a source of information not only for hydrologists, but also for agronomists, foresters, environmental protectionists, meteorologists, and for anybody who is interested in the applications of soil hydrology to water-resource formation and management to preserve water resources in adequate quantitative and qualitative conditions for future generations. Bratislava, Slovakia

Viliam Novák Hana Hlaváčiková

Acknowledgements

The authors would like to acknowledge the Slovak National Grant Agencies VEGA Project No. 2/0055/15 and APVV Project No. 15-0497, as well as the Project ITMS 26240120004 ‘Centre of excellence for integrated flood protection of land’ supported by the Research and Development Operational Program funded by the European Regional Development Fund (ERDF) for supporting their scientific work. Some equipment that provided data for this book was obtained within the framework of the Project ITMS 26220120062 ‘Centre of excellence for the Integrated River Basin Management in the Changing Environmental Conditions (CEIMP)’ supported by the Research and Development Operational Program funded by ERDF.

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Contents

1

2

3

Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Soil Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Conceptual Approaches in Soil Hydrology . 1.2 Soil and Its Role in SPAC . . . . . . . . . . . . . . . . . . . . 1.3 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Water Vapour . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Carbon Dioxide . . . . . . . . . . . . . . . . . . . . 1.4 Plant Canopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Basic Physical Characteristics of Soils . . . . . . . . . . . . 2.1 Porous Media, Capillary-Porous Media and Soil. Are There Differences? . . . . . . . . . . . . . . . . . . . 2.2 Soil as a Three-Phase System . . . . . . . . . . . . . . 2.3 Pedogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Soil Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Soil-Texture Analysis . . . . . . . . . . . . . 2.5 Soil Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Rock Fragments in Soil . . . . . . . . . . . . . . . . . . . 2.7 Clay Minerals . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Soil Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Chemical and Mineralogical Properties of Soil . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Physical Properties of Water . . . . . . . . . . 3.1 Soil Water and Soil Solution . . . . . 3.2 Air in Water and Its Solubility . . . . 3.3 Water Density, Compressibility and

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3.4 Water-Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Viscosity of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Soil-Water Interface Phenomena . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Wetting of the Solid-Phase by Liquids . . . . . . . . . . . . . . . 4.2 Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Soil as a Bundle of Capillary Tubes . . . . . . . . . . 4.2.2 Height of Capillary Rise . . . . . . . . . . . . . . . . . . 4.2.3 Decrease of Water-Vapour Pressure in Capillary Tubes Above the Menisci . . . . . . . . . . . . . . . . . 4.2.4 Capillary Phenomena and Their Influence on Soil-Water Retention . . . . . . . . . . . . . . . . . . 4.3 Adsorption and Desorption of Water on Soil Surfaces . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Soil-Water Content and Its Measurement . . . . . . . . . . . . . . . . . 5.1 Quantitative Expression of Soil-Water Content . . . . . . . . . 5.2 Measurement of Soil-Water Content . . . . . . . . . . . . . . . . . 5.2.1 Gravimetric Method for Measurement of Soil-Water Content . . . . . . . . . . . . . . . . . . . . 5.2.2 Soil-Moisture Probes . . . . . . . . . . . . . . . . . . . . . 5.2.3 Neutron Method . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Capacitance Method . . . . . . . . . . . . . . . . . . . . . 5.2.5 Electrical Resistance Method . . . . . . . . . . . . . . . 5.2.6 Time-Domain Reflectometry (TDR) Method . . . . 5.2.7 Frequency-Domain Reflectometry (FDR) Method 5.2.8 Geophysical Methods . . . . . . . . . . . . . . . . . . . . 5.2.9 Remote-Sensing Methods . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soil-Water Potential and Its Measurement . . . . . . . . . . . . . . . . 6.1 Energy of Soil Water . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Soil Water Potential of a Capillary Porous Medium (Soil) . 6.3 Quantitative Expression of Soil-Water Potential . . . . . . . . 6.4 Components of the Total Soil-Water Potential . . . . . . . . . 6.4.1 Gravitational Component of the Total Soil-Water Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Matric (Moisture) Potential of Soil Water . . . . . . 6.4.3 Pneumatic (Pressure) Soil-Water Potential . . . . . 6.5 Total Soil-Water Potential . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Measurement of Soil-Water Potential Components . . . . . .

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6.6.1 Piezometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Tensiometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 74 76

Soil-Water Retention Curve . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Definition of the Soil-Water Retention Curve . . . . . . . . . 7.2 Hysteresis of Soil-Water Retention Curves . . . . . . . . . . . 7.2.1 Main Branches of Soil-Water Retention Curves 7.3 Soil-Water Retention Curves and Their Analytical Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Hydrolimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Measurement of Soil-Water Retention Curves . . . . . . . . . 7.5.1 Tension Methods . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Pressure Methods . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Combined Method for SWRC Estimation . . . . . 7.5.4 Psychrometric Method . . . . . . . . . . . . . . . . . . 7.5.5 Adsorption and Desorption Methods . . . . . . . . 7.6 Estimation of Soil-Water Retention Curves from Pressure-Chamber Data . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soil-Water Movement in Water-Saturated Capillary Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Two Concepts of the Quantification of Soil-Water Movement in Saturated Capillary Porous Media (Soils) . . 8.1.1 Darcian (Macroscopic) Approach . . . . . . . . . . . 8.1.2 Model Porous Medium . . . . . . . . . . . . . . . . . . 8.2 Darcy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Macroscopic (Darcian) and Porous (Real) Water-Flow Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Distribution of Pressure During Water Flow Through Various Measuring Devices . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Horizontal Soil Sample . . . . . . . . . . . . . . . . . . 8.4.2 Vertically Positioned Soil Sample, Downward Water Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Vertically Positioned Soil Sample, Water Flows Bottom-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Hydraulic Conductivity of Soil Saturated with Water Measured in the Laboratory . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Simple Laboratory Method for Measurement of Saturated-Soil Hydraulic Conductivity . . . . . 8.6 Hydraulic Conductivity and Properties of Soil . . . . . . . . 8.6.1 Darcian and Non-Darcian Flow . . . . . . . . . . . . 8.6.2 Air Entrapped in Soil Pores . . . . . . . . . . . . . . .

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8.6.3

Temperature and Saturated-Soil Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4 Flow of Water in Layered Saturated Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Field Measurement of Saturated-Soil Hydraulic Conductivity Above Groundwater Table in the Field . . . 8.7.1 Measurement of Saturated-Soil Hydraulic Conductivity in the Field Above Groundwater Table at Variable Hydraulic Gradient . . . . . . . 8.7.2 Measurement of Saturated-Soil Hydraulic Conductivity in the Field Above Groundwater Table at Constant Hydraulic Gradient . . . . . . 8.8 Pedotransfer Functions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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Water in Unsaturated Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Differences Between Water Movements in Saturated and Unsaturated Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Basic Equations of Water Movement in Unsaturated Soil . . 9.2.1 Darcy–Buckingham Equation . . . . . . . . . . . . . . . 9.2.2 Equation Describing Water Movement in Unsaturated Soil: The Richards Equation . . . . . . . 9.3 Basic Characteristics of Water Flow in an Unsaturated Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Measurement and Calculation of the Unsaturated-Soil Hydraulic Conductivity Function k(hw) . . . . . . . . . . . . . . . . 9.4.1 Measurement of Unsaturated Hydraulic Conductivity of Soils . . . . . . . . . . . . . . . . . . . . . 9.4.2 Unsaturated Hydraulic Conductivity of Soil, Calculated by Analysis of Soil-Water Content and Soil-Water Matric Potential Profiles . . . . . . . . 9.4.3 Calculation of the Function k = f(hw) Using Soil-Water Retention Curve and Saturated-Soil Hydraulic Conductivity . . . . . . . . . . . . . . . . . . . . 9.5 Water Movement at Low Soil-Water Content and Diffusivity of Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Diffusion of Water Vapour in Soil . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Infiltration of Water into Soil . . . . . . . . . . . . . . . . . . . . . . . 10.1 Infiltration into Homogeneous Soil . . . . . . . . . . . . . . . 10.1.1 The Basic Characteristics of Infiltration . . . . 10.2 Infiltration into Soil from Rain or Sprinkling Irrigation 10.3 Ponding Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.4

The Influence of Initial Soil-Water Content and Rain Rate on Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Infiltration and Entrapped Air . . . . . . . . . . . . . . . . . . . . 10.6 Soil-Water Content Profiles During Infiltration . . . . . . . . 10.7 Infiltration Calculation According to Green and Ampt . . . 10.7.1 Horizontal Infiltration . . . . . . . . . . . . . . . . . . . 10.7.2 Vertical Ponding Infiltration . . . . . . . . . . . . . . 10.7.3 Pressure Head at the Infiltration Front . . . . . . . 10.7.4 Ponding-Time Calculation by the Green–Ampt Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Infiltration Curves Expressed by Empirical Equations . . . 10.9 Analytical Expression of Unsteady Infiltration into Homogeneous Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Infiltration into Nonhomogeneous Soil . . . . . . . . . . . . . . 10.10.1 Infiltration into Layered-Soil Profiles . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Redistribution of Water in Homogeneous Soil . . . . . . . . . 11.1 Basic Characteristics of Water Redistribution in Soil . 11.2 Water Movement in Soil During Redistribution . . . . 11.3 Quantitative Analysis of the Redistribution Process . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Interaction of Groundwater and Soil Water . . . . . . . . . . . . . 12.1 Types of Groundwater and Soil-Water Interactions . . . . 12.2 Water-Flow Direction Between Groundwater and an Unsaturated Zone of Soil . . . . . . . . . . . . . . . . . 12.3 Uptake of Water to the Soil from Groundwater . . . . . . . 12.4 Empirical Equations to Calculate Flow of Water from Groundwater to the Soil . . . . . . . . . . . . . . . . . . . 12.5 Flow of Soil Water to Groundwater—Internal Drainage of Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Groundwater Drainage . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Drainage-Design Equation Describing Steady Groundwater Flow to Drains . . . . . . . . . . . . . 12.6.2 Drainage-Design Equation of Unsteady Water Flow to Drains . . . . . . . . . . . . . . . . . . . . . . . 12.7 Drainable and Wettable Porosity . . . . . . . . . . . . . . . . . 12.8 Risk of Soil Salinization by Groundwater and Surface Irrigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Transport of Water in the Soil–Plant–Atmosphere Continuum (SPAC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Evaporation as a Physical Process . . . . . . . . . . . . . . . . . . . 13.2.1 Water Evaporation . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 The Basic Characteristics of Water Evaporation . . 13.2.3 Evaporation from the Water Table and Bare Soil . 13.3 Water Transport in Bare Soil During Evaporation . . . . . . . . 13.3.1 Calculation of Water Movement in Bare Soil During Evaporation . . . . . . . . . . . . . . . . . . . . . . . 13.4 Transpiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Soil-Water Movement During Transpiration . . . . . 13.4.2 Soil-Water Uptake Patterns by Plant Roots . . . . . . 13.4.3 Distribution of Root-Extraction Patterns Evaluated from Field Measurements . . . . . . . . . . . . . . . . . . 13.5 Methods for Evapotranspiration Estimation . . . . . . . . . . . . . 13.5.1 Method of Evapotranspiration Estimation from the Energy-Balance Equation of the Evaporating Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Evaporation Calculation from a Wet Surface: The Penman Equation . . . . . . . . . . . . . . . . . . . . . 13.6 Calculation of Plant-Canopy Potential Evapotranspiration . . 13.6.1 Evapotranspiration Calculation: The Penman–Monteith Equation . . . . . . . . . . . . . . . . . 13.7 Reference Evapotranspiration and Plant-Canopy Evapotranspiration (FAO Method) . . . . . . . . . . . . . . . . . . . 13.7.1 Reference Evapotranspiration . . . . . . . . . . . . . . . . 13.7.2 Plant-Canopy Evapotranspiration . . . . . . . . . . . . . 13.8 Calculation of Actual Evapotranspiration from Potential Evapotranspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 Structure of Potential Evapotranspiration . . . . . . . . . . . . . . 13.10 Daily and Annual Courses of Evapotranspiration and Its Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.11 Evapotranspiration Estimation by the Eddy-Correlation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.12 Calculation of Potential Evapotranspiration by Empirical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.12.1 Linacre Equation . . . . . . . . . . . . . . . . . . . . . . . . 13.12.2 Ivanov Equation . . . . . . . . . . . . . . . . . . . . . . . . . 13.12.3 Tichomirov Equation . . . . . . . . . . . . . . . . . . . . . 13.13 Evaporation from Various Evaporating Surfaces . . . . . . . . . 13.13.1 Evaporation in Slovak Territory . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Transport of Solutes in Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Basic Processes of Solute Transport in Soils . . . . . . . . . . . . 14.2 Concentration of Dissolved Compounds in Solution . . . . . . 14.3 Transport of Dissolved Compounds in Soils . . . . . . . . . . . . 14.3.1 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Hydrodynamic Dispersion . . . . . . . . . . . . . . . . . . 14.4 Equation of Soil-Solution Transport . . . . . . . . . . . . . . . . . . 14.5 Péclet Number and Identification of Transport Mechanisms of Dissolved Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Outflow and Breakthrough Curves . . . . . . . . . . . . . . . . . . . 14.6.1 Outflow Curves . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Breakthrough Curves . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Water and Energy Balance in the Field and Soil-Water Regimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Soil-Water Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Water and Energy Balance of the Land . . . . . . . . . . . . . . 15.3 Soil-Water Regimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Soil-Water Regimen and Its Diagnostics . . . . . . . . . . . . . . 15.4.1 Soil-Water Regimen Classification . . . . . . . . . . . 15.4.2 Hydrological Classification of the Soil-Water Regimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Soil-Water Regimen Diagnostics and Biomass Production . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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236 238 238 239 242

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16 Swelling and Shrinking Soils . . . . . . . . . . . . . . . . . . . . . . . 16.1 Cracks Porosity and Soil-Water Content . . . . . . . . . . 16.2 Specific Volume and Specific Surface of Soil Cracks 16.3 Formation and Kinetics of Soil Cracks . . . . . . . . . . . 16.4 Soil Characteristics Influenced by Soil Swelling and Shrinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Infiltration of Water into Soils with Cracks . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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17 Stony 17.1 17.2 17.3 17.4 17.5 17.6

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Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Features of Stony Soils . . . . . . . . . . . . . . Stony Soils Classification . . . . . . . . . . . . . . . . . . Stony Soils Occurrence . . . . . . . . . . . . . . . . . . . . Representative Elementary Volume of Stony Soils Sampling of Undisturbed Stony Soils . . . . . . . . . . Physical Characteristics of Stony Soils . . . . . . . . .

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17.6.1 Stoniness of Stony Soils . . . . . . . . . . . . . . . . . . 17.6.2 Stony Soils’ Bulk Density . . . . . . . . . . . . . . . . . 17.6.3 Stony Soils’ Porosity . . . . . . . . . . . . . . . . . . . . 17.7 Hydrophysical Characteristics of Stony Soils . . . . . . . . . . 17.7.1 Stony Soils’ Volumetric Water Content . . . . . . . 17.7.2 Stony Soils’ Water Retention . . . . . . . . . . . . . . . 17.7.3 Hydraulic Conductivity of Stony Soils . . . . . . . . 17.7.4 Effective Hydrophysical Characteristics of Stony Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7.5 Soil-Water Content of Stony Soils Measurement 17.8 Water Flow in Stony Soils . . . . . . . . . . . . . . . . . . . . . . . . 17.8.1 Modelling of Water Flow in Stony Soils . . . . . . 17.8.2 More Complex Modelling Approach to Water Flow in Stony Soils . . . . . . . . . . . . . . . . . . . . . 17.8.3 An Example of Measured Characteristics of the Stony Soil and Estimation of Input Data for a Deterministic Simulation Model (a Case Study) . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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267 269 269 270 270 270 272

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273 275 275 276

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283 283 284 284 285 287 287 288

18 Water Repellent Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Water Repellency of Soils and Its Identification . . . 18.2 Characteristics of Soil-Water Repellency . . . . . . . . 18.2.1 Severity of Soil-Water Repellency . . . . . . 18.2.2 Persistence of Soil-Water Repellency . . . . 18.2.3 The Repellency Index . . . . . . . . . . . . . . . 18.3 Water Repellent Compounds in Soils . . . . . . . . . . . 18.4 Water Repellency and Soil-Water Characteristics . . 18.5 The Effects of Soil-Water Repellency on Soil-Water Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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19 Soil Air and Its Dynamics . . . . . . . . . . . . . . . . . . . . . 19.1 Soil Aeration and Plant Respiration . . . . . . . . . 19.2 Composition of Soil Air . . . . . . . . . . . . . . . . . 19.3 Transport of Soil Air . . . . . . . . . . . . . . . . . . . . 19.3.1 Convective Flow of Soil Air . . . . . . . 19.3.2 Diffusion of Soil Air . . . . . . . . . . . . . 19.4 Movement of Oxygen from Soil to Plant Roots References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

20 Soil Temperature and Heat Transport in Soils . . . . . . . . . . . . 20.1 Soil Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1.1 Soil-Temperature Regimen . . . . . . . . . . . . . . . 20.2 Soil-Heat Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Transport of Heat in Soil by Conduction . . . . . 20.3 Soil-Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 Thermal Diffusivity of Soil . . . . . . . . . . . . . . . . . . . . . . 20.5 Soil-Water Transport Under Non-isothermal Conditions . 20.6 Temperature and Water Properties . . . . . . . . . . . . . . . . . 20.6.1 Water Density and Temperature . . . . . . . . . . . . 20.6.2 Water Viscosity and Temperature . . . . . . . . . . 20.6.3 Surface Tension of Water and Temperature . . . 20.7 Hydrophysical Characteristics of Soil and Temperature . . 20.7.1 Hydraulic Conductivity of Saturated Soil and Temperature . . . . . . . . . . . . . . . . . . . . . . . 20.7.2 Soil-Water Retention Curves and Temperature . 20.8 Soil-Water Movement Under Non-isothermal Conditions . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Modelling of Water Flow and Solute Transport in Soil . . . . . 21.1 Modelling in Soil Hydrology . . . . . . . . . . . . . . . . . . . . . 21.2 Governing Equations of Water and Solute Transport in Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Characteristics of the Soil-Plant-Atmosphere Continuum: Model Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Main Model Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Governing Equations and Their Solutions . . . . . . . . . . . . 21.5.1 Initial and Boundary Conditions . . . . . . . . . . . 21.6 Water Flow Modelling in Heterogeneous Soils . . . . . . . . 21.6.1 Dual-Porosity Model . . . . . . . . . . . . . . . . . . . . 21.6.2 Dual-Permeability Model . . . . . . . . . . . . . . . . . 21.7 Calibration, Verification and Validation of Soil-Water Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 Overview of Some Soil-Water Flow Models . . . . . . . . . . 21.9 Current Trends and Future Challenges Concerning Water-Flow Modelling in the Unsaturated Soil Zone . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Symbols

Variables are presented in metric (SI) units. In the text, variables’ dimensions are given even in generalised units: (L length, M mass, T time, and K temperature). a c c(hw) ci cpa cw d de e eo g h ha hb hg hk ho hw i k l

Relative volume of air in soil (aeration) (−) Solute concentration (kg m−3); specific heat capacity of soil (J kg−1 K−1) Specific soil-water capacity (m−1) Specific heat capacity of ice (J kg−1 K−1); concentration of ith component of solute in a unit volume of soil solution (kg m−3) Specific heat capacity of air at constant pressure (J kg−1 K−1) Specific heat capacity of water (J kg−1 K−1) Saturation deficit of air (Pa); particle diameter (m) Effective height of a plant canopy (m) Water-vapour pressure (Pa) Saturated water-vapour pressure (Pa, N m−2) Acceleration of gravity (m s−2) Pressure head (m) Pressure head corresponding to the anaerobiosis point of soil-water content (m) Pressure head corresponding to the entrance pressure of air into soil pores (bubbling pressure) (m) Pressure head of the gravitational component of the total potential of soil water (m) Height of water-capillary rise (m) Thickness of the water layer above the soil surface (m) Matric (moisture) potential of soil water (head with a negative sign) (m) Cumulative (total) infiltration (m) Hydraulic conductivity of soil unsaturated with water, unsaturated conductivity (m s−1) Empirical parameter of van Genuchten’s equation expressing tortuosity and connectivity of pores (−)

xix

xx

lr m md mbd mrfd mw n pa pk q qa qc qd qh qr qs qso r ra rc rs s sp srs t tp u u* v vc vi vp vz w z zk zo zr A Aco C Ci

Symbols

Specific length of roots (m m−3) Shape parameter of van Genuchten’s equation (−) Mass of a dry-soil sample (kg) Mass of dry solid phase of stony soil together with rock fragments (kg) Mass of dry-rock fragments (kg) Mass of water in a soil sample (kg) Shape parameter of van Genuchten’s equation (−) Air pressure (Pa, N m−2) Capillary pressure (Pa, N m−2) Darcian water-flux rate (m s−1), specific humidity of air (kg kg−1) Absolute humidity of air (kg m−3) Rate of soil airflow (kg m−2 s−1) Rate of air diffusion in soil (kg m−2 s−1) Rate of heat flow in soil (W m−2) Rate of oxygen consumption by plant roots (kg m−2 s−1) Specific humidity of air just above the evaporating surface (kg kg−1) Specific humidity of air saturated with water vapour (kg kg−1) Relative humidity of fair (−); radius (m) Aerodynamic resistance of air layer above the evaporating surface (m−1 s) Canopy resistance to water-vapour flow (m−1 s) Stomata resistance to water-vapour flow (m−1 s) Specific soil surface (m2 kg−1) Soil covering; relative covering of soil by a plant canopy (m2 m−2) Specific surface of roots (m2 m−3) Time (s) Ponding time (s) Rate of airflow, wind velocity (m s−1) Friction velocity of air (m s−1) Rate of liquid flow (m s−1) Steady rate of infiltration (m s−1) Infiltration rate (m s−1) Average rate of water flow in soil pores (m s−1) Rain rate (intensity) (m s−1) Soil-water content in mass units (mass soil-water content) (kg kg−1) (vertical) Coordinate (m) Critical depth below soil surface at which water-flow rate to soil from groundwater can be neglected (m) Roughness of surface (m) Root system depth (m) Area (m2) Specific crack area of soil; soil crack area per unit area of soil surface (m2 m−2) Specific volumetric heat capacity of soil (J m−3 K−1) Concentration of ith component of solute in mass units in a unit soil volume (kg m−3)

Symbols

D D(h) Da Das De Dv Dl Dlh Dsl Dwl Doa Dos Dow Dva Dvs E Ee Eep Ep Et Etp ETc ETo F G H Hf I Jw K Kb Kc Kef Kf Kp Krs L Lco Lf Ls

xxi

Turbulent transport coefficient of air (m s−1) Soil-water diffusivity (m2 s−1) Coefficient of molecular diffusion of an air component in air (m2 s−1) Coefficient of molecular diffusion of air in soil (m2 s−1) Effective coefficient of hydrodynamic dispersion of a particular ion in soil (m2 s−1) Water-vapour diffusivity in soil (m2 s−1) Water-vapour diffusivity of liquid water in soil (m2 s−1) Coefficient of hydrodynamic dispersion (m2 s−1) Effective diffusion coefficient of particular ions in soil (m2 s−1) Coefficient of molecular diffusion of particular ions in water (m2 s−1) Coefficient of molecular diffusion of oxygen in air (m2 s−1) Coefficient of molecular diffusion of oxygen in soil (m2 s−1) Coefficient of molecular diffusion of oxygen in water (m2 s−1) Coefficient of molecular diffusion of water vapour in air (m2 s−1) Coefficient of molecular diffusion of water vapour in soil (m2 s−1) Evapotranspiration (mm); evapotranspiration rate (m3m−2 s−1) Evaporation rate (m3 m−2 s−1) Potential evaporation rate (m3 m−2 s−1) Potential evapotranspiration rate (m3 m−2 s−1) Transpiration rate (m3 m−2 s−1) Potential transpiration rate (m3 m−2 s−1) Potential evapotranspiration rate of a particular plant canopy (not reference evapotranspiration) (m3 m−2 s−1) Reference (canopy) evapotranspiration rate (m3 m−2 s−1) Force (capillary, gravity) (N) Heat-flux rate to the soil through a soil surface (W m−2) Turbulent (sensible) heat flux (heat-flux rate from an evaporating surface into the atmosphere) (W m−2; J m−3 s−1); hydraulic head (m) Effective pressure head at the infiltration front (m) Plant-canopy interception (mm; m3 m−2 s−1) Steady rate of water flow (m s−1) Hydraulic conductivity of saturated soil (saturated-soil hydraulic conductivity) (m s−1) Effective hydraulic conductivity of saturated stony soil (m s−1) Crop coefficient (−) Effective hydraulic conductivity of saturated layered soil (m s−1) Hydraulic conductivity of saturated fine earth of stony soil (m s−1) Soil permeability (m2) Relative hydraulic conductivity of saturated stony soil (Kb/Kf) (−) Latent heat of evaporation (liquid water–water vapour) (J kg−1); length (m) Length of cracks per unit soil-surface area (m m−2) Infiltration front depth below soil surface (m) Latent heat of ice sublimation (J kg−1)

xxii

Symbols

Lt LAI O P Pb Pc Pd Pf Pn Prf Q Rl Rm Rn Rs Rv S S(z)

Latent heat of ice melting (J kg−1) Leaf-area index (m2 m−2) Surface run-off, surface-run-off rate (mm; m3 m−2 s−1) Soil porosity (m3 m−3) Effective porosity of stony soil (m3 m−3) Soil crack porosity (m3 m−3) Drainable porosity, specific yield (m3 m−3) Porosity of fine-earth fraction of stony soil (m3 m−3) Wettable porosity (m3 m−3) Rock fragments’ porosity of stony soil (m3 m−3) Discharge or inflow of water (m3 s−1) Long-wave radiation rate (W m−2) Relative mass of rock fragments (stoniness) (kg kg−1) Net radiation (W m−2) Short-wave radiation rate (W m−2) Relative volume of rock fragments (stoniness) (m3 m−3) Sorptivity (m s−½); soil saturation (−) Inflow (outflow) of water into (from) a unit volume of soil per unit time (m3 m−3 s−1) Specific surface of soil cracks (m2 m−3) Effective soil saturation (−) Potential (maximum) rate of water uptake by roots (m3 m−3 s−1) Temperature (°C, K) Volume; soil volume (m3) Volume of air in soil (m3) Total (bulk) volume of stony soil (m3) Volume of cracks in soil (m3) Specific volume of soil cracks (m3 m−3) Volume of fine-earth fraction of stony soil (m3) Rock fragments’ volume in stony soil (m3) Volume of dry solid phase of a soil sample (m3) Volume of water in a soil sample (m3) Volume of water in stony soil (m3) Volume of water in a fine-earth fraction of stony soil (m3) Volume of water in rock fragments of stony soil (m3) Biomass production, yield (t ha−1) Precipitation total; precipitation rate (mm; m3 m−2 s−1) Irrigation-water total; irrigation rate (mm; m3 m−2 s−1) Albedo of surface (reflection coefficient) (−); parameter of van Genuchten’s equation expressing SWRC (s−1) Permittivity (m−3 kg−1 s4 A2) Psychrometric constant (m−1 kg s−2 K−1) Dynamic viscosity of water (m−1 kg s−1) Contact angle, wetting angle (-) Thermal conductivity of soil (W m−1 K−1)

Scs Se Sp T V Va Vb Vc Vcv Vf Vrf Vs Vw Vbw Vfw Vrfw Y Z Zz a e c gw u kðhÞ

Symbols

ka ki kw h hawc hb hf hfc hh hla hmac hr hbr hrf hs hbs hsf hwp qa qad qb qbb qbf qrfb ql qp qr qs qv qw rw s t Cw wt

xxiii

Thermal conductivity of air (W K−1) Thermal conductivity of ice (W K−1) Thermal conductivity of water (W K−1) Volumetric soil-water content (m3 m−3) Plant available water capacity (m3 m−3) Effective (total) volumetric soil-water content of stony soil (m3 m−3) Volumetric soil-water content of stony-soil fine earth (m3 m−3) Volumetric soil-water content corresponding to the field capacity of soil (m3 m−3) Hygroscopicity coefficient (adsorption water capacity) (m3 m−3) Volumetric soil-water content corresponding to the point of limited availability of water to plants (m3 m−3) Monomolecular soil-water capacity (m3 m−3) Residual volumetric soil-water content (m3 m−3) Effective residual volumetric soil-water content of stony soil (m3 m−3) Volumetric soil-water content of stony-soil rock fragments (m3 m−3L3 L−3) Saturated volumetric soil-water content (m3 m−3) Effective volumetric soil-water content of saturated stony soil (m3 m−3) Volumetric soil-water content of a saturated stony-soil fine-earth fraction (m3 m−3) Volumetric soil-water content corresponding to wilting point (m3 m−3) Density of (moist) air (kg m−3) Density of dry air (kg m−3) Density of dry soil (bulk density) (kg m−3) Effective density of dry stony soil (kg m−3) Density of dry fine-earth fraction of stony soil (kg m−3) Density of dry-rock fragments of stony soil (kg m−3) Density of liquid (water) (kg m−3) Density of plant canopy (number of plants per unit of soil-surface area) (m−2) Density of dry roots in soil (kg m−3) Specific density of soil (particle density) (kg m−3) Density (bulk density) of water vapour (kg m−3) Density (bulk density) of liquid water (kg m−3) Surface tension of water (N m−1) Tangential tension (N m−2); tortuosity of pores (−) Kinematic viscosity of water (m2 s−1) Parameter characterising water transport between two domains in dual-porosity model (s−1) Energy of soil water (total soil-water potential) (J kg−1)

Chapter 1

Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

Abstract Soil is a part of the Soil–Plant–Atmosphere Continuum (SPAC); water and energy are transported mostly through all three parts of the SPAC system, but in the case of bare soil or the water table, the SPAC system is reduced to the Soil–Atmosphere Continuum (SAC) or to the Water-Table–Atmosphere Continuum (WAC). This chapter contains basic information about all three subsystems of SPAC. Conceptual approaches to the soil-water and energy transport are defined, as well as the representative elementary volume of soil (REV) needed to measure soil characteristics. The basic characteristics and properties of water (water vapour), soil, plant canopy and atmosphere are presented and their role in water and energy transport in the SPAC system is discussed. Soil water is always a low concentrated solution, but its physical properties are not changed significantly, and therefore it can be assumed to be clean water. The solute concentration is considered to evaluate the increased risk of soil salinization. The role of carbon dioxide in the SPAC system and its role in climate change are discussed.

1.1 Soil Hydrology Soil hydrology is a part of the science of hydrology. The word hydrology is composed of two Greek words: hydros (water) and logos (word, science), thus it is the science concerning water. Hydrology is the science of the movement, distribution and quality of water and its interaction with the environment. The product of the science of hydrology is a system of generalised, systematically organized information that expresses quantitatively the movement of water in nature since it depends on the properties of an environment. The space of the movement of water is the hydrosphere. The hydrosphere is the space on both sides of Earth’s surface consisting of the geoderma and the atmosphere, which contains water in all the three phases. Part of the hydrosphere is the water in oceans, icebergs, rivers, lakes, even the water below Earth’s surface (subsurface water) and water in the atmosphere.

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_1

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2

1 Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

We will focus our attention on the hydrology of land, through that of water in an atmosphere, surface and subsurface water, and to that of water in living organisms (plants and animals). The water content of plants and animals is negligible in comparison to the other parts of hydrosphere, but water transport through plants to the atmosphere (transpiration) is crucial from the point of view of the water-and-energy balance of the land. In biomass production, transpiration is a key process to secure food as a first element of the food chain of the living organisms of Earth. Let us designate the part of the hydrosphere, located just above the land surface, as a part of the atmosphere. The atmosphere is a relatively thin layer just above the Earth’s surface. Its thickness is not constant, but it changes with position above the geoid and changes with time, but its thickness is about 200 km (Chrgijan 1978). Water formations (silver clouds) have been observed 80 km above Earth’s surface. Silver clouds are a mixture of solid particles and ice crystals. Therefore, the height of 80 km above Earth can be assumed as the upper boundary of the hydrosphere. For us, the most important quantity of water contains the part of the atmosphere denoted as the troposphere. It is approximately a 17-km-thick layer of atmosphere. Clouds in the troposphere are the source of precipitation. About the half of the atmospheric mass (and of most water) is located in 0–5-km layer of air. Water in the atmosphere represents only 0.001% of Earth’s total water in the hydrosphere, but its extreme dynamics is responsible for water transport in the environment, as a part of the water cycle. The part of the hydrosphere containing surface and subsurface water is only a few kilometres thick. Rivers, lakes, water reservoirs, icebergs and snow compose the part of the hydrosphere called surface water. It is interesting to note that the “sweet water” of lakes, water reservoirs and wetlands amounts to approximately the same quantity of water as is in the atmosphere in the water-vapour phase, but it is about a hundred times the water content of all the rivers of Earth. The same is true for subsurface water; the Earth’s subsurface contains about 100 times more water than all the rivers of Earth (Brutsaert 1982). The high velocity of water flow in rivers is also an important factor. The relatively small amount of water flowing in the rivers can transport a great quantity of water from land to the oceans due to its relatively high velocity. However, the high rate of water flow (about 100,000-times higher than the groundwater-flow velocity) and small retention capacity of rivers can be responsible for droughts and floods. Therefore, the field of hydrology studying surface water, so-called surface hydrology, is very important for sustaining life on Earth. Subsurface water as a part of hydrosphere can be divided into groundwater and soil water. Groundwater is continuously filling all the pores in the soil and water in pores is of positive pressure (i.e. higher than atmospheric pressure). The top of the groundwater layer is the so-called “free water” table and is at atmospheric pressure. Groundwater, confined from the top by an impermeable layer of soil, can be of higher than atmospheric pressure at the interface between groundwater and the impermeable layer of soil. This groundwater is denoted as artesian water. The part of the hydrosphere with pores filled completely with water is denoted as the space (zone) saturated with water or as the saturated zone.

1.1 Soil Hydrology

3

Soil pores above the groundwater table are only partially filled with water, and part of the porous space is filled with air. This space is denoted as the unsaturated space (zone) of soil, (unsaturated soil, the vadose zone). Air in the soil is crucial for biomass production. The thin, fertile upper layer of Earth is denoted as soil. It is the environment for plants and creates the interface between the atmosphere and pedosphere. Soil transforms mass and energy fluxes between both subsystems. Therefore, the energy and mass fluxes in soil are crucial for life on Earth. To study water and energy transport in soil, specific methods of measurement, quantification and generalisation of transport processes are needed. To study those processes, it is even necessary to apply boundary scientific fields like meteorology, biology, mathematics, physics, geology, hydraulics and plant physiology. Therefore, within the science of hydrology, the particular branch soil hydrology was established. Soil hydrology is the science concerning the retention and transport of water and energy in the soil, as a part of the Soil—Plant—Atmosphere Continuum (SPAC) . The transport of water in the soil and between the soil and atmosphere and soil and groundwater can be divided into individual processes: precipitation and its transformation into intercepted water (water intercepted by plants’ surface), runoff, infiltration and evapotranspiration (evaporation, transpiration). Soil water, extracted by plant roots, is transformed into transpiration; part of the water is accumulated by plant bodies and participates in photosynthesis. Nutrients are transported by water to plants and distributed within plants. Water can enter the soil from groundwater or it can to flow into groundwater if there is abundant water (Fig. 1.1). Groundwater usually flows in quasi-horizontal direction to rivers or creeks; they drain groundwater during low levels of water in the recipients, or they are recharge groundwater during periods of high levels of water table. Quantification of water-and-energy transport in the SPAC system is complicated by the existence of soil as a three-phase system: solid, gaseous and liquid phases. Water can occur simultaneously in all three phases: the solid (ice), liquid and vapour phase since it as the only natural material with continuous phase transformations. The most important consumer of energy on Earth is evapotranspiration. The energy of the Sun increases soil temperature significantly, so the processes of water and energy transport in soil are in principle non-isothermal processes. Another problem in the quantification of such processes is the non-conservative behaviour of plants. This means that plants adapt to changing environmental conditions and change their properties accordingly. Those phenomena will be described later.

1.1.1 Conceptual Approaches in Soil Hydrology The objective of science is to understand processes in the system under study, to garner quantitative information about processes and to generalize them to be able to quantify (numerically) those processes in various conditions. The same approach is used in the science of soil hydrology. The objective is to understand, generalize and quantify the movement of water and dissolved compounds in soils. A specific

4

1 Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

Fig. 1.1 Water transport processes in the soil–plant–atmosphere continuum (SPAC)

feature of soil as a capillary porous medium is the movement of water in small, irregularly distributed and interconnected pores within the reach of surface forces at the interface of the solid–liquid phase. In principle, the movement of soil water can be analysed on several scales: (a) Microscopic scale describing water movement in spatially distributed porous systems. Theoretically, it is possible to formulate the system of transport equations describing water movement in a system of pores and by their solution to calculate water movement in particular pores and, by their integration, to calculate the water transport rate through porous media. This approach (denoted as microscopic) cannot be applied in real soils because information about the geometrical properties of porous systems cannot be evaluated. However, large progress has been made last years in using visualizing techniques like computed tomography combined with image processing and scanning electron microscopy for studying a soil porous system. The advantage of microscopic approach is its clarity and physically sound interpretation of transport and retention phenomena. It is used in “model” soils composed of capillary tubes of various diameters or spheres. Results of such an approach can help us understand even more complicated water/transport processes in real soils. (b) Macroscopic scale, indicated as “Darcian”, is based on estimations of measurable soil characteristics. It was named for Darcy (1856), who used this approach to measure the capacity of sand filter. Typical of the Darcy (or macroscopic) approach is the measurement of the soil characteristics of macroscopic soil sam-

1.1 Soil Hydrology

5

ples. The resulting “macroscopic” soil characteristics are integrative soil properties, characterizing the soil sample as a whole and without particular attention to individual pores. Knowing macroscopic soil characteristics, it is possible to use simple phenomenological (or macroscopic) transport equations describing quantitatively soil/water flow rates. One such equation is the so-called Darcy equation. The term “macroscopic scale” in soil hydrology is related to the so-called “representative elementary volume” (REV) . To measure soil characteristics, the soil sample volume must be large enough to be representative of the particular soil. The simple definition of representative elementary volume is: the REV has to be of such volume of a soil to preserve the value of its measured characteristics (porosity, hydraulic conductivity, bulk density) when increasing the soil-sample volume. On the contrary, decreasing soil volume below REV, soil characteristics do not represent the soil. Soil characteristics measured in this way are strongly influenced by properties of individual pores, or structure nonhomogeneities. Figure 1.2 demonstrates the relationship between soil porosity and soil-sample volume. This can be seen in the case of a soil-sample volume smaller than the volume of pores; then, such soil volume porosity is P  1 [initial section of the relationship P  f (V )]. On the contrary, if the soil sample is filled with the solid phase of a soil only, then P  0 [minimum of the relationship P  f (V )]. When increasing the soil-sample volume, the influence of individual pores and individual soil particles on soil porosity decreases. When reaching the critical volume of a soil stipulated as the representative elementary volume (REV), the soil-sample volume will not influence measured soil-sample characteristics. Therefore, the soil-sample volume on which soil characteristics in macroscopic (Darcian) scale are measured has to be larger (or to be equal) to the REV. To measure the characteristics of relatively homogeneous soil, REV of 100 cm3 (Kopecky cylinder) can be successfully used. But the REV of soils with macropores or stony soils is significantly larger. For the stony soil in the High Tatra Mountains, an REV of 1 m3 must be used (Novák and Kˇnava 2011). (c) Megascopic scale is used to analyse water and energy transport of the catchment (or field) dimension (Kutílek and Nielsen 1994).This scale involves the influence of vertical and horizontal nonhomogeneities on water and energy transport of a relatively large soil area (usually expressed in km2 ); soil depth is expressed in meters. The methods of soil-characteristic measurement remain macroscopic (Darcian). The catchment (or field) can be distributed to pedotops (the smallest pedogeographical units). The macroscopic approach is used to determine soil characteristics and their variability in the vertical and horizontal directions. The variability of soil characteristics in the vertical direction will likely be deterministic, determined by soil horizons properties, but the variability of soil characteristics in horizontal directions is usually stochastic. From the results of individual measurements in the scale of pedotops, representative soil characteristics are valuable information about the soil and can be used as input data for the simulation models of water and energy transport of the area.

6

1 Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

Fig. 1.2 Definition of the representative elementary volume (REV) of a porous medium represented by the relationship between porosity (P) and homogeneous porous medium sample volume (V )

1.2 Soil and Its Role in SPAC The role of soil and its importance for mankind can be understood properly if we analyse it as a part (subsystem) of the soil–plant–atmosphere continuum (SPAC). Soil as a part of SPAC is interacting with geological structures below as well with the subsystem atmosphere above the soil surface. Between all the three subsystems, simultaneous transport of water and energy occurs. Because the SPAC system produces biomass, its importance for mankind is crucial. The soil is the only subsystem of SPAC that can be substantially influenced by people. It makes possible managing the soil water and nutrient regime and thus increasing the biomass production on Earth and securing the existence of mankind. The properties and processes intrinsic to the SPAC system and its influence on plant production are the specific topics of the science called agrophysics (Glinski et al. 2014). The SPAC system is a complicated one, because it is composed of three subsystems with quite different physical and chemical properties. Hillel (1982) defined the SPAC system as an “integrated, dynamic system in which simultaneous mass and energy transport processes occur”.

1.2 Soil and Its Role in SPAC

7

The main challenge problem of understanding and quantifying this system is the so-called “non-conservatism”of the plant subsystem as a component of SPAC. Plant properties are not conservative (stable), but plants interact with the environment and they adapt to it. Plant properties are changing with time, depending on mass-andenergy transport processes in the SPAC system. Soil is generally assumed to be a conservative component of the SPAC system. This assumption is only approximately valid because pedogenesis, i.e. the process of soil development, operates continuously too, but soil properties change very slowly in comparison to the rapid changes of plant and atmosphere properties. There are many definitions of soil, but the definition “soil is the upper, weathered part of the Earth which is suitable environment for plant growth” is the clearest and simplest. There are many porous media, but the soil as a porous medium is specific to secure plant growth and their proper development. Soil stabilizes plants and contains the necessary water, nutrients and oxygen because a soil having such properties can be the proper medium for plant growth. Other types of porous media, like subsoils, lying just below the soil root zone, are not suitable for plant growth. Photosynthesis is one of the most important processes in the biosphere; it transforms mass and energy into biological, organic objects–namely, biomass. For photosynthesis, carbon dioxide is taken from the atmosphere and the necessary energy of the Sun is used, plus water (solution) flow from the soil through plants to the atmosphere (transpiration) is necessary. Nutrients are dissolved in water, and they are transported to plants. For biomass production, respiration in the soil root zone is necessary to oxidize organic matter, simultaneously producing carbon dioxide and energy. Energy is used for plant growth, and part of it is dissipated to the environment. The quantity of carbon dioxide as a product of respiration in the soil root zone is less than 10% of the carbon dioxide consumed in photosynthesis. Plants are net producers of oxygen and consumers of carbon dioxide. Therefore, plant canopies are often considered as “Earth’s lungs”. Since oxygen for respiration is part of soil air, soil aeration is necessary for plant growth. The majority of crops can only grow in water-unsaturated soil, therefore knowledge of the quantification of mass and energy processes is a necessary condition to increase biomass production.

1.3 Atmosphere The atmosphere is the gaseous cover of the Earth, about some hundred kilometres thick; but up to 95% of atmospheric mass is concentrated in the air layer of altitudes of 0–16.3 km. The density of the atmosphere decreases with height above the Earth surface exponentially. Its total mass is only approximately 1 × 10−6 of the total Earth mass. The components of the atmosphere can be divided into four groups (Chrgijan 1978):

8

1 Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

The most important gases: nitrogen (N2 ) with a relative content 78.08%; oxygen (O2 ) 20.9%, and argon (Ar) 0.98%. Water vapour comprises a relatively small concentration, but its importance for the Earth is substantial; therefore it too belongs among the most important components of the atmosphere. The pressure of the water vapour at 20 °C is about 2% of atmospheric pressure, at temperature 0 °C it is 0.6% and at temperature −30 °C it is only 0.05% of atmospheric pressure. Water vapour content in an atmosphere is within limits 1 g cm−2 in Antarctica) up to 5 g cm−2 in tropical areas. Water-vapour pressure in Central Europe, north of Vienna, is about 2.5 g cm−2 (Chrgijan 1978). These amounts express the equivalent water layer contained in the atmosphere, in centimetres. Chemically stable gases with low concentration in the atmosphere: carbon dioxide (CO2 ) participates in photosynthesis, with continually increasing concentration ranging from 336 parts per million (ppm) up to 407 ppm in the year 2017. Concentration of CO2 primarily increase due to fossil-fuel burning (Kutílek and Nielsen 2010). Other gases in this group are ozone, methane, hydrogen and rare gases. Chemically unstable gases (free radicals) with low concentrations but important chemical activity: hydroxyl group (OH− ), atomic oxygen (O2− ) and chlorine (Cl− ). Aerosols are solid and liquid particles of small diameter such as salts, mineral dust and drops of solutes. Water vapour and carbon dioxide are the components of the atmosphere that will be analyzed in detail in this work. Water as a component of life is relatively homogeneously distributed in the hydrosphere. Carbon dioxide, a part of photosynthesis, is the so-called “greenhouse gas”. It influences long-wave radiation transport through the atmosphere and contributes to the Earth’s temperature increase. The quantitative contribution of the carbon dioxide concentration to the temperature of the atmosphere is still not known. Air temperature changes are probably the result of combined extraterrestrial and planetary effects (Kutílek and Nielsen 2010). The basic air properties are listed in Table 1.1.

1.3.1 Water Vapour The atmosphere is the gaseous component of the biosphere, and water vapour is the gaseous phase of hydrosphere. Part of the atmosphere is the gaseous phase of soil, below the soil surface, in the aeration zone. The gaseous phase of soil is necessary in oxidation processes (respiration), which take part in biomass production. The low degree of soil aeration and resulting low oxygen concentration are leading to reduction processes, to products toxic for plants and can results in their wilting. Water vapour as one of the phases of water can be quantitatively expressed by the following characteristics. Specific humidity of air qs is the mass of water vapour per unit mass of moist air; it is dimensionless. It can be expressed by the ratio of the water-vapour density (ρ v ) and density of moist air (ρ a ) (ML−3 , kg m−3 ):

1.3 Atmosphere

9

Table 1.1 Basic physical properties of liquid water, ice and air (Kikonin 1976) Liquid water σ × 10−3 (kg s−2 )

ηw × 10−3 (kg m−1 s−1 )

L × 106 (J kg−1 )

T (° C)

ρw (kg m−3 )

λw (W m−1 K−1 )

–20

–

–

–

2.549

4354

–

–10

997.94

–

2.600

2.525

4271

–

0

999.87

75.6

1.787

2.501

4218

0.561

5

999.99

74.8

1.516

2.489

4202

0.573

10

999.73

74.2

1.306

2.477

4192

0.586

15

999.13

73.4

1.138

2.466

4186

0.594

20

998.23

72.7

1.002

2.453

4182

0.602

25

997.08

71.9

0.8903

2.442

4180

0.611

30

995.68

71.7

0.7975

2.430

4178

0.619

35

994.06

70.3

0.719

2.418

4178

0.628

40

992.25

69.5

0.6531

2.406

4178

0.632

cw (J kg−1 K−1 )

Air T (°C)

eo (hPa)

ρ v × 10−3 (kg m−3 )

ρ a (kg m−3 )

ρ a /ρ w (−)

υ a × 10−6 (m2 s−1 )

cpa

λa (W m−1 K−1 )

(kJ kg−1 K−1 )

–10

2.9

2.36

1.34

0.00134

10.4

1.005

–

–5

4.3

3.41

1.317

0.00131

11.3

1.005

–

0

6.2

4.85

1.29

0.00129

13.3

1.005

0.0243

10

12.5

9.39

1.247

0.00124

13.61

1.005

0.0250

20

23.8

17.29

1.204

0.00120

15.11

1.005

0.0257

25

32.3

23.05

1.184

0.00118

15.33

1.005

0.0261

30

43.3

30.38

1.165

0.00117

15.55

1.005

0.0264

40

75.2

51.18

1.127

0.00113

15.97

1.005

0.0271

50

125.8

83.05

1.097

0.00110

16.37

1.005

0.0279

Ice T (°C)

ρ v (kg m−3 )

L t × 106 (J kg−1 )

L s × 106 (J kg−1 )

ci (J kg−1 K−1 )

λi (W m−1 K−1 )

–20

917

0.2889

2.838

1940

2.44

–10

917

0.3119

2.837

2000

2.32

0

917

0.3337

2.834

2060

2.834

ρ w —water density, ρ a —air density, ρ v —water vapour density, σ —surface tension of water, ηw —dynamic viscosity of water, L—latent heat of evaporation, L t —latent heat of ice melting, L s —latent heat of ice sublimation, cw —specific heat capacity of water, cpa —specific heat capacity of air at constant pressure, ci —specific heat capacity of ice, λw —thermal conductivity of water, λi —thermal conductivity of ice, λa —thermal conductivity of air, eo —saturated water vapour pressure, υ a –kinematic viscosity of air

qs 

ρv ρv  ρa (ρv + ρad )

(1.1)

where ρ ad is the density of dry air (ML−3 , kg m−3 ). Absolute humidity of air (qa ) is the mass of water vapour (mv ) per unit volume of moist air (V v ), (ML−3 , kg m−3 ): qa 

mv Vv

(1.2)

Partial pressure of water vapour (e) denotes the pressure of water vapour as a component of the atmosphere; it is often expressed in Pascal (Pa) or hectoPascal

10

1 Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

(hPa). The pressure of any individual component of the atmosphere is a result of its own “partial” pressure. The sum of partial pressures of all components of the atmosphere is atmospheric pressure, which can be readily measured. Air saturated with water vapour (containing the maximum quantity of water vapour at the actual temperature) is saturated water-vapour pressure, usually denoted as (eo ). Saturated water-vapour pressure is a function of air temperature and can be expressed by the empirical equations proposed by Clausius–Clapeyron or Magnus. Clausius–Clapeyron equation can be written as:   L T − T01 (1.3) eo  eo1 exp RTo1 T where eo , eo1 is saturated water-vapour pressure at the temperature T , (hPa) and saturated water-vapour pressure at air temperature T 01 (air temperature is expressed in K), L is latent heat of a phase-change liquid water–water vapour (J kg−1 ) and R is the gas constant (J kmol−1 K−1 ). Water-vapour deficit, the saturation deficit of air (d) (L−1 MT−2 , kg m−1 s−2 , Pa), is defined as the difference between saturated water-vapour pressure at the actual temperature (eo ) (L−1 MT−2 , kg m−1 s−2 , Pa) and the actual water-vapour pressure at the same temperature (e) (L−1 MT−2 , kg m−1 s−2 , Pa): d  eo − e

(1.4)

The relative humidity of air (r) (−) is the ratio of the water-vapour pressure at the actual temperature (e) and the saturated water-vapour pressure at the actual temperature (eo ): r

e eo

(1.5)

The specific air humidity qs (kg kg−1 ) can be converted to the water-vapour pressure (e) (hPa), or vice versa. The empirical equation can be used (Havlíˇcek et al. 1986): qs  0.622

e ≈ 0.622 × 10−3 e 1000 − 0.037e

(1.6)

1.3.2 Oxygen Oxygen produced in photosynthesis is needed for the respiration in the root system of plants. The role of living organisms in the oxygen balance of the Earth is not significant; the decisive role is played by plants. Oxygen is also consumed by plants and living organisms in water. The presence of oxygen in water is important for the

1.3 Atmosphere

11

self-cleaning of water by oxidation processes and thus for maintaining an acceptable quality of the water. Soil air contains less oxygen in comparison to the atmospheric air (it is consumed by respiration) and more carbon dioxide (the product of respiration). Those differences are usually small, due to the intensive rate of gas exchanges between soil air and the atmosphere.

1.3.3 Carbon Dioxide Carbon dioxide (CO2 ) is one of the key chemical participants in photosynthesis, therefore its presence is crucial in the biomass-production process. Its presence in the atmosphere has been relatively stable, but it is significantly increasing, mostly due to the combustion of fossil fuels. Intensification of this process during the last century has led to the increasing concentration of CO2 in the Earth’s atmosphere—from 336 parts per million (ppm) up to 407 ppm. Carbon dioxide (CO2 ) is one of so called “greenhouse gases” that is fostering the so-called “greenhouse” effect, i.e. increasing the Earth temperature by reflecting the long–wave radiation radiated by the Earth’s surface back to the Earth. The reason for this phenomenon is the heating of the relatively big molecules of CO2 which radiate this captured energy in all directions, even back to the Earth. The atmosphere acts like a greenhouse. One positive effect of the increase in CO2 concentration is expected higher temperatures of the boundary layer of the atmosphere and so a lower need of energy for heating. Another positive effect of increased CO2 in the atmosphere is an expected higher rate of photosynthesis, which can lead to an increase of biomass production. Therefore, global warming (if it continues) can even have positive features for mankind (Kutílek and Nielsen 2010). The expected negative effects of the global warming can be icebergs melting, the resultant increase of oceans levels and positive dilation of oceans due to the temperature increase which can be dangerous for coastal areas. An important source of CO2 is the respiration of plants. Carbon dioxide (CO2 ) is not the only one of the atmospheric components that participates in the greenhouse effect. Even water vapour is strongly influencing the greenhouse effect. It is theorized that the most important phenomena influencing global change are interplanetary effects (Kutílek and Nielsen 2010).

1.4 Plant Canopy The family of plants, usually called the plant canopy, is known not only as the first element of the nutrient chain of Earth´s living organisms, but also as an important element in the water cycle of the hydrosphere. A specific feature of the plant canopy

12

1 Soil as a Part of the Soil–Plant–Atmosphere Continuum (SPAC)

as a part of the system soil–plant–atmosphere continuum (SPAC) is its variability during the growth period and its adaptability to the changing conditions in SPAC. The topic of plant physiology is described in specialized literature (Lambers 1998), and here it will be presented only briefly. Plants will here be quantitatively described by their characteristics, enabling quantification of the role of plants (and the plant canopy) in the water cycle of the hydrosphere. To do so, plants (canopy) properties will be described as a part of the SPAC system by the quantitative (phytometric) characteristics sometimes denoted as plant-canopy architecture parameters (Ross 1975). These characteristics describe the geometric structure of the canopy and their changes during the vegetation period. The individual plants of the same canopy differ by dimensions of their aboveground and below-ground parts. Therefore, canopy characteristics are statistical parameters, and they are the results of numerous measurements of individual plants. For example, the height of individual plants of many individuals was measured; so the canopy height is the average height of those plants (zp ). Thus defined, canopy height is not the height of some particular plant, but it is good representation of plant-canopy height. To calculate evapotranspiration and its components, as well as the interception of water, only a limited number of phytometric characteristics of the canopy are needed. The most important plant-canopy characteristics are: Canopy height (zp ) is the average height of numerous plants (L, m). Canopy density (ρ p ) is the number of plants per unit of soil surface (L−2 , m−2 ). Leaf area index, LAI (ω), is the sum of all unilateral green parts of plants (transpiring parts) per unit soil-surface area. Only one side of the area of green leaves and green stems are accounted for. LAI is a dimensionless characteristic; it is one of the most important characteristic of a canopy. Values of LAI can vary over a wide range, depending on the plant-ontogenesis stage. For bare soil, LAI  0, but broadleaved trees can reach an LAI > 20. The LAI of cereals can reach a value of 6.5, alfalfa up to 12. The LAI of maize can vary in the range 3–3.2, the LAI of sugar beet can reach a value of 6 and the LAI of potatoes can reach 5. Those maximum values of LAI change from zero to their maximum values during the vegetation period (L2 L−2 , m2 m−2 ). Soil covering (sp ) is the ratio of the vertical projection of the plant canopy area and the area of the soil surface on which the canopy has been projected (L2 L−2 , m2 m−2 ). Crop growth rate, (CGR) is the dry biomass production rate per unit time, per unit soil surface area (M L−2 T−1 , kg m−2 s−1 ). Root system depth (zr ) is the maximum depth of roots (L, m). It depends on the plant and other environmental properties; a decisive factor is the soil–water regimen. Root-system depth of majority of crops is within the range 1–2 m. Roots density (ρ r ) is the dry mass of roots in the unit soil volume. Its values range from a maximum (just below the soil surface) up to value of zero just below the root system depth. In homogeneous soil, the root density distribution can be expressed by an exponential function (ML−3 , kg m−3 ). Roots specific length (lr ) is the sum of the roots length in a unit volume of soil (L L−3 , m m−3 ).

1.4 Plant Canopy

13

Roots specific surface (srs ) is the area of roots’ surface in a unit volume of soil (L2 L−3 , m2 m−3 ). The above phytometric characteristics of the plant canopy are only the basics. Detailed information can be found in specialized literature (Slavík 1974; Matejka and Huzulák 1989; Masaroviˇcová et al. 2002; Novák 2012).

References Brutsaert W (1982) Evaporation into the atmosphere. Theory, history and applications. D. Reidel Publishing Company, Dordrecht Chrgijan ACh (1978) Physics of atmosphere I. II, Gidrometizdat, Leningrad (In Russian) Darcy H (1856) Les fontaines publique de la ville de Dijon. Dalmont, Paris Glinski J, Horabik J, Lipiec J, Slawinski C (eds) (2014) Agrophysics, processes, properties, methods. Inst Agrofiziky, Lublin (In Polish) Havlíˇcek V et al (1986) Agrometeorology. SZN, Prague (In Czech) Hillel D (1982) Introduction to soil physics. Academic Press, New York Kikonin NK (ed) (1976) Tables of physical characteristics. Handbook, Atomizdat, Moscow Kutílek M, Nielsen D R (1994) Soil hydrology. Catena, Cremlingen–Destedt Kutílek M, Nielsen DR (2010) Facts about global warming. Catena Verlag, Reiskirchen Lambers H (1998) Plant physiological ecology. Springer, New York Masaroviˇcová E, Repˇcák M et al (2002) Plant physiology. Comenius University, Bratislava (In Slovak) Matejka F, Huzulák J (1989) Analysis of plant canopy microclimate. Veda, Bratislava. (In Slovak with English abstract) Novák V (2012) Evapotranspiration in the Soil-Plant-Atmosphere System. Springer Science + Business Media, Dordrecht Novák V, Kˇnava K (2011) The influence of stoniness and canopy properties on soil water content distribution: simulation of water movement in forest stony soil. Eur J Forest Res 131:1227–1735 Ross JK (1975) Radiation regimen and architectonics of plant canopy. Gidrometizdat, Leningrad (In Russian) Slavík B (1974) Methods of studying plant–water relations. Academia, Prague

Chapter 2

Basic Physical Characteristics of Soils

Abstract Soil as a specific type of porous media known as a capillary porous medium is described, and its basic physical properties are defined. Because soil is a three-phase system, composed of solid, liquid and gaseous components, the basic properties of all three phases are described and quantified. Definitions of soil porosity, specific bulk density, bulk density, aeration, soil water content and soil specific surface are presented. Soil texture is defined and methods of soil texture (mechanical composition) determination are described (sieving, sedimentation method, pipette method, hydrometric method, laser-diffraction method) and their specific features are briefly mentioned. Pedogenetic factors and processes and their role in soil pedogenesis of various soil types are described, as well as their role in the formation of soil physical properties. The influence of clay minerals in soil physical properties formation and mechanisms of their interaction with soil water are demonstrated.

2.1 Porous Media, Capillary-Porous Media and Soil. Are There Differences? A porous medium is a system of pores (voids), confined by the solid phase. Pores are usually interconnected to allow liquid and air transport through this medium. The solid phase of a porous medium is denoted as the soil matrix. As an example of porous media, soil, wood, textile, building materials like concrete, brick and organic matter can be mentioned. It can be stated that the majority of materials around us can be denoted as porous media. The ability of a porous medium to conduct liquid depends on the properties of pores. The most important properties are pore dimensions and their distribution, their interconnections and properties of the solid-phase surface. The most important porous medium on the Earth is soil. As has been mentioned, it is fertile, and therefore suitable for plant growth. Soil is a medium that help supply plants with water, nutrients and oxygen. The subsoil, with a similar distribution of pores, usually does not contain the nutrients (and very often not even oxygen)

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_2

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needed for plant growth. But, through the process of pedogenesis, the subsoil can be transformed into soil. Soil is a type of porous medium denoted as capillary porous medium—with interconnected capillary pores (Lykov 1953). Capillary porous media contain interconnected pores of dimensions that enable the movement of water by capillary forces. There are porous media with interconnected, but relatively large, pores (diameters of some millimetres) in which capillary forces are not significant; water through such pores is moving by the force of gravity, i.e. downward. In some porous media, pores are not interconnected and do not transport fluids. Soil pores are always filled with the liquid or gas phase. As an effect of intermolecular forces, water in contact with the solid phase wets it and covers the solid phase with a thin layer of water. Under some special conditions (dry soil particles are covered with a layer of organic matter), soil can be temporarily water repellent or can exert limited (decreased) hydrophilicity. This effect is usually temporary because, in permanent hydrophobic porous media, plants cannot grow. In porous media saturated with water, all the pores are filled with water, and only the interface between the solid and liquid phases exists. The thin water layer within the reach of solid phase surface forces is formed between the solid phase (charged negative) and liquid phase (water). Because water is a polar liquid, its molecules are dipoles; they are oriented along the electric field of the solid phase of the soil surface and attracted to the solid phase of soil by the positive part of the water molecule. This thin layer of water is attracted to the solid phase of soil by weak, electrostatic, so-called van der Waals forces. First, a monomolecular layer of water is strongly attracted by the solid surface of soil, therefore, being relatively immobile. The next water layers are attracted to the soil solid phase by weaker forces, and they can move. The water layer containing two sublayers (immobile and mobile) is known as the “electrical double layer”. Soil unsaturated with water (unsaturated zone) is a typical state of soil during the vegetation period. In this state, the pores are only partially saturated with water and even contain some air. Then, there are two phase interfaces: the interface between the water and solid phase of soil (soil matrix) and the interface between water and soil air. Capillary forces, acting in unsaturated soil together with gravitational forces (gravitational forces were the only outer forces acting in saturated soil) make the movement of soil water possible, even upwards to the plant roots, against gravity. Detailed information about soil physics is reported by Hartge and Horn (1991, 2016). Clean liquid water does not exist in nature, rather, in all cases it is a solution containing solute, part of which are nutrients. Usually, soil water implicitly means a soil solution. The physical characteristics of soil solution (density, surface tension, viscosity) are practically the same as those of clean water, therefore, for a soil solute, the term soil water and the characteristics of clean water are normally used. Soil solution (denoted as soil water) concentration is usually less than 1.0 g l−1 . An exception is the movement of highly concentrated solutions in salt-affected soils. Then, dissolved matter can significantly change the physical characteristics of the soil solution, which must be considered when quantifying the transport of concentrated liquids in soil.

2.2 Soil as a Three-Phase System

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2.2 Soil as a Three-Phase System Soil is a three-phase system; it is composed of the solid, liquid and gaseous phases. The liquid phase (as was mentioned) is not clean water, but a solution containing the nutrients necessary for soil fertility. Soil as a medium for plant growth can produce biomass if there are all three phases. We can recall that the gaseous phase of soil (soil air) is necessary for respiration of the roots; the liquid phase is a component of plants and necessary for photosynthesis, as well as a transporter of water and nutrients to and through plants. The majority of water entering plants is flowing through the plant and transpires into the atmosphere, thus creating conditions for plant functioning. The solid phase of soil (soil matrix) is stabilizing the plant, and it enables to retain the liquid and gaseous phases of soil. Soil volume V is the sum of all the three soil components: V  Vl + Vs + Va

(2.1)

where V , V l , V s , V a is volume of soil, volume of liquid phase of soil, volume of solid phase (soil matrix) and volume of gaseous phase of soil (L3 , m3 ). Subscripts denote phases in accordance with internationally accepted standards. These are the first letters of English words denoted liquid (l), solid (s) and air (a). This can simplify the notation in the literature. The soil and its components are shown schematically in Fig. 2.1. It is a cross section of hypothetical porous medium, unsaturated with water, and demonstrates the relative volumes (or masses) of all three soil components. Dividing Eq. (2.1) by the soil volume, we can write: θ+

Vs +a 1 V

(2.2)

Fig. 2.1 Soil as a three phase system. 1—solid phase, 2—liquid phase (water), 3—gaseous phase (air). Soil components are expressed in volumetric (V ) and mass (m) units. Indexes a, l, s represent the gaseous, liquid and solid phases of soil. V p is the volume of soil pores. A is the representative elementary volume (area)—REV; B is the non-representative volume (area) of a soil

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θ

Vl V

(2.3)

where θ is volumetric soil-water content (volume of water in a unit volume of soil, L3 L−3 , m3 m−3 ). It represents the relative volume of soil that is filled with water. Relative volume of solid phase of soil ss (−): ss 

Vs V

(2.4)

a is the relative volume of soil air (aeration) (−): a

Va V

(2.5)

Sum of volumetric soil-water content and relative volume of soil air (aeration) is porosity P: P a+θ

(2.6)

Porosity means relative volume of soil pores, i.e. the ratio of pores and total volume of soil, is dimensionless. Soil phases can even be expressed by their mass. Then: m  ml + m s + ma

(2.7)

where m is mass of soil, ml , ms , ma are mass of the liquid phase (water), solid and gaseous phase (air); their dimensions are in mass units (M, kg). Density of soil solid phase (particle density) ρs (ML−3 , kg m−3 ) (mass of unit volume of dry solid phase of soil) can be expressed by the equation: ρs 

ms Vs

(2.8)

Density of the mineral soil solid phase oscillates around 2.60 g cm−3 , which is approximately the density of silicon. The differences in soil densities can oscillate in the range of ±0.1 g cm−3 , depending on the soil substrate. Organic soil densities are usually significantly lower. Density of soil ρ b (mass of solid phase in unit volume of dry soil), neglecting the mass of soil air, can be expressed by: ρb 

ms V

(2.9)

Density of soil (in comparison to density of soil solid phase) ranges over a wide scale: organic soils with densities less than 0.1 g cm−3 , andosols (less than 0.9 g cm−3 ), then uncompacted loamy soils (from 1.1 g cm−3 ) up to 1.8 g cm−3 for

2.2 Soil as a Three-Phase System

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compacted, heavy soils, usually subsoils. Relatively high bulk densities of sandy soils correspond to their relatively low porosity and the solid phase is usually comprised of quartz. Combining the above equations, the following equation for porosity P can be obtained: P 1−

ρb ρs

(2.10)

From Eq. (2.10), the well-known fact follows: the higher is the soil density, the lower can be observed the soil porosity, and vice versa. Relatively low porosity of compacted soils means lower retention capacity for air and water, and so they are therefore less suitable for plant growth. Another important soil characteristic is specific soil surface (s), defined as the ratio of soil solid-phase surface (As ) and soil solid-phase mass (ms ): s

As ms

(2.11)

The dimension of specific soil surface is (L2 M−1 , m2 kg−1 ).

2.3 Pedogenesis Soil is a product of the process known as pedogenesis. Pedogenesis is a process of soil formation, determined by pedogenetic factors. The result of pedogenesis (soil formation by pedogenetic processes) is the soil type; soil properties are the result of pedogenetic factors and pedogenetic processes. The most important pedogenetic factors are the following: Parent material is the basic material for pedogenesis. It is weathered, or unweathered rock material, or organic matter. Physical and chemical properties of developed soil are significantly influenced by the properties of the parent material. Climate is the next important pedogenetic factor because the temperature and water content of the parent material strongly influence pedogenetic processes. Plants and living organisms are an inevitable part of pedogenesis. They cause the disintegration of sterile rocks and transform organic matter to humus in the process of humification. Among the first inhabitants on the substrate are bacteria, then lichens and finally vegetation. Landscape properties, especially slope, exposition and height above sea level are important pedogenetic factors. Anthropogenic activity is a pedogenetic factor, especially in sites exploited over thousands of years. In the modern era, heavy machinery is used; herbicides and pesticides are applied, thus changing the physical and chemical properties of soils.

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The basic pedogenetic processes are: Weathering of rocks is the basic pedogenetic process. Solid rock changes into a polydisperse system. It is not only a physical process of solid rock division, but also chemical weathering by which even the chemical composition of the substrate changes. Humification is a process of the transformation of dead organic matter to humus. Humus’ importance for soils is basic. It is the source of nutrients for plants, but it is important to create and preserve soil structure. The specific pedogenetic process is peat formation. This process happens in wet environments and anaerobic conditions; peat soil contains more than 50% organic matter. Typical of peat soil is the higher increment of soil organic mass than losses by disintegration as a part of pedogenesis. Existing classification schemes are based on prevailing types of pedogenetic processes. The basic element of this kind of soil classification is the soil type. Particular soil types (even by their typical soil profiles) can usually indicate some physical and chemical soil characteristics, as well as its optimal use. There are many soil classification systems; practically every country uses its own. The most well-known are two soil classification systems: the U.S. Department of Agriculture (USDA) and International Soil Science Society (ISSS) classification systems. There are small differences in classes according to both classification systems. Here we describe pedogenetic processes only briefly. The details of this topic are presented in publications such as Hraško and Bedrna (1988), Soil Survey Division Staff (1993), Jury and Horton (2004), Hillel (2004), Fulajtár (2006) and Brady and Weil (2008).

2.4 Soil Texture The term soil texture refers to the size range of soil-solid primary particles and their relative distribution according to their size classes. Soil texture is often referred to as the mechanical composition of the soil. It is expressed by the relative mass of the solid particles of a defined size range to total mass of the solid phase of the sample. Relative mass of solid particles of a defined range of size (they belong to a defined “category”, as this range is usually known) is the basis for soil classification according to their texture. There are two basic soil-texture classes: – Fine earth, fine soil (solid particles smaller than 2 mm) – Rock fragments (skeleton, solid particles larger than 2 mm) Fine earth is usually divided according to their particles dimension into three fractions: sand, silt and clay. The boundaries among them differ according to various classification systems. Soil Science Society (ISSS) classification system of soil fractions according to their diameter denotes particles smaller than 0.002 mm as

2.4 Soil Texture

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clay, silt particles are in the range 0.002–0.02 mm and sand particles are of diameter range 0.02–2.0 mm. Soil particles larger than 2 mm in diameter are defined as gravel, larger rock fragments are called stones, then cobbles and boulders. USDA classification system differs from ISSS classification by the soil particles diameter at the boundary between silt and clay. USDA classification defines this boundary diameter as 0.05 mm in comparison to 0.02 mm in ISSS classification system. A comparison of various classification systems can be found in Hillel (1982).

2.4.1 Soil-Texture Analysis Soil-texture analysis is a procedure for the evaluation of particle-size distribution, also known as mechanical analysis of soil. There are various methods of soil-texture analysis; its use is determined mostly by the soil texture of the soil to be analyzed. Before mechanical analysis begins, it is necessary to separate primary soil particles, which are often aggregated in soil structure units, like aggregates or clods. Separated individual particles should be dispersed in aqueous solution. Separation (dispersion) of soil particles Soil particles as parts of soil structure units should be dispersed into separate individual soil particles. Dispersed soil samples then undergo so-called mechanical analysis. There are a few recommended methods for soil preparation for analysis, which can be found in the literature. Methods can be a combination of shaking, stirring, boiling or ultrasonic vibration, in combination with a dispersing agent (e.g., sodium hexametaphosphate, calgon). Removal of organic matter is usually achieved by oxidation with hydrogen peroxide (Hillel 1982; Glinski et al. 2011). Sieving Sieving is frequently used; it is the simplest method for mechanical analysis of soil. Separation of soil particles into size groups is carried out by passing the suspension through a graded series of sieves. At first, rock fragments (skeleton) with diameters higher than 2 mm are separated. Fine earth (soil particles with diameters less than 2 mm) is passed by a graded series of sieves down to a mesh diameter of 0.1 mm. Soil particles smaller than 0.1 mm are usually electrostatically attracted and create groups of particles. Therefore, the rest of dispersed soil particles undergo analysis by other methods. The most frequently used methods for particle-size analysis (below the diameter range of meshed sieves) are sedimentation methods. Actually, mechanical analysis by the laser-diffraction method can be used in some cases. Sedimentation methods Methods for sedimentation are based on Stoke’s law, which states that the sedimentation velocity of spherical particles of identical density under the influence of gravity in the aqueous solution depends on the particle diameter only. Knowing the time

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interval from the beginning of sedimentation and the position of the particle, it is possible to calculate its diameter. All sedimentation methods use Stokes law. In this equation, soil particles are assumed to have a spherical shape and the flow of water around the particles is laminar and, since soil particles are much bigger than water molecules, not influenced by intermolecular forces (the so-called Brown’s motion influence on the sedimentation process can be eliminated). Stoke’s equation is usually written as: v

(ρs − ρw )d 2 g 18η

(2.12)

where v is the velocity of suspended particles’ sedimentation (L T−1 , m s−1 ), ρ s is soil-particle density (M L−3 , kg m−3 ), ρ w is liquid-water density (M L−3 , kg m−3 ), d is soil-particle diameter (L, m), g is acceleration of gravity (L T−2 , m s−2 ) and η is dynamic viscosity of liquid water (M L−1 T−1 , kg m−1 s−1 ). Sedimentation velocity can be expressed as v = h/t, where h is the depth below the suspension (water) table level (L, m), where sedimented soil particles of diameter d are observed in time t. Substituting sedimentation velocity into Eq. (2.12), the time t for a soil particle with diameter d to reach the depth h is: t

18ηh (ρs − ρw )d 2 g

(2.13)

According to Radcliffe and Šim˚unek (2010), Stoke’s equation is applicable to the sedimentation of soil particles of diameters less than 0.06 mm. The most frequently used sedimentation methods area the pipette method and the hydrometer method. Pipette method Pipette method is based on taking a small volume of suspension by pipette at a depth h at time t. The sedimentation starts with the suspension homogenization by stirring. Knowing water properties and particles bulk densities, the diameter of soil particles corresponding to the estimated sedimentation velocity can be calculated by Eq. (2.13). The volume of suspension in the sedimentation cylinder has to be large enough not to influence the accuracy of results caused by significant changes of the suspension volume by pipetting. Hydrometer method Hydrometer method is a relatively simple one, but yields good results. Its principle is based on the Stoke’s equation and on the measurement of decreasing densities of suspension as a function of time, due to soil-particle sedimentation. Using a hydrometer specially designed for this procedure, the density of suspension is measured. Using this characteristic, the relative contents of soil-particle sizes in the soil sample is calculated. Details about sedimentation methods are described in the specialized literature (Klute 1986).

2.4 Soil Texture

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Laser-diffraction method This modern method for the estimation of particle-size distribution is based on the measurement of light diffraction by soil-solid particles. Results of such measurements are used to calculate the diameter of particles that diffract the light. When analyzing soil texture by the laser-diffraction method, the following effects should be considered: (a) diffraction on the outer surface (contour) of the particle, (b) reflection from the particle surface, (c) absorption of light by the particle and (d) refraction of the laser light by the particle. Soil particle-size distribution according to their diameter is calculated using a theoretical model of the mentioned phenomena and measured data. The weak point of this method is its limited range of applicability; it is recommended to be used for particles smaller than 0.025 mm, but the evaluation of clay-particle content is significantly underestimated. It is recommended not to be used for analysis containing significant proportions of clay particles (Yang et al. 2015). Its advantage in comparison to other methods is that it is relatively fast (Glinski et al. 2011).

2.5 Soil Type Soil-type classification is based on the ratio of textural fractions in a given soil. Textural fractions are of three separate components: sand, silt and clay. As was mentioned, there are various definitions of soil textural fractions. Two of the most accepted classification schemes are: the U.S. Department of Agriculture (USDA) and International Soil Science Society classification (ISSS). Both classification schemes are briefly summarized in Table 2.1. Soil type itself is important information about soil properties. Knowing the soil type, it is possible to evaluate approximately soil hydraulic, agrotechnic and mechanical properties. Simple, but widely used classification of soils is based on soil-texture analysis. The expressions “heavy soil” and “light soil” have nothing in common with their

Table 2.1 Soil fractions and their diameters according to USDA and ISSS classification Classification scheme USDA ISSS Soil separate

Diameter range (mm)

Clay

<0.002

<0.002

Silt Sand Gravels Cobbles Stones Boulders

0.002–0.05 0.05–2.0 2–75 75–250 250–600 >600

0.002–0.02 0.02–2.0

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weights or densities. It is well-known that soils with relatively high densities (sandy soils) are called “light” soils, but soils containing a majority of clay particles, with relatively high porosity and low bulk density, are declared as “heavy” soils. This type of classification is mainly related to their mechanical properties, tillage properties and typical features of soil–water regimens. Sandy soils are usually tilled relatively easily across the wide range of soil–water contents because their particles easily separate too. Clayey soils can be tilled within a narrow range of soil–water contents only, due to the cohesion of its primary particles, mostly clay minerals; so the time intervals permitting tillage are limited. The bulk densities of heavy soils significantly increase with drying, as well as their exhibiting significant deformation, creating the net of soil cracks and “pedons” which are hard, impermeable soil blocks. During dry intervals, clods form in heavy soils, and during the wet parts of the season they are impermeable and difficult to till.

2.6 Rock Fragments in Soil Rock fragments are soil fractions with diameter larger than 2 mm; these particles are separated by sieving with a 2-mm mesh. According to USDA (2017) definition, stony soils have to contain more than 15% of rock fragments by volume. Stony soils and their properties will be analyzed in detail in Chap. 17.

2.7 Clay Minerals Clay minerals influence the physical behaviour of the soil decisively, since they exhibit the greatest specific surface area. Clay minerals are active in physico-chemical processes. Their large specific surfaces are mostly negatively charged and therefore active in water adsorption. Such soils increase in volume, i.e. swell. When hydrated, clay particles (micelles) lose water (they are drying) and the soil volume decreases, they shrink. In the literature, it is possible to find detailed information about clay minerals (Kutílek and Nielsen 1994), but we are interested mostly in the influence of clay minerals on particular properties of soils, containing clay minerals. Hydrated colloidal particles with mostly negatively charged surfaces are neutralized by positively charged molecules of a soil solution, or water. Water molecules are dipoles; therefore they are oriented by the positive part of dipoles to the negatively charged soil surface. Particle surfaces and the neutralizing cations (or positive part of dipoles) form an electrostatic double layer. Figure 2.2 demonstrates the concentration of ions distribution in the proximity of the particle surface. The range of surface forces is relatively narrow; it ranges approximately over a few layers of soil solution (water) molecules and depends on the nature of the clay mineral’s surface. The surface forces are indirectly proportional to the sixth power of distance from the

2.7 Clay Minerals

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Fig. 2.2 Positively and negatively charged ions concentration distribution in soil solution (c) and distance from surface of soil-solid phase (positively charged) (x). Ions concentration in the bulk solution is co

soil surface (Skopp 2000). The closest water layer to the soil surface (monomolecular layer) is attracted relatively firmly, but the attractive forces become smaller with the distance from the solid-phase surface, and water molecules become movable. This distribution of water-molecules attraction forces in proximity of soil surface is result of two phenomena: electrostatic (Coulomb) forces attract water molecules and Brownian (chaotic) motion homogenizes ions concentration. Micelles as a part of soil can coagulate (flocculate), which mean joining together and forming clusters, flocks or clumps of a larger size. On the contrary, micelles can be dispersed (peptization) in a continuous phase (water). Micelles coagulate usually during increases of soil-solute concentration, and, during concentration decrease, dispersion usually occurs. The specific surface of colloids is in the range 10–400 m2 g−1 (Skopp 2000), but the specific surface of organic matter is usually higher and in the range 500–800 m2 g−1 . Soil surface is generally negatively charged, and its properties are quantitatively expressed by the term cation-exchange capacity—CEC, which is the quantitatively expressed potential of the solid phase of soil to bind positively charged ions and even water dipoles. Exchange capacity of soil for anions (anion–exchange capacity—AEC) is substantially smaller. Colloids of organic matter—in comparison to clay minerals—possess higher cation-exchange capacity (CEC). Therefore, humus is a key component of soil and significantly influences its properties.

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2.8 Soil Structure The arrangement and organization of primary soil particles is called soil structure. Primary soil particles join together to form secondary soil particles. Properties of those secondary soil elements and their arrangements determine soil structure. Secondary soil structure elements are aggregates, clods and pedons (Radcliffe and Šim˚unek 2010). Aggregates and clods can be identified in the soil plowing layer; they are formed spontaneously, depending on the soil properties, mineralogical composition of the soil and, most of all, on the quality and quantity of soil organic matter. Clods form mostly artificially, i.e., by plowing of soil with high water content and then by soil drying. Pedons could be identified in the space below the soil plowing layer. Pedons are secondary, three dimensional soil elements, separated mostly by vertical soil cracks. Soils can be classified as structural and nonstructural soils. Nonstructural soils do not contain aggregates. They are mostly coarse, sandy soils with a low content of organic matter. Primary particles of nonstructured soils are easily separated. Nonstructural soils can be even soils with a high content of fine particles, but this is a rare case. Aggregates of structural soils can be divided according to their sizes, shapes and interactions between them. The shapes of aggregates can be classified as follows: platy, prismatic and angular. Why is the soil structure so important in hydrologic processes and soil-production processes? The main reason is that soil structure strongly influences soil porosity and the pore-size distribution. Inter-aggregate pores are mostly large pores, called macropores. Soil-water movement in macropores is mostly driven by gravity, and capillary forces in macropores are usually not dominant. Macropores, whose diameter is usually assumed to be above 1 mm (height of capillary rise in macropores is usually less than 3.0 cm) can be represented not only by inter-aggregate pores, but even by spaces created by dead roots, by soil fauna (e.g., by worms) and soil cracks. Macropores enable fast infiltration of rain or irrigation water into soil, decreasing the risk of erosion and enabling easier tillage. On the other hand, macropores enable the fast propagation of pollution, nutrients and other solutes to the area below the root zone and eventually to the groundwater, thus polluting them. Macropores enable proper aeration and respiration of roots, so proper soil structure is a precondition for its hydrologic and production functions. Sustainable, proper soil structure relies on soil-aggregate stability, i.e., its resistance against decomposition during heavy rain. Organic matter (humus) in soil is a precondition for soil-aggregate stability. Soil structure is one of the most important physical properties of soil. It influences soil porosity and soil water movement. To define and quantify soil structure and its relationship to soil properties is not an easy task (Radcliffe and Šim˚unek 2010).

2.9 Chemical and Mineralogical Properties of Soil

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2.9 Chemical and Mineralogical Properties of Soil Chemical properties of soils (nutrients content and their availability, exchange capacity, soil reaction) depend mostly on the mineralogical composition of soil. Mineralogical composition of soil substrate determines the texture of the soil and other soil properties. Sand or silt as soil components are usually quartz particles with a relatively small specific surface. Soils containing significant amount of clay (aluminosilicates, like kaolinite, vermiculite and smectite) with the dimensions of nanoparticles (10–2000 nm), are typified by their high specific surface. For example, a cube of dimension 1 cm has a surface of 6 cm2 . Dividing this cube into small cubes with 10-nm edges, the specific surface will then be 600 m2 , so it will have increased by a factor of 100. That is why clay minerals with plate structures feature such great specific surfaces. Plants are known to need only the 16 elements (nutrients) necessary for plant growth (Donahue et al. 1983). The essential nutrients are hydrogen (H), oxygen (O) and carbon (C) in the water and in air. Phosphorus (P), potassium (K), sulphur (S), calcium (Ca), iron (Fe), magnesia (Mg), boron (B), copper (Cu), manganese (Mn), zinc (Zn), molybdenum (Mo) and chlorine (Cl) contained in the soil; and nitrogen (N) in both the air and the soil. All these elements (except nitrogen) are products of soil-substrate weathering. Nearly all plant nutrients are used in ionic forms. Animals consume the same nutrients as plants, with the addition of sodium, cobalt, selenium and iodine, but with the exception of boron.

References Brady NC, Weil RE (2008) The nature and property of soils, 14th edn. Pearson Prentice Hall, Upper Saddle River, NJ Donahue RL, Miller RW, Shickluna JC (1983) Soils. An introduction to soils and plant growth, 5th edn. Prentice-Hall Inc., Engelwood Cliffs, New Jersey Fulajtár E (2006) Physical properties of soils. VÚPOP, Bratislava (In Slovak) Glinski J, Horabik J, Lipiec J (eds) (2011) Encyclopedia of agrophysics. Springer, Science & Business Media BV, Dordrecht, Netherlands Hartge KH, Horn R (1991) Einführung in die Bodenphysik. Ferdinand Enke Verl, Stuttgart, Germany Hartge KH, Horn R (2016) Essential soil physics. An introduction to soil processes, functions, structure and mechanics. In: Horton R, Horn R, Bachmann J, Peth S (eds), Schweitzerbart Sci Publ, Stuttgart, Germany Hillel D (1982) Introduction to soil physics. Academic Press, New York, London Hillel D (2004) Introduction to environmental soil physics. Elsevier, Academic Press, Amsterdam Hraško J, Bedrna Z (1988) Applied pedology. Príroda, Bratislava (In Slovak) Jury WA, Horton R (2004) Soil physics. Wiley, NY, p 384 Klute A (ed) (1986) Methods of soil analysis. Part 1. Physical and mineralogical methods, 2nd edn, Madison, Wisconsin, USA Kutilek M, Nielsen DR (1994) Soil hydrology. Catena, Cremlingen—Destedt Lykov AV (1953) Transport phenomena in capillary-porous media. Gostechizdat, Moscow (In Russian)

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Radcliffe DE, Šim˚unek J (2010) Soil physics with HYDRUS, modeling and applications. CRC Press, Boca Raton Skopp JM (2000) Physical properties of primary particles. In: Summer ME (ed) Handbook of soil science. CRC Press, Boca Raton, pp A3–A18 Soil Survey Division Staff (1993) Soil survey manual. Soil conservation service. USDA Handbook 18 USDA (2017). Determination of grain size distribution. http://www.nrcs.usda.gov/wps/portal/nrcs/ detail/soils/survey/office/ssr10/tr/?cid=nrcs144p2_074845#item1b. Accessed 20 June 2017 Yang X, Zhang Q, Li X, Jia X, Wei X, Shao M (2015) Determination of soil texture by laser diffraction method. Soil Sci Soc Am J 79:1556–1566

Chapter 3

Physical Properties of Water

Abstract Water covers more than two third of Earth’s surface and thus is the most prevalent substance on Earth. It is also the only liquid on Earth occurring simultaneously in all three phases: ice, liquid water and water vapour. Its dynamics are linked to the great expenditure of energy to keep in motion ‘the water cycle and thus stabilize Earth’s climatic conditions. Water’s occurrence and its movement are preconditions for biomass production. In this chapter, the physical properties of water are quantitatively described to enable a calculation of its transport in the hydrosphere. Moreover, the molecular structure of water that determines water’s physical and transport properties is treated. Soil water is characterized as a solution of solute low concentration, and the solubility of gas in water is defined and quantified. Water density, compressibility and expansion, as well as water’s surface tension and viscosity are defined, and their dependence on temperature is explained.

3.1 Soil Water and Soil Solution Water is the only substance in nature that occurs simultaneously in all three phases. The chemical formula of water is H2 O, which means that each molecule consists of two atoms of hydrogen and one atom of oxygen. In nature, there are three isotopes of hydrogen and three isotopes of oxygen, which means that a water molecule can contain various numbers of neutrons and various mass of molecules. By combinations of various isotopes of hydrogen and oxygen, 18 different types of water can be created. The most common combination is water composed of isotopes 1 H and 16 O (so-called protium 1 H16 2 O) which constitutes the greatest part of all water molecules (99.98%). Other important combination of hydrogen and oxygen ions are deuterium 3 16 (2 H16 2 O), with a proportion of natural water of 0.0015% and tritium ( H2 O). The use of deuterium and tritium in nuclear technology is significant: deuterium is used to moderate neutrons, and tritium is used as an indicator of water flow. Two atoms of hydrogen and one atom of oxygen create one water molecule, and they are adhered by a strong, covalent bond. The hydrogen atom is positively charged, and oxygen atom is charged negatively. The outer electron shell of hydrogen lacks © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_3

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3 Physical Properties of Water

Fig. 3.1 Molecules of carbon dioxide (CO2 ) and water (H2 O). Bonds among atoms are shown by dashed lines

one electron, but oxygen’s outer shell lacks two electrons, therefore one atom of oxygen can combine with two atoms of hydrogen, thus creating water molecule. An important factor is the arrangement of the water molecule in a triangular form (Fig. 3.1), with an angle about 105o between the two hydrogen atoms and oxygen atom. The dimension of a water molecule is 2.76 × 10−10 m; it is one of the smallest molecules of all. This means that the water molecule represents a dipole, i.e. water molecules are polar; they orient themselves in the electric field of mostly negatively charged solid-soil surfaces. Comparing the water-molecule structure to the structure of carbon dioxide (CO2 ), the difference in their arrangement is easily visible. CO2 is a neutral (non-polar) molecule, but a water molecule can orient itself to charged objects, even to another water molecule. Electrostatic forces exist between the positively charged side of a dipole of one water molecule and the negatively charged side of the water molecule; both molecules are attracting. Such bonds are called “hydrogen bonds”, and they are relatively weak in comparison to covalent bonds. Each water molecule can interact with four water molecules and thus clusters form. Liquid water is a relatively cohesive substance, with a relatively high density in comparison to ice. Water dipoles below 0 °C pack into a solid structure, but one that is less dense than liquid water; therefore ice floats in liquid water. As temperature increases, molecules of water oscillate and destruct the firm ice structure, their arrangement is becoming denser. The highest density of liquid water is at 3.9 °C. The increasing motion of water molecules, due to the increasing temperature, the hydrogen bonds, become less effective in comparison

3.1 Soil Water and Soil Solution

31

Fig. 3.2 Temporal cluster of liquid-water molecules (H2 O). For the transformation of water molecules into clusters, so-called “hydrogen bonds” are responsible (according to Hillel 1973)

to the heat-induced motion of the water molecules so its density decreases with temperature. The basic characteristics of water are listed in Chap. 1 (Table 1.1). Temporary clusters by hydrogen bonds are continuously formed and dissipated. Therefore, liquid water exists as a medium of quickly forming and dissipating clusters of interacting molecules. Figure 3.2 depicts liquid water as a cluster of interacting water molecules. Structure, which resembles the structure of ice, is partially preserved in liquid water even after the melting of ice (Slayter 1967; Hillel 1973). Energy of sublimation (latent heat of sublimation) is L s  2.834 × 106 J kg−1 , but the latent heat of thawing is only L t  3.34 × 105 J kg−1 . Therefore, a substantial part of the bond energy between liquid molecules is preserved even after thawing. Separate water molecules with minimum interaction forces can be found in water vapour. Hydrogen-bond energy is responsible even for the high value of water’s specificheat capacity. It is the energy needed to increase the temperature of one g of liquid water by 1 °C. Water’s specific heat capacity is the highest of all liquids. The high value of water’s specific heat capacity (c  4.18 J g−1 K−1 ) is the reason for the extremely high heat capacity of water bodies on Earth and thus the relative stability of climatic conditions. Water and wet surfaces cover a substantial part of Earth’s surface; because the change of temperature requires a great input of energy, the Earth experiences temperature change only slowly. This is the precondition of life on the Earth, which is not the case for other planets of our solar system. Liquid in soil, denoted as soil water, is not pure water but rather a solution. Soils, to be fertile, have to contain nutrients, but the concentration of dissolved substances is usually lower than 1 g L−1 . According to the US Salinity Laboratory Staff (1954), a soil solute concentration of 3 g L−1 is the lowest boundary of saltaffected soils. The concentration range of solute substances of salt-affected soils is 3–15 g L−1 . To increase solute-substance concentration from zero to 10%, the surface tension of the liquid increases approximately 2%. This is not a significant increase,

32

3 Physical Properties of Water

but such a relatively highly concentrated soil solutions is a rare case. Soil-solute concentration changes—due to the application of industrial fertilizers—have only minor effects on the physical properties of the soil solute, and this influence can be neglected when calculating soil-solute movement. Therefore, the influence of soilsolute concentration on the physical characteristics of a soil solution can usually be neglected, and the soil solution can be characterized by the physical characteristics of water. The presence of dissolved substances in water increases its surface tension and boiling-point temperature, while decreasing its freezing point. This is due to the stronger bonds among molecules of the solute (in comparison to water); to break them, application of additional energy is needed. This phenomenon is used in road maintenance by salt application. The added salt in water decreases its freezing-point temperature; ice is thawing in some sub-zero temperature range.

3.2 Air in Water and Its Solubility Soil in contact with the atmosphere contains components of dissolved air. The solubility of gas (air) in water increases with air pressure and decreases with temperature. Gas solubility can be expressed by the ratio of the dissolved air quantity and the quantity of water. The solubility of gases in water can be expressed in mass (sam ), or volumetric (sav ) units: ma mw Va  Vw

sam 

(3.1)

sav

(3.2)

where ma is the mass of dissolved gas in water of mass mw ; V a is the volume of gas dissolved in volume of water V w ; the solubility of gases in water (sam ), (sav ) are dimensionless. The solubility of gases (sa ) is proportional to their partial pressure pi (Henry’s law): sa  si pi

(3.3)

where si is the coefficient of solubility of i-th component of gas in water; it is a function of temperature. Solubility of air in water (i.e. the maximum quantity of air soluble in water at some temperature and atmospheric air pressure) is sam = 0.020 g kg−1 at the temperature 20 °C. The solubility of O2 at the same temperature is sam = 0.009 g kg−1 , but the solubility of carbon dioxide (CO2 ) in water at the temperature of 10 °C is sam = 22 g kg−1 (at maximum CO2 concentration). The maximum volume of air in water can reach 0.016 (1.6%) of water volume at the temperature of 20 °C.

3.3 Water Density, Compressibility and Expansion

33

3.3 Water Density, Compressibility and Expansion Heat motion of water molecules is partially eliminated by the hydrogen bonds among them. Hydrogen bonds are the reason for the relatively small water-density change with temperature. The highest water density is at the temperature of 3.9 °C. Water density in the temperature range 4–50 °C does not change significantly; it is in the range 1.000–0.988 g cm−3 . Density of the liquid phase of water is relatively stable over a wide range of temperatures. The fact that the maximum water density occurs at the water temperature 3.9 °C is of critical importance for life on the Earth. The relationship between liquid-water density and temperature is shown in Table (1.1). The low compressibility of water depends on the small water-density change over a wide range of temperature. The compressibility of liquid water is defined by the relative change of water volume per increase in unit pressure: V  β V p

(3.4)

where V is the water-volume change due to pressure change p (Pa) of water of initial volume V (L3 , m3 ). The coefficient of compressibilityβ is a relative change of water volume by unit pressure change. The compressibility of water at 0 °C is a very low 5.1 × 10−10 Pa−1 , so the volume of water will change about half a million parts of initial water volume by a pressure increase of 100 kPa; this is approximately atmospheric pressure. At the depth of 4 km below sea level, where the pressure is 40 MPa, the relative water volume change is approximately 0.018. The compressibility of water is very low compared to other liquids. But, the compressibility of steel is about 100 times lower than the compressibility of water. Water, like all other substances, changes its volume with temperature. The coefficient of thermal expansion is defined by the relative increase of liquid volume by unit temperature change. The coefficient of thermal expansion changes with temperature; the higher the temperature, the higher is the coefficient of thermal expansion. The coefficient of thermal expansion at 20 °C is 1.4 × 10−4 (dimensionless); with a water-temperature change 1 °C, a water volume of 1.0 m3 changes its volume by about 0.14 L, i.e. a water layer of 0.14 mm. Therefore, analyzing the soil-water transport, soil-water expansion with temperature can be neglected and liquid water can be treated as a non-deformable medium.

3.4 Water-Surface Tension The phenomenon of surface tension of water can be observed at the liquid water–gas (air) interface. Visually, the surface/tension phenomenon appears as a thin membrane on the water surface. A metal paper clip carefully set on the water surface will not sink but floats on the water surface, deforming it. The specific surface tension is defined as the force within a unit length of the water membrane cross section:

34

3 Physical Properties of Water

σ 

F l

(3.5)

where σ is the surface tension (of water) (MT−2 , Nm−1 ), and F is the force (LMT−2 , N) acting per unit length of water/table cross section l (L, m). The method for measurement of surface tension is based on the above definition. Water-surface tension σ is a function of water temperature (it decreases with temperature increase); at 20 °C, σ  0.072 Nm−1 . For comparison, mercury surface tension (in contact with air) is approximately 7 times greater. On the contrary, surface tension of oil is about 3 times lower than the surface tension of water. Table 1.1 shows how water-surface tension depends on temperature. Surface tension can be significantly influenced by substances diluted in water; it can be increased or decreased, depending on the interrelation between the molecules of water and molecules of the diluted substance. Higher cohesion forces between molecules of a diluted substance and the molecules of water, in comparison to the cohesion forces between only molecules of water, increase the surface tension of such a solution. On the contrary, lower cohesion forces between molecules of diluted substance and molecules of water, in comparison to the cohesion forces between molecules of water, decrease the surface tension of such a solution. The latter phenomenon is typical of interactions between organic matter and detergents, and it is used to separate them in cleaning processes. The surface-tension phenomenon can be explained by various modes of interactions between the intermolecular forces in the water volume and in the surface layer of water. Liquid-water molecules in the volume are attracted equally from all directions by cohesive forces, but molecules on the surface are attracted in the liquid direction only, but forces of attraction in the air direction are significantly smaller; so the resulting force is directed to the liquid (water). The liquid is therefore “pressed” together, and the surface “membrane” is formed. The surface-tension phenomenon is of great importance for water retention in porous media (soils). Surface tension is a precondition for capillary phenomena in soils, which will be analyzed later (Chap. 4).

3.5 Viscosity of Water The relative motion of water molecules is resisted by cohesion forces among them. Let us illustrate this phenomenon by the water-layer motion along the solid plate. Let the thickness of water layer be z. Friction forces between the solid plate and water cause a decrease of water-flow velocity in the direction to the solid plate. The highest flow velocity will be on the water surface if the friction between water and air is neglected. The lowest velocity will be observed at the contact between water and solid plate. Thus, the velocity profile in the z layer of water depends on liquid properties and on the flow velocity.

3.5 Viscosity of Water

35

Liquid-water flow can be interpreted as the flow of the thin interacting layers. Forces of interactions between thin layers of liquid (water) are denoted as internal friction: τ

Fs du η A dx

(3.6)

where τ is the shearing stress between liquid (water) layers (lamina) (L−1 MT−2 , Nm−2 ) under acting force F s (LMT−2 , N), on area A (L2 , m2 ). The ratio du/dx is the liquid-water velocity gradient perpendicular to the direction of water flow; η is the dynamic viscosity of liquid (water) (L−1 MT−1 , kg m−1 s−1 ). Following from Eq. (3.6), the viscosity of liquid water is the shearing stress force acting on a unit area of contact between two layers of moving liquid (water) at unit velocity gradient in the direction perpendicular to the direction of water flow. Or simply, the viscosity of liquid (water) is the proportionality factor between shearing stress and the velocity gradient du/dx. The term kinematic viscosity ν (L2 T−1 , m2 s−1 ) is often used; it is defined by the ratio of dynamic viscosity η and liquid-water density ρ w : ν

η ρw

(3.7)

Viscosity depends on the liquid type, its temperature and its pressure. The dependence of viscosity on pressure is not significant and can be neglected. The relationship between viscosity and temperature is significant and cannot be neglected. When increasing a liquid’s temperature by about 3 °C, its viscosity decreases about 10%. The relationship between the kinematic viscosity of water and its temperature can be expressed by the empirical equation: ν

1.78 × 10−6 1 + 0.0337T + 0.000221T 2

(3.8)

where ν is kinematic viscosity of water (m2 s−1 ), and T is water temperature (°C). The liquid of lower viscosity is flowing faster than the liquid with higher viscosity. Low viscosity means relatively weak interactions between adjacent liquid-water layers and lower tangential forces. The low-viscosity liquid is characterized by higher fluidity, which is the inverse of viscosity. The viscosities of glycerine and oil are higher than water, but the viscosities of acetone and gasoline are lower. See Table 1.1 to see how viscosity depends on temperature. The characteristics of water and gas can be found in Kikonin (1976).

36

3 Physical Properties of Water

References Hillel D (1973) Soil and water, physical principles and processes. Academic Press, New York, London Kikonin NK (ed) (1976) Tables of physical characteristics. Handbook, Atomizdat, Moscow Slayter RO (1967) Plant—water relationships. Academic Press, New York, London US Salinity Laboratory Staff (1954) Diagnosis and improvement of saline and alkaline soil. U.S. Govt. Printing Office, Washington, D.C., USDA Handbook No. 60

Chapter 4

Soil-Water Interface Phenomena

Abstract Capillarity in porous media (soils) is the basic physical phenomenon responsible for soil-water retention. Without this solid-water interface phenomenon, soil water is transported downward by the force of gravity, thus moving water from precipitation through the top layer of soil to groundwater. Water retention and plant growth would be profoundly limited. In this chapter, capillarity as an important physical phenomenon is described, and the height of the capillary-rise formula is presented. A simplified model of soil as a bundle of capillary tubes is used to explain soil-water behaviour in real soils. The influence of the shape of water menisci on water-vapour pressure just above menisci and capillary-condensation phenomenon are explained. Adsorption and desorption isotherms and the influence of clay minerals on adsorption and desorption phenomena are also mentioned.

4.1 Wetting of the Solid-Phase by Liquids A drop of water falling on a solid, dry surface can cover it with a thin layer of water, or it can preserve its shape on the impact site. Depending on solid-surface properties, such formation of water can be characterized by the wetting angle (Fig. 4.1), whose range can be 0° ≤ ϕ ≤ 180°. Ideal wetting is observed when the wetting angle is zero (ϕ  0), for wetting angle less than 90°, the solid surface is characterized as hydrophilic, and at wetting angles ϕ ≥ 90°, the solid surface is hydrophobic. Simply put, this phenomenon can be explained by various adhesive and cohesive forces (interaction forces between water molecules and molecules of solid phase). If adhesive forces between the solid and liquid phases are higher than the cohesive forces between liquid molecules, the liquid is wetting the solid phase, but if cohesive forces (between water molecules) are higher than adhesive forces between liquid and solid phase, water is not wetting the solid phase.

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_4

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38

4 Soil-Water Interface Phenomena

Fig. 4.1 Water drops contact angle ϕ on various solid surfaces: a ideally wetted surface (ϕ  0°); b (ϕ < 90°), hydrophilic surface of solid phase; c (ϕ > 90°) water-repellent surface; d (ϕ  180°), is an ideally hydrophobic surface

Fig. 4.2 Capillary tubes dipped in various liquids. Liquid (a) with contact angle ϕ  90°; capillary height is zero; liquid (b) wetting the capillary tube surface (like water) with contact angle ϕ < 90°, its elevation height in capillary tube is hk ; liquid (c) is not wetting capillary tube (mercury) ϕ > 90°, the meniscus depression in capillary tube is hd (capillary depression)

4.2 Capillarity In a capillary tube dipped in water, water will rise to the height indirectly proportional to the tube’s diameter. A capillary tube is tube of small diameter, where capillary forces are comparable or more significant than the force of gravity. Usually, the capillary tube has a diameter below 1 mm (the height of capillary rise of water is then 29.8 mm); pores with diameters larger than 1 mm are called macropores, where the force of gravity is usually dominant. The water table in a capillary tube is not “flat” but of a curved shape, called a meniscus (Fig. 4.2). The meniscus and its shape is the result of water–surface tensions at the contact between water and air, air and the glass tube and water and the glass tube. The contact angle between the water and solid phase can be calculated from the equilibrium equation written for the contact of a water meniscus with the solid surface of the capillary tube (Fig. 4.5):

4.2 Capillarity

39

σsg  σsl + σlg cos ϕ

(4.1)

where ϕ is the contact angle of water and the solid phase, and σ sg , σ sl and σ lg are the three interface tensions between surfaces denoted by particular subscripts. Then, the contact angle can be expressed by the equation: cos ϕ 

σsg − σsl σlg

(4.2)

where σ is surface tension (or energy of interaction between both phases); l—liquid phase, g—gas (air), s—solid phase. A positive value for Eq. (4.2) means wetting of the solid surface by the liquid; if the liquid is water, such a surface is “hydrophilic”; the negative value of this equation means a non-wetting (hydrophobic) solid surface (ϕ ≥ π/2). Among hydrophilic solids there are majority of substances in ecosystem (soils, carbonates, silicates). Metals, graphite and some organic matter are among hydrophobic substances. Some soils in a dry state are hydrophobic too, but that is a rare case. Temporarily hydrophobic soils (water repellent) will be analyzed in Chap. 18. The contact angle is zero (ϕ  0) if the cosines of the contact angle equal one (the numerator of Eq. (4.2) equals zero)—in which case, wetting is ideal. Calculation of soil-water movement is usually based on the assumption of zero contact angles. In reality, contact angles between soil and water are usually higher, but this approximation is acceptable.

4.2.1 Soil as a Bundle of Capillary Tubes Soil is a three-phase system: it is a complicated porous space filled with immiscible substances like water and air. To characterize quantitatively the geometrical characteristics of a porous medium, the porous medium is often simplified. One of possibilities is assuming that a porous medium is composed of solid spheres of various diameters. Another idealization of the soil can be the bundle of capillary tubes of various diameters. The simplest model of soil composed of a capillary-tube system has many advantages for understanding and quantification of basic principles of soil-water retention and movement. Such a type of model is especially suitable when calculating the capillary rise in soil or for quantification of the capillary forces’ influence on soil-water retention and movement. In a capillary tube of radius r dipped in water (Fig. 4.3), water in the capillary tube will rise to the height hk above the water table. In equilibrium state, the weight of water in the tube is in equilibrium with the capillary force; its quantity depends on the contact angle between the water and capillary tube and on the radius of capillary tube. According to Laplace, the pressure difference across the curved meniscus (below and above the meniscus) is called the capillary pressure pk , which can be expressed by the equation:

40

4 Soil-Water Interface Phenomena

Fig. 4.3 Linear distribution of water pressure in capillary tube above the free water-table level (pk ) and below the water level (p); water pressure above the water table is negative, below the water table is positive

pk 

2σw R

(4.3)

where pk is capillary pressure (L−1 MT−2 , kg m−1 s−2 ); σ w is surface tension of water (as a function of temperature) (MT−2 , kg s−2 ), and R is meniscus radius of curvature (L, m). Capillary pressure is the force acting on a unit area of capillary-tube cross section due to the capillary phenomenon. The water pressure in a capillary tube just below the meniscus is negative, and the air pressure at the water-table level is atmospheric (usually taken as zero pressure); but in a capillary tube dipped below the free water table (as well as in the water around the capillary tube), it is positive. The pressure distribution along the capillary tube is linear. At the zero contact angle of a meniscus, the radius of curvature R equals the capillary tube radius r. If the contact angle is other than zero, the radius of curvature R is other than the capillary tube radius r. Capillary tubes shown in Fig. 4.4 are filled with liquid of various wetting (contact) angles. For contact angle ϕ > 0 (Fig. 4.4a), the curvature of the meniscus can be calculated by the equation: R

r cos ϕ

(4.4)

where r is the capillary tube radius (L, m). The contact angle ϕ  0, for which the meniscus radius of curvature R equals the capillary tube radius r, is shown in Fig. 4.5b. The pressure in the water drop (or in

4.2 Capillarity

41

Fig. 4.4 Shapes of menisci in capillaries containing wettable (a) and not wettable (b) liquid. 1—liquid, 2—gas, 3—solid phases

Fig. 4.5 Menisci curvatures (R) in capillaries with not ideally wetting liquid ϕ > 0° (a); ideally wetting liquid is shown in (b); ϕ  0°. Menisci curvature (R) of ideally wetting liquid equals to the capillary radius (R  r)

the bubble) of radius r is twice that of the capillary pressure in the capillary tube of the same radius because the drop is under pressure from all sides in comparison to the meniscus in the tube (see Eq. 4.3): pk 

4σ R

(4.5)

There is one peculiarity in the contact-angle problem. “Contact angle” is usually measured at equilibrium state; but in the capillary tube it depends on the velocity of the meniscus movement. There are significant differences between contact angles of moving menisci in capillary tubes, and their value depends even on the direction of the meniscus movement. The higher the meniscus velocity in a capillary tube is observed, the higher is the contact angle, when the meniscus is advancing. But, if the meniscus is retreating, the higher the velocity of retreat is, the smaller the contact angle is observed. The soil-water movement velocity is usually so small that this phenomenon is not significant. But it can be significant, if the water movement (discharge of water) in a soil sample is measured by the meniscus advance, usually in a

42

4 Soil-Water Interface Phenomena

horizontally positioned capillary tube. Then, the different contact angles can produce additional capillary forces, which can (at small gradients of soil-water potential) significantly influence the results of measurement.

4.2.2 Height of Capillary Rise When one dips a capillary tube into water at atmospheric pressure, water in the tube will rise; its movement will be stopped at some height hk above the free water level. Then, the capillary force (trying to move water along the tube) is in equilibrium with gravitational force, which acts downward, i.e. in the opposite direction to the capillary forces. The maximum height reached by water in a capillary tube is called the height of capillary rise or capillary height hk (L, m). The capillary height can be calculated using equations expressing the equilibrium between liquid-water weight in a capillary tube F g (MLT−2 , kg m s−2 ) and the capillary force F k (MLT−2 , kg m s−2 ): Fg  π r 2 h k ρw g

(4.6)

Fk  2π r σw cos ϕ

(4.7)

At equilibrium, this can be written: Fk  Fg

(4.8)

To substitute Eqs. (4.6) and (4.7) into Eq. (4.8) and after some rearrangements, the final equation for the height of capillary rise hk as it depends on capillary tube radius r is: hk 

2σw cos ϕ ρw g r

(4.9)

where ρ w is the water density (ML−3 , kg m−3 ); σ w is water-surface tension (MT−2 , N m−1 ); g is the acceleration of gravity (LT−2 , m s−2 ). The height of capillary rise is indirectly proportional to the angle of contact between the liquid and solid phases (the smaller is the contact angle, the higher is the capillary height) and indirectly proportional to the radius of capillary tube (Eq. 4.9). Therefore, the smaller the capillary tube radius is the higher is the capillary height. The relationship between the capillary height (hk ) and radius of capillary tube (r) for two realistic contact angles is depicted in Fig. 4.6. According to Czachor et al. (2010), effective angles of contact between soil and water are in between 0° and 21°. Substituting water characteristics ρ w and σ w at temperature T  20 °C, (Table 1.1) and angle of contact ϕ  0 into Eq. (4.9), the capillary height (hk ) can be expressed by the formula:

4.2 Capillarity

43

Fig. 4.6 Capillary height hk in capillary tube and the radius of capillary tube r, for two contact angles; ideal wetting ϕ  0° and contact angle close to the real soils ϕ  20°

hk 

14.8 r

(4.10)

where r is the radius of capillary tube in mm; the height of capillary rise hk is in millimetres too. The capillary height of water in a capillary tube of 1 mm in diameter and at an ideal contact angle will rise up to 29.8 mm. Therefore, pores with diameters greater than 1 mm are often denoted as noncapillary pores (or macropores), and pores with smaller diameters are denoted as capillary pores, importance of which increases with decreasing radii of pores.

4.2.3 Decrease of Water-Vapour Pressure in Capillary Tubes Above the Menisci Water-vapour pressure just above the meniscus in the capillary tube depends on meniscus curvature. Just above the concave meniscus, equilibrium water-vapour pressure is lower than that above the flat surface. An example of a concave meniscus is the water meniscus in a capillary tube (Fig. 4.4a). The decrease in saturated watervapour pressure above the concave meniscus pv can be expressed by the modified equation of Thomson (1871): pv 

ρv pk ρl − ρv

(4.11)

where ρ v , ρ l are densities of saturated water vapour and density of liquid water (ML−3 , kg m−3 ); pk is the capillary pressure (ML−1 T−2 , kg m−1 s−2 ), (Eq. 4.3); and pv is the change (decrease) of saturated water vapour pressure above the meniscus (ML−1 T−2 , kg m−1 s−2 ).

44

4 Soil-Water Interface Phenomena

The decrease in water-vapour pressure above the concave meniscus is relatively small; the fraction of the right side of Eq. (4.11) at standard temperature is approximately 1 × 10−5 . Even this small decrease in water-vapour pressure above the concave meniscus is responsible for the phenomenon called capillary condensation; it is condensation of water on menisci of great curvature in soil pores. The Thomson (Kelvin) equation can be applied even to water drops or bubbles in water. Just above the water drops, the water-vapour pressure is higher than above a flat surface; it means water vapour from drops is evaporating more intensely than from a flat water surface. The water-vapour pressure in air bubbles in water is smaller than above the flat surface. In an air bubble of 1 mm diameter, the difference between water-vapour pressure in the bubble and in the atmosphere is small: 288 Pa. But the difference in water-vapour pressures in an air bubble of diameter 3 μm and above the flat water table is a remarkable 96 kPa.

4.2.4 Capillary Phenomena and Their Influence on Soil-Water Retention Capillary phenomena in soils are crucial to understanding the movement and retention of liquid water in soils and porous media. Capillary and gravitational forces in soils unsaturated with water are the dominant forces responsible for transport and retention of water in soil. Capillary forces are the main forces that hold water in soil: without their existence, rain or irrigation water will be moved downward by gravity quickly. Without capillary forces, only a thin layer of water can cover the solid phase of soil (by adsorption of water vapour from atmosphere). This (adsorption) water is not readily available for plants because it is attracted to the solid surface by forces that cannot be overcome in the root system of plants. Capillary forces act on interfaces of liquid and gaseous phases; they are the result of the adhesion of the solid and liquid phases and cohesion between molecules of the liquid phase (water). Capillarity is the physical phenomenon in tubes of small diameter (capillary tube) resulting in capillary rise (elevation) or capillary depression by capillary forces (Fig. 4.2). Capillary tube is the tube of small diameter where capillary forces are dominant. The water table (meniscus) in a capillary tube dipped in water is above the free water table into which capillary tube has been dipped. Height of capillary rise is the maximum height of capillary rise of a particular liquid in a particular capillary tube; then, capillary and gravitational forces are in equilibrium.

4.3 Adsorption and Desorption of Water on Soil Surfaces

45

4.3 Adsorption and Desorption of Water on Soil Surfaces Adsorption (desorption) by the solid phase of soil is a spontaneous formation of a gaseous or liquid layer on the solid surface by the action of intermolecular forces. This interfacial phenomenon is followed by heat production (adsorption) or heat consumption (desorption). The heat production is due to the loss of energy of adsorbing molecules by their fixing on the solid surface, while the heat consumption is needed to mobilize the molecules fixed by a solid surface. Quantitatively, the energy produced when adsorbing a unit amount of liquid (water) is known as the heat of wetting. The adsorption of water upon a solid surface is generally of an electrostatic nature. Dry soil in contact with wet air (air containing water vapour) increases in weight. The quality of soil enabling the water vapour to be adsorbed on its surface is called hygroscopicity, and water adsorbed in this way is hygroscopic water. Adsorption (desorption) isotherm is the relationship between water-vapour mass adsorbed, by a unit soil mass and relative partial pressure of water vapour above the soil determined at a constant temperature. The quantity of adsorbed water by soil depends on the relative humidity of the air. The higher relative humidity of the air is the higher quantity of immobilized water by soil in an equilibrium state. If the relative humidity of an air increases (step by step), the quantity of adsorbed water (and the soil-water content) will increase too, and by this procedure can be estimated the relationship called the adsorption isotherm. The procedure of soil-water desorption starts with wet soil; then, by decreasing the relative humidity of the air above the soil sample, the desorption isotherm can be estimated. Such relationships are called isotherms because the temperature strongly influences adsorption (desorption) process, and such procedures should be performed under isothermal conditions, usually at constant temperature, T  20 °C. They are usually of sigmoid shape and the adsorption–desorption isotherms exhibit hysteresis. Examples of adsorption isotherms of clay minerals are illustrated in Fig. 4.7 (Kutílek and Nielsen 1994). Adsorption (desorption) phenomena are strongly influenced by a soil’s mineralogical composition, especially by the type of clay minerals, then by exchangeable cations, organic matter content. Usually, minerals having the highest specific surface will exhibit the highest adsorption capacity. For example, the maximum adsorption capacity of kaolinite is 0.04 (per unit of mass), and the illite 0.1 and montmorillonite adsorption capacity can be up to 0.3 of its dry weight. Therefore, adsorption phenomena are even important for retention of water in soil. Adsorption (desorption) isotherms are usually determined by measurement of soil-sample (or samples of other materials) water content in an air-tight container to maintain a constant water-vapour pressure. Constant water-vapour pressure in the container is the equilibrium water-vapour pressure above the appropriate solution. After reaching the equilibrium state (it can even take months), the soil samples are weighed. Then, the measurement continues to the next step, with a higher watervapour pressure in the container. To measure the desorption isotherm of a soil, solutes with a progressively smaller pressure of the water-vapour equilibrium above them are used.

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4 Soil-Water Interface Phenomena

Fig. 4.7 Adsorption isotherms (mass soil-water content w and relative water-vapour pressure (r) on montmorillonite (1), illite (2) and kaolinite (3)) (according to Kutílek and Nielsen 1994)

Table 4.1 Specific surfaces of some soils and their constituents (Kutílek and Nielsen 1994)

Soil (constituent)

Specific surface (m2 g−1 )

Kaolinite Montmorillonite Illite Soils Humus in chernozem Fine sand

4–10 150–500 50–80 20–50 900 0.1

To know the adsorption isotherm is important to estimate the specific surface of the soil. The principle of this procedure is based on the information that the heat of wetting is released mainly by the first, monomolecular layer of water absorption. Therefore, the key is to estimate the soil-water content corresponding to the adsorbed monomolecular layer (wm ). Value wm can be estimated as the equilibrium soil-water content corresponding to the relative water-vapour pressure p/p0  0.2 above the solute of 58% H2 SO4 at T  20 °C. Assuming the densest arrangement of water molecules on the solid-phase surface (i.e. a water molecule can occupy an area of 0.01 nm2 ), then, the formula to calculate specific surface is: s  3610 wm

(4.12)

If the soil-water content wm is expressed in mass units (g g−1 ), then the specific surface s is given in (m2 g−1 ). Specific surfaces of some typical soils are listed in Table 4.1.

References

47

References Czachor A, Doerr SH, Lichner L (2010) Water retention of repellent and subcritical repellent soils: new insights from model and experimental investigations. J Hydrol 380:104–111 Kutílek M, Nielsen DR (1994) Soil hydrology. Catena, Cremlingen–Destedt Thomson W (1871) On the equilibrium of vapour at a curved surface of liquid. Phil Mag Ser 4 42(282):448–452

Chapter 5

Soil-Water Content and Its Measurement

Abstract Soil-water content is the basic state characteristic of soil; it expresses the relative quantity of water in the soil. Soil-water content can be expressed as the ratio of water amount and the amount of soil. The most frequently used term is volumetric soil-water content, which is the ratio of water volume and soil volume containing water. Mass soil-water content is expressed as the ratio of water mass and mass of dry soil containing water. Measuring soil-water content is the basic procedure in soil research. Methods for the measurement of soil-water content are briefly described. The basic method is the gravimetric method, a measuring procedure involving only weighing; it is also the reference method of soil-water content measurement, therefore it is described in detail and its weak points (soil—profile destruction and its nature as a discontinuous type of measurement) are discussed. Wide varieties of methods with continuous output are briefly described; neutron method, capacitance method, electrical resistance method, time-domain reflectometry (TDR), frequency-domain reflectometry (FDR), and geophysical and remote-sensing methods.

5.1 Quantitative Expression of Soil-Water Content Soil-water content—sometimes the term wetness is used (Hillel 1982)—is expressed normally as a relative value, i.e. as the ratio of water amount in the soil sample and the amount of soil (soil-sample volume or soil-sample dry mass) (Kutílek and Nielsen 1994). This term can be confusing because “soil-water content” can be interpreted even as the content of water in a soil layer. The most common expression of soil-water content is volumetric soil-water content, which is the ratio of water volume (V w ) and soil volume (V ) that contains this water (see Chap. 2). Usually, volumetric soil-water content is expressed as the ratio of water volume (V w ) per unit of soil volume (V ). This term can be expressed as water concentration in the soil. Volumetric soil-water content is usually denoted by the Greek symbol θ : θ

Vw V

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_5

(5.1) 49

50

5 Soil-Water Content and Its Measurement

where θ is volumetric soil-water content, dimensionless (−); or (cm3 cm−3 , L3 L−3 ),V w and V are the volume of water in a volume of soil; they are frequently expressed in cm3 (L3 ). Volumetric soil-water content is the relative volume of soil filled with water. If the volumes are expressed in cm3 , then volumetric soil-water content equals how many cubic centimetres of water contain one cubic centimetre of soil. Volumetric soil-water content (of non-swelling soils) is in the range of 0 ≤ θ ≤ 1. Soil-water content can be expressed even as the ratio of the mass of water mw (M, kg) and mass of dry soil md that contain the mass of water mw , (M, kg). This is the mass soil-water content w: w

mw md

(5.2)

Dry-soil mass md is expressed by the mass of soil dried (up to the constant weight) at 105 °C. Mass soil-water content is dimensionless, often expressed by grams of water per one gram of dry soil (or in kilograms of water per kilogram of dry soil). In practice, mass soil-water content is not used frequently, because the soil mass depends on soil density and involves other, not immediately needed, information. As will be shown later, knowing the mass soil-water content is a necessary step for calculating volumetric soil-water content by the gravimetric method. Comparing Eqs. (5.1) and (5.2), the relationship between volumetric and mass water content can be obtained so: θ w

mw ρb  ρw V ρw

(5.3)

where ρ b and ρ w are the bulk densities of soil and water (ML−3 , kg m−3 ); the most frequently used dimension is (g cm−3 ). The useful expression for relative soil-water content is a term called soil saturation, S; it is the proportional part of the pores that are filled with water: S

θ θs

(5.4)

where θ s is volumetric soil-water content of soil saturated with water. It equals soil porosity (P) if soil water does not contain dissolved or entrapped air. Another expression of saturation (effective saturation) was proposed by van Genuchten (1980): Se 

θ − θr θs − θr

(5.5)

where θ r is residual soil-water content. The term “residual” is not defined exactly. It is the soil-water content at which soil hydraulic conductivity is so small that it can be neglected. This characteristic is useful when calculating soil hydraulic conductivity

5.1 Quantitative Expression of Soil-Water Content

51

as a function of soil-water content by the Genuchten–Mualem method, which will be analyzed in Chap. 9. The term soil-water content can even mean the water content (in cm3 , or m3 ) of a defined soil layer, or it can be expressed by the depth of the water layer in the soil layer and is usually denoted by the symbol W . In fact, it is the thickness of the soil-water layer in a soil layer of depth d: W  θ.d

(5.6)

The dimension of W is the same as dimension of soil-layer thickness d (L, m). The soil-layer thickness d is in meters, so the thickness of the water layer in soil layer is in meters too. The water content of the soil layer of 1.0-m thick of volumetric soil-water content θ  0.35 cm3 cm−3 is W  0.35 × 1.0  0.35 m (or 35 cm). This value is close to the field capacity of silty soil. The soil-water content of an inhomogeneous soil profile W , composed of n layers with thickness d i , can be calculated so: W 

n 

θi di

(5.7)

i1

5.2 Measurement of Soil-Water Content Soil-water content is a basic state quantity of soil (together with soil-water potential), therefore soil- water content measurement is the basic procedure in soil research. Actually, the soil-water content can be measured by variety of methods, but no one of them can be used universally. The basic (sometimes called an absolute method) is the so-called gravimetric method because the only procedure to quantify soilwater content is weighing, and it is used as a reference method. Calibration in the gravimetric method is not needed, in comparison to other methods for estimation of soil-water content.

5.2.1 Gravimetric Method for Measurement of Soil-Water Content It is the simplest and still the most frequently used measurement method for soilwater content. Its calculation is based on the definition of the mass soil-water content w (Eq. 5.2). To calculate it, it is necessary to know the mass of water mw , the mass of dry soil md and the mass of the container with soil sample mt : w

mw m sw − m sd mass o f wet soil − mass o f dr y soil   md m sd − m t mass o f dr y soil − mass o f container

52

5 Soil-Water Content and Its Measurement

The mass of the wet soil (msw ) is estimated by weighing the wet soil in appropriate container (or in metal cylinder containing the soil sample). The mass of dry sample msd can be estimated by weighing a sample that has been dried out at the 105 °C (up to the constant weight); the mass of the empty container is also weighed. Mass soil-water content does not offer direct information about the ratio of pores filled with water, because the soil bulk density can change over a wide interval, from 0.6 g cm−3 (organic soils) up to 1.8 g cm−3 (dense mineral soils). In reality, the mass soil-water content w = 0.3 can be interpreted as volumetric soil-water content in the range of 0.2–0.6 for the same mass soil-water content, but for varying soil bulk densities. Because the volumetric soil-water content θ is direct information regarding the ratio of pores filled with water, usually this characteristic is needed (Eq. 5.1). In the field, the disturbed soil is usually sampled and stored in air-tight containers; its volume is not known. Then, the mass soil-water content w is evaluated first, and then, (using Eq. 5.3) the volumetric soil-water content θ is calculated. To calculate θ , the soil bulk density ρ b and water density ρ w are needed. Water density is a function of temperature, but, using the constant value ρ w = 1.0 g cm−3 , the error will be negligible. Soil densities have to be measured. The most used method is sampling of soil by metal cylinder (or by another container of known volume); then the sample is dried and weighed. The mass of the cylinder (container) is subtracted from the mass of the dried soil sample with container, and the resulting value is divided by the container (cylinder) volume. The volume of a standard cylinder is 100 cm3 . Comment to gravimetric method 1. Soil-water content changes over time, and with depth and in space. Changes in soil-water content are usually at a maximum at the soil surface and in the upper soil layer. Soil samples have to be large enough to be representative, but not too big as to lose information about the distribution of soil-water content along the sample and in the horizontal direction. A soil volume of 100 cm3 (standard cylinder) of a height of about 5 cm is probably the maximum volume of a soil sample to calculate the “point” value of volumetric soil-water content. This volume of soil sample is suitable even for estimating the soil density of relatively homogeneous soil. 2. To estimate reliably the value of soil-water content in the field due to spatial variability of soil water content (due to the variability of soil density, soil-water conductivity and its retention capacity) more soil samples should be taken. The more the samples, the more representative are the results. The reliability of results is proportional to the number of soil samples. Three soil samples are assumed to be the minimum; if one sample was not sampled properly, or not representative, then the two close values of soil-water content can be used. 3. Results of measurement have shown the significant changes in soil-water content exist mostly in the upper, one-meter-thick soil layer, and therefore soil sampled in this layer could give representative information about the changes in soil-water content during the vegetation period. The distribution of soil-water

5.2 Measurement of Soil-Water Content

4.

5.

6.

7.

8.

53

content with depth can vary in soils with a high groundwater table so that the dynamics of soil-water content depends on groundwater-table movement and on water-table oscillation in nearby rivers or lakes. The gravimetric method for measurement of soil-water content is a destructive method; thus the field soil is disturbed by each soil sampling. The soil profile is continually disturbed thus completely changing the original soil-pores distribution. Holes, macropores of artificial origin, can substantially change the soil-water regimen. Therefore, to monitor soil-water content at one site, it is recommended to use one of the existing continual non-destructive methods, like the TDR method. Disturbed soil samples are usually sampled in marked glass or metal containers of known weight. To minimize the weighing error, it is recommended to sample a minimum of 20 g of soil. Using cylinders of 100 cm3 , the weight of such a soil sample is about 100 g, and the relative weighing error is acceptable. The same sample can be used to estimate soil bulk density or to measure saturated hydraulic conductivity of a soil sample; both data can be estimated via the same soil sample. A commonly made error in the sampling is that sampled soil volume is smaller than the volume of cylinder. Another error can be due to an unnatural soil surface in the cylinder. The knife used to finish the upper and bottom surfaces of the sample in the cylinder can change the pores distribution on both cross sections and thus their hydraulic conductivity. Soil samples should be closed for transportation to prevent evaporation and soil water change of the samples. Cylinders or containers with soil are weighed with an accuracy of 0.01 g (msw ). Examples of measured and calculated data are shown Table 5.1, as well as the relevant site location, time of sampling, number of cylinders or containers and the depth of sampling. Cylinders or open containers are put into a drying oven. Soil samples are dried at the 105 °C (up to constant weight). Soil samples of volume 100 cm3 need 24 h to dry out; a disturbed soil sample in an open container is dried out for about 5–8 h. Then, the dry soil samples are weighed (their mass is denoted by symbol msd ). It is recommended to put (hot) soil samples into containers with desiccators to prevent secondary adsorption and thus the change of sample weight. This can important especially for heavy soils or during the measurement of adsorption/desorption isotherms. To estimate soil-water content of relatively dry soil,

Table 5.1 Calculation of soil density (ρ b ), volumetric soil-water content (θ), mass soil-water content (w). Most pri Bratislave site, Slovakia. August 4, 1993 Sampling Cylinder msw (g) msd (g) depth (z) (cm) number

mt (g)

ρb (g cm−3 ) w (g g−1 ) θ (cm3 cm−3 )

30–35

75.76

1.69

66

266.39

245.14

0.125

0.21

54

5 Soil-Water Content and Its Measurement

it is recommended to use glass containers with tight lids so as to minimize the errors of measurement by water evaporation from the soil sample. 9. By drying soil samples at 105 °C, the soil volume and its structure changes; therefore, the sample cannot be used for further analysis. Then, the empty container can be weighed; the weight of the empty container (or cylinder) is denoted by symbol mt . It is useful to store dry, weighed soil samples until the final analysis has been done to be able to check them in case of possible errors during the process of measurement and calculation. Then, the samples are at hand for eventual repeated weighing. 10. Finally, the mass soil-water content w will be calculated first (Eq. 5.2); knowing the soil bulk density ρ b and density of water ρ w  1.0 g cm−3 , the volumetric soil-water content θ can be estimated. This procedure is illustrated in Table 5.1. The gravimetric method is a direct and destructive procedure, not allowing continuous measurement of soil-water content at one point. Nowadays, there are many different non-direct and non-destructive methods available. They are based on the measurement of other soil properties that are related to soil-water content. Among such methods are: soil-moisture probes measuring soil electromagnetic properties, geophysical methods and radiometers. The preconditions of their application depend even on the scale at which they are applied. A review of the contemporary methods can be found in Vereecken et al. (2008). Here, only some, especially the principles of soil-water content measurement, will be mentioned. Details can be found in the manuals for particular devices or in the specialized literature.

5.2.2 Soil-Moisture Probes Soil-moisture probes are permanently placed in the soil, and therefore they provide almost continuous (in small time-step interval) soil-moisture measurements that are their advantage. On the other hand, they need specific calibration (a soil-specific and in some cases also a sensor-specific one). They vary according to their principle of measurements.

5.2.3 Neutron Method This method is not often used mainly because of the risk of irradiation; in some countries, the use of neutron probe is prohibited. But during the previous decades, this method was used frequently (Kutílek and Nielsen 1994). Another reason is the availability of alternative, relatively cheap and accessible methods (like TDR, FDR), which are risk free. The principle of the neutron-probe method is measuring the number of neutrons with lowered energy (slow, thermalized neutrons) returning back to the neutrons

5.2 Measurement of Soil-Water Content

55

detector (counter), being emitted before by the source of high-energy (fast) neutrons. The energy of the returning neutrons was deflected (scattered) and their energy decreased mostly due to collisions with hydrogen atoms that exist in soil water. The source of the fast neutrons is usually americium (241 Am) and beryllium (Be). Alpha particles emitted by americium are captured by the beryllium, which emits fast neutrons. Fast neutrons are radially penetrating the soil and collide elastically with the molecules of soil elements. Heavy nuclei of elements (Al, Si, and O) reflect the fast neutrons, so they lose only small part of their energy; but by collisions with hydrogen nuclei (with the same mass as the emitted neutrons) they slowdown, losing part of their energy and dissipating it in the soil. A fraction of the slow (thermalised) neutrons returns to the neutron counter, that is located near the neutrons source (probe). The number of detected slow neutrons per unit time is proportional to the number of water molecules in the soil. Hydrogen molecules are not in the soil water only, but even as a component of organic compound; therefore it is necessary to estimate the relationship between the number of returning slow neutrons and the soil-water content. This relationship constitutes the calibration curve. Equipment (neutron-moisture meter) to measure soil-water content by neutron probe includes a source of neutrons (probe), a detector for the slow (returning) neutrons, a control panel (display) and a container absorbing neutrons (shield), emitted continuously during the life of the neutron source. For example, the half-life of the radium–beryllium source is about 1620 years, so its emitting rate is stable. During measurement, the emitting probe is released from the container and lowered into the access tube. The access tube is made of material that does not influence the neutron flow significantly. Iron and aluminium are usually used. Neutrons thermalise approximately inside a sphere of diameter 20–30 cm (depending on the properties of soil and the probe). The result of measurement is the soil-water content of the sphere’s volume, close to the average value. This is the reason why the soil-water content of the upper (ploughing) layer of soil cannot be measured by this method.

5.2.4 Capacitance Method This non-destructive method measures soil-water content indirectly, by measuring soil dielectric capacity. The principle of this method is the known fact that soil capacitance between two electrodes (the capacity of a condenser) depends on the soil’s relative permittivity, influenced mostly by the permittivity of water. Relative permittivity depends on soil density, its temperature and chemical composition, too. Therefore this method needs calibration for a particular soil. Condenser capacitance can be expressed by the equation: C  ε0 εr

A d

(5.8)

56

5 Soil-Water Content and Its Measurement

where C is condenser capacitance (L−2 M−1 T4 I2 , m−2 kg−1 s4 A2 ), A is the area of electrode (one side) (m2 ), d is distance between electrodes (m), and εr , ε0 are relative permittivity and vacuum permittivity (L−3 M−1 T4 I2 , m−3 kg−1 s4 A2 ). The most frequently used configuration of electrodes is on the surface of the cylindrical probe, with a gap between them. Electromagnetic (EM) field in soil between both electrodes is strongly influenced by soil properties, mostly by water, so the capacity of this condenser changes with soil-water content. The capacitance probe can be moved vertically so that the profile of the soil-water content can be estimated. The low price of the equipment is an advantage of this method, but the necessity of calibration for particular soil is its disadvantage. The method is extremely sensitive to the air gap between the probe and access tube; this phenomenon is important especially for swelling /shrinking soils. Therefore, application of this method is limited.

5.2.5 Electrical Resistance Method Soil as a three-phase system is composed of two non-conductive parts—gaseous phase (air) and solid phase; water is electrically conductive. This means the electric conductivity depends on soil-water content. Electrical conductivity of soil increases with its water content (electrical resistance decreases). The problem is that electrical conductivity of soil increases with increasing temperature, too; the relative soil electrical conductivity increases about 0.01 with a soil-temperature increase of about 1 °C. The typical shape of the relationship between electrical conductivity of soil and its water content is exponential. The electrical conductivity (and resistance, as well) change linearly for the range of small water contents, but, for higher soil-water content, this change is small and nonlinear, therefore the sensitivity of this method at this range is low (Fig. 5.1). The measuring equipment consists of two electrodes embedded in the soil; they may be plates, screens, wires and an electrical conductivity meter. To keep electrodes at a constant distance, electrodes are usually embedded in porous blocks, most often made of plaster. After installing such blocks into the soil, a relatively long time (days) is needed to reach equilibrium between the water in the soil and in the plaster blocks. An equilibrium state of soil-water potentials should be reached; therefore this method in reality is a method for soil-water potential measurement. To use this method to measure soil-water content, calibration for a particular soil is needed. This method of measurement is sensitive to hysteresis, i.e. the equilibrium soil-water content depends on the manner of soil-water movement, i.e. if the soil water content is increasing or decreasing.

5.2 Measurement of Soil-Water Content

57

Fig. 5.1 Electric conductivity of soil (1/R) and volumetric soil-water content (θ). Linear relationship in the range of low soil-water content (I) is followed by the range of low sensitivity of measuring devices at higher soil-water contents (II)

5.2.6 Time-Domain Reflectometry (TDR) Method Time-domain reflectometry, briefly TDR as a method of soil-water content measurement is based on the recognition that the soil electrical and magnetic properties affect the propagation of electromagnetic (EM) waves through the soil. The “reflectometry” was first used to identify the location of metal-wire interruption by measuring the time interval between emissions of EM signals and their identification after reflecting from the point of wire interruption. Knowing the time between the EM signal emission and its return, the distance from the emitter of EM to the point of wire interruption can be estimated. The EM signal velocity is v, the time of EM propagation to the end of the wire and back is t and then the distance from emitter to the point of wire interruption is l: l

vt 2

(5.9)

The relationship between impulse-propagation velocity v and the dielectric permittivity εr of the material around the wire is: c v√ εr

(5.10)

where c is the velocity of light in a free space (c  300 × 106 m s−1 ), and εr is apparent relative permittivity (dimensionless) (−), which quantifies the influence of

58

5 Soil-Water Content and Its Measurement

surrounding material on electrical field propagation. Relative permittivity is the ratio of surrounding permittivity to the permittivity of vacuum: εr 

ε ε0

(5.11)

where ε, ε0 are the permittivities of the surrounding medium and of the vacuum. Combining Eqs. (5.9) and (5.10) the relative permittivity εr can be expressed by the equation:  εr 

ct 2l

2 (5.12)

Knowing the time t (travel time) and relative permittivity of the surrounding media εr , the length of wire to the point of its interruption l can be calculated from Eq. (5.12). Application of TDR method to soil-water content measurement This method is described in detail by Ferré and Topp (2002) and Radcliffe and Šimunek (2010). The principle of TDR method application to the soil-water content measurement is based on the knowledge that the relative permittivity of water (in the range above the 10 MHz EM-field frequency) is much higher than the relative permittivity of other soil components. Water’s relative permittivity is approximately 80, the relative permittivity of the solid phase of mineral soils is in the range 3–7, and the relative permittivity of air is εr = 1. This means that the EM-impulse velocity is influenced mostly by water. A molecule of water is oriented in an EM field, i.e. polarized, and this depends on the EM-field’s properties. Energy consumed to “orient” water dipoles slows down the EM field-propagation velocity. The more water dipoles surround the wire (or metal rod) the smaller is the EM-field velocity propagation. To measure soil-water content, the original equipment for estimating wire interruption was modified. To increase the sensitivity of the equipment, two or three metal rods (length in the range of 10–30 cm) are inserted into the soil and connected to the impulse generator and equipment for estimating the travel time between impulse generation and return (Fig. 5.2). The resulting soil-water content is displayed on the screen. In the equipment, calibration relationship, i.e. relationship between volumetric soil-water content θ and relative permittivity of a soil, εr is involved. The relative permittivity can be calculated by Eq. (5.12) when travel time t and the lengths of the metal rods L are known. The relative permittivity εr is substituted into Eqs. (5.13) or (5.14), and the volumetric soil-water content θ can thus be calculated. Those calculations are done by the device, which is an integral part of it. Topp et al. (1980) published the equation sufficient for calculating the volumetric soil-water content of the wide range of soil textures: θ  −5.3 × 10−2 + 2.92 × 10−2 εr − 5.5 × 10−4 εr2 + 4.3 × 10−6 εr3

(5.13)

5.2 Measurement of Soil-Water Content

59

Fig. 5.2 Type of sensors used to measure soil-water content by TDR and FDR methods. 1—wire, 2—metal rods for insertion into the soil, 3—handle

Topp and Reynolds (1998) proposed this relatively simple equation to calculate θ: √ θ  0.115 εr − 0.176

(5.14)

The relative differences between the results obtained by both equations are less than 0.01. Both are applicable for the majority of soil types. But, calibration is needed especially for organic soils, soils of small densities and for soils containing a high content of clay minerals. Actual TDR equipment for measurements of soil-water content is reliable and inexpensive, therefore suitable for wide application. Their advantage is rapid, nondestructive measurement, enabling continuous electronic output and recording, so therefore they are widely used. Their disadvantage is the need for calibration, especially under nonstandard conditions. In reality, calibration of the equipment is often neglected, instead relying on factory standard calibration. Metal rods should be carefully installed to provide the securely tight contact between soil and rods. It is not an easy task in swelling (and shrinking) soils. They are not suitable for stony soils because of problems with inserting relatively long rods into the soil.

5.2.7 Frequency-Domain Reflectometry (FDR) Method Frequency-domain reflectometry (FDR) method is another frequently used method to estimate soil- water content. In the actual literature, it is often referred to as the capacitance method or capacitance/FDR technique (Bogena et al. 2007; Vaz et al. 2013). The FDR sensor emits electromagnetic waves and measures the frequency of reflected waves, which correlate with soil-water content (Hamed et al. 2006). An advantage of the FDR method is the need for only short metal rods (approximately

60

5 Soil-Water Content and Its Measurement

6 cm). The FDR sensors operate at lower frequencies (~20–300 MHz) than TDR (GHz), they are lower cost, easily inserted into the soil, but more sensitive to effects of salinity, temperature and soil textural variations (Vaz et al. 2013). Due to the smaller dimension of the device, the measurement volume is also smaller than that of TDR. Therefore, it is less representative, and more sensors are needed to be installed at the location. Electromagnetic sensors (EM) like TDR, FDR or other similar devices can measure soil-water content continuously at various depths, as well as at various points. Comparison of various electromagnetic sensors and examples of their calibration can be found, e.g. in Jones et al. (2005), Czarnomski et al. (2005), Blonquist et al. (2005), Bogena et al. (2007), Rosenbaum et al. (2010) and Vaz et al. (2013).

5.2.8 Geophysical Methods Geophysical methods have been primarily developed to measure geological discontinuities of bedrocks. Later they were found to be sensitive to soil-water content, so they now are frequently used to estimate soil-water content and also hydrophysical properties and their spatial variability on the field scale. Geophysical methods measure electrical conductivity (EMI—electromagnetic inductance) or resistivity (ERT—electrical resistivity tomography) of porous media. Both properties are strongly influenced by soil salinity, type and content of clay minerals, porosity and soil temperature (Popp et al. 2013). They are also influenced by rock fragments in a soil (especially mountainous); therefore calibration of the method for a particular soil is needed. Ground-penetrating radar The principle underlying the method is evaluation of surface-soil layer properties based on analysis of the high-frequency electromagnetic waves (3–30 GHz) radiated; reflected waves are detected by a detector. This method is used in agriculture for fast evaluation of the spatial variability of soil-water content of the surface-soil layer. More details concerning geophysical methods can be found in Butler (2005).

5.2.9 Remote-Sensing Methods Remote sensing methods belong to a group of contact-free measurement techniques. This includes passive microwave radiometers, synthetic aperture radars, scatterometers and thermal methods. They can operate on the ground but also from airborne or spaceborne platforms (Vereecken et al. 2008). Their main limitations are problems with spatial averaging of measured data and the short penetration depth, so they are sensitive to soil-moisture variations in a thin surface layer.

5.2 Measurement of Soil-Water Content

61

Remote-sensing methods in combination with other soil-moisture measurement methods can provide valuable complex information about soil-moisture dynamics on larger scales with high temporal and spatial resolution.

References Blonquist JM, Jones SB, Robinson DA (2005) Standardizing characterization of electromagnetic water content sensors: Part 2. Evaluation of seven sensing systems. Vadose Zone J 4:1059–1069 Bogena HR, Huisman JA, Oberdörster C, Vereecken H (2007) Evaluation of a low-cost soil water content sensor for wireless network applications. J Hydrol 344:32–42 Butler DK (2005) Near-surface geophysics. In: Butler DK (ed) Society of exploration geophysicists. Tulsa, Oklahoma, USA, p 732 Czarnomski NM, Moore GW, Pypker TG, Licata J, Bond BJ (2005) Precision and accuracy of three alternative instruments for measuring soil water content in two forest soils of the Pacific Northwest. Can J For Res 35:1867–1876 Ferré PA, Topp GC (2002) Time domain reflectometry. In: Dane JH, Topp GC (eds) Methods of soil analysis: part 4, physical methods. SSSA, Madison, WI, pp 434–446 Hamed Y, Samy G, Persson M (2006) Evaluation of the WET sensor compared to time domain reflectometry. Hydrol Sci J 51:671–681 Hillel D (1982) Introduction into soil physics. Academic Press, New York Jones SB, Blonquist JM, Robinson DA, Rasmussen VP, Or D (2005) Standardizing characterization of electromagnetic water content sensors: part 1 methodology. Vadose Zone J 4:1048–1058 Kutílek M, Nielsen DR (1994) Soil hydrology. Catena Verl, Reiskirchen Popp S, Altdorff D, Dietrich P (2013) Assessment of shallow subsurface characterization with non-invasive geophysical methods at the intermediate hill-slope scale. Hydrol Earth Syst Sci 17:1297–1307 Radcliffe DE, Šim˚unek J (2010) Soil physics with HYDRUS, modeling and applications. CRC Press, Boca Raton Rosenbaum U, Huisman JA, Weuthen A, Vereecken H, Bogena HR (2010) Sensor-to-sensor variability of the ECH2 O EC-5, TE, and 5TE sensors in dielectric liquids. Vadose Zone J 9:181–186 Topp GC, Reynolds WD (1998) Time domain reflectometry: a seminal techniques for measuring mass and energy in soil. Soil Tillage Res 47:125–132 Topp GC, Davis JL, Annan AP (1980) Electromagnetic determination of soil water content: measurement in coaxial transmission lines. Water Resour Res 16:574–582 van Genuchten MT (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Vaz C, Jones S, Meding M, Tuller M (2013) Evaluation of standard calibration functions for eight electromagnetic soil moisture sensors. Vadose Zone J. https://doi.org/10.2136/vzj2012.0160 Vereecken H, Huisman JA, Bogena H, Vanderborght J, Vrugt JA, Hopmans JW (2008) On the value of soil moisture measurements in vadose zone hydrology: a review. Water Resour Res. https:// doi.org/10.1029/2008WR006829

Chapter 6

Soil-Water Potential and Its Measurement

Abstract Soil-water potential expresses quantitatively the energy of soil water (of water in porous medium); the difference between soil-water potentials of two points is necessary a condition to transport soil water from the point of higher to the point of lower soil-water potential. In this chapter, soil-water potential is defined and expressed as a negative value because of binding forces between soil and water. Total soil-water potential and its components (gravitational, matric and pneumatic) are defined, and principles of methods of their measurement are described. The principles underlying piezometer measurement of positive pressure potentials and tensiometer measurement of soil-matric potentials are analyzed, and the techniques of their application in the field are discussed.

6.1 Energy of Soil Water Soil water can move between two points of porous media if there is the difference of potentials between them (the second condition is the non-zero hydraulic conductivity of soil). Water potential is defined as the energy of a unit quantity of soil water at a certain point. The difference between water potentials at two points initiates the soil-water movement from the point where the energy of water is higher, to the point where it is lower. The rate of movement of soil water is proportional to the waterpotential difference. The proportionality coefficient, characterizing a porous medium (soil) property, is called the conductivity of the porous medium. Physics recognizes two forms of energy: potential and kinetic. Kinetic energy is proportional to the square of the water’s velocity; potential energy is proportional to the position of a certain point relative to the reference level. Kinetic energy represents not only the energy of macroscopic movements of water, i.e. its flow, but even the microscopic (heat) motion of water molecules. Microscopic movement of molecules is usually chaotic, oscillatory movement, and its rate is proportional to the temperature. The movement of molecules stops at absolute zero (−273 °K). This type of movement in isothermal conditions usually does not initiate macroscopic movement and usually is neglected. © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_6

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Mechanical energy is the sum of kinetic (W k ) and potential (W p ) energy and initiates mechanical movement. Kinetic energy of a body W k (L2 T−2 , J kg−1 ) characterizes its mechanical movement and is measured by the work done by the body of mass m, moving by velocity v, up to a full stop. Wk 

mv2 2

(6.1)

Potential energy W p (L2 T−2 , J kg−1 ) is a part of the mechanical energy; it depends on the configuration of the system, i.e. on the relative positions of the system components and on their position in the outer force field. It is measured by the work done by the forces (internal and external) transiting parts of the system from its actual configuration to another, reference configuration, where the energy of the system is assumed to be zero. The reference (zero) configuration is chosen arbitrarily; therefore it is possible to measure the potential-energy changes only, but not the absolute value.1 Potential energy W p of a body of mass m in a gravitational field at height z above the reference level can be expressed by the equation: Wp  m · g · z

(6.2)

where m is a mass of a body (M, kg); g is acceleration of gravity (LT−2 , m s−2 ); and z is the height above the reference level (L, m). Potential energy of a system W p can be expressed by the equation: W p  W p,out + W p,in

(6.3)

where W p,out is potential energy of a body (water) due to outer forces, not part of the system itself (gravitational forces), and W p,in is internal potential energy of the system due to interaction between components of the system (internal forces). Internal energy of the system depends on the state (pressure, temperature) of the system only, not on the outer force fields. The state and movement of water in capillary porous medium (soil) is determined mostly by the potential energy of water differences between points of this medium. Why? The rate of water movement in soil saturated with water is usually very small; even in sandy soils, it is slower than 1.0 m d−1 (1 × 10−5 m s−1 ); but, in soil unsaturated with water, it is usually slower than 1.0 cm d−1 (1 × 10−7 m s−1 ). Kinetic energy of a unit quantity of flowing water (saturated soil) is W k  5 × 10−11 J kg−1 , and in 1 Potential

energy of soil water (soil-water potential) always refers to the reference level; it is socalled “standard water”. It is clean water (not containing dissolved compounds) in a gravitational field, at atmospheric pressure and standard temperature. The reference level is usually the free water level.

6.1 Energy of Soil Water

65

unsaturated soil W k  7 × 10−15 J kg−1 . Potential energy of a unit quantity of water at height 1.0 m above the reference level is W p  10 J kg−1 , the same in both cases, because it does not depend on the water velocity. The difference between potential and kinetic energy of water under given conditions is more than eleven orders of magnitude; therefore, the kinetic energy of water flowing even in saturated soil is usually neglected. As an example, consider the potential energy of water at height z  1 m; this energy equals the kinetic energy of the flowing water at velocity 4.5 m s−1 (390 km d−1 ); this is approximately twice of maximum velocity of the Danube River in Vienna or Bratislava.

6.2 Soil Water Potential of a Capillary Porous Medium (Soil) The concept of soil-water potential is one of the most important ideas in soil hydrology. This concept was introduced by Buckingham (1907) as a hypothetical quantity expressing the soil-water energy. To formulate his famous equation (today it is known as the Darcy–Buckingham equation), he used a formal analogy to the equations expressing the transport of heat (Fourier equation) and electric current (Ohm equation). It is interesting that he probably did not know the equation expressing the flow velocity in saturated porous medium (Darcy equation). Water will flow in an unsaturated porous medium if there is a difference between the potentials at two points in the direction of decreasing value of soil-water potential, or (which is the same) against the gradient of the potential. Buckingham did not explain in his publication how this potential introduced should be measured. It is clear that heat is transported when there is a difference in temperature between two points; therefore, the potential of transport involves the temperature of the body. The potential of electric current is voltage (or electric-charge concentration); for water flow in open channels, saturated porous medium or tubes, water potential it is water pressure. The water flows due to differences in water pressure between two points. The difference of water pressure (pressure drop) per unit of length in the flow direction is the gradient of pressure. The flux of water is directly proportional to the hydraulic gradient. What is the potential of water flow in an unsaturated capillary porous medium (soil)? What kinds of forces are holding water in the capillary porous media (soil) unsaturated with water? The wet walls of old buildings are an often observed phenomenon. They are wet all year long. Why? Let us fill with dry sand a transparent tube of large diameter and a height of some meters with mesh at its bottom (Fig. 6.1). Let us put it into a container with water of constant height h1 . What will happen? Water will rise quickly up the tube (cylinder) to the level corresponding to the water table in the container. It can be observed as the upward movement of the dark area in the sand. After that, the advance of water

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Fig. 6.1 The cylinder filled with soil dipped in a container with water (a). Piezometers (1, 2) measure positive pressures of water, tensiometer (3) measures negative soil water pressure. Soil-water matric potential distribution (b) and corresponding soil water content profile (c). At equilibrium, the soil-water content distribution is identical to soil-water retention curve

uptake will not stop, but will be slower. The contrast between dark (wet) and dry part of the sand will become smaller; and, after some hours (or days), the contrast at the wetting front will not be observable and the water uptake rate will be negligible. Then, the soil-water distribution along the tube with the sand is in a (quasi) equilibrium state (Fig. 6.1c). This is approximately the state, corresponding to the water-content distribution observed on the wet walls of the house. It is known that the potential energy of water increases with height z above the reference level, and water should flow downward. But this was not observed, and water is held in the sand for a long time. What is the force (or forces) holding the water at a significant height above the water table? The equilibrium distribution of soil-water content in the tube filled with sand is depicted in Fig. (6.1c). The reason for this distribution of soil-water content along the sand sample must be the energy of attraction between water and the solid phase of soil not dependent on the position above the reference level, but only on forces of interaction between water and solid phase. Those interactive forces are in equilibrium with the potential energy of water, increasing proportionally with the height above the reference level. Experience shows the decreasing equilibrium soil-water content above the reference level. This means that the forces of attraction between solid phase of soil and water are not able to attract more water than was measured at the particular height in an equilibrium state. Equilibrium distribution of soil-water potential above the water table is shown in Fig. (6.1b); it corresponds equilibrium soil-water content distribution in Fig. (6.1c). Water pressure in soil below the water table in the container is positive (maximum pressure height is at the bottom of the tube with sand) and can be measured by a piezometer (1), which is an instrument to measure positive pressures. The positive pressure below the water table increases linearly, and soil is saturated with water (its water content is saturated soil-water content θ s ) . Piezometer (2) above the water table (in an unsaturated soil) does not show any pressure, i.e. the air pressure equals

6.2 Soil Water Potential of a Capillary Porous Medium (Soil)

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the atmospheric pressure, and there is no connection between the water in the sand and in the piezometer. In unsaturated soil with a negative pressure of soil water, a piezometer does not work (Fig. 6.1b). Negative pressure of water in soil pores can be measured by an instrument called a tensiometer (Fig. 6.1a, 3). The principle of the tensiometer is the equilibrium between soil water and water in the tensiometer via a semiconductive membrane (or porous desk). The semiconductive membrane (or porous plate) is saturated with water and conductive for water but not for air. Understanding and explanation of the term soil-water potential is not easy because it is influenced by various force fields. The total energy of soil water is the result of forces like adsorption forces between the liquid and solid phase of soil, capillary forces and the force of gravity. Capillary forces depend on water surface tension which depends on the liquid temperature; a high concentration of dissolved compounds can strongly influence physical properties of a soil solution and the capillary forces as well. The energy of soil water as a result of the action of all actual force fields is the total soil-water potential. Total soil-water potential is the sum of components of a particular soil-water potential.

6.3 Quantitative Expression of Soil-Water Potential The soil-water potential can be expressed in various ways, depending on the dimension of the unit quantity of water in which soil-water potential is expressed. The frequently used quantity units are: mass, volume and weight. 1. Energy of soil water (J) per unit mass of water (kg): ψt 

J kg

(6.4)

2. Energy of soil water (J) per unit volume of water (m3 ): ψt 

J Nm N  3  2  Pa m3 m m

(6.5)

The unit of energy is the Joule (J); the unit of force is the Newton (N). Pressure is defined as a force per unit area and its unit is the Pascal (Pa). 3. Energy of soil water (J) per unit of weight (ρw g); (hydraulic head): ψt 

Nm J  m ρw g N

(6.6)

where ρ w g is the weight of a unit volume of water, expressed in force units, in Newton’s (N). As can be seen in Eq. (6.6), the total potential of soil water is expressed in length units and is denoted as the water-pressure head. This form of soil-water potential expression is used most frequently because is easy to

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understand, and its meaning is physically clear. To analyze a porous medium saturated with water, pressure heads are always positive, but for porous media unsaturated with water, pressure heights (or pressure heads) are always negative; therefore, sometimes the incorrect term “soil suction” or “suction pressure” are used. It is simply negative pressure; it means the force vector is oriented out of the unit area under consideration, not to the unit area as it is under positive pressure. Simply speaking, a negative pressure head is not pressing the soil water, but “pulling” it. To express soil-water potential in other units is easy. The relationship between soil-water potential expressed by the pressure of water column p and by the pressure head h is: ψt  p  ρw g h ψt h ρw g

(6.7) (6.8)

where ρ w is water density (kg m−3 ).

6.4 Components of the Total Soil-Water Potential 6.4.1 Gravitational Component of the Total Soil-Water Potential Bodies on the Earth are attracted to the Earth’s centre by a force called gravitational. Gravitational force equals the body’s weight. Gravitational potential of soil water is defined as the energy difference of a unit quantity (volume or mass) of clean water at standard conditions (water table) and at the height z above the standard level in the gravitational field of the Earth. Usually, the symbol ψ z is used. Let us consider the equilibrium soil-water content profile above the reference water table (see Fig. 6.1c) and assume the unit quantity of water is at a height z above the water-table level. The water is clean, i.e. it does not contain dissolved compounds, its temperature is constant, air pressure in the soil is atmospheric; all the forces acting on the water are in equilibrium. There are two kinds of forces in equilibrium: gravitational forces and forces of attraction between the solid phase of soil and the water. The energy of attraction between the solid phase of soil and water at some height z is called the matric potential of soil water (sometimes the term moisture potential is used). This component of soil-water potential is in equilibrium with gravitational forces, or with gravitational component of total soil-water potential. Let us denote the matric potential of soil water by the symbol ψ w . Let us try to express it in the language of mathematics. To move a unit quantity of water m from the reference level z0 to the level z, the work done can be expressed

6.4 Components of the Total Soil-Water Potential

69

by the formula m.g.(z − z0 ), and the gravitational component of the total soil-water potential (briefly soil-water potential) will increase by this quantity. Because the reference level with zero matric (moisture) potential is z0 , the gravitational component of soil-water potential (gravitational potential) ψ z of unit mass of soil water can be expressed by the equation: ψz  m g(z − z 0 )/m

(6.9)

The gravitational potential of soil water can be expressed even for a unit volume of water V  1: ψz  m g(z − z 0 )/V  ρw g(z − z 0 )

(6.10)

where ψ z is gravitational component of soil-water potential (gravitational potential) expressed in Newtons per meter squared (ML−1 T−2 , Nm−2 ), or in Pascals (Pa), z is the height above the reference level (L, m), z0 is the height of the reference level (L, m), m is unit mass of water (M, kg), g is acceleration of gravity (LT−2 , m s−2 ), and ρ w is water density (ML−3 , kg m−3 ). Gravitational potential per unit of water weight can be expressed as: ψz  m g

(z − z 0 )  (z − z 0 ) mg

(6.11)

The system is in equilibrium, therefore the matric (moisture) potential expressed by the pressure head of water is the difference between the reference level and the level at which the matric potential of soil water is being evaluated. If, the reference level height will be assigned as “zero”, z  0: ψz  z

(6.12)

Then, the value of matric potential of soil water expressed by the pressure height (pressure head) is equal to the height above the reference level where the analyzed water unit is located. The soil/water content profile must then be in equilibrium, i.e. it does not change in time.

6.4.2 Matric (Moisture) Potential of Soil Water This component of total soil/water potential characterizes the interaction of the solid phase of soil and water in the porous system. The term matric potential is preferred to the traditional term moisture potential because it better characterizes the action of the force fields on soil water. The adjective “matric” has its origin in the word “matrix” or “matric” which means soil solid-phase spatial arrangement, something like “soil skeleton”. Porous space is the area surrounded by the solid phase of a soil. Size and

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shape of soil pores are estimated by the dimensions and arrangement of primary solid particles. Those pores are named micropores in contrast to the macropores comprised of biopores, cracks and interaggregate voids. Their dimensions are bigger in comparison to the micropores. The generally accepted opinion is that gravitational forces are dominant in macropores, but capillary and adsorption forces are dominant in micropores (Kutílek and Nielsen 1994). Matric (moisture) potential of soil water can be defined as the difference between the energy of a unit quantity of soil water at standard conditions (soil saturated with water) and the energy of water in unsaturated soil and therefore influenced by the forces of interaction between solid and liquid phases of water. Matric (moisture) potential of soil water can be denoted by the symbol ψ w . Matric potential is defined for unsaturated soil only and is always negative. The reason of the negative sign of the matric potential is that the work ψ w has to be done by external forces to transport a unit quantity of water from the z level to the standard level z0 . If the energy is expended moving water to the reference level, then (according to the convention) the soil-water potential is negative. In porous media saturated with water (soil), the soil-water potential is positive as measurable by a piezometer, and water flows spontaneously against the direction of the pressure gradient, thus doing the work.

6.4.3 Pneumatic (Pressure) Soil-Water Potential Pneumatic soil-water potential as a component of total soil-water potential characterizes the influence of various air pressures at various points of soil on the soil-water movement. Pneumatic soil-water potential can be of importance if the air pressure in pores is other than atmospheric pressure. This situation is observed frequently because atmospheric air pressure changes continuously and is never the same, even during the same day. These changes are small in comparison to the matric potential values, and soil-water movement caused by the atmospheric pressure changes is small and can be neglected. But, the significant influence of different soil air pressures on soil water-movement can be observed during ponded infiltration. As water infiltrates into soil, the air pressure before the infiltration front increases, and the infiltration rate decreases. The pressure of entrapped air before the infiltration front can reach a value higher than the matric potential of the soil water at the infiltration front, so entrapped air can move (bubble) through the layer of infiltrated water and escape the soil. The pressure height corresponding to the “bubbling” pressure is hb . This phenomenon (bubbling) can be easily observed when irrigating plants in flower pot; under field conditions, it is rare because ponding infiltration in the field is rare too. Pneumatic potential ψ a is defined as the difference in energy of a unit quantity of water (usually in mass units) under standard conditions (free water level, atmospheric pressure, isothermal conditions) and in unsaturated soil, where the air pressure in pores is Psa and is different than atmospheric pressure Pa :

6.4 Components of the Total Soil-Water Potential

ψa  Psa − Pa

71

(6.13)

The pneumatic potential of soil water in pressure-head (height) units ha (L, m) can be expressed by the equation: ha 

Psa − Pa ρw g

(6.14)

The pneumatic potential component of total soil-water potential is not used frequently, and it is usually limited to ponding infiltration. But, as will be shown later, application of air pressure to the soil sample makes possible evaluation of the soilwater matric potential, corresponding to the pneumatic potential applied to the soil sample. This phenomenon is used in the pressure method for soil-water retention curves measurement.

6.5 Total Soil-Water Potential Total water potential of a soil unsaturated with water is simply the sum of its components. Generally, total soil-water potentialψ t can be expressed as: ψt  ψa + ψz + ψw

(6.15)

All components are expressed in pressure units, but they can also be written in pressure head (height) units: ht  ha + hz + hw

(6.16)

Pneumatic potential of soil water is rarely of importance; therefore total soilwater potential is usually expressed as the sum of matric and gravitational potential components; the total soil-water potential using those two components is used for the majority of cases: ht  hz + hw

(6.17)

Matric potential is usually the most important component of the total soil-water potential, especially at relatively low levels of soil-water content, when the other components of soil-water potential are low. The gravitational component of the total soil-water potential is important especially at relatively high soil-water contents. When quantifying water flow in saturated porous media (soil), the gravitational component of total soil water potential is dominant. Matric potential is not defined in saturated porous media (soil). In older literature other components of the total soil-water potential can be found. One of them is “osmotic” potential, the second “overburden” potential. “Osmotic”

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potential characterizes quantitatively water transport from a solute of higher concentration through a so-called “semipermeable” membrane, to the solute of lower concentration. The term “semipermeable” membrane here is meant the membrane conductive for water molecules, but not for molecules of dissolved compounds. This “semipermeable” membrane does not occur in the soil, therefore it does not influence soil water movement, instead it is located at the interface of soil-plant roots. Osmosis as a process will be described later. The term “overburden” potential means the influence by mechanical pressure on the soil (heavy machinery) of the soil porous system. The porous system, as changed under this pressure, constitutes a different porous medium with different matric potential. Therefore, it is meaningless to introduce a new term for this component of soil-water potential. Soil temperature and its variations can influence the properties of soil water and soil air. These differences based on temperature changes can strongly influence the soil’s hydrophysical characteristics and its transport processes (Novák 2012). But to define something like “temperature soil-water potential” is not practical; the better way is to express the temperature influence on the soil-water transport by the soil’s hydraulic characteristics (soil-water retention curves, hydraulic conductivities of soil water) as a function of temperature. Concentrations of dissolved compounds in principle can influence the properties of soil water (density, viscosity, surface tension) and thus the soil-solution transport. As was mentioned previously, observed concentrations of dissolved compounds are normally so low as not to significantly influence soil matric potential and the movement of soil water. In extreme cases (soil temperature, solute concentration), soil matric potential can be expressed as a function of temperature and solute concentration. This is the way to simplify understanding and calculation of soil-water transport.

6.6 Measurement of Soil-Water Potential Components Direct measurement of total soil-water potential is technically impossible; therefore, components of total soil water potential are measured. The most important component is soil-water matric (moisture) potential. The measurement method depends on the measurement conditions (field or laboratory) and on the range of matric potentials to be measured.

6.6.1 Piezometer Piezometers are simple equipment to measure pressure below the water-table level, usually below the groundwater table. Measured pressures are positive. Hydraulic heads H measured by the piezometer in the groundwater are close to hydrostatic heads, because groundwater-flow velocity is usually small. Hydrodynamic compo-

6.6 Measurement of Soil-Water Potential Components

73

Fig. 6.2 Piezometer indicates the position of the water table below soil surface (groundwater table, GWT). Hydraulic height H is the sum of the pressure height h and the height between reference level and the bottom of the piezometer z, which depends on the arbitrarily chosen reference level, where z  0

nent of the pressure head can be of importance when measuring hydraulic head in rivers or open channels. In Greek “piezein” means pressure, and the word piezometer denotes an instrument to measure pressure. A piezometer is a tube inserted into prepared hole in soil; the tube has both ends open for water/air flow (Fig. 6.2). The bottom opening of the piezometric tube must be below the groundwater table. When placing piezometer into the soil, the water will flow in the tube from the bottom up to the stable position of the water level in the tube; then the water table in the piezometer corresponds to the water table surrounding the device. The upper end of the piezometric tube should not be airtight so as to allow the air in the tube to freely escape. The pressure head at the level of the piezometer’s bottom is h (see Fig. 6.2). Hydraulic head H is defined by the sum of the pressure head (h) and the height of piezometer’s bottom above the reference level (z): H h+z

(6.18)

Reference level z  z0 can be arbitrarily chosen; in the case of its location at the bottom of the piezometer, Eq. (6.18) can expressed as H  h. By piezometric measurement, not only can the groundwater-table level (GWT) be estimated, but also the direction of GW flow. More piezometers should be installed, located around the point of interest. After a stationary state of groundwater level in piezometers is reached, GWT levels are estimated and isolines of pressure drawn. The groundwater-flow direction will be perpendicular to the isolines. The GWT level can change during measurement, so the water-table levels in the piezometers will change correspondingly, but with some delay; the time lag between GWT changes in the field and in piezometers should be taken into account.

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6.6.2 Tensiometer The term “tensiometer” reflects the use of the device; the word “tension” means socalled negative pressure or tension. It is the device most frequently used to measure soil matric potential. The underlying principle is illustrated in Fig. 6.3. A porous, usually ceramic sensor (1) is saturated with water; its pores should be filled with water during the measurement of soil matric potential, i.e. measured negative pressures should be always higher (less negative), than bubbling pressure hb , when water is expelled from the pores. Otherwise the tensiometer cannot work or accurately measure matric potential. A manometer (2) is the part of a tensiometer. When equilibrium is reached between soil- water tension and the tension in the tensiometer, the manometer shows a negative pressure (or pressure height) hw . A commercial tensiometer (Fig. 6.4) is composed of a glass or plastic tube (4); at one (bottom) end is a ceramic, glass or metal sensor (1). The second end of the tensiometer is closed by stopper (3). The porous ceramic sensor should be saturated with water so that liquid water can flow through it. The tensiometer, fully filled with water, is inserted into the soil along the pre-drilled hole. Care should be taken to make good contact between the porous sensor and the soil. Prior to tensiometer insertion into the soil, the water in the tensiometer is at atmospheric pressure, but after its

Fig. 6.3 Tensiometer, the device for measurement of soil-water matric potential. Porous body of the sensor (1) is in tight contact with soil. Manometer filled with water (2) in equilibrium indicates the soil-water matric potential (negative pressure height) hw

6.6 Measurement of Soil-Water Potential Components

75

Fig. 6.4 Commercially produced tensiometer with metal manometer (2). Tensiometer is filled with water (4) and closed by stopper (3). The area around the sensor (1), marked by dashed line, indicates the area of measurement. To measure the soil-water matric potential, indicated by the manometer (2), it is necessary to add (positive) pressure height z to estimate an accurate value of matric potential in the measured area

insertion into the soil, water will flow to the soil, via the ceramic sensor, because soil water is usually at lower negative pressure. By reaching an equilibrium state (the negative pressure gauge of manometer does not change), it is possible to measure the matric potential of soil water. The disadvantage of tensiometric measurements of soil-water matric potential is the relatively narrow range of measured matric potentials of soil water, thus limiting its application. Theoretically, the measured potentials should be in the range (0 up to −100 kPa), but practically, it is 0 − (−60) kPa only. The reason is usually caused by problems related to soil-sensor contact, the tightness of individual parts of the tensiometer, and even the escape of diluted air from the water in the tensiometer when water pressure changes. A tensiometer can be applied in soils with relatively high values of matric potential of soil water, which limits its use to relatively wet soils. Tensiometers as equipment for soil matric potential measurement are still improved. Furthermore, new devices like e.g. dielectric water potential sensors can now provide larger measured range up to −100,000 kPa.

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References Buckingham E (1907) Studies in the movement of soil moisture. US Dept Agric Bur Soils, Bull 38 Kutílek M, Nielsen DR (1994) Soil hydrology. CATENA, Cremlingen–Destedt Novák V (2012) Evapotranspiration in the soil-plant-atmosphere system. Springer Science + Business Media, Dordrecht

Chapter 7

Soil-Water Retention Curve

Abstract Soil/water retention curve (SWRC) is the relationship between soil/water matric potential and volumetric soil-water content at equilibrium above the reference (zero) level represented by the free water table at atmospheric pressure. SWRC is the basic soil characteristic needed as input data for simulation models. SWRCs have a hysteresis-like character, i.e. this relationship is different for drying and wetting processes. Methods for SWRC measurement are described (pressure method, method of hanging column, psychrometric method, adsorption and desorption methods), as well as methods for their evaluation from measured data. Analytical expressions of the SWRC are presented and evaluations of useful “hydrolimits” like monomolecular water capacity, wilting point, available water capacity and field capacity using SWRC data are described.

7.1 Definition of the Soil-Water Retention Curve As was shown in the previous chapter, soil-water matric potential (or the energy of attraction between soil water and the solid phase of soil) depends on soil-water content. The lower is the soil-water content, the stronger are the attraction forces between both phases. The equilibrium distribution of soil-water content above the reference level (free water-table level) is known as the soil-water retention curve (Fig. 6.1c). The energy of attraction between soil water and the solid phase of soil is expressed by Eq. (6.12); then, the mentioned energy of attraction is in equilibrium with the potential energy of water. Potential energy of water above the reference level can be expressed by pressure head hw which equals the height of the unit quantity of water above the reference level z; (hw = z). Pressure-head distribution above the free water-surface level is linear, and pressure heads are negative. It is important to measure soil-water retention curves (SWRC) by one of two types of methods. During the drying process (drainage), measurement should start close to a saturated state of the soil; the wetting

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_7

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process starts with dry soil. Because—as it will be shown—the relationship (SWRC) hw = f (θ ) is not unambiguous, it depends on the type of the process (drying or wetting) and on the soil-water content at which the SWRC measurements begin.

7.2 Hysteresis of Soil-Water Retention Curves From the definition, the soil-water retention curve can be estimated by simultaneous measurement of soil-water content (θ ) and matric (moisture) potential of soil water (hw ) in equilibrium, i.e. water is not in motion. To perform such a measurement of soil-water matric potential (by tensiometer or psychrometer) and corresponding soil-water content in the field, the results are characterized by pairs of data (θ i , hwi ), where i means the i-th pair of measured data. Presenting results of the measurement, surprisingly no single and simple curve can be found, but the data points extend over an area delimited by two curves (functions) presented in Fig. 7.1 and denoted by “d” and “w”. The relationship hw  f (θ ) is not unambiguous, but to each particular soil-water content can correspond a wide spectrum of soil-water matric potentials; and on the contrary, to a particular soil-water matric potential corresponds a wide range of soil-water contents. Why is this so? Hysteresis, i.e. the dependence of function on the direction of the process, is a frequently observed phenomenon in physics. To measure the SWRC, the relationship hw  f (θ ) depends on the way that equilibrium has been reached. If the starting point of SWRC measurement is soil saturated with water (θ i = θ s ), where θ s is volumetric soil-water content, then the matric potential of soil water (and SWC, as well) will

Fig. 7.1 Soil-water matric potential hw and volumetric soil-water content θ. Pairs of possible values (θ i , hwi ) are confined by main drying (d) and main wetting (w) branches of the soil-water retention curve. Scanning curves of SWRC (wi , di , and s) are shown too

7.2 Hysteresis of Soil-Water Retention Curves

79

Fig. 7.2 Height of capillary rise hv a and height of water level in a capillary tube at draining hp b in capillaries with regularly changing cross section

be measured in the process of drying (from points A to B; Fig. 7.1), the main drying (desorption) branch of the SWRC will be measured. Starting the measurement with dry soil (point B, curve (w)) by wetting up to point C, the main wetting (sorption) branch of the SWRC can be estimated by consecutive measurements of SWC and soil-water matric potentials at the discrete points at equilibrium state. As can be seen, the loop created by both main branches (“d” and “w”) of the SWRC is usually not closed, but there is a gap between points A and C; this is probably the entrapped air when the porous space is saturated. From the above analysis, it follows that the position of a point (θ i , hwi ) in the hysteresis loop depends on the type of process (wetting, drying), on the initial soilwater content at the starting point of measurement and on the previous history of water-transport processes. If point D has been reached by the drying process, then, the soil will start to wet, and values (θ i , hwi ) will change along the curve wi . The drying process, starting after the wetting phase (point E), will follow curve d i . After a rain event (point F), the relationship hw  f (θ ) will be characterized by the curve “s”, as a scanning curve, etc. It seems to be complicated at first sight, but taking into account the hysteresis phenomenon, soil-water transport processes can be calculated with acceptable accuracy. The hysteresis of SWRC can be neglected in specific cases. What are the reasons for hysteretic behaviour in the relationship hw  f (θ )? The most important are non-regular changes of porous space (capillaries) in which water flows (Fig. 7.2), and the resulting various capillary forces indirectly proportional to capillary diameter. The various contact angles of drying and wetting soil result in different capillary forces, too. Those differences are the reasons for varying locations of the soil-water front in soils. Figure 7.2 represents two capillary tubes with regularly varying diameters. The radius of capillary tube varies in the range r 1 − r 2 . Figure 7.2a illustrates the rising water in the capillary tube; it can reach the height hv , but when the meniscus decreases, it will stop at height hp above the reference level (Fig. 7.2b). The difference between the heights hp − hv is called “Haines jump”, according to Haines (1930), who was the first to describe the hysteresis of the soil-water retention curve. Therefore, during the drainage process, the soil’s retention is higher than during the wetting process,

80

7 Soil-Water Retention Curve

which is in agreement with measurements. The heights of water tables hv and hp in capillary tubes with radius r 1 and r 2 can be expressed by the equations: hv 

2σw cos ϕ 2σw cos ϕ ≤ hp  ρw gr2 ρw gr1

(7.1)

7.2.1 Main Branches of Soil-Water Retention Curves To secure reproducibility and comparability of different soil-water retention curves (SWRC), the relationships hw  f (θ ) are usually measured during the drainage process, starting with soil saturated with water (this is the preferred case), or during soil wetting, then starting with air-dry soil. Both SWRC main branches are typical of a particular soil; the shape and position of the SWRC reflect porous space dimensions, and porous space dimensions reflect soil texture. SWRC drainage (drying) branches of some typical soils from the territory of Slovakia are presented in Fig. 7.3. It shows the curves of soils with various textures; extremely heavy soil with large ratios of clay minerals, containing very small pores (Zemplínske Hradište, the Eastern Slovakia Lowland) up to the sandy soil (Láb site, the Western Slovakia). The SWRC of nearly monodispersed glass sand Fig. 7.3 reflects this “monodispersity” and appears to be a “step-like” function. To understand the physical sense of the SWRC, let us analyse simple physical models of a soil. Three types of porous-media models made of capillary tubes are presented in Fig. 7.4. A model porous medium made of capillary tubes of identical diameters is depicted in Fig. 7.4a; the height of capillary rise (Eq. 7.1) is hk = hb , where hb is the pressure head needed for air to enter into the capillary tubes and to move water out; it can be called the bubbling pressure. Therefore, above the height hb , the model soil will be dry because the radius of capillary tube does not allow water to rise above the height hb . This is the case in a monodispersed porous system, where all the pores are identical with hypothetical diameter of pores d k . The SWRC close to this type of porous media is presented in Fig. 7.3, line 1, characterising monodispersed sand. The model soil composed of capillary tubes of two different diameters is seen in Fig. 7.4b. The SWRC of such a model soil has a two-step form (so-called bimodal porosity). This type of soil comprises soil with two types of pores; it can be compared to soil with relatively big inter-aggregate pores and smaller intra-aggregate pores. The most frequently occurring type of porous medium (soil) has the SWRC curve in Fig. 7.4c. This model soil is composed of capillary tubes with diameters changing continuously, as is commonly seen in field soils. The SWRC of such a soil is continuous function starting at a matric potential expressed by the pressure height hb . The soil layer, saturated with water (or close to saturation) is a capillary fringe; its thickness is hb . The SWRC presented in Fig. 7.3 characterizes soils with pores of continually changing dimensions.

7.2 Hysteresis of Soil-Water Retention Curves

81

Fig. 7.3 Drying branches of soil-water retention curves hw  f (θ) of some typical soils of Slovakia and diameter of pores corresponding to the particular soil-water matric potentials hw . 1—Glass sand, 2—Láb (the Záhorská Lowland), loamy sand, 3—Kostolište (the Záhorská Lowland), sandy soil, 4—Záhorská Ves (the Záhorská Lowland), loamy soil, 5—Trnava (the Podunajská Lowland), loamy soil, 6—Most pri Bratislave (Great Rye Island), loamy soil, 7—Zemplínske Hradište (the Eastern Slovak Lowland), clay

Another soil characteristic is the specific soil-water capacity (c(w)  dθ /dhw ). By its shape, the specific soil-water capacity reflects the soil pores’ size-distribution function. As an example, the specific soil-water capacity of the soil sampled from the site Láb (Fig. 7.3, line 2) is presented in Fig. 7.5 as a function of a pressure head hw . It is the derivative of the function hw  f (θ ). It follows from Fig. 7.5 that the critical soil-water content changes are observed in the range of pressure heights (−100 < hw < −10 cm). The maximum value of this function corresponds with the pore diameters most frequently occurring. A small change of pressure head is necessary to change from wet to dry state of such soil. Negative-pressure heights at which soil-water content is measured usually span five orders, so the values of soil-water matric potentials expressed by the negative pressure heights are usually presented on a logarithmic scale. Analogically with the symbol pH in chemistry, the symbol pF was introduced to represent the decadal logarithm of the absolute value of negative pressure height. Soil-water retention curves in this format are sometimes presented as “pF” curves.

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7 Soil-Water Retention Curve

Fig. 7.4 Models of capillary porous media made of capillary tubes and corresponding soil-water retention curves. a Model of a soil composed of capillaries of the same diameter and corresponding soil-water retention curve (SRWC), b Bimodal model of a soil composed of capillary tubes of two diameters and its SWRC. c Continuously changing diameters of capillary tubes and its SWRC

Soil-water retention curves are temperature dependent. The temperature dependence of water surface tension is the main reason of this dependence. The higher the soil temperature is, the smaller surface tension and capillary height is observed (Novák 1975). Therefore, every SWRC should be tagged with the temperature at which it was measured.

7.3 Soil-Water Retention Curves and Their Analytical Expression

83

Fig. 7.5 Specific water capacity of sandy soil as a function of soil-water matric potential (c(w)  dθ/dhw = f (hw )). Soil from site Láb (the Záhorská Lowland)

7.3 Soil-Water Retention Curves and Their Analytical Expression Soil-water retention curves are a part of basic soil hydrophysical characteristics. SWRC contain information about water-retention capacity of porous medium (soil) and, using SWRC, available soil-water content of plants can be estimated. The analytical expression of the SWRC function (hw  f (θ )) is one necessary input data for mathematical models of soil-water movement as a part of the soil–plant–atmosphere system (SPAS). Soil-water retention curves (Fig. 7.3) are usually of sigmoidal shape. Van Genuchten (1980) proposed characterizing SWRC by an analytical function flexible enough to describe a variety of sigmoidal-shaped functions. This equation is currently the standard function that is used as an input to the mathematical simulation models. This equation can be written as: m  1 θ − θr  (7.2) Se (h)  θs − θr 1 + (α|h w |)n Parameters α, n, m are known as “van Genuchten’s” parameters. Their values are usually in the range 1 ≤ n ≤ 10, 10−3 ≤ α ≤ 10−1 , 0 ≤ m ≤ 1. One can write m  1 − 1/n; where θ , θ r , θ s are volumetric soil-water content, residual soil-water content and soil-water content of soil saturated with water. The term “residual” soil-water content was previously used; it characterises soil-water content in which hydraulic conductivity is small enough to be neglected. The parameter n depends on the soil texture and determines the “shape” of SWRC; parameter α characterises the “position” of the SWRC in two dimensional space. Universality and flexibility of van Genuchten’s equation are the main reason for its worldwide use. The basic characteristics of various SWRCs (shown in Fig. 7.3) are listed in Table 7.1, as well as van Genuchten’s parameters of the related SWRCs.

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7 Soil-Water Retention Curve

Table 7.1 Basic characteristics of soils and van Genuchten’s parameters of soil-water retention curves of typical Slovak soils (Fig. 7.3) Site/soil type

θs (cm3 cm−3 ) θr (cm3 cm−3 ) K (cm d−1 )

α (cm−1 )

n (–)

Glass sand Láb (sandy loam)

0.385 0.375

0.005 0.010

2448.0 676.0

0.028 0.034

7.778 2.070

Kostolište (sandy soil)

0.295

0.012

218.0

0.016

1.752

Záhorská Ves (loam) 0.420

0.020

10.0

0.014

1.362

Trnava (loam)

0.357

0.035

39.6

0.086

1.122

Most pri Bratislave (loam)

0.465

0.045

17.0

0.030

1.161

Zemplínske Hradište 0.505 (clay)

0.060

1.0

0.012

1.092

An older, but still used, analytical function expressing the part of the SWRC in the interval below the soil-water matric potential corresponding to the “bubbling pressure” hb , is the exponential function (Gardner et al. 1970): h w  aθ b

(7.3)

where a and b are parameters of soil-water retention curve estimated empirically. Brooks and Corey (1964) proposed the equation in the form: Se 

θ − θr  θs − θr



hb hw

λ (7.4)

where S e is the effective saturation of soil, hb is matric potential corresponding to the entrance of air into the porous space (bubbling pressure −50 cm ≤ hb ≤ −20 cm) and λ is parameter in the range 2–5. Both Eqs. (7.3) and (7.4) enable expressing analytically the region of SWRC for ranges of soil-water matric potential less than the “bubbling pressure”; (hw ≤ hb ). For pressure heads, hw ≥ hb , S e  1, (Eq. 7.4).

7.4 Hydrolimits Soil-water retention curves are basic data providing information about the energy state of soil water and its availability to plants. For ontogenesis, plants need water and nutrients, but also air for roots’ respiration and all these substances exist in soil. Soil water is attracted in the soil by forces that are smaller than those that the roots attract. Values of soil-water content or matric potentials of soil water, which characterize the state and availability of soil water to plants, are called “hydrolimits” or “soil-water constants”; these terms denote the approximate character of those characteristics.

7.4 Hydrolimits

85

Hydrolimits are not exactly (physically) defined. But, they are useful in practice and are frequently used. The most important hydrolimits are: monomolecular water capacity, adsorption water capacity, wilting point, field capacity, point of limited availability (to plants), plant available water capacity, saturated soil-water capacity. Monomolecular water capacity (θ mac ) is the soil-water content (volumetric) corresponding to the monomolecular layer of water adsorbed by the solid phase of soil; its thickness corresponds to the dimension of one water molecule (2.72 × 10−10 m). The corresponding absolute value of the soil-water matric potential is hw = 106.36 cm. It is assumed to form on the soil in the airtight container above the solution of 58% concentrated H2 SO4 , with a relative humidity above of r = 0.2 in an equilibrium state. This hydrolimit is used to calculate the specific surface of the soil. Adsorption water capacity (θ h ) is expressed in volumetric soil-water content; it is the maximum quantity of water adsorbed by the solid phase of soil from the ambient atmosphere with relative humidity r  0.98 (above the solution of 10% H2 SO4 ). The absolute value of soil-water matric potential corresponding to the adsorption water capacity is hw  104.87 cm. Heat of wetting is released for soil-water contents below θ h . Heat of wetting is the energy lost by water molecules when they fix at the solid phase of soil. Adsorption water capacity indicates water affinity to the solid phase of a soil. The permanent wilting point (θ wp ) is the soil-water content (or it can be expressed by soil-water matric potential) at which plants wilt permanently and, after reaching this point, no further irrigation is effective. The permanent wilting point is characterized by the absolute value of the soil-water matric potential hw  104.18 cm. The permanent wilting point is the oldest hydrolimit (Briggs and Shantz 1912) and, it was measured using the results of pot experiments. Plants were grown under ideal conditions; transpiration was not limited by soil water. At some stage of plants ontogenesis, irrigation was stopped. When the symptoms of wilting were observed, the soil-water content (average) was averaged and denoted as permanent wilting point. If plants grow in non ideal conditions, they are able to adapt to these conditions, and they do not wilt even when reaching the so-called permanent wilting point. In the literature, there is information about plant canopies showing no symptoms of wilting at a soil-water matric potential of −3 MPa. So the hydrolimit of permanent wilting point corresponds to the methodology used by Briggs and Shantz (1912). The wilting point is the soil-water content at which a plant is actually wilting permanently; its value depends on the soil-water regime and on the stage of the plants’ ontogenesis. But, the term permanent wilting point can be understood as the “safe” upper boundary at which the wilting of plants could occur. Nevertheless, this hydrolimit is used often as a criterion of lowest boundary of soil-water availability to plants. Generally speaking, the wilting point is usually estimated as the soil-water content strictly

86

7 Soil-Water Retention Curve

linked to the soil-water matric potential −hw  104.18 cm. The modern view on the relation between biomass production and soil-water content will be described later. Field capacity (θ fc ) characterises the maximum soil water (soil-water content) which can be maintained in soil up to some number of days. As will be shown later, soil can be saturated with water for only a short period. After intensive rain or ponding (full soil-water capacity, saturated soil) , water in macropores will drain fast, but water in micropores drains relatively slowly. Such observations formed the basis for introducing the term field capacity that is the maximum soil water (soil-water content) available to plants due to slow drainage (or evaporation). The difference between the terms field capacity and saturated soil-water content is the water content in macropores, which will drain quickly. Field capacity is measured in the field when the soil surface has ponded: after soil saturation, the soil surface is covered to prevent evaporation. As the soil-water content redistributes, it is measured every day; the soilwater content, which changes relatively slow, is denoted as the field capacity. In the literature, it is possible to find various soil-water matric potentials corresponding to the field capacity that allow its estimation from the SWRC, but to use such data is risky. Field capacity varies for different soils with varying textures, so such values are indicative only. Therefore, to estimate field capacity, field measurements are strongly recommended. Point of limited availability (of soil water for plants) (θ la ) characterises the soilwater content or soil-water matric potential when water flow to the roots is limited and biomass production starts to decrease. It is assumed the value θ la is constant and was calculated from the difference between the field capacity and wilting point:   θla  θwp + a θ f c − θwp

(7.5)

The coefficient of proportionality is in the range 0 ≤ a ≤ 1, depending on the transpiration rate; the value normally used is a  0.6 (Kutílek 1978). It is important to note that the point of limited availability indicates a decrease of transpiration below the potential value (maximum possible or potential transpiration under given conditions); it is related to the decreased photosynthesis rate and to the biomass production (Hanks and Hill 1980; Vidoviˇc and Novák 1987). The point of limited availability of water to plants is therefore a critical parameter. The term θ la is identical to critical soil-water content θ c1 (Chap. 13) for which estimation method is described. Plant available water capacity (θ awc ) is the interval of soil-water contents of the soil root layer available to plants. This usually is measured in the interval of soil-water contents between the field capacity (θ fc ) and the permanent wilting point (θ wp ): θawc  θ f c − θwp

(7.6)

The ranges of hydrolimits of various soils are listed in Table 7.2; the higher values are typical for heavy soils, the lower for light (sandy) soils.

7.5 Measurement of Soil-Water Retention Curves

87

Table 7.2 Hydrolimits and their soil-water content intervals Hydrolimit

Volumetric soil water content (cm3 cm−3 )

Monomolecular adsorption water capacity (θ mac )

0.003–0.03

Adsorption water capacity (θ h )

0.01–0.2

Wilting point (θ wp )

0.02–0.3

Point of limited availability of water (θ la )

0.04–0.35

Field capacity (θ fc )

0.1–0.4

Full water capacity (θ s )

0.25–0.60

7.5 Measurement of Soil-Water Retention Curves Soil-water retention curves, i.e. the relationships between soil-water matric potential and volumetric soil-water content hw = f (θ ), are important characteristics of variably saturated soils. As was mentioned, this relationship is of an hysteretic character. There are two basic (drying and wetting) “main” branches of the relationship hw  f (θ ) and theoretically infinite number of so-called “scanning” curves, located within the hysteresis loop; they depend on the type of water movement. Which part of the relationship hw  f (θ ) should be measured to be useful in quantifying soil-water transport? The main branches of the relationship hw  f (θ ) are the most important bits of information. The periods of precipitation or irrigation during which the soil-water content increases are relatively short in comparison to the periods without precipitation, when soil-water content decreases, i.e. the soil is drying. The average number of rainy days in the Danube Lowland during the summer months is seven; but the number of days without precipitation is three-times greater, therefore the drying of soil is a dominant process in the decisive period of plant growth. This phenomenon and the technically much simpler way of drying main branches of SWRC measurement are the reason why the drying branches of SWRC are usually measured. Application of main drying branches of SWRC as input data into simulation models have led to the acceptable errors of soil-water movement modelling during the entire vegetation period (Novák and Gallová 1998). Speaking about the SWRC, usually the main drying branches are implied if there is no specific indication. The soil-water matric potential of some soil-water content depends also on the soil temperature; so it is necessary to measure SWRCs at a constant temperature, usually 20 °C. Therefore, the laboratory in which SWRCs are measured should be air-conditioned. There are a few methods for estimating SWRC; they can be categorised as tension, pressure, adsorption (desorption) and psychrometric methods.

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7 Soil-Water Retention Curve

7.5.1 Tension Methods To measure soil-water retention curves by the tension method, a negative pressure on the soil sample is applied, frequently by “hanging” column of water (therefore this method is often denoted as “hanging column method”) (Fig. 7.6). Soil water that attracts by forces smaller than F  ρ w .g.hw will flow out of the soil. Knowing the negative pressure height hwi and corresponding volumetric soil-water content θ i , it is possible to collect a series of points (hwi , θ i ) to calculate the drying branch of the soil-water retention curve. The advantage of the tension method of SWRC measurement is the possibility to estimate relatively accurate values of data (hwi , θ i ) within the range of small negative-pressure heights −10 up to −150 cm. This advantage of the tension method is also a disadvantage; the range of the application of negative-pressure heights is theoretically limited to −900 cm, but practically the limit is less than −200 cm. It is limited by the height of the laboratory in which the vertically moved hanging column is situated. See the apparatus to measure SWRC by the tension method (tension-plate apparatus) in Fig. 7.6. Tension-plate apparatus (usually made of glass) contains a semipermeable porous (glass or ceramic) plate (2) with an input pressure height of gas (bubbling pressure) that is lower than the lowest applied pressure height hw . This assures that pores in the plate are small enough to prevent air from entering into and through the porous plate during measurement. The soil sample (1), usually placed in a cylinder of 100 cm3 , is put on the porous plate, saturated with water. It is recommended to put filtration paper or another hydrophilic porous membrane below the soil sample to prevent the soil from sticking to the porous plate. Saturation of the soil sample by water is done by increasing the

Fig. 7.6 Tension-plate apparatus to measure soil-water retention curves. Soil sample (1) is on the porous plate conductive for water only (2); water outflows to water reservoir (3). Soil-sample water content during the equilibrium state corresponds to the negative pressure (tension) hw

7.5 Measurement of Soil-Water Retention Curves

89

water table in the container nearly the top of the sample. After a few hours (depending on the soil properties: sandy soils are saturated quickly, but heavy soils need more time) when soil sample is assumed to be saturated, the water table in the container will be lowered below the porous plate level (usually to −2 cm below the plate surface), allowing the soil sample to reach an equilibrium state. Then, the soil sample (together with its metal cylinder) is weighed with an accuracy of 0.01 g. The soil sample is carefully placed on the porous plate, and tension (negative-pressure height) is increased by changing the container’s vertical position until a new equilibrium state is reached. By changing of water-table level in the container with water and by monitoring the outflow quantity, the values (hwi , θ i ) are obtained. The retention container with water is usually put on balance and it is possible to measure the water outflow from the sample until equilibrium state was reached, corresponding to the pressure height |hw |. The measurement is complete when the final equilibrium is reached (for a maximum applied negative-pressure height in absolute values). The soil sample is weighed, oven-dried and finally weighed. Then, the soil-water content is evaluated by the gravimetric method. Using the retrospective method, knowing quantities of outflow water for every pressure step, volumetric soil-water contents can be calculated and pairs of data (hwi , θ i ) estimated. Then, the SWRC can be drawn. As was mentioned previously, the temperature of the laboratory should be controlled during the measurements of the hydrophysical characteristics of soil. The higher the temperature (of soil water) is, the lower the soil-water content (corresponding to a particular soil-water matric potential) of the soil sample is observed (Novák 1975). The recommended standard temperature is 20 °C. It is recommended to cover the container, thus preventing evaporation from soil samples. The cover should not be airtight, to allow for the pressure of container and ambient space to equalize. A sand tank (Fig. 7.7) is a frequently used modification of the tension method. The function of the semipermeable plate (in the tension method) is replaced by the fine sand or clay layer (2), with a bubbling pressure higher than the absolute values of the negative pressures applied to the samples during measurement. To allow horizontal flows of water from soil samples (1), the sand layer of the higher hydraulic conductivity (3) is positioned below the upper fine sand layer. The measuring system is covered up, to prevent evaporation, but the cover should not be airtight. Pressure height (negative) is usually measured from the bottom (or from the average height) of the soil sample to the water layer in the water container. The advantage of the sand tank is the possibility to measure the SWRC of many soil samples simultaneously.

90

7 Soil-Water Retention Curve

Fig. 7.7 Sand tank, modification of the tension-plate apparatus for soil-water retention curve measurement. Soil samples (1) are on the layer of fine porous material (fine sand, clay) (2). Highly conductive material (sand) below allows the water drainage by the outflow tube (3) to the water reservoir with water table at the distance hw below the soil samples’ bottom

Fig. 7.8 Pressure-plate apparatus to measure soil-water retention curves. Soil sample in cylinder (1) is located on the ceramic semipermeable plate (2) in pressure container (3), under air pressure pwi (larger than atmospheric pressure pa ) (4). Water outflows by opening (5)

7.5.2 Pressure Methods The pressure method for the measurement of soil-water matric potential is based on positive air-pressure application (pressures higher than atmospheric) on the sample of porous media or soil. Water in the soil pores, which is attracted to the soil matrix by a force less than the applied air pressure pw  ρ w .g.hw , flows out of the soil. The equipment for measurement by the pressure method is the pressure-plate apparatus, which is sketched in Fig. 7.8. Measurement of SWRC by the pressure-plate method Figure 7.9 is the technical design of pressure-plate apparatus to measure the SWRC. A semipermeable ceramic plate is placed into a pressure container (usually made of

7.5 Measurement of Soil-Water Retention Curves

91

Fig. 7.9 The technical design of a pressure-plate apparatus. The metal pressure container is pressurised by the compressor. Water from porous media (soil) flows through the porous plate, and then to an atmosphere (a). Detailed outflow part is in (b)

metal)—soil samples are usually placed into metal cylinders. Samples are saturated with water and then drained at a small negative-pressure head hw = −2 cm, usually in a sand tank, before placing them into the pressure container. Then, after weighing the sample, the pressure container is sealed to be airtight. The air (from the compressor) is regulated by the system of regulators, enable maintenance of constant air pressure in the pressure container for long periods. Water flows from soil samples at constant air pressure through the porous plate and leaves the apparatus, as can be seen in Fig. 7.9. When an outflow of water from the apparatus is no longer observed (for sandy soil, it can take a few hours, for heavy soils even some weeks), air from the pressure container is released; the soil samples are weighed and carefully returned to the pressure plate. To preserve good contact of soil samples and porous plates, a slight wetting of the pressure-plate surface is recommended. Slight “wetting” of the surface in the process of the drying main branch of the SWRC secures good hydraulic contact between soil sample and porous plate during the next pressure step. Then, the next equilibrium state between soil water in the sample and acting pressure should be reached. The whole process is repeated step by step. Once having reached their final equilibrium state, soil samples are weighed and their final soil-water contents is calculated by the gravimetric method. Once the pairs of negative-pressure heights and volumetric soil-water contents (hwi , θ i ) have been recorded, the main drying branch of the soil-water retention curve can be drawn. The range of pressures applied to the soil samples depends on the soil type and on technical parameters of the apparatus used. The air pressures in the pressure-plate apparatus are in the range of p  5 kPa (hw  −50 cm) up to 1.5 MPa (hw  −15,000 cm). The abovementioned range of pressures can be handled by various modifications of pressure-plate apparatuses. Technical parameters of currently used pressure-plate

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7 Soil-Water Retention Curve

equipment usually allow dividing the pressure interval into two segments: 5 − 500 kPa (−5000 cm ≤ hw ≤ −50 cm) and pressure interval above 500 kPa (hw < −5000 cm). Usually, the SWRC cover intervals of pressure hw > −5000 cm; the interval of lower values of soil-water matric potentials are usually handled by extrapolating the measured section of the SWRC. Pairs of data (hwi , θ i ) for small soil-water contents can be estimated by adsorption or desorption methods, and ultimately by the psychrometric method. The limited accuracy of gas-pressure stabilization in pressure-plate apparatus at low pressures allows starting SWRC measurement at a pressure p = 5 kPa (hw  − 50 cm), with air pressure increasing in steps no less than 5 kPa, up to the pressure p  20 kPa (hw = −200 cm), because in this interval of pressures soil-water content changes can be relatively significant. Later, the pressure step 50 kPa is suitable, but it depends on soil type. Small pressure steps are suitable for sandy soils, but for heavy soils higher pressure steps are a better choice. The appropriate pressure steps during the measurement can be estimated from the SWRC of soils in Fig. 7.3.

7.5.3 Combined Method for SWRC Estimation To estimate pairs of data (hwi , θ i ) during wetting and drying processes, using the same soil samples, the apparatus named “Tempe’s pressure cell” was designed (named after Tempe, Arizona, USA). It is pressure-plate chamber which enables: (a) to increase or decrease air pressure in pressure chamber and (b) to measure outflowing or inflowing water quantities from or into the soil sample. Inflow (outflow) is measured by a calibrated, horizontally oriented measuring tube; this makes it possible to identify equilibrium states.

7.5.4 Psychrometric Method After time, the unsaturated soil placed in the airtight container at the constant temperature can reach a state of equilibrium between the soil water and atmosphere. Water-vapour humidity surrounding the soil sample is a function of the potential energy of the soil water. The lower the energy of water is (it decreases with soilwater content), the fewer molecules of water can escape into the atmosphere and the lower is the air humidity. This information is used to measure the soil-water matric potential. The relationship expressing the relative humidity of air (r) as a function of the soil-water matric potential ψ w (MPa) is known as the Kelvin equation. For 20 °C it can be written: −ψw  3.17 × 102 log r or

(7.7)

7.5 Measurement of Soil-Water Retention Curves Table 7.3 Soil-water matric potentials Ψ w in equilibrium with air humidity above the saturated solutions of four solutes

Saturated solution

93 Ψ w (MPa)

K2 SO4

1.90

CaSO4 .5H2 O

2.88

KCl NaCl

21.80 38.90

−ψw  137.2 ln r

(7.8)

This method of soil-water matric potential measurement is based on measurement of the relative air humidity r above the isolated soil sample, at an equilibrium state; thermocouple psychrometers are used. One of the many modifications for the measuring apparatus enables measuring the relative air humidity r just above the soil sample located in a small container (after the equilibrium is reached). Another modification of this type of equipment is to isolate the soil sample from the space in which psychrometer is installed by a semipermeable membrane (permeable for water) and then, to measure relative humidity of the air and using the above equations to calculate soil-water matric potentials. A thermocouple is a basic part of the thermocouple psychrometer. The thermocouple is a double junction of two dissimilar metals (copper and constantan are the most frequently used). If the two junctions are subjected to different temperatures, they will generate a voltage difference. This difference is proportional to the temperature difference—thus the term thermocouple. If a voltage difference is applied to the junctions, the temperature difference will result. The temperature of junctions depends on the direction of the electric current. To use the thermocouple as a psychrometer, i.e. to be able to measure air humidity, one junction has to be kept at a constant temperature and the second junction placed in the space above the soil sample which is being measured. Switching on the electric current, condensation of water vapour will occur on the colder part of the two-metals junction; a thin layer of water will cover the wire surface. Switching off the electric current, the water starts to evaporate, and the temperature of this part of the junction decreases (the latent heat of evaporation is consumed) , and the thermocouple starts to generate an electric current. During the period of constant evaporation, the electric current will also be constant and dependent on the evaporation rate. Knowing the air humidity above the soil sample that is in equilibrium with the soil, the soil-water matric potential can be calculated (Eq. 7.7). Since soil water is usually not clean water, but a solution, the relative humidity of the air above the soil sample is depressed by the dissolved compounds in the solution; and the influence of osmotic potential of the solution on soil-water matric potential is measured. Generally, this influence is relatively small and can be neglected (Radcliffe and Šim˚unek 2010). The psychrometric method is suitable to measure low values of soil-water matric potentials in the range ψ w ≤ −1 MPa. Table 7.3 summarizes soilwater matric potentials in equilibrium with air humidity above the most frequently used saturated solutions.

94

7 Soil-Water Retention Curve

7.5.5 Adsorption and Desorption Methods Adsorption and desorption methods are physically based on the same principle as the psychrometric method. Even the ranges of both methods application are similar. In the adsorption and desorption methods, the relative air humidity above the soil sample (in an airtight container with constant humidity) is known, as well as the soil-water matric potential. Once an equilibrium state between the soil sample and relative humidity (in airtight container) is reached, soil-water content (estimated by the gravimetric method) corresponding to known soil-water matric potential is estimated; then data pair (hwi , θ i ) is known. Characteristic values of soil-water matric potentials, corresponding to the relative humidity of air in equilibrium with the soil, are listed in Table 7.3. The advantage of this method in comparison to the psychrometric one is the possibility to estimate pairs of values (hwi , θ i ) of absorption or desorption in the main branch of the soil-water retention curve, depending on the type of the process—soil samples can be drying or wetting. But, the differences at such extremely small values of soil-water matric potentials are usually not significant due to hysteresis.

7.6 Estimation of Soil-Water Retention Curves from Pressure-Chamber Data The result of SWRC measurement by the pressure method is the mass of the wet soil sample in the sampling cylinder (msw ), corresponding to the applied pressure in the pressure chamber (pw ). These values are estimated in an equilibrium state, i.e. when the outflow from the pressure chamber at the applied pressure stops; see Fig. 7.5. The mass of wet soil in equilibrium conditions, together with the mass of the cylinder (msw ), correspond to the applied pressure (pw ) expressed by the pressure head (hw ) are collected Table 7.4a. In this case, the SWRC from the site Borovce (the experimental site of the Research Institute of Plant Production, Piešˇtany, Slovakia) were chosen.

Table 7.4a Pressure heads hw and corresponding mass of wet soil samples (together with container) msw , mass soil-water contents w and volumetric soil-water contents θ. Soil sample No. 99, the site B1, Borovce, VÚRV Piešˇtany, Slovakia hw (cm)

−2.5

−50

−100

−300

−1000

−7.4 × 104

msw (g)

266.7

254.39

249.23

244.9

242.3

–

0.321

0.233

0.196

0.165

0.146

–

θ (cm3 cm−3 ) 0.447

0.323

0.273

0.23

0.204

0.0379

w (g

g−1 )

7.6 Estimation of Soil-Water Retention Curves …

95

Table 7.4b Characteristics of the soil sample (No. 99) needed to calculate soil-water content. Soil sample volume (volume of the cylinder) V , mass of dry sample with the cylinder msd , mass of the cylinder mt and soil bulk density ρ b . Van Genuchten’s parameters of the SWRC α, n, θ s and θ r are presented too No. of V sample (cm3 )

msd (g) mt (g)

ρb (g cm−3 )

α (cm−1 ) n (–)

θr θs (cm3 cm−3 ) (cm3 cm−3 )

99

221.91 82.59

1.39

0.078

0.02

100

1.22

0.447

The measurement was stopped at the pressure height hw  −1000 cm, the soil sample was dried out at the temperature 105 °C, and then it was weighed. The dry-soil weight together with sampling cylinder is denoted as msd (Table 7.4b). The weight of empty cylinder is mt (Table 7.4b). Then, the mass soil-water content w can be calculated, corresponding to the pressure head hw : w

m sw − m sd m sd − m t

(7.9)

Weights msd and mt are constant for a particular soil, only the weights of wet soil (with cylinder) msw change. The numerator of Eq. (7.9) is the weight of water in soil sample; denominator is the mass of dry soil, both expressed in grams (g). Volumetric soil-water contents θ , corresponding to the pressure heads hw are calculated according to the formula: θ

m sw − m sd ρb w ρw ρw V

(7.10)

where ρ b is density of the soil sample (g cm−3 ), and it can be calculated from the equation

Fig. 7.10 Soil-water retention curve of a loamy soil approximated by the equation according van Genuchten (Eq. 7.2). Parameters of SWRC are in Table 7.4b. Soil sample No. 99, the site B1, Borovce near Piešt’any, Slovakia

96

7 Soil-Water Retention Curve

ρb 

m sd − m t V

(7.11)

where V is volume of soil (usually it is volume of the sampling cylinder, V = 100 cm3 ). The calculated soil bulk densities ρ b are in Table 7.4b, and volumetric soil-water content θ corresponding to the particular pressure heights hw are shown in Table 7.4a. The last column of this table are the data (hw , θ ) estimated by the adsorption method to characterize SWRCs over a wide range of pressure heights. The soil-water retention curve approximated according to van Genuchten (Eq. 7.2) is in Fig. 7.10; parameters of van Genuchten’s equation α, n, θ r and θ s are in Table 7.4b. The same procedure can be applied when calculating the SWRCs from results of measurement by the hanging-column method.

References Briggs LJ, Shantz H (1912) The relative wilting coefficient for different plants. Bot Gar (Chicago) 53:229–235 Brooks RH, Corey AT (1964) Hydraulic properties of porous media. Hydrology paper 3, Colorado St. University, Fort Collins Gardner WR, Hillel D, Benyamini Y (1970) Post irrigation movement of soil water; I. Redistribution. Water Resour Res 6:851–861 Haines WB (1930) Studies in the physical properties of soil: V. The hysteresis effect in capillary properties and the modes associated therewith. J Agric Sci 20:97–116 Hanks RJ, Hill RW (1980) Modeling crop responses to irrigation in relation to soils, climate and salinity. International Irrigation Information Center, Publ no. 6, Bet Dagan, Israel Kutílek M (1978) Pedology for water management. SNTL–ALFA, Prague (In Czech) Novák V (1975) Non-isothermal flow of water in unsaturated soils. J Hydrol Sci 2:37–52 Novák V, Gallová C (1998) The influence of different hydrophysical characteristics on the results of seasonal course of soil water content modeling. J Hydrol Hydromech 46, 6:436–450 (In Slovak, with English abstract) Radcliffe DE, Šim˚unek J (2010) Soil physics with HYDRUS. Modeling and applications. CRC Press, Taylor & Francis Group, Boca Raton, USA van Genuchten MT (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Vidoviˇc J, Novák V (1987) Yield of maize and evapotranspiration. Rostlinní výroba 33:663–670 (In Slovak with English abstract)

Chapter 8

Soil-Water Movement in Water-Saturated Capillary Porous Media

Abstract The movement of water through capillary porous media (soils) saturated with water is quantitatively described by the Darcy equation; its application to water flow and water-potential distribution for various configurations of soil samples is described. The Darcian approach to saturated soil-water movement assumes the porous space is fully saturated with water; the soil pores do not change their dimensions, and soil temperature is constant. Hydraulic conductivity and permeability of soils saturated with water are defined and a wide variety of its measurement methods is presented. The frequently used methods for the measurement of the saturated hydraulic conductivity of soil samples in the laboratory are described in detail. Field methods for the measurement of saturated hydraulic conductivity above the groundwater table are presented and discussed. Pedotransfer functions to evaluate hydraulic conductivity from available data are briefly discussed.

8.1 Two Concepts of the Quantification of Soil-Water Movement in Saturated Capillary Porous Media (Soils) All organic and majority of inorganic compounds are porous media. Specific feature of soils (among many types of porous media) are their properties that enable plant growth. Physical laws, describing the state and dynamics of soil water, are the same for all capillary porous media, even for soil. In this chapter, it is assumed that in a stable porous system of soils that soil water is clean (no dissolved matter) and incompressible. Soil water flows isothermally; soil-water properties are stable. Soil is homogeneous and saturated with water; it contains only two phases: the liquid phase of soil water and solid phase of soil matrix. The porous space of natural porous media is complicated and variable so the quantitative description of porous space is practically impossible. To describe quantitatively water movement in such a system, it is suitable to use one of the two concepts described later.

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_8

97

98

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

8.1.1 Darcian (Macroscopic) Approach A characteristic feature of the Darcian (integral, macroscopic) approach to soil-water movement is quantification of the effect of unknown porous system on soil water and its movement by the measurement of soil-water velocity through the soil sample, containing a theoretically infinite number of soil pores. The effective, average soil water-flow velocity through the great amount of soil pores is measured. The cross section of pores is not known, and the measured flow of water is divided by the cross section of the soil sample, through which soil water moves. The properties of soil porous system are also not known, but measured flow rates also reflect the properties of the porous system. The soil-water rate estimated in this way is called the Darcian flow rate, after Henry Darcy (Darcy 1856). In this concept, it is not important how many pores of which diameters participate in the flow. Mathematically, using this approach to the rate of flow through the porous system can be expressed by the equation: v

V At

(8.1)

where v is the macroscopic (Darcian) water-flow rate through the soil (L3 L−2 T−1 , m3 m−2 s−1 ), V is the volume of water (L3 , m3 ) crossing the area of the soil cross section A perpendicular to the flow direction (L2 , m2 ), and t is time (T, s). This Darcian rate of flow does not characterize the real water flow rate; it is assumed the water flow through the whole area of soil cross section A. It is assumed the soil porosity P can be approximately expressed by the ratio of soil pores cross section area Ap and soil cross section A: P

Ap A

(8.2)

Then, the average rate of water flow in soil pores saturated with water (vp ) is: vp 

v P

(8.3)

It can be seen that the average rate of water flow in soil pores is higher than the macroscopic (Darcian) flow rate (vp > v) because soil porosity is always less than one. It should be noted that the so-called “pores rate” is not the real rate of water flow in some particular pores. Because soil-pore diameters vary over a wide range of values, the water velocities in the pores vary. But this macroscopic approach has been used successfully more than 150 years.

8.1 Two Concepts of the Quantification of Soil-Water Movement …

99

8.1.2 Model Porous Medium The next approach to describe water movement in porous medium (soil) is to replace complicated—and therefore not easily recognizable—porous space by simple models of porous media (physical models) that can characterize transport properties of soil. The simplest and often used is the model of a soil composed of capillary tubes. A distribution of capillary-tube diameters can characterize a particular soil. Before applying such a model, it is necessary to analyze the type of soil-water flow regimen, because various flow regimens (laminar, turbulent) are expressed by different relationships between the rate of water flow and flow potential. From analysis of water flow in the previous section, it follows that the soil water usually flows at low rates, and the soil-water regimen can, in the majority of cases be characterized by small Reynolds numbers (Re < 1). Then, the soil-water flow regimen will be laminar, and the rate of flow vp through the tube with radius r can be expressed by the equation known as the Poiseuille equation (Hillel 1982; Kutílek and Nielsen 1994): vp 

r 2 p ρw g r 2 h  8η l 8η l

(8.4)

where vp is the average rate of flow in a tube (LT−1 , m s−1 ), r is the radius of the tube (L, m), Δp is the pressure difference (L−1 MT−2 , Pa) on the tube length Δl (L, m), η is dynamic viscosity of water (ML−1 T−1 , Pa s), ρ w is water density (ML−3 , kg m−3 ), Δh is the difference of pressure heights of water (L, m) on the tube length Δl (L, m), and g is the acceleration of gravity (MT−2 , m s−2 ). Let us take a unit volume of a soil model V , made of n capillary tubes with radius r and length L  1.0 m. The porosity P of such a model soil is: P

n πr 2 L V

(8.5)

The specific surface of such a porous medium can be expressed by the equation: s

n 2πr L V

(8.6)

If the porosity P and the specific surface of the soil s are known, then the above equations help to calculate the representative, hypothetical, radius of the model-soil capillary tubes r, representative of a soil saturated with water: r

2P s

(8.7)

The macroscopic rate of water flow in soil pores can be calculated with Eqs. (8.3)–(8.6):

100

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

v

ρw g P 3 h 2η s 2 l

(8.8)

The family of soil and water characteristics on the right side of Eq. (8.8) can be denoted by the symbol K, which is the coefficient of hydraulic conductivity of the soil saturated with water (saturated hydraulic conductivity) (L T−1 , m s−1 ): K 

ρw g P 3 η 2s 2

(8.9)

h KI L

(8.10)

Then, Eq. (8.8) can be rewritten as: vK

where I is hydraulic gradient, the change of pressure height h on the length ΔL (dimensionless). The group of soil characteristics in Eq. (8.8) can be denoted by symbol K p : Kp 

P3 2s 2

(8.11)

This term is denoted as soil permeability K p (L2 , m2 ); then the hydraulic conductivity of a soil K can be expressed as: K  Kp

ρw g η

(8.12)

Permeability K p is characteristic of porous space of soil only and does not depend on the properties of the flowing liquid. It is a universal characteristic of porous medium; its dimension is area, and it could be simply expressed as the soil pores cross section area through which liquid flows. Equation (8.12) is of great practical importance because, knowing the permeability of porous media, the conductivity of this media for any liquid can be calculated (also knowing its density and viscosity). In this way, the conductivity for liquids and gases can also be calculated.

8.2 Darcy Equation Laminar flow can be quantitatively expressed by the Navier–Stokes equation. But the application of this equation to water flow in porous medium is practically impossible because a quantitative description of porous space is needed. To calculate soil-water flow parameters in porous media (soil), one uses the empirical relationship proposed by Darcy (1856), who generalized results of his measurement of water flow through a sand filter. The town Dijon in France was supplied with

8.2 Darcy Equation

101

Fig. 8.1 A scheme of the system for a flow rate through vertically positioned soil-sample measurement; water is flowing up–down. This type of apparatus was used by Darcy (1856)

water from sources about 12 kilometres distant. Before distribution of the water, it was necessary to filter the water. Darcy proposed a simple sand filter (as shown in Fig. 8.1). Its capacity was estimated by measuring the water volume passing through the filter (V ) during time (t). The thickness of the sand filter (L) was measured, as well as the area of its cross section (A). The positions of the water table above (h1 ) and below the sand layer (h2 ), were measured, too. It was estimated that the discharge of water through the filter depends on the pressure-height difference H between the inflow and outflow cross sections. This resulting discharge of water an be expressed by the equation in which discharge is proportional to the pressure-height difference H and to the filtration time t, with the sand-filter cross section area A indirectly proportional to the sand-filter thickness L: V∝

A t H L

(8.13)

The proportionality coefficient expressing the properties of the filter material was named as the coefficient of hydraulic conductivity of saturated porous medium (soil) (briefly, saturated hydraulic conductivity); sometimes it is called the coefficient of filtration. The term coefficient of filtration is not recommended, but this was the original term of saturated hydraulic conductivity because, in Darcy’s experiment, water was really filtrated by sand. This equation was later named Darcy’s equation (law) and can be written in the form: QvAK

H A L

(8.14)

102

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

The flow rate through the unit area of a sand filter (A  1 m2 ) can be expressed as: vK

H L

(8.15)

where Q is the water discharge through the filter (L3 T−1 , m3 s−1 ), A is an area of filter cross section perpendicular to the flow direction (L2 , m2 ), v is Darcian (macroscopic) flow velocity through the sand filter (L3 L−2 T−1 , m3 m−2 s−1 ), or (LT−1 , m s−1 ), K is proportionality coefficient, named the coefficient of hydraulic conductivity of porous media saturated with water; its dimension is the same as for the rate (LT−1 , m s−1 ), ΔH is the difference between water tables at the inflow and outflow of the sand filter (difference of pressure heights) (L, m), and L is the thickness of filter (layer of sand) (L, m). Darcy’s equations for one dimension x in differential form can be written: v  −K

dh dx

(8.16)

The formula for three dimensions is: v  −K grad h

(8.17)

The sign “minus” at the right side of the equation means that liquid (water) will flow in the direction of decreasing potential, or in opposite direction to the slope of the pressure line, or against the pressure gradient. Potential is here expressed by the pressure head h.

8.3 Macroscopic (Darcian) and Porous (Real) Water-Flow Rate The Darcian rate of flow is denoted by v. Darcian rate of flow is not the real rate of water flow through the porous system; it is the average rate of flow through the whole cross section of the filter. The real average rate of water flow can be calculated using Eq. (8.3). As was described previously, the Darcy’s equation is the result of numerous measurements of water flow rate through the sand filter. This equation (frequently called Darcy law) is not a natural “law”, because its validity is limited to laminar flow, i.e. it is not valid for turbulent flow at relatively high flow rates. The original publication of Darcy (1856) also contains tabulated results of measurements; Presenting those results graphically, it can be stated that at high hydraulic gradients the slopes of the relationship v  f (I) (where I is the hydraulic gradient, or the slope of the pressure line) do not increase proportionally to the hydraulic gradient, but they are smaller

8.3 Macroscopic (Darcian) and Porous (Real) Water-Flow Rate

103

than expected and are positioned below the line v  f (I). Water flow at high rates are in the area of the turbulent regime, where hydraulic losses are not proportional to the flow rate, but hydraulic losses increase exponentially with the flow rate; and measured rates of flow are below the linear relationship v  f (I). Darcy neglected this nonlinear range and formally expressed the relationship v  f (I) by a linear function. His “engineering” approach was found very useful because the majority of cases of subsurface water flow have laminar conditions. There are some classes of water flow in saturated conditions where Darcy’s equation cannot be applied, like in water transport in coarse porous media with macropores, in water flow under great hydraulic gradient or in industrial applications of water flow in porous media.

8.4 Distribution of Pressure During Water Flow Through Various Measuring Devices 8.4.1 Horizontal Soil Sample Figure 8.2 represents an apparatus for measuring the saturated water flow rate through the horizontally positioned soil sample of length L. Both ends (inflow and outflow) are connected to pools with stable water-table levels a, b, but the water tables in the pools are on different levels; so the rate of water flow is steady. If the reference level is located at the soil sample axis, then the water level in water pools are h1 and h2 . The difference between them is ΔH  h1 − h2 . Piezometers show the pressure heights in the soil sample; at the inflow and outflow profiles, the pressure heights are the same as in water pools because the hydraulic resistances of inflow (outflow) tubes are negligible in comparison to the hydraulic resistances of the soil sample. Pressure heights of water in pools (related to the reference level) are: h1 

p1 p2 ; h2  ρw g ρw g

(8.18)

Bernoulli equation (pressures are expressed by pressure heights): h1  h2 + hs

(8.19)

h s  h 1 − h 2  H

(8.20)

The energy difference between the levels expressed by the pressure-head differences h1 and h2 is consumed as a driving force to transport water between those levels. It can be also be denoted as hydraulic loss hs . The linear distribution of pressure along the soil sample (Fig. 8.2) is shown by dashed line.

104

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

Fig. 8.2 A scheme of the apparatus for a flow rate through the horizontally positioned soil-sample measurement; the linear distribution of pressure heads is shown

8.4.2 Vertically Positioned Soil Sample, Downward Water Flow Let the reference level of the soil sample (Fig. 8.1) be at the bottom of the soil sample. The Bernoulli equation, expressed by the pressure heights units, can be written: L + h1  h2 + hs

(8.21)

h s  (L + h 1 ) − h 2  ΔH

(8.22)

The hydraulic loss (hs ) during the water flow expressed by the pressure heights is the difference between the water table levels at the inflow and outflow cross sections.

8.4.3 Vertically Positioned Soil Sample, Water Flows Bottom-Up This configuration of equipment is frequently used to measure hydraulic conductivity of saturated porous medium in laboratory conditions (Fig. 8.3). The advantage of this configuration is decreased risk of air entrapment in soil pores; it is more frequent when water flows downward. During the water flow from the bottom up, air is pressed out of pores and by this procedure less water is entrapped in soil pores in comparison to the opposite direction of water flow. The reference level is identical with the bottom, inflow cross section of the soil sample. Then, the Bernoulli equation expressed by the pressure height units can be written for both cross sections:

8.4 Distribution of Pressure During Water Flow Through Various …

105

Fig. 8.3 A scheme of the system for a flow rate through the vertically positioned soil-sample measurement; water is flowing bottom–up

h1  L + h2 + hs

(8.23)

h s  h 1 − (L + h 2 )  ΔH

(8.24)

Even in this case, the hydraulic loss can be expressed by the difference between water tables at the inflow and outflow profiles. To calculate hydraulic conductivity of saturated soil, the water-table difference, rate of water flow and length of the soil sample are substituted into the formula (8.15); then the saturated hydraulic conductivity can be calculated.

8.5 Hydraulic Conductivity of Soil Saturated with Water Measured in the Laboratory Hydraulic conductivity of soil saturated with water is one of the basic hydrodynamic characteristics of porous media; it must be estimated in the field and laboratory conditions. In the laboratory, the hydraulic conductivity K should be measured on

106

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

Fig. 8.4 A scheme of the falling-head system for saturated hydraulic conductivity measurement of a soil

undisturbed soil samples; their volume should be larger than the so-called representative elementary volume (REV). Under the term REV, it is understood that for the minimum volume of soil the characteristics of which do not change with an increasing volume of the soil sample, but they change as the soil volume decreases. For homogeneous soils, a cylinder (or soil sample) of 100 cm3 usually fulfils the REV criteria. But an REV of soil containing macropores, aggregates, fissures, or rock fragments has to be larger. The REV of such non-structural soils must be a factor of 10–100-times larger than the standard 100 cm3 (Kutílek and Nielsen 1994), or the hydraulic conductivity should be measured in the field. In the laboratory, the saturated hydraulic conductivity of soil K can be measured by one of the methods presented in Figs. 8.1, 8.2 and 8.3. It is recommended to use simple apparatus with a changing hydraulic gradient through the soil sample. This type of method is called the falling hydraulic head method or, simply, the falling head method (Fig. 8.4). After soil-sample saturation by water from the bottom that fills measuring tube with water; measurement of the necessary parameters needed to calculate K can be started. Water will flow through the soil sample cross section A2 (perpendicular to the flow direction) at the initial pressure-head difference h1 up to the final pressure head h2 . The pressure head difference h1 − h2 is reached during the time interval Δt  t 1 − t 2 . The area of the tube’s cross section is A1 , and the elementary volume of water in the measuring tube is A1 .dh. The water discharged through the soil sample during elementary time dt equals:

8.5 Hydraulic Conductivity of Soil Saturated with Water Measured …

A1 dh  A2 v dt

107

(8.25)

And the rate of flow v from Eq. (8.25) is substituted into the Darcy equation v  −K

h L

(8.26)

Separating the variables according to h and t and to integrating the equation within the limits h1 , h2 and t 1 , t 2 , the final equation to calculate the saturated hydraulic conductivity K is: K 

h1 A1 L ln A2 t h 2

(8.27)

To calculate K, it is necessary to know: the initial (h1 ) and final water level (h2 ) in measuring tube, the time interval corresponding to the pressure height drop from h1 to h2 , t, the length of the soil sample L, the area of the soil sample’s cross section (A2 ) and the tube’s cross section (A1 ). It is important to minimize any entrapped air in the sample by slowly saturating it from the bottom by deaerated (pre-boiled) water; distilled water is not recommended because of the dilution of minerals from the soil and the subsequent change in properties. The bottom-up saturation of the soil sample is critical because this limits the air entrapment that can significantly decrease the soil permeability by limiting the “active” cross section of pores. Also important is the isothermal condition, the best being at a standard temperature 20 °C. If the temperature in the laboratory differs, the value of K can be recalculated to the standard temperature by the procedure described in Sect. 8.6.3.

8.5.1 Simple Laboratory Method for Measurement of Saturated-Soil Hydraulic Conductivity In the literature, numerous laboratory methods for the measurement of soil-saturated hydraulic conductivity can be found. One that can be recommended is a simple method of K measurement, which is one of the modifications so-called falling head methods. The equipment is presented in Fig. 8.5; the difference between this method and the previously described one in Sect. 8.4 is its simplicity. The metal cylinder containing the soil sample is connected tightly by rubber to the second identical, but empty, metal cylinder. The cylinder is put on the metal mesh to collect the free-water outflow from the sample. Both cylinders are placed vertically into the container with water, maintaining a constant water-table level at the bottom (outflow) end of the double cylinder.

108

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

Fig. 8.5 A simple falling-head system for saturated hydraulic conductivity measurement of a soil sample (K) composed of two cylinders; soil sample is in the bottom cylinder. Cylinders are tightly connected by rubber band; the bottom cylinder with metal mesh is in a container with constant water-table level

Before the measurement, the soil sample is slowly saturated with water, by putting it into a larger container in which the water table can increase slowly, up to the time when the water table appears in the upper cylinder and is filling it. The eventually partly empty volume of the upper cylinder can be filled carefully with water up to the top, not disturbing the soil surface in the bottom cylinder. Then, the time interval t of the water-table drop in the upper cylinder h is measured. It is recommended to repeat the measurement three times (minimum) to avoid gross errors. The height of the cylinder L is known, and heights h1 and h2 are measured. The water-table level in the bottom pool is constant level during the measurement. The saturated-soil hydraulic conductivity K can be calculated by the modified Eq. (8.27): K 

h1 L ln t h 2

(8.28)

The input data, procedure of calculation and results are gathered in Table 8.1. It is recommended to measure the saturated hydraulic conductivity of soil and soilwater retention curve (SWRC) on the same soil sample. Therefore, after measurement of the data needed to calculate K, the same sample can be used to measure SWRC. This procedure can avoid the situation of measuring K and SWRC on different soil samples of the same size, which can have different properties. To measure soil-water contents and soil-water matric potentials needed to calculate SWRC, the soil sample will be dried out at 105 °C. This procedure was described in Chap. 5. By the drying of the soil samples, their properties were changed, and they cannot be used in further measurement.

t1 (min)

2:36

Sample number

99

3:09

t2 (min) 33

t (s) 9.75

h1 (cm) 6.3

h (cm) 3.45

h2 (cm)

3.55

L (cm)

0.112

K (cm min−1 )

Table 8.1 Example of measured data by falling-head method needed to calculate saturated hydraulic conductivity of a soil sample K

1.61

K (m d−1 )

8.5 Hydraulic Conductivity of Soil Saturated with Water Measured … 109

110 Tab.8.2 Range of saturated hydraulic conductivities of texturally varied soils

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

Soil texture

K (cm d−1 )

Sand Sandy loam

100–500 30–100

Loam Clay loam

10–50 5–30

Silt clay

1–10

Clay

0.1–1

8.6 Hydraulic Conductivity and Properties of Soil Hydraulic conductivity of saturated porous medium (soil), defined by Darcy’s equation (Eq. 8.17), is a characteristic of porous media. This is valid only if the flowing liquid is water, which is expressed by the word “hydraulic”. Generally, the adjective “hydraulic” usually denotes a property related to the artificial, technical structure which contains water or in which water flows. From the basic properties of capillary porous media (meaning media containing continuous capillary pores), the proportionality of the flow rate to the dimensions of soil pores at the same pressure difference between inflow and outflow cross section follows. Therefore, the highest saturated hydraulic conductivity is expected in coarse porous media (gravel, coarse sand). The smallest K is expected for fine soils (loam, clay). This was proved by measurements in the field and laboratory. Typical values of K for soils with different textures are shown in Table 8.2. Darcy’s equation (Eq. 8.16) shows that the water-flow rate through the soil sample (Fig. 8.4) is proportional to the water-table difference between inflow and outflow profiles. This difference, divided by the length of soil sample L, is the hydraulic gradient, and the Darcian flow rate is proportional to the hydraulic gradient. The rate of flow decreases proportionally to the decrease in the hydraulic gradient (Fig. 8.4). The relationship between soil-water flow rate through the soil sample v and the hydraulic gradient Δh/L should be linear. Two linear relationships v  f (I) for sandy soil and for clay are presented in Fig. 8.6. Saturated hydraulic conductivity K is the slope of the linear relationships v  f (I) (see Eq. 8.16), therefore the higher slope of the relationship v  f (I) for sandy soil means a higher saturated hydraulic conductivity of sandy soil; on the contrary, the smaller slope of this relationship for clayey soil means lower saturated hydraulic conductivity of clayey soil in comparison to the sand. Saturated hydraulic conductivity of soil often depends on the flow direction through the soil. Sediments (soils are often formed on sediments) are composed of oriented solid particles, and so their saturated hydraulic conductivity is usually different in various directions of water flow. Such soil is called anisotropic in comparison to isotropic soils with the same values of saturated hydraulic conductivities in any direction. Anisotropy of saturated hydraulic conductivities of soils is often influenced by the presence of oriented cracks, fissures or macropores created by dead

8.6 Hydraulic Conductivity and Properties of Soil

111

Fig. 8.6 Water-flow rate in an unsaturated soil (v) and hydraulic gradient I of a sandy and a loamy soil; relationships are linear

roots. In such soils, the vertical water flow direction prevails; therefore, it is possible to quantify soil-water flow by application of saturated hydraulic conductivities measured on soil samples (or in the field) measured by vertically flowing water. Soil should be sampled to make K measurement in vertical flow direction possible.

8.6.1 Darcian and Non-Darcian Flow To apply Darcy’s equation to calculate soil-water movement, or to measure saturated hydraulic conductivity of soils, it is necessary to keep within the limits of the Darcy’s equation validity. The flow of water through the soil have to be “Darcian”, i.e. rates of water movement should be small enough (flow regimen should be laminar) and the inertial terms in the Navier-Stokes equation should be negligible. From this follows the linearity of the relationship v  f (I). So called non-Darcian flow is expressed by nonlinearity of the relationship v  f (I) occurring during the turbulent water-flow regimen, or in soils in which the porous system (soil matrix) changes its dimensions due to deformation caused by hydrodynamic forces; then the relationship v  f (I) can be nonlinear (Novák 1972).

8.6.2 Air Entrapped in Soil Pores Entrapment of air in soil pores is often a problem when measuring saturated hydraulic conductivity in the laboratory, or during infiltration tests in the field. When saturating the soil sample, it is difficult to avoid air entrapment in soil pores. Entrapped air in

112

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

pores works like a stopper to decrease the measured saturated hydraulic conductivity and retention capacity of soil. It is possible to minimize the influence of entrapped air on the saturated hydraulic conductivity by using a slow rate of saturation; the next principle to be observed is soil-sample saturation from the bottom by deaerated (preboiled) water. Water always contains dissolved air, and its quantity is proportional to the water temperature. It is not recommended to use “fresh” water directly from the water-supply system because during outflow significant aeration takes place.

8.6.3 Temperature and Saturated-Soil Hydraulic Conductivity The physical properties of water change with temperature. Soil temperature can significantly change the saturated hydraulic conductivity of soil K due to the changing water viscosity with temperature. The significant change of water surface tension is not important because, in soil saturated with water, water–air interfaces should not occur. To eliminate the influence of temperature changes on K, it is important to measure the saturated hydraulic conductivity at a constant temperature; the best choice is the standard temperature 20 °C. If this cannot be done, it is necessary to prevent temperature fluctuation and to measure actual temperature in the laboratory or in the field. Then, the estimated K 1 at the temperature T 1 can be recalculated to K 2 at a standard temperature T 2 according to Eq. (8.12), expressed for two temperatures with corresponding dynamic viscosities of water (η1 , η2 ): η2 K1  K2 η1

(8.29)

From the known K 2 and the temperature T 2 at which it was measured, K 1 for a standard (or chosen) temperature T 1 can be calculated from Eq. (8.29).

8.6.4 Flow of Water in Layered Saturated Porous Medium Soil profile is usually composed of soil layers with various characteristics. They are usually not separated strictly, but their properties change fluently. The common case is the soil profile composed of two layers: a ploughing layer with relatively low density and high hydraulic conductivity and a relatively dense and less conductive sublayer. The vertically nonhomogeneous soil profile should be considered during ponding infiltration tests. Let us take a soil composed of two layers (Fig. 8.7), characterized by saturated hydraulic conductivities K 1 , K 2 , and layer thicknesses L 1 , L 2 . The difference between water tables at inflow and outflow profiles of such a soil profile is h. Other characteristics are manifested in Fig. 8.7.

8.6 Hydraulic Conductivity and Properties of Soil

113

Fig. 8.7 A scheme of the falling-head system for saturated hydraulic conductivity measurement of a two-layered soil sample

The water-flow velocity v through the particular layers is identical and can be expressed by the Darcy’s equations: h − h2 L1 h2 v  −K 2 L2

v  −K 1

(8.30) (8.31)

To express pressure heads h and h2 and after some rearrangement, the flow rate through the two-layered soil can be written: v−

h L1 K1

+

L2 K2

(8.32)

The identical procedure can be used to calculate the flow rate through saturated soil composed of n-th soil layers. The effective saturated hydraulic conductivity of a soil (K ef ) composed of n layers can be calculated from the equation: n Li (8.33) K e f  n1 L i 1 Ki

where K ef is not simply the arithmetic average of individual saturated hydraulic conductivities of soil layers but the weighted average, involving even thicknesses of individual soil layers, as well as their saturated hydraulic conductivities. Then, the water-flow rate v through the soil composed of n layers can be calculated from the equation: v  −K e f

h L

(8.34)

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8 Soil-Water Movement in Water-Saturated Capillary Porous Media

where h is the difference between water tables of the inflow and outflow cross sections, and L is total thickness of soil profile.

8.7 Field Measurement of Saturated-Soil Hydraulic Conductivity Above Groundwater Table in the Field 8.7.1 Measurement of Saturated-Soil Hydraulic Conductivity in the Field Above Groundwater Table at Variable Hydraulic Gradient Figure 8.8 is the schematic diagram of the measurement apparatus. The auger hole is carefully filled with water; a few auger-hole volumes of water should infiltrate into the soil to saturate it and to minimize the influence of capillary forces on infiltration. It can take a few minutes up to some tens of minutes, depending on the soil properties. Then, the measurement can begin as is shown in Fig. 8.8. The two positions of the water table in the auger hole are measured, as well as the time intervals corresponding to those positions (h1 , t 1 ), (h2 , t 2 ). Ritzema (1994) proposed equation to calculate K:     h 1 + 0.5r 0.5r ln K  t h 2 + 0.5r t  t2 − t1 (8.35) where K is hydraulic conductivity of soil saturated with water, (cm s−1 ), r is the radius of auger hole (cm), h is the height of water table in the auger hole (cm), and t is time (s). Equation (8.35) was developed assuming the gradient of soil-water potential equals one; i.e. the gravitational-potential component is the only driving force.

Fig. 8.8 A scheme of the falling-head system for saturated hydraulic conductivity measurement of a soil above the groundwater table in the field

8.7 Field Measurement of Saturated-Soil Hydraulic Conductivity …

115

8.7.2 Measurement of Saturated-Soil Hydraulic Conductivity in the Field Above Groundwater Table at Constant Hydraulic Gradient Figure 8.9 is a diagram of the measurement set-up for this method. A constant watertable level in the auger hole is maintained by a Mariotte bottle; the rate of infiltration into soil is measured. When the steady-state infiltration rate has been reached (it can take 15–30 min, depending on the soil properties), the rate of steady infiltration v, auger hole radius r and the height of the water table in the auger hole above its bottom h is substituted into the Glover equation (Amoozegar and Warrick 1986); the saturated hydraulic conductivity of a soil is then calculated:   sinh−1 (h/r ) − (r 2 / h 2 + 1)0.5 + r/ h (8.36) K v 2π h 2 where K is the hydraulic conductivity of soil saturated with water, (cm s−1 ), v is the water-infiltration rate from the auger hole (cm3 s−1 ), r is the radius of auger hole (cm), h is the height of water table in the auger hole (cm), and t is time (s). Boersma (1965) published the equation to calculate saturated hydraulic conductivity of soil (often called the auger-hole method):   ln (h/r ) − (h 2 /r 2 − 1)0.5 − 1 (8.37) K v 2π h 2

Fig. 8.9 A scheme of the system for soil-saturated hydraulic conductivity measurement at constant pressure head in the auger hole above the groundwater table

116

8 Soil-Water Movement in Water-Saturated Capillary Porous Media

where K is the hydraulic conductivity of soil saturated with water, (cm h−1 ), v is the water-infiltration rate from the auger hole (cm3 h−1 ), r is the radius of the auger hole (cm) and h is the height of the water table in the auger hole (cm). The different dimensions of some terms in Eqs. (8.36) and (8.37) should be mentioned. Equation (8.37) can be used only if h > r and the vertical distance between the auger hole bottom and the depth of impermeable layer hu is within the range 3 h > hu > h. To more easily measure K by the described method, a special apparatus (Guelph permeameter) was designed. It is often used in a field conditions (Elrick and Reynolds 1992) and this method is also often called “Guelph method” of K measurement. The advantage of this method is the ability to measure K at a chosen depth below the soil surface.

8.8 Pedotransfer Functions Direct measurement of soil hydraulic characteristics in the field (hydraulic conductivities, soil-water retention curves) is technically complicated and time consuming. Therefore, many researchers have sought simpler estimation methods that use known (or more easily measured) soil properties. Pedotransfer functions (PTFs) are relationships between (unknown) soil hydraulic characteristics and known, mostly textural, characteristics of a soil. PTFs are usually developed using a multiple regression of empirically estimated parameter values for basic soil characteristics such as the grain-size distribution, organic-matter content and bulk density. PTFs are used to calculate saturated hydraulic conductivities. Soil matric potentials corresponding to some typical values of soil “hydroconstants” like wilting point and field capacity are evaluated using empirically developed PTFs. The relative simplicity of using PTFs is attractive, so they are frequently used. But their application has some limitations. Firstly, they are recommended for sites where PTFs were developed based on empirical data. They are also recommended only in the case when it is not possible to measure soil hydraulic characteristics directly in the field and in the laboratory. Those parameters gained from PTFs represent a rough range of parameters values that can be very different than real ones must be taken into account. How to calculate PTFs and some examples of their development have been published by Tietje and Tapkenhinrichs (1993), Cornelis et al (2001), and Pachepsky and Rawls (2004).

References Amoozegar A, Warrick AW (1986) Field measurement of saturated hydraulic conductivity. In: Klute A (ed) Methods of soil analysis, Part 1. Agronomy monograph series no 9, 2nd edn. ASA and SSSA, Madison, Wisconsin, pp 735–770

References

117

Boersma L (1965) Field measurement of hydraulic conductivity above a water table. In: Black CA (ed) Methods of soil analysis, Part 1, No 9, Series Agronomy. ASA, Madison, Wisconsin, pp 234–252 Cornelis WM, Ronsyn J, Meirvenne MV, Hartmann R (2001) Evaluation of pedotranfer functions for predicting the soil moisture retention curve. Soil Sci Soc Am J 65:638–648 Darcy H (1856) Les fontaines publiques de la ville de Dijon. V Dalmont, Paris Elrick DE, Reynolds WD (1992) Infiltration from constant head well permeameters and infiltrometers. In: Topp GC, Reynolds WD, Green RE (eds) Advances in measurement of soil physical properties: Bringing theory into practices, SSSA Special Publ 30. SSSA, Madison, Wisconsin, pp 1–24 Hillel D (1982) Introduction to soil physics. Academic Press, New York Kutílek M, Nielsen DR (1994) Soil hydrology. Catena Verl, Reiskirchen Novák V (1972) Hysteresis of flux-gradient relations for saturated flow of water through clay materials. J Soil Sci 23:248–253 Pachepsky Y, Rawls WJ (eds) (2004) Development of pedotransfer functions in soil hydrology. In: Developments in soil science, vol 30. Elsevier Science Ritzema HP (1994) Subsurface flow to drains. In: Ritzema HP (ed) Drainage principles and applications. ILRI, Publ 16, Wageningen, the Netherlands, pp 283–294 Tietje O, Tapkenhinrichs M (1993) Evaluation of pedotransfer functions. Soil Sci Soc Am J 57:1088–1095

Chapter 9

Water in Unsaturated Soil

Abstract Water in soil usually fills soil pores only partially, therefore soil unsaturated with water is the prevailing state of soil. The soil unsaturated with water is a necessary condition for the majority of the plants that grow in the environmental conditions of Europe. The diagnosis and prognosis of the soil-water regimen is necessary to manage the soil-water content to reach maximum yields. This chapter defines the basic hydrophysical characteristics of an unsaturated zone of soil; the relationship between soil-water matric potential and volumetric soil-water content (so-called retention curve) and soil hydraulic conductivity as a function of soil-water matric potential and methods of their estimation are described. The Darcy–Buckingham equation is presented, as well as the equation of Richards that describes transport of liquid water in unsaturated soil. The transport of water vapour in the unsaturated soil is quantified by soil-water diffusivity, as well as by the coefficient of water-vapour diffusion of the soil, which is quantitatively described too.

9.1 The Differences Between Water Movements in Saturated and Unsaturated Soils In soil saturated with water (saturated soil), all pores are filled with water; in the soil unsaturated with water (unsaturated soil), some of its pores are occupied by air. The consequence of this difference is the existence of phase interfaces and the existence of capillary forces, which are the main forces governing water retention and conductivity of unsaturated soils but complicate the quantification of the water state and movement in an unsaturated soil in comparison with the saturated one (Fig. 2.1). Dealing with the soil as the upper layer of the Earth in which plant roots are located, soil saturated with water is a rare case and can be observed only temporarily, during ponding infiltration or heavy rain. Soil can be saturated by water even in the case of groundwater rising into the root zone, by infiltration from rivers and channels into soil or by flooding of the soil. Typical state of natural soil is unsaturated soil, the environment necessary for growing most plants. In the typical European climate, © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_9

119

120

9 Water in Unsaturated Soil

practically all the crops need unsaturated condition in the soil root zone (aeration zone, vadose zone) as an environment for plants ontogenesis. Photosynthesis produces chemical energy which is stored in organic solutes (assimilates), but its energy has to be released by respiration; then it can be used for plant growth. Respiration is a process of oxidation; soils have to contain air. Therefore, unsaturated soil is an essential condition for plant growth. There are some crops (like rice) which derive oxygen from water in which it is dissolved. In the soil saturated with water, porosity P equals the saturated soil-water content θ s ; so it can be written: P  θs

(9.1)

where θ s is volumetric soil-water content of saturated soil (saturated soil-water content) (−). In reality, even the saturated soil contains small amounts of air dissolved in the water; its quantity depends on air pressure and temperature. Part of soil air is entrapped in pores. Later, the validity of Eq. (9.1) is assumed. For unsaturated soil, it can be written: P θ +a

(9.2)

where a is soil aeration, i.e. the air volume in a unit volume of soil (−): a

Va V

(9.3)

It can be written: V  Va + Vs + Vw

(9.4)

where V , V a , V s , V w are partial volumes of soil, air, the solid phase of soil and water in soil (M3 , m3 ).

9.2 Basic Equations of Water Movement in Unsaturated Soil 9.2.1 Darcy–Buckingham Equation Buckingham (1907) proposed the generalised equation to express the rate of water flow in an unsaturated zone of soil. It is a basic thermodynamic equation, expressing the water-flow rate as proportional to the gradient of soil-water potential. This equation, known today as Darcy–Buckingham equation can be written as:

9.2 Basic Equations of Water Movement in Unsaturated Soil

v  −k(h w )grad h

121

(9.5)

This equation was proposed as an analogy to the well-known relationships such as Ohm’s law, Fourier’s law, Fick’s law, or Darcy’s equation. Buckingham defined the total soil-water potential h (it is expressed as a pressure head) as a thermodynamic term without an instrumentalist definition, i.e. without a proposal how to measure or evaluate it. The hydraulic conductivity of unsaturated soil k(hw ) is not a constant quantity as is the hydraulic conductivity of soil saturated with water, rather it depends on the soil-water content or soil-water potential of unsaturated soil. The negative sign on the right side of Eq. (9.5) means that the water-flow direction is opposite to the direction of a gradient of total soil-water potential, or soil water flows from areas of higher total soil-water potential to the areas of lower values of total soil-water potential. Of course, flow velocity cannot be negative value. Because total soil-water potential h in Eq. (9.5) is expressed in units of pressure heads (L), the flux-rate dimension is (L3 L−2 T−1 ). Hydraulic conductivity k(hw ) is of the same dimension as flow rate. It should be noted that hydraulic conductivity k(hw ) is a function of matric (moisture) potential of the soil, which represents the energy of interaction of the soil solid phase-water; other components of the total soil-water potential (gravitational potential) determine flow rate, but not the hydraulic conductivity of the soil. The functional relationship k(hw ) is not unambiguous, because the retention curve, expressed by the relationship hw  f (θ ), is also not unambiguous, but it is of hysteretic character.

9.2.2 Equation Describing Water Movement in Unsaturated Soil: The Richards Equation The differential equation quantitatively describing soil-water dynamics in an unsaturated zone of soil can be obtained by a standard procedure, using the Darcy–Buckingham equation, expressing soil-water flow rate as proportional to the gradient of soil-water potential (Eq. 9.5) and the continuity equation. Water balance of all the inflows and outflows of water of a unit volume of the soil (the volume of the soil equals one) can be written in generalised form: ∂θ  −div v ∓ S ∂t

(9.6)

where θ is volumetric soil-water content (−), t is time (T), v is water flow rate (LT−1 ), and S is water outflow (negative sign) or water inflow (positive sign) from or to soil volume under consideration (LT−1 ). The often used negative term is usually indicated as the root extraction rate. The sign of the right side of Eq. (9.6) indicates the opposite directions of soil-water content change and soil-water flow. If the divergence (or water flux from the soil volume) is positive (water outflows from the soil), soilwater content of the soil volume decreases, and if water flux is directed into the soil

122

9 Water in Unsaturated Soil

(water-flow rate is negative, therefore the divergence is negative too), the soil-water content of the soil volume is of opposite sign, i.e. is increasing. One dimensional continuity equation describing the vertical movement of water can be written: ∂v ∂θ − ∓ S(z) ∂t ∂z

(9.7)

The soil-water (total) potential characterises the interaction of soil and clean water, so h comprises two components: matric potential hw (as a consequence of phase interfaces in an unsaturated soil) and gravitational component hg  z (positive direction of z axis is upward): h  hw + hg  hw + z

(9.8)

Introducing h into Eq. (9.5): v  −k(h w )grad(h w + z)

(9.9)

To obtain the general form of the partial differential equation describing the flow of water in an unsaturated soil, the continuity equation is introduced into Eq. (9.9): ∂θ  div[k(h w )grad(h w + z)] ∓ S ∂t

(9.10)

Equation (9.10) can be rewritten in the form that was first published by Richards (1931); this equation is known as the Richards equation: ∂θ ∂k(h w )  div[k(h w )grad(h w )] + ∓S ∂t ∂z

(9.11)

The signs on the right side of the equation of continuity and the Darcy–Buckingham equation are negative, while the resulting signs on the right side of both Eqs. (9.10, 9.11) are positive. The gravitational component of the total soil-water potential is defined in the vertical direction only; derivatives in other directions are zero. Equation (9.11) is often used to quantify one-dimensional soil-water flow in the vertical direction z (as a governing equation of the simulation models) in the form:   ∂ ∂k(h w ) ∂h w ∂θ  k(h w ) + ∓ S(z, t) (9.12) ∂t ∂z ∂z ∂z The Richards equation is usually used in the form in which the left side of Eq. (9.12) is expressed as a derivative of the soil matric potential with time. Because the soil-water matric potential is a function of soil-water content (the soil-water retention curve), the left side of Eq. (9.12) can be written as follows: ∂θ ∂h w ∂h w ∂t

(9.13)

9.2 Basic Equations of Water Movement in Unsaturated Soil

123

The first term on the right side of Eq. (9.13) is the slope of the retention curve and is called specific soil-water capacity c(hw ); it is the change in soil-water content of a unit volume of soil due to a unit change of soil-water matric potential (here expressed in pressure heads). Then: ∂θ c(h w )  ∂h w   ∂ ∂k(h w ) ∂h w ∂h w c(h w )  k(h w ) + ∓ S(z, t) ∂t ∂z ∂z ∂z

(9.14) (9.15)

Equation (9.15) represents the basic form of the Richards equation in so-called “potential” form; it can be expressed also in a form, like Eq. (9.12), often used in simulation models of soil-water flow in homogeneous soils and under isothermal conditions. To solve this equation, it is necessary to know the soil-water retention curve (SWRC), hydraulic conductivity of soil as a function of soil matric potential, roots extraction patterns as a function of vertical coordinate and time and initial and boundary conditions. This problem will be analysed in Chap. 21.

9.3 Basic Characteristics of Water Flow in an Unsaturated Soil Unsaturated soil-water movement can be illustrated easily by the case of water flow through the horizontal soil sample (Fig. 9.1). The advantage of the horizontally positioned unsaturated soil sample is elimination of gravitational forces; this type of water movement is influenced only by the difference of matric (moisture) component of the soil-water potential applied on both ends of the soil sample.

Fig. 9.1 The model illustrating measurement of unsaturated-soil hydraulic conductivity in horizontal soil column

124

9 Water in Unsaturated Soil

The soil sample in the cylinder filled with soil is closed on both sides by so called “semipermeable” plate. The water saturated “semipermeable” plate is permeable for liquid water only, and air cannot pass through it because the pores are full of liquid water and held by capillary forces. This is valid for an air-pressure range below the so-called “bubbling” pressure, i.e. air pressure above which liquid water is forced to empty some of the pores and air can flow (bubbling) through the porous desk. Usually the “semipermeable” element can be made of porous membranes, ceramic plate and even of metal. In the cylinder containing soil, an opening should be made to equilibrate the air pressures on both sides of the cylinder containing the soil sample. When changing the positions of vessels on both sides of the cylinder with soil, the volume of water (soil-water content) changes, and the volume of water has to be replaced by air, which is flowing into the sample through the opening in the cylinder. To start the process, the soil sample in the cylinder (Fig. 9.1) is saturated with water at first by levelling both vessels with water above the cylinder. To saturate the soil sample, the air must escape the soil via an opening, followed by water after the saturation of the soil sample. Then, the water table in both vessels will be levelled to establish the difference of (negative) pressure heights Δh  h1 − h2 . Water will flow from the higher value of soil water matric potential h2 (higher matric potential is smaller in absolute value) to the smaller value of the matric potential h1 . The flow ¯ rate v will be assigned to the average pressure height h. Figure 9.2 is schematic of apparatus to measure water-flow rate through waterunsaturated soil. The soil sample (1) is placed in the horizontal cylinder (2) with an opening for air to escape (3) and equilibrate air pressure following soil-water content changes during water flow. The soil sample is located between two semipermeable desks saturated with water (4). The apparatus is fixed between two metal ends (5), with bolts (6). The contact areas between the cylinder (2), ceramic plates (4) and endings (5) are sealed by rubber rings (7). Both endings (5) contain the inflow, outflow (9) and air-escape tubing (8, 11), necessary for fill the apparatus and measuring flow in the system. The cylinder containing the soil sample is fixed to the apparatus (Fig. 9.1). The soil sample will be filled with water through the tubing (8), until water flows out via the air escape (3). Tubing (11), as a part of metal endings, function as an air escape for the space filled with water (12); then, openings are closed. The next step of the process is the levelling of water-table vessels to positions h1 and h2 . It is important to keep the difference between water levels in both vessels small; the smaller the difference is, the better is the representativeness of the measurement. The difference Δh  10 cm is acceptable. The outflow of water is measured usually by continuous weighing of the sample with a sensitivity 0.01 g. After establishing the difference Δh, the soil sample is drained at both ends at first. Then, the system will reach equilibrium and water will flow at a constant rate through the sample to the vessel with higher (in absolute value) negative pressure h1 . The Darcy–Buckingham equation can be written in the form: ¯ v  −k(h)

h ¯ h2 − h1  −k(h) L L

(9.16)

9.3 Basic Characteristics of Water Flow in an Unsaturated Soil

125

Fig. 9.2 A scheme of the equipment for water-flow measurement through an unsaturated soil sample

where v is constant water-flow rate through the soil sample corresponding to the ¯ and it can be expressed by the equation: average pressure height h; v

V A.t

where V is volume of water (L3 , m3 ) that flowed through the soil cross-section area A (L2 , m2 ), during the time interval Δt (T, s). The unsaturated hydraulic conductivity of the soil, using results of measurement on this equipment (Fig. 9.1) can be calculated using Eq. (9.17): ¯  k(h)

V L vL  h At h

(9.17)

Devices (Figs. 9.1 and 9.2) are used to measure the hydraulic conductivity of unsaturated soil; in this case, it is a function of hw ; (k  f (hw )), but it can be expressed also as a function of soil-water content (k  f (θ )). Soil-water retention curves are ambiguous functions, as they are of hysteretic character. This means that even the k  f (hw ) is ambiguous (Fig. 9.3). But, the relationship k  f (θ ) is unambiguous (Fig. 9.4) because there is only one hydraulic conductivity corresponding to the particular soil-water content; but, the particular soil-water matric potential could be identified by the family of soil-water contents. The unambiguous relationship k  f (hw ) can be used if there is a process of wetting or drying only, i.e. drying or wetting branches of the soil-water retention curves are followed. To measure water flow using the equipment (Fig. 9.1), preserving the average value of matric potential h¯ but increasing the difference Δh, the water-flow rate increases proportionally to Δh. Figure (9.5), showing the relationship v  Δh/L corresponding to various average values of soil-water matric potential (pressure height) h. The ¯ The lower the relationships v  f (Δh/L) are linear, but different for any value of h. ¯ value h is, the smaller the slope of this relationship. The derivative (slope) of the relationship v  f (Δh/L) is the hydraulic conductivity of unsaturated soil k  f (hw ) corresponding to the value h¯ (Fig. 9.5).

126

9 Water in Unsaturated Soil

Fig. 9.3 Hydraulic conductivities of unsaturated sand and clay (k) as functions of matric potential of soil water hw . Horizontal cross sections of vertical axis are saturated hydraulic conductivities. Relationships k  f (hw ) are of hysteretic character

Fig. 9.4 Hydraulic conductivities of unsaturated sand and clay (k) as functions of volumetric soil-water content θ. Crosses indicate saturated hydraulic conductivities. Relationships k  f (θ) are unambiguous

Fig. 9.5 Water-flow rate in unsaturated soil v and slopes of pressure height Δh/L for various average values of negative pressure heights h¯

The hydraulic conductivities of unsaturated sand and clay k as functions of matric potential of soil water hw are indicated on Fig. (9.3). Higher hydraulic conductivities of sand at higher (less negative) matric potentials than for clay can be seen. The saturated hydraulic conductivities K of coarse soils, like sandy soil, are higher than K of heavy soils, like clay (Table 8.2). Saturated hydraulic conductivities K are the intersections of the relationships k  f (hw ) and vertical, k axis. Various situations can be seen at relatively low soil-water contents (and matric potentials) of such soils. The relatively big pores of sandy soils empty quickly, thus decreasing their hydraulic

9.3 Basic Characteristics of Water Flow in an Unsaturated Soil

127

conductivity, while clayey soils with mostly small pores decrease their hydraulic conductivity slowly. Their hydraulic conductivity at low soil-water contents is usually higher because there is an existing continuous system of small pores, which conducts water. At the low soil-water contents of sandy soils, water in such soils usually creates isolated areas of “cuticular” water at the contacts between sand grains. As was mentioned, the relationships k  f (hw ) are of hysteretic character, which is shown schematically by the dashed lines in Fig. 9.3. The horizontal part of the relationship k  f (hw ) represents the range of soil-water content close to saturation, just above the zero soil matric potential (or just above the groundwater table, Fig. 7.3), which is indicated as a “capillary fringe”. Figure 9.4 graphs the hydraulic conductivities of sand and clay unsaturated with water as a function of volumetric soil-water content θ , (k  f (θ )). The saturated hydraulic conductivities K are indicated by crosses. To calculate soil-water movement (e.g. by simulation models), functional relationships k  f (hw ) are used because the potential of soil-water movement is the total soil-water potential(soil-water matric potential is one of its components). Soil-water content is not soil-water movement potential because transport of water from areas of lower soil-water content to the areas of higher soil-water content can be observed. Its rate depends on the distribution of total soil-water potential.

9.4 Measurement and Calculation of the Unsaturated-Soil Hydraulic Conductivity Function k(hw ) Methods of unsaturated-soil hydraulic conductivity estimation can be divided into methods of measurements and calculation and field and laboratory methods. Measurement methods of hydraulic conductivities generate data close to reality, but its measurement is time consuming due to small rates of soil-water movement, especially for low soil-water contents. Small flow rates at low soil-water contents can lead to significant measurement errors. Methods of hydraulic conductivities measurement (infiltration method, calculation of k by analysis of soil-water contents and soil matric potential profiles) are therefore used mostly to verify calculated functions k  f (hw ). Therefore, calculation methods of function k  f (hw ) have been developed, which can estimate this function relatively simply, with acceptable accuracy for wetting and drying processes. It has been shown that the approach originally proposed by Childs and Collis-George (1950), later modified by Mualem (1976), using the equation of the soil-water retention curve proposed by van Genuchten (1980), leads to the function k  f (hw ) applicable to the simulation models describing soil-water movement in an unsaturated soils.

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9 Water in Unsaturated Soil

9.4.1 Measurement of Unsaturated Hydraulic Conductivity of Soils The direct method of k estimation is based on the measurement of the differences between soil matric potentials of two soil sections (hw ) and Darcian flow rate (v). With the measured data, the unsaturated hydraulic conductivity k(hw ) can be calculated by substituting measured values into the Darcy–Buckingham equation in differential form as a function of the average value of soil matric potential h w   h w +1 (9.18) v  −k(h w ) z This simple method of unsaturated hydraulic conductivity measurement is the so-called infiltration method proposed independently by Budagovskij (1955) and later by Youngs (1964). This method of k(θ i ) and k(hw ) measurement is based on the assumption that, if steady infiltration (see Fig. 9.6) is driven by the force of gravity only and the gradient of soil-water content and soil matric potential during steady infiltration is zero (Δhw /z)  0, then vi  k (hwi ). The infiltration rate in the upper part of the soil profile vi is constant, corresponding to the constant soilwater content along the soil sample θ i or hwi . Therefore the infiltration rate equals the unsaturated hydraulic conductivity of soil corresponding to both θ i and hwi . See Fig. 9.6 for a schematic of the measuring device. Infiltrating water into the soil at a constant rate, the soil-water content profiles corresponding to the infiltration rate will be formed. After the infiltration front has reached the bottom of the soil profile, just above the capillary fringe, the area of constant soil-water content θ 1 , corresponding to the steady infiltration rate v1 , will be formed. Then, the infiltration rate can be increased to value v2 , corresponding to the soil water content θ 2 . This procedure can be repeated until enough paired values (vi , θ i ) are obtained.

Fig. 9.6 Infiltration of water into cylinder with a soil and soil-water content profiles, corresponding to three water infiltration rates

9.4 Measurement and Calculation of the Unsaturated-Soil Hydraulic …

129

This method can be used in the field and laboratory conditions, and it is suitable especially for relatively conductive soils, like sand or relatively light soils. Another variation of the above described method is the application of a pressureplate infiltrometer to infiltrate water into unsaturated soils at negative pressures. Pressure heights up to −10 cm can be applied, and the infiltration rate is measured by Marriott bottle. When a steady state of infiltration rate has been reached, the corresponding hydraulic conductivity k  f (hw ) can be estimated. The advantage of this method is the possibility to eliminate macropores of given dimensions (determined by the negative pressure head used) and to estimate hydraulic conductivity of micropores only. This (pressure-plate infiltrometer) method is much easier to perform and therefore is frequently used.

9.4.2 Unsaturated Hydraulic Conductivity of Soil, Calculated by Analysis of Soil-Water Content and Soil-Water Matric Potential Profiles The unsaturated-soil hydraulic conductivity calculation method based on the analysis of soil-water content and the soil matric potential profiles known for at different times is often named the “instantaneous profile method” (Watson 1966). It can be applied in field and laboratory conditions for bare soil. Water extraction by roots can be classified as a “short circuit” between the soil and atmosphere; therefore, application of instantaneous profile method for soils with plant canopy is not recommended. Evaporation from the soil surface can be usually neglected when using the instantaneous profile method. This method is based on application of the Darcy–Buckingham law (Eq. 9.18). To calculate k(hw ), it is necessary to know the average value of soil-water flow (Darcian) and the average total soil-water potential slope during the measurement interval Δhw /Δz. The slope of the matric potential can be used if the gravitational component of total soil-water potential can be neglected, which is often the case in unsaturated soils. The calculation procedure for unsaturated hydraulic conductivity k(hw ) is shown in Fig. 9.7. The requirements include two soil-water content profiles θ  f (z) for two times of measurement t 1 and t 2 , and two corresponding profiles of soil-water matric potentials hw  f (z). The average flow rate across the horizontal plane z  (z2 + z1 )/2 can be calculated from the soil-water content change below level z divided by a time interval Δt  (t 2 − t 1 ): ⎤ ⎡ L  L 1 ⎣ v(z)  θ1 (z)dz − θ2 (z)dz ⎦ (9.19) t z

z

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9 Water in Unsaturated Soil

Fig. 9.7 Calculation of hydraulic conductivity of an unsaturated soil, using two subsequent profiles of volumetric soil-water content θ and soil-water matric potential hw

The first integral represents the soil-water quantity at time t 1 at depth interval (z, L), where L is the rooting depth; the second integral represents the soil-water quantity in the same depth interval but at time t 2 ; and their difference is the water quantity that has flowed through level z. Divide this quantity by the time interval Δt, the average soil-water flow velocity through the level z during Δt can be calculated. The average value of soil-water matric potential slope Δhw /z can be estimated by direct measurement using a tensiometer, or more often by calculation, using soil-water content and the soil-water retention curve if there is a drying process only (drying branch). It should be mentioned that signs of the matric potentials in Eq. (9.20) are negative. The average value of soil-water matric potential can be calculated by the equation:   1 h w11 − h w12 h w21 − h w22 h w  − (9.20) z z 2 2 Now, substituting the calculated data into the Darcy–Buckingham equation in differential form, we obtain:  k hw 

v h w z

+1

(9.21)

The resulting unsaturated-soil hydraulic conductivity corresponds to the average value of soil-water matric potential h w . This procedure can be repeated for various depth z (see Fig. 9.7) to calculate k for various hw .

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131

This method is basic, but it is used mainly to verify unsaturated hydraulic conductivities k evaluated by other methods; this is because the process is time consuming and requires field measurements, but it gives realistic values for undisturbed soil.

9.4.3 Calculation of the Function k  f(hw ) Using Soil-Water Retention Curve and Saturated-Soil Hydraulic Conductivity Measurement of unsaturated-soil hydraulic conductivity function is time consuming and can be realised in only narrow intervals of matric potential of soil water. This interval is usually close to the saturated soil-water content because, at relatively low soil-water content (and matric potential), water-flow rates are small and difficult to measured. Therefore, a few proposals have appeared to calculate k  f (hw ), based on the distribution of soil pores according to their diameters as calculated using soil-water retention curves (Childs and Collis-George 1950; Marshall 1958). The basic progress in k  f (hw ) function calculation was reached by a combination of the calculation method proposed by Mualem (1976) and the analytical function expressing soil-water retention curve proposed by van Genuchten (1980). This method, known as the Mualem–Genuchten method, is often used to calculate k  f (hw ) as a function to be applied in simulation models. The analytical function expressing k  f (hw ) can be written as:

 m 2 1 k(Se )  K (Se )l 1 − 1 − (Se ) m

(9.22)

where k(S e ) is the unsaturated-soil hydraulic conductivity, as a function of effective soil saturation: m  θ (h w ) − θr 1  (9.23) Se (h)  θs − θr 1 + (α|h w |)n where α, n, m belong to van Genuchten’s soil-water retention parameters—usually in the range 1 ≤ n ≤ 10; 10−3 ≤ α ≤ 10−1 ; 0 ≤ m ≤ 1 (see Chap. 7). As an approximation, the equation m  1 − 1/n can be used. θ , θ r , θ s are soil-water content, residual soil-water content and saturated soil-water content, respectively (see Chap. 7). The empirical parameter l characterises the pores’ interconnections (l  0.5 is recommended). Generally, the relationship k  f (hw ) can be characterised by an exponential function. Therefore, to the small soil-water content changes, significant changes in soil hydraulic conductivity correspond. The empirical functions k  f (hw ) used previously are usually also of an exponential type. Often used function was proposed by Gardner (1956):

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9 Water in Unsaturated Soil

k  K exp c(h w − h b )

(9.24)

where hb is the pressure needed to enter air into a system of pores (bubbling pressure), its pressure height approximately equal to the capillary-fringe height, c is empirical parameter 0.01 ≤ c ≤ 0.04. hw in Eq. (9.24) has a negative value.

9.5 Water Movement at Low Soil-Water Content and Diffusivity of Soil To calculate soil-water movement in unsaturated soil, especially at low soil-water contents, where water-vapour flow cannot be neglected, the Richards (1931) governing equation is written using a diffusivity coefficient, or diffusivity (Eqs. 9.29 and 9.30). In this form of Richards equation, the potential of water movement is the volumetric soil-water content. It can be used to quantify non-hysteretic transport of soil water flow of low soil-water contents. Because the soil-water matric potential hw is an unambiguous function of volumetric soil-water content θ during non-hysteretic water transport in soil (drying or wetting only), it can be written: ∂h w ∂θ ∂h w  ∂z ∂θ ∂z

(9.25)

The Darcy–Buckingham equation, neglecting the gravitational component of total soil-water potential (this approximation can be used at small soil water contents), can be written: v  −k(θ )

∂h w ∂θ ∂h w  −k(θ ) ∂z ∂θ ∂z

(9.26)

Childs and Collis-George (1950) introduced the term diffusivity: D(θ )  −k(θ )

∂h w ∂θ

(9.27)

Then, using Eqs. (9.27), (9.26) can be written in the form: v  −D(θ )

∂θ ∂z

(9.28)

“Diffusivity” is defined for homogeneous soil and non-hysteretic soil-water transport. This term is sometimes confusing because of its similarity to term “diffusion”. But, the term “diffusivity” has nothing in common with the process of diffusion. Diffusion is transport of molecules in the direction of decreasing concentration of the considered compound; the mechanism of this movement is molecular diffusion,

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133

i.e. the energy of colliding molecules is the driving force. There are more collisions from the side with higher concentration of molecules in comparison to the number of collisions from the side of lower concentration of molecules; therefore the resulting movement of molecules is to the side of lower mass concentration. This process can be quantitatively described by Fick’s equation, which will be treated later. Because “diffusivity” has the same dimension as coefficient of diffusion (L2 T−1 ), authors used some modification of the word “diffusion”–diffusivity. Using diffusivity as it is expressed in Eq. (9.27), the Richards equation (9.15) can be written as:   ∂ ∂θ ∂k(θ ) ∂θ  D(θ ) + ∓ S(z, t) (9.29) ∂t ∂z ∂z ∂z For relatively dry, bare soil, the gravity term, as well as the root extraction term, can be neglected, and then Eq. (9.29) can be rewritten as:   ∂ ∂θ ∂θ  D(θ ) (9.30) ∂t ∂z ∂z The relationship between diffusivity D and volumetric soil-water content θ for sandy loam soil (Western Slovakia) is shown in Fig. 9.8. Liquid-water transport is dominant at higher soil-water contents, expressed by the diffusivity of the liquid phase (DL ), but water-vapour movement is dominant at low soil-water contents (DV ), with a local maximum at soil-water contents below the hygroscopicity number. To quantify soil-water transport at relatively low soil-water contents, where watervapour transport is significant, Eq. (9.30) can be recommended because the function k  f (hw ) does not involve water-vapour transport. But, water-vapour movement in the soil-water content range above the hygroscopicity number is usually negligible in comparison to the liquid-water movement. When modelling the soil-water regime, it is correct to use hydraulic conductivities k  f (hw ), describing liquid-water movement only.

9.6 Diffusion of Water Vapour in Soil Air (and water vapour as an air component) can be transported by two mechanisms: by convection, if there is a water-vapour pressure gradient, or by molecular diffusion, if there is a gradient of water-vapour concentration. Water-vapour transport by convection is a rare case because the water-vapour pressure difference between two points is usually very small. Water-vapour movement by molecular diffusion is a more common case, but it is assumed to less significant. Molecular diffusion of water vapour (also other compounds) can be quantitatively expressed by Fick’s law. This law is expressed by the equation, formally similar to those of Ohm, Fourier, or Darcy:

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9 Water in Unsaturated Soil

Fig. 9.8 Diffusivity D and volumetric soil-water content θ of a sandy soil. DL and, DV are diffusivities of liquid water and water vapour (Šutor and Novák 1968)

qv  −Dvs (θ )

∂ρv ∂x

(9.31)

where qv is the molecular diffusion rate of water vapour in soil, (M L−2 T−1 , kg m−2 s−1 ), Dvs is the coefficient of molecular diffusion of water vapour in soil, (L2 T−1 , m2 s−1 ) and ρ v is water-vapour density, (ML−3 , kg m−3 ). The coefficient of molecular diffusion of water vapour in soil Dvs can be expressed as: Dvs  Dva τ (P − θ )

(9.32)

where Dva is the coefficient of molecular diffusion of water vapour in air (Dva  1.89 × 10−5 m2 s−1 , at T  20 °C); τ is so called tortuosity, i.e. correction on the transport path curvature, τ  0.66; and P is soil porosity. The continuity equation can be expressed as: ∂qv ∂θ − ∓ S(z) ∂t ∂z

(9.33)

where S(z) can express adsorption or desorption of water vapour in a unit volume of the soil. Water-vapour density at constant temperature (isothermal conditions) is a function of soil-water content only (ρ v  f (θ )), and it can be written:

9.6 Diffusion of Water Vapour in Soil

135

∂ρw ∂θ ∂ρw  ∂x ∂θ ∂ x

(9.34)

Similarly, like soil-water diffusivity DL , even the diffusivity of water vapour DV can be expressed by the equation: DV (θ )  Dvs

∂ρw ∂θ

(9.35)

Then, the resulting equation describing soil-water content change as a function of time due to molecular diffusion of water vapour is:   ∂θ ∂ ∂θ  DV ∓ S(z) (9.36) ∂t ∂x ∂z As can be seen in Fig. 9.8, the maximum value of the coefficient of water-vapour diffusion in soil DV is in the region of very small soil-water content; where relative air humidity is less than one, the main area of pores is occupied by air. This mechanism of water-vapour transport can be significant at low soil-water contents, or under nonisothermal conditions. But, the transport of oxygen by molecular diffusion to the aeration zone is of primary importance to create the conditions for roots respiration.

References Buckingham E (1907) Studies on the movement of soil moisture. Bulletin 38, USDA Bureau of Soils, Washington DC Budagovskij AI (1955) Water infiltration into soils. Nauka, Moscow (in Russian) Childs EC, Collis-George N (1950) The permeability of porous materials. Proc R Soc Lond A 210:392–405 Gardner WR (1956) Calculation of capillary conductivity from pressure plate outflow data. Soil Sci Soc Am Proc 20:317–320 Marshall TJ (1958) A relation between permeability and size distribution of pores. J Soil Sci 9:1–8 Mualem Y (1976) A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour Res 15:513–522 Richards LA (1931) Capillary conduction of liquids through porous mediums. Physics 1(5):318–333 Šutor J, Novák V (1968) The effect of temperature gradient on movement of soil water. In: Transactions. 9th international congress of soil science, vol 1. ISSS, Angus and Robertson, Adelaide, pp 85–93 van Genuchten MT (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Watson KK (1966) An instantaneous profile method for determining the hydraulic conductivity of unsaturated porous materials. Water Resour Res 2:709–715 Youngs EG (1964) An infiltration method for measuring the hydraulic conductivity of unsaturated porous materials. Soil Sci 97:307–311

Chapter 10

Infiltration of Water into Soil

Abstract Infiltration is the process of water entry into soil; it is the basic process supplying the soil root zone with water. Infiltration as a process can be divided into two basic types: rain infiltration and ponding infiltration (infiltration from ponds on the soil surface). Infiltration (and cumulative infiltration) curves of both types of infiltration are described and expressed by empirical and analytical functions proposed by Philip. Special attention is devoted to the Green and Ampt approach to quantify infiltration because of its simplicity and clear physical interpretation of the process. The influences of the rain rate, entrapped air, initial soil-water content and layered soil profile on infiltration dynamics are discussed. Methods for the estimation of the soil hydraulic conductivity using the infiltration process in the field are described as well.

10.1 Infiltration into Homogeneous Soil Infiltration of water into soil is the process of water entry into soil through the soil surface, most frequently (but not necessarily), in the vertical direction, downward. Water can infiltrate into soil from rain; this is the dominant case, then during irrigation, from temporary ponds, thawing snow, from rivers or artificial water reservoirs. From a hydrologic point of view, the most important source is infiltration from rain. If the precipitation rate is higher than the infiltration rate, water can accumulate on the soil surface, forming temporary ponds, or part of the rain water can form surface runoff. The surface-runoff formation is usually a dangerous phenomenon. Erosion destructs soil and transports it to rivers via flowing water which can contribute to dangerous floods. Therefore, understanding and quantification of the infiltration process, as it depends on environmental properties, is the necessary condition of the regulation of landscape water regimen. The aim is to maximize infiltration to use the retention capacity of the soil, minimize runoff and prevent soil erosion. Physically, water can infiltrate from rain, from irrigation sprinkling equipment and from ponds of water on the soil surface, so-called ponding infiltration. During © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_10

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10 Infiltration of Water into Soil

rain infiltration, water usually does not cover the soil surface in a continuous layer of water; water infiltrates from separate drops of water and thus allows the release air from the soil. Ponding infiltration occurs when the soil surface is covered by continuous layer of water. Infiltration can be either steady or unsteady. Steady infiltration is characterized by a steady (constant) rate of infiltration, but unsteady infiltration rate changes with time, usually decreasing with time. Infiltration is usually treated as infiltration into homogeneous soil, which is a good approximation in most cases. But, in some cases (preferential ways occurrence, layered soils), it is necessary to take into account nonhomogeneities of soils, as will be explained later. Detailed descriptions of the infiltration process and methods for its measurement and quantification are presented in the monograph by Angulo-Jaramillo et al. (2016).

10.1.1 The Basic Characteristics of Infiltration The infiltration rate vi is expressed by the water quantity (volume) infiltrating into soil through a unit area of soil surface and during a unit of time t: vi 

V dV ≈ Adt At

(10.1)

where vi is infiltration rate (L3 L−2 T−1 , cm3 cm−2 s−1 ), V is the volume of water infiltrated into the soil (L3 , cm3 ) during the time t (T, s) and soil-surface area A (L2 , cm2 ). The infiltration rate is often expressed by the thickness of the water layer infiltrated during the unit time interval (mm s−1 ). Cumulative infiltration of water into soil i is the quantity (volume) of water infiltrated into soil through a unit soil-surface area during the time interval from the beginning of infiltration t. It can be expressed as: i

V (t) A

(10.2)

where V (t) is the volume of water infiltrated into soil during the time interval t from the infiltration beginning (L3 , cm3 ). The dimension of cumulative infiltration i is normally the length, i.e. it is expressed as the thickness of the water layer infiltrated into soil from the beginning of infiltration (L, cm). The relationship between the infiltration rate and cumulative (total) infiltration can be expressed by the equations: t vi dt

i 0

(10.3)

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139

Fig. 10.1 Infiltration curve vi  f (t) and cumulative infiltration curve i  f (t) of a homogeneous soil from rain or trickle irrigation

Fig. 10.2 Infiltration curve vi  f (t) and cumulative infiltration curve i  f (t) of a homogeneous ponded soil

vi 

di dt

(10.4)

The infiltration rate vi , which depends on the time t i from the infiltration beginning, is called the infiltration curve (vi = f (t)). The dependence between cumulative infiltration i and time t from the start of infiltration is the cumulative infiltration curve (i  f (t)). Both relationships are illustrated in Figs. 10.1 and 10.2.

10.2 Infiltration into Soil from Rain or Sprinkling Irrigation Precipitation is the source of fresh water on the Earth. The substantial part of the precipitation falling on the Earth infiltrates into soil. Surface runoff is a rare case in temperate zones and is observed during heavy rains only. The surface runoff is estimated to be a few percent of total runoff. Infiltration of water from rain results from individual drops, randomly distributed on the soil surface. The soil surface is

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10 Infiltration of Water into Soil

not ponded, therefore air from the soil can escape; the air quantity entrapped in the soil during rain or sprinkling irrigation is smaller than during ponded infiltration. The infiltration rate vi during the rain of a rate vz smaller than the maximum possible infiltration rate is the same as rain rate, vi  vz (Fig. 10.1), and the entire rain volume infiltrates into a soil. The maximum possible infiltration rate into the soil (sometimes called the infiltration capacity) depends mainly (for a particular soil) on initial soil-water content. Therefore, it is not a constant value. After some time, the rate of infiltration will decrease, and water ponds form on the soil surface. This time is called the ponding time, and it is denoted by symbol t p . Of course, the ponding time for a particular soil is not constant, but varies, depending on the rain rate and initial soil-water content (distribution). The infiltration rate decreases continuously, up to the constant rate denoted as vc , which is approximately equal to the soil’s saturated hydraulic conductivity (vc ÷ K). Usually some of the pores contain entrapped air, which decreases the soil-water content and infiltration rate, therefore the soil is usually not fully saturated and the “real” soil’s saturated hydraulic conductivity should be higher than measured by this method. Moreover, water can infiltrate at the rate vc theoretically for a very long time; this is usually denoted as the soil saturated hydraulic conductivity. During the majority of rain events, ponding or surface runoff is not observed, because rain rates are usually lower than saturated hydraulic conductivity of soils. Saturated hydraulic conductivity of majority of agricultural soils is higher than 20 mm h−1 ; only small numbers of rain events in temperate climate zone are at higher rates. In meteorology, the term “heavy” rain denotes rains at rates in the range of 8–40 mm h−1 . Rain events of such rates are rare, and their duration is usually shorter than one hour. Soil-surface ponding is often observed on alluvial deposits of rivers (e.g. East Slovakia Lowlands), where soils are composed of fine sediments with very low values of saturated hydraulic conductivities (K < 1.0 cm d−1 ). In such locations, the slopes of the soil surface are usually very small, and surface runoff is not significant. Alluvial soils are usually ponded during the springtime so that tillage should be postponed, which is the reason for relatively low crop yields. Two horizontal, linear sections of the rain infiltration curve (Fig. 10.1) are identified. The first section is at the initial stage of infiltration (0, t p ) where rain and infiltration rates are the same vi  vz . The other linear section of the infiltration curve is identified at time interval (t ≥ t c ), where the infiltration rate is steady (vi  K). The cumulative infiltration function represents those two linear, horizontal sections by the linear sections of constant slope of the function i = f (t). The hatched area (Fig. 10.1), for time interval (t > t p ), represents water volume accumulated on the soil surface, or forming surface runoff. Relationships vi = f (t i ) and i  f (t) are important information because, using them, the water quantity to be infiltrated into the soil during the indicated time can be calculated. Then, the proper irrigation rate to prevent soil ponding can be estimated. Those relationships can be estimated by the rain simulators, which can regulate simulated rain rates for safe infiltration.

10.3 Ponding Infiltration

141

Fig. 10.3 Infiltration curves of ponded bare soils of various textures

10.3 Ponding Infiltration Ponding infiltration is the process of water entry into soil through its surface when the soil is fully covered by a layer of water. Ponding infiltration can be observed during a flood, when soil surface is ponded or after intensive rain, when low permeable soil accumulates water in temporary ponds. The infiltration curve vi  f (t i ) and corresponding cumulative infiltration curve i  f (t) of ponding infiltration are sketched in Fig. 10.2. The relationship vi  f (t i ) is typical, with a rapid decrease in infiltration rate vi at the beginning of infiltration process; the smaller the initial soil-water content is, the faster decrease of infiltration rate is observed. After some time t c , the ponding infiltration rate becomes steady (constant); its value will be close to the saturated hydraulic conductivity of saturated soil vc ÷ K. The reason for such a infiltration rate course is the large hydraulic gradient at the infiltration front at the beginning of infiltration process; after its decrease the infiltration rate is determined mostly by the gravitational component of the soil-water potential; its gradient equals one (Eq. 9.18). Figure 10.3 illustrates the ponding infiltration curves of three dry bare soils: sand, silty loam and clayey soil. The higher the saturated hydraulic conductivity of soil is, the higher an infiltration rate is observed. Finally, the infiltration rates are constant, approximately corresponding to the value of saturated hydraulic conductivity of soil K. Table 8.2 shows some typical values of saturated hydraulic conductivities of various types of soil, equal to the steady-state infiltration rates (vi = vc ) (Hillel 1973). Infiltration rates of water into soil are dependent not only on the type of soil, but also on the properties of the surface layer of the soil. This surface layer of soil can be significantly influenced by plant canopy, tillage, soil-surface micromorphology, mulching, and soil cracks or by soil crust. Soil crust is usually the top layer of soil of high density and low hydraulic conductivity. It is usually formed by saturation of the soil surface, its peptisation and by following fast drying.

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Fig. 10.4 Cumulative curves of ponded infiltration into a silt loam with different surfaces

The influence of four surfaces on cumulative infiltration curves of water into silt loam are depicted in Fig. 10.4. Grass canopy, mulch (made of residuals of plants), wheat and bare soil show significant influence on the infiltration rate. Grass canopy, together with its root system forms a system of macropores, allowing rapid movement of water during infiltration. In such a case, there is no dominant soil texture, but soil structure, formed by the permanent grass canopy ant its root system. A relatively low infiltration rate into the soil with a wheat canopy can be explained as a result of tillage techniques. The use of heavy machinery usually increases the subsoil density and decreases its hydraulic conductivity, thus decreasing the infiltration rate. The relationships vi = f (t) and i = f (t) are usually the results of infiltration test. Infiltration rings are usually applied to measure infiltration curves (Figs. 10.5 and 10.6). The infiltration rate is measured at a constant water-table level in the cylinder, until a constant rate of infiltration is reached. It is assumed vc ÷ K. Water from the rings infiltrates into relatively small area, so air before the infiltration front can escape horizontally from the space, and it can be assumed that only a small decrease of infiltration rate results from using this method. The double-rings infiltration method for K measurement gives better results than the single-ring method because the flow of water from the outer ring partially eliminates side leaking. Then, the area of the inner cylinder is close to the real infiltration area. This is valid mostly for relatively conductive soils (sandy and sandy loam), where the gravitational component of total soil-water potential dominates. In heavy soils, the leaking of water in the horizontal direction is more significant. The result is that, during the infiltration into light soils, the infiltration profile is prolonged in the vertical direction, as would be expected. The infiltration profile of heavy soil is close to the sphere (ring in the two-dimensional presentation), which can lead to a significant error in the K calculation (Fig. 10.13). An infiltration test (often ponding infiltration from double rings) to calculate K is used frequently; it is simple to perform in the field and easy to calculate. Saturated

10.3 Ponding Infiltration

143

Fig. 10.5 Characteristic shape of infiltration front during water infiltration from a ring into soil; water infiltrates also laterally, behind the ideal “dashed” area

Fig. 10.6 Characteristic shape of infiltration front of water infiltrated from central and outer cylinder; ideal profile (dashed line) is only slightly modified (outer dashed line) during infiltration from double-ring infiltration set

hydraulic conductivity K evaluated by this method are close to the real values. Alternative methods of K estimation in the field (various modifications of the auger-hole method, piezometric method, pumping test) require complicated devices and more complicated processing methods for the measurement results.

10.4 The Influence of Initial Soil-Water Content and Rain Rate on Infiltration The shape of the infiltration curve for particular soil vi  f (t) depends also on the initial soil-water content θ i . For ponding infiltration, the difference between initial soil-water content and the saturation soil-water content determines the infiltration rate. During ponding infiltration, the maximum soil-water content equals the saturation θ s , therefore, the initial soil-water content is decisive in determining the infiltration rate. The infiltration rate into soil saturated with water (θ i  θ s ) is minimal and equals the saturated hydraulic conductivity of the soil (vi  K). This is the principle of K measurement in the field.

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Fig. 10.7 Infiltration curves vi  f (t) and ponding time (t pi ) for a constant rain rate for various initial soil-water content θ i . Rain rate is vz  .2 mm min−1 , initial volumetric soil water contents are θ i  0.2, θ i  0.3 and θ i = 0.4 cm3 cm−3 . Loamy soil (according to Rubin 1966)

Fig. 10.8 Infiltration curves vi  f (t) and ponding time (t pi ) corresponding to various sprinkling irrigation rates vz  0.2, 0.3, and 0.4 mm min−1 , respectively. Initial soil-water content was θ i  0.265 cm3 cm−3 . Loamy soil (According to Rubin 1966)

The influence of initial soil-water content on infiltration rate from rain is illustrated in Fig. 10.7. It can be seen that the ponding time t p (when ponds start to form on the soil surface) becomes shorter with higher initial soil-water content. The reason is clear: the higher is the difference between initial soil-water content θ i and soil-water content corresponding to the rain (or sprinkling irrigation) rate θ 0 , the higher is the difference in soil-water matric potentials between both soil-water contents and the higher the infiltration rate. Shape of the infiltration curve of particular soil vi = f (t) depends on the rain rate vz (Fig. 10.8). The highest the rain rate is, the sooner the ponds on the soil surface will form. In actuality, the ponding time is indirectly proportional to the rain rate for identical initial soil-water content θ i .

10.5 Infiltration and Entrapped Air Theoretical analysis of water infiltration into soil assumes escape of air through the porous space before the infiltration front, and infiltration process is not influenced by the entrapped air. This assumption is usually valid during rain infiltration or infiltration from sprinkling irrigation, if their rates are lower than the saturated hydraulic

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145

Fig. 10.9 Infiltration rate of water into soil with entrapped air which can escape (1), and with one that is compressed before infiltration front, but after reaching “bubbling” pressure does escape (2)

conductivity of soil. But ponding infiltration could be significantly influenced by the presence of air in the soil. Air presence in the soil pores always decreases infiltration rates. Typical examples of the influence of entrapped air on infiltration kinetics are: Limited escape of air from the soil During infiltration, the infiltration front compresses air before the front, so air cannot escape into the atmosphere. As the infiltration front advances, the air pressure in front of its advance increases, and the infiltration rate decreases. After reaching the air pressure before the infiltration front becomes higher than the air pressure needed to remove water from soil pores (bubbling pressure), air escapes from the pores, and the infiltration rate increases immediately. This phenomenon known as the “pot effect” is observable after irrigating plants in pots. Infiltration curves of ponding infiltration, with and without air escape from the soil, are sketched in Fig. 10.9. The differences are significant. Entrapped air Even soil saturated with water contains air entrapped in isolated pores, thus decreasing the retention capacity of the soil, as well as its hydraulic conductivity and finally infiltration rate. This effect is difficult to eliminate, but it can be minimized by proper procedures of soil saturation, as was mentioned before. Císlerová et al. (1988) demonstrated the role of entrapped air on steady-state infiltration and identified the decrease of the steady-state infiltration rate in recurrent ponding infiltration experiments. It is assumed that, in repetitive ponding, additional air is entrapped and thus the “effective” porosity and steady infiltration rates decrease too. Different air pressures in soil pores can be the reason of the local transport of water in the soil. Dissolved air in soil water Soil water contains some percent of dissolved air, the quantity of which depends on the temperature of the soil water and on air pressure. The concentration of air dissolved in soil water is usually below 5% of water volume (Swiecicki 1967). The

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Fig. 10.10 The soil-water content profiles during infiltration. Typical soil-water content distribution during infiltration (a) and realistic shape of the infiltration front with fingers of infiltrated water (b)

volume of air dissolved in soil increases with temperature and with an air-pressure decrease, thus the changing retention capacity and hydraulic conductivity of soil. Air dissolved in water can enter soil during sprinkling irrigation; such water is usually highly aerated. In the laboratory, it is recommended to use deaerated water (preboiled or deaerated by decreased air pressure in the container with water), but not distilled water.

10.6 Soil-Water Content Profiles During Infiltration Previously, the kinetics of infiltration from rain or ponds into homogeneous soil (infiltration curves) were analyzed. It is important to have a picture of the infiltration into the soil, especially the soil-water content and soil-water matric potential profiles since they depend on soil properties. Penetration of water into the soil during infiltration can be observed in the field or laboratory as an advance of a dark area (area of relatively high soil-water content) into an area of lower soil-water content (SWC). The interface between dark (higher SWC) and light (lower SWC) areas is the infiltration front. The relatively dry area of a soil profile represents the initial soil-water content. The rate of advance of the infiltration front depends mainly on the difference between initial soil-water content and water content of the relatively dark area, i.e. the area with infiltrating water, to which correspond differences in soil-water matric potentials, the driving force of infiltration together with gravity. Figure 10.10 shows a typical soil-water content profile during water infiltration into homogeneous soil of initial soil-water content θ i . The soil-water content of the soil surface during rain infiltration is usually less than the saturated SWC (θ 0 < θ s ). The soil-water content of the surface-soil layer during ponding infiltration and in so-called “transit” areas equals the saturated soil-water content θ 0  θ s . The soilwater content of the thin surface layer of soil during ponding infiltration can be higher

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147

than estimated saturated soil-water content in the transit zone; the long interaction of the soil-surface layer with water can change the structure of the upper soil layer, so this thin layer of soil develops different parameters. The soil-water content of the transit area of infiltration profile is stable. The fluently changing SWC profile at the infiltration front is the profile of average SWC (Fig. 10.10). In reality, there are some dry areas and some fully saturated ones; “fingers” are formed even in relatively homogeneous soil. A much more dramatic picture of infiltration can be observed in nonhomogeneous soils. The calculated soil-water content profiles during vertical rain infiltration into silt loam from Trnava site (Western Slovakia) are presented in Fig. 10.11. The initial soil-water content is θ i = 0.03 cm3 cm−3 . Two SWCs at the soil surface were applied resulting in two SWC profiles series. The soil-water content profiles are close to the “box” shape-like profiles; i.e. soil-water content changes at the infiltration front occur within a small range of soil depths. This is due to low initial soil-water content. The infiltration rate is proportional to the difference of the upper soil layer’s soil-water content (determined mainly by the rain rate) and the initial soil-water content (Fig. 10.11). The higher is the rain rate, the higher are the infiltration rate and the rate of infiltration-front advance. The soil-water content of the transit zone corresponds to the infiltration rate (and to the rain rate). It is valid until the rain rate becomes smaller than the maximum possible infiltration rate (infiltration capacity). Figure 10.11 presents two soil-water content profiles of transit zones of soil (θ 1 = 0.45 cm3 cm−3 and θ 2  0.325 cm3 cm−3 ), the infiltration rates differing by a factor of 50! It means the infiltration rate is proportional to the rain rate (at the same initial SWC), but the SWC difference between transit zone and initial soil-water content θ i on infiltration rate dominates. The driving force of soil-water movement is a gradient of soil-water potential (total); during the initial phase of infiltration, it is the gradient of the soil-water matric potential; therefore it is interesting to know its distribution across the soil depth during infiltration. Figure 10.12 shows the calculated distributions of soilwater matric potential hw during ponding infiltration for three time intervals from the infiltration beginning t 1 < t 2 < t 3 . It also illustrates the continuous decrease of hydraulic gradients at the infiltration front over time. Infiltration profiles also show that soil-water content profiles are not more box-like types, and infiltration rate becomes smaller. Finally, the rate of infiltration will be constant and equal to the soil saturated hydraulic conductivity K (horizontal section of the infiltration curve, Fig. 10.1) when the infiltration front is deep, or it reaches the groundwater table. Then, the driving force for the infiltration becomes the gravitational component of the soil-water potential and the slope of the line h  f (z) will be one; in Fig. 10.12, it is indicated by a dashed line. The previous section dealt with rain or ponding infiltration, where infiltrating water was uniformly distributed over the soil surface. During water infiltration from the small soil-surface areas (drop irrigation) or from the linear source (furrow irrigation), the water distribution during irrigation is nonuniform. The locations of infiltration fronts during infiltration from furrows for three time intervals from the infiltration beginning (t 1 < t 2 < t 3 ) for sand, loam and clay are shown in Fig. 10.13. For clay, the

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Fig. 10.11 Calculated soil-water content distribution profiles during rain infiltration into a loam. Initial condition of soil-water content was θ i = 0.03 cm3 cm−3 and two soil-surface layers’ water contents corresponding to various rain rates were θ 1  0.45 cm3 cm−3 (1), and θ 2  0.325 cm3 cm−3 (2), respectively. Time intervals from the infiltration onset are shown for particular profile (Trnava site, Western Slovakia)

Fig. 10.12 Calculated soil-water matric potential profiles of hw = f (z) during the ponding infiltration of water and three time intervals from the infiltration onset; t 1 < t 2 < t 3 . Dashed line is the final distribution of the total soil-water potential of the silt loam soil

infiltration front has a cylindrical form; this means water infiltrates in all directions at (approximately) the same rates, and the gravitational component of the soil-water potential does not influence infiltration significantly. The greater soil pores are, the more significant the role of gravitational component of soil-water potential is and the further downward is the infiltration front prolonged.

10.6 Soil-Water Content Profiles During Infiltration

149

Fig. 10.13 Schematic positions of infiltration fronts during furrow infiltration, for three time intervals from the infiltration onset (t 1 < t 2 < t 3 ) and three soil types: sandy, loamy and clayey soil Fig. 10.14 Infiltration curves of chernozem soil with soil crust (1) and without soil crust (2)

Rain drops impact the soil surface and disintegrate soil aggregates; this increases the local soil-water content and then the soil swells and peptizes. Soil can dry, and a thin, dense soil layer with low hydraulic conductivity—the soil crust—can form on the soil surface. Soil crust decreases infiltration and also evaporation rate. Figure 10.14 contains infiltration curves of silt soil with soil crust (1) and without soil crust (2). Soil crusts do not generally form on sandy soils.

10.7 Infiltration Calculation According to Green and Ampt Infiltration is the first and the most important process of water entering the soil and of soil-water resource formation. Water resources formed in this way are used during plant ontogenesis. One of the first, physically based methods of infiltration quantification was proposed by Green and Ampt (1911). Their method is simple and easily interpreted. Today, the Green and Ampt method is used rarely; but its simplicity and physical interpretability enables understanding of the fundamentals of the infiltration process, and therefore it is frequently presented in the modern literature.

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Fig. 10.15 Idealized soil-water content distribution (box-like profile) during infiltration according the Green and Ampt model (1911)

The infiltration-front advance in the soil is interpreted by the Green and Ampt theory as a piston-like movement (Fig. 10.15), pushed by the pressure of ponded water on the soil surface (expressed by the pressure height h0 ) and pulled down by the force acting at the infiltration front (H f ). This force H f depends on the difference between the soil-water content of the transit (transmission) range of the infiltration profile θ 0 and initial soil-water content θ i . The soil-water content of the transit zone of soil during ponding infiltration equals the saturated soil-water content (θ 0  θ s ). The gravitational component of infiltrated water, expressed by the depth of infiltration front (L f ), acts downward. The initial soil-water content θ i is constant with soil depth. The ponding infiltration rate according to Green and Ampt (1911) can be expressed by the equation: vi  K

h0 − H f + L f Lf

(10.5)

where vi is the infiltration rate (cm3 cm−2 s−1 ; cm s−1 ); h0 is the thickness of the water layer on the soil surface (cm); H f is the effective pressure height (negative) at the infiltration front, acting downward (cm); and L f is the depth of the infiltration front below the soil surface (cm). Let the soil surface not be covered by a continuous layer of water during the rain or sprinkling (h0  0); the soil-water content of the transiting part of the soilwater content profile is less than the saturated hydraulic conductivity of the soil (θ 0 < θ s ), and hydraulic conductivity of the transit zone is less than the saturated hydraulic conductivity (k(θ 0 ) < K). Then, the rate of infiltration can be expressed by the equation:   −H f + L f −H f  k(θ0 ) +1 (10.6) vi  k(θ0 ) Lf Lf where k(θ 0 ) is hydraulic conductivity of unsaturated soil corresponding to the transit zone’s soil-water content θ 0 (cm s−1 ).

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151

The infiltration rate in the horizontal direction, where the gravitational component of soil-water potential can be neglected, can be expressed by the equation:   −H f (10.7) vi  k(θ0 ) Lf The rate of infiltration will be less than in the case of vertical infiltration; the quantity one is omitted from the right side of Eq. (10.7), characterizing the gravitational component of the total soil-water potential. The ponding infiltration rate in the vertical direction will be higher than the rate of rain infiltration or horizontal infiltration because, on the right side of Eq. (10.7), the pressure height of the water layer on the soil surface h0 will be added (remember the negative sign of the pressure height H f ; by substituting it into Eq. (10.8), resulting in a positive sign):   h0 − H f +1 (10.8) vi  k(θ0 ) Lf Equation (10.5) is the result of a simple idea about water infiltration into soil. As has been shown, this approach became popular due to its simplicity and physical intuitiveness. Jury and Horton (2004) applied the approach of Green and Ampt to develop an equation to calculate soil hydraulic conductivity as a function of soil-water matric potential of the transit zone of infiltration, knowing the cumulative infiltration in time t (Fig. 10.15):   i(t) K (h 0 )t  i(t) − hθ ln 1 + hθ where θ  θ0 − θi , h  h 0 − H f θ i is the initial volumetric soil-water content, θ 0 is the constant soil-water content of the depth range between the soil surface and infiltration front; h0 is soil-water matric potential corresponding to the soil-water content θ 0 , H f is the soil-water potential at the infiltration front; it can only be estimated by infiltration experiments for a particular soil (Morel-Seytoux 2008).

10.7.1 Horizontal Infiltration The rate of horizontal infiltration of water into soil is possible to calculate from Eq. (10.7); cumulative (total) infiltration can be expressed as (see Fig. 10.15):

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i  L f (θ0 − θi )  L f Δθ

(10.9)

where θ 0 is the soil-water content of the soil-surface layer; it equals approximately the soil-water content in the transit zone; during ponding infiltration it equals the saturated soil-water content; the infiltration rate vi can be expressed by the Eq. (10.4): vi 

dL f di  Δθ dt dt

(10.10)

The equation developed as a combination of Eqs. (10.7) and (10.10) can be integrated within limits (0, L f ) and (0, t): L f 0

  t  −H f k(θ0 ) dt L f dL f  θ

(10.11)

0

The location of the infiltration front L f can be then expressed as:   21   −H f t L f  2k(θ0 ) θ

(10.12)

Cumulative infiltration can be expressed by substitution of L f into Eq. (10.9):   1 1 i  L f Δθ  2k(θ0 ) −H f Δθ 2 t 2

(10.13)

Infiltration rate vi can be expressed by the derivation of Eq. (10.13):  1 2 −1 k(θ0 )  di  −H f Δθ t 2 vi  dt 2

(10.14)

For horizontal ponding infiltration (infiltration from a river or channel into soil), the following equations can be written: 1 1   i  2K −H f Δθ 2 t 2  1 2 −1 K vi  −H f Δθ t 2 2

(10.15) (10.16)

From these equations follows the proportionality of the infiltration rate vi to the hydraulic conductivity of the transit zone of soil and to the soil-water content difference between the transit zone of soil and initial soil-water content. This proportionality decreases with time (it is indirectly proportional to the square root of time from the beginning of infiltration). Because the infiltration rate is always positive value, cumulative infiltration increases proportionally with the square root of time from the infiltration beginning.

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153

The above equations can be applied also for vertical infiltration when the ratio (−H f /L f ) is negligible in comparison to the influence of gravity on infiltration. It is usually valid during the initial stage of infiltration, when (−H f ) is some orders higher than the influence of gravity on infiltration.

10.7.2 Vertical Ponding Infiltration Equations describing vertical ponding infiltration using Green and Ampt approach can be developed in a similar way as was done for horizontal infiltration. Equation (10.5) can be rewritten in the form: vi  K

[(h 0 − H f ) + L f ] Lf

(10.17)

By combining Eqs. (10.17) and (10.10) and after separation of variables t and L f , it can be written that: t

θ dt  K

0

L f 0

Lf dL f (h 0 − H f ) + L f

(10.18)

After integration within mentioned limits, the resulting equation expresses the relationship between the time elapsed from the infiltration beginning t and the infiltration-front position L f . The infiltration curve vi  f (t), can be calculated if the necessary soil characteristics, infiltration boundary and initial conditions are known: t

 θ  L f − (h 0 − H f ) ln (h 0 − H f ) + L f K

(10.19)

For rain infiltration without ponding, h0  0 and k(θ ) < K; Eq. (10.19) can be then simplified to: t

 θ  L f + H f . ln L f − H f k(θ )

(10.20)

Both equations can be solved by iterative methods. For various values of L f (and constant values of θ , k(θ ) and H f ,), the equation can be solved until the required accuracy between both sides of the equation is reached. If the infiltration front is in deep soil horizons, or if the groundwater is recharging (L f > H f ), so that an infiltration front does not exists, then H f  0, the soil-water content of the transit zone of soil is θ 0 , and Eqs. (10.19) and (10.20) will be reduced to the form:

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t

θ i Lf  k(θ0 ) k(θ0 )

(10.21)

From Eq. (10.21) also follows the hydraulic conductivity of soil corresponding to the soil-water content of transit zone of soil during the steady vertical infiltration in a homogeneous soil profile equals the infiltration rate k(θ 0 )  vi .

10.7.3 Pressure Head at the Infiltration Front To calculate infiltration parameters by the Green and Ampt method, it is necessary to know the pressure height at the infiltration front H f . Morel-Seytoux and Khanji (1974) proposed the equation of H f calculation that describes the infiltration into wet soil: 1 Hf  K

0 k(h w )dh

(10.22)

h wi

where hwi is soil-water matric potential (m), expressed by its pressure head, corresponding to the initial soil-water content θ i . Substituting the exponential function k(hw ) from Eq. (9.24) into Eq. (10.22), and after integration, the following relationship can be written: Hf  −

1 c

(10.23)

where H f values are usually in the range (−10) cm (coarse soils) up to (−100) cm (fine soils). According to Morel-Seytoux et al. (1996), H f can be expressed using the parameters of the soil-water retention curve of van Genuchten’s (1980) equation α, and m  1 − 1/ n: Hf 

0.046m + 2.07m 2 + 19.5m 3  1 + 4.7m + 16m 2 α

(10.24)

From the previous analysis, it follows that the just described procedures of H f calculation are suitable to quantify rain and ponding infiltration into relatively light (sandy) soils. They are not suitable for infiltration of rain of long durations and low rates, where an assumption of “box”-like soil-water content profiles during infiltration is not fulfilled, and this procedure could lead to significant errors.

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155

Fig. 10.16 Schematic diagram to calculate ponding time (t p ). Rain (rain rate vz ) infiltration curve (curve 2) and ponding infiltration curve (curve 1). Both hatched areas are of the same size

10.7.4 Ponding-Time Calculation by the Green–Ampt Approach Soil surfaces can be ponded (or surface runoff will form) if the infiltration rate is lower than the rain rate. Soil ponding is a harmful phenomenon because it prevents the soil aeration that is a necessary condition for most crop growth. Soil ponding during some days is usually the reason for crops’ wilting. Extreme rain rates (or irrigation) could be the reason for surface-runoff formation and the ensuing soil erosion. Secondarily, soil aggregates on the soil surface peptize and soil crust is formed after soil-surface drying. Ponded soil surfaces also limit soil tillage. Ponding should be limited by increasing the infiltration capacity of soil and by appropriate rates and durations of irrigation. To do this, it is necessary to know ponding time t p , (the time interval from the infiltration beginning, up to the time of pond formation). An infiltration test by a rain simulator with regulated rain rates and durations is the best method for estimating ponding time. The ponding time t p is identified when ponds on the soil surface appear or surface runoff is observed. The higher the rain rate is, the sooner the ponding appears. Ponds on the soil surface will not appear when the infiltration rate vi is less than the saturated hydraulic conductivity of the soil K; then vi = vz . An infiltration test is not an inexpensive procedure and needs necessary instrumentation, therefore some approximate calculation methods have been proposed to estimate ponding time as it depends on soil properties and rain rates. One simple method for ponding-time estimation (Kutílek 1978) will be described, assuming that ponding will appear at time t p when the cumulative infiltration from rain with the rate vz equals the cumulative infiltration from ponded soil (Fig. 10.16). The infiltration rate from a ponded soil surface is, at the initial stage of infiltration, higher than the rain infiltration; but ponding infiltration decreases fast because the effective pressure height at the infiltration front also decreases quickly. If we assume equality of infiltrated-water quantity from both ponding and rain infiltration during time t p , then one can write:

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t p

t p vz (t)dt 

0

vi (t)dt

(10.25)

0

    h0 − H f + 1 tp vz t p  v¯i t p  K Lf

(10.26)

where v¯i is an average infiltration rate during the time interval t p . The left side of Eq. (10.26) (cumulative infiltration of rain) can be expressed according to the Fig. 10.16: vz t p  L f (θs − θi )

(10.27)

L f is obtained from Eq. (10.27), then substituted into Eq. (10.26). Then the equation to calculate t p is:

  h0 − H f t p  K θ (10.28) vz (vz − 1) A constant rain rate is assumed; if the rain rate changes significantly, then it is necessary to divide the rain event into intervals with different, but constant rain rates. Equation (10.28) demonstrates the ponding time proportionality to the saturated hydraulic conductivity of soil and to the soil-water contents’ difference θ = θ s –θ i , and thus even to the force H f (expressed by the negative pressure height) acting at the infiltration front. Ponding time becomes smaller with increasing rain rates. These conclusions agree with reality.

10.8 Infiltration Curves Expressed by Empirical Equations Infiltration rates and time from the infiltration beginning (infiltration curves) are measured in the field, most often by infiltration tests using the double-rings method. Usually the result sought aim by infiltration tests is an estimation of saturated hydraulic conductivity, assuming a homogeneous soil profile. Results of measurement are pairs of data (t i , hi ); t i , are the time intervals during which the water layer hi infiltrates the soil. Using those data, the cumulative infiltration curve i  f (t) or infiltration curve vi  f (t), shown in Fig. 10.1, can be drawn. The infiltration rate is calculated using Eq. (10.1). Usually, during the initial stage of infiltration, discrete pairs of data (vi , t i ), characterize this stage of infiltration poorly (because of the quickly changing infiltration rate); it is suitable to express the infiltration curve by the continuous function vi  f (t). By integrating the infiltration curve, it is possible to obtain the cumulative infiltration curve. And even more important is the possibility to extrapolate the function vi  f (t) for a relatively long time from

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157

the infiltration beginning (usually this time interval is not measured) to estimate the saturated hydraulic conductivity of soil. Empirical equations of infiltration curves should be understood as tools for measured data representation, especially during the short and long time intervals from the infiltration beginning. Many functions have been proposed to describe the infiltration curves’ shape. We will present some of them next. Kosˇtjakov (1932) proposed the following form of exponential function to characterize the infiltration curve: vi  vi1 t −α

(10.29)

This cumulative infiltration function is the integral of Eq. (10.29), according to time: i

vi1 (1−α) t 1−α

(10.30)

where vi1 is a parameter, numerically equal to the infiltration rate at the end of the first time unit (usually one minute is used). This value is used to fix infiltration curves at the initial stage of infiltration; α is an empirical parameter dependent on soil properties; its value is usually in the range 0.2 < α < 0.8. In general, saturated hydraulic conductivity of soil K is calculated using the infiltration curve of vertical infiltration for conditions t → ∞, vi → K. However, Eq. (10.29) shows t → ∞, vi → 0, therefore the equation of Kosˇtjakov is not suitable to evaluate K. But, it is suitable for early stage of infiltration. This equation was probably the first analytical formulation of an infiltration curve, and it can be applied to horizontal infiltration. Mezencev (1948) proposed the following form of the Kosˇtjakov equation: vi  vc + At −β 1 i  vc t + At (1−β) 1−β A  vi1 − vc

(10.31) (10.32) (10.33)

. where vc is the steady-infiltration rate, after long time interval of infiltration vc  K and β is an empirical coefficient. A frequently used infiltration equation is that proposed by Horton (1940): vi  vc + (vi1 − vc ) exp(−γ t) 

1 i  vc t + (vi1 − vc ) 1 − exp(−γ t) γ

(10.34) (10.35)

where γ is an empirical coefficient determining the shape of the infiltration curve. Infiltration curves are often expressed by the Green–Ampt equation (Eq. 10.5) and equation proposed by Philip (Eq. 10.54), which will be presented later.

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Philip (1957) proposed an equation which is the result of an analytical solution of the Richards equation. Green and Ampt (1911) developed infiltration-curve equation utilizing simple physical principles that characterize infiltration surprisingly well. Empirical equations describe the shape of infiltration curves (e.g. by Eqs. 10.34 and 10.35) well, but it is only a formal characterization of infiltration curves without any physical meaning.

10.9 Analytical Expression of Unsteady Infiltration into Homogeneous Soil The kinetics of the infiltration front is important information for estimating water flow into the soil root zone and soil-water content profiles during infiltration. Soil-water content distribution below the soil surface can be measured by application of the modern methods of continuous measurement (TDR, FDR). To calculate soil-water content distribution during infiltration, knowing initial and boundary conditions and basic soil characteristics is easier, faster and less expensive; complicated measuring devices are not needed. This is one of the reasons why science and scientific methods are being developed to make such evaluation easier and to quantify unmeasurable processes. This calculation can be done by using some of the many commercially available simulation models. But, in some cases, it is easier to use existing simple analytical solutions, enabling calculation of soil-water content distributions with acceptable accuracy. Philip (1957, 1969) was the first who presented the theory of water infiltration into homogeneous soil based on an analytical solution of the Richards equation (Eq. 9.29); for one dimension, it can be written as:   ∂ ∂θ ∂k(θ ) ∂θ  D(θ ) + (10.36) ∂t ∂z ∂z ∂z The simplest case of vertical infiltration is infiltration into a semi-infinite profile of homogeneous soil; this means the soil layer should be thick enough not to be reachable by the infiltration front during the process of infiltration under consideration. Constant soil-water content along the soil profile is assumed (Fig. 10.15). The initial condition (constant initial soil-water content θ i ) is: t  0, z ≥ 0, θ  θi

(10.37)

The boundary conditions are: (a) Constant initial soil-water content at the soil surface during infiltration: t > 0, z  0, θ  θ0

(10.38)

10.9 Analytical Expression of Unsteady Infiltration into Homogeneous Soil

159

(b) Soil-water content before the infiltration front is constant and equal to the initialsoil water content: t ≥ 0, z → ∞, θ  θi

(10.39)

Analytical solution of Eq. (10.36) is possible for D  const. only, or for a linear function D(θ ). The function D(θ ) is usually strongly nonlinear and Eq. (10.36) is named after Fokker–Planck; its analytical solution is not known. Equation (10.36) can be solved assuming some simplifications. For the development of infiltration theory, the key proposal was made by Philip (1957), and his theory of infiltration maintains its significance even now, when numerical methods make possible the simple and fast solution of the Richards equation. Philip solved the simplified equation without gravity term (Eq. 10.36):   ∂θ ∂ ∂θ D(θ ) (10.40)  ∂t ∂z ∂z To solve this equation, Philip applied the so-called Boltzmann transformation η  zt −0.5 ; then, Eq. (10.40) can be transformed into a linear differential equation with two variables θ , η. This means the soil-water content profiles during infiltration are functions of variable η only; for all times from the onset of infiltration, only one curve θ  f (η) will be obtained, because the time is involved in the transformation applied. Results of measurement and simulations have shown that actually measured soilwater content profiles during infiltration into homogeneous soil can be transformed according to the Boltzmann transformation (Šim˚unek et al. 2000). Let us apply the Boltzmann transformation to Eq. 10.40. Then, instead of a nonlinear, partial differential equation, this linear differential equation is obtained:   d dθ η dθ  D (10.41) − 2t dη dη dη The transformed boundary conditions are: η  0, θ  θ0 , η → ∞, θ  θi

(10.42)

Equation (10.41) is multiplied by d and then is integrated within boundaries (θ i , θ z ), where θ z is soil-water content at depth z below the soil surface and dθ /dη  0. The resulting equation is: θz θi



dt ηdθ  −2D(θz ) dη

 (10.43) θz

where D(θ z ) is diffusivity, corresponding to the soil-water content θ z .

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Philip (1957) solved the equation without the gravitational term, and then the influence of gravity on infiltration was accounted for. The solution of Eq. (10.43) is expressed by the infinite series: 1

3

z  η0 (θ )t 2 + η1 (θ )t + η2 (θ )t 2 . . .

(10.44)

η0 (θ ), η1 (θ ) and η2 (θ ) are functions depending on soil properties. The first term of the series (Eq. 10.44) is the solution of Eq. 10.43 without gravity term (horizontal infiltration). Cumulative infiltration i can be expressed: 1

3

i  a0 t 2 + a1 t + a2 t 2 . . .

(10.45)

Coefficients a0 , a1 , a2 depend on the characteristics of the soil and infiltration process. By differentiation of Eq. (10.45) according to time, the infiltration rate vi is obtained: vi 

1 −1 3 1 a0 t 2 + a1 + a2 t 2 + · · · 2 2

(10.46)

As was shown by Philip, for short-time intervals from the beginning of infiltration, the first two terms of the series (10.45) and (10.46) can be applied to express the vertical infiltration rate, as well as a cumulative infiltration: 1

i  St 2 + At 1 1 vi  St − 2 + A 2

(10.47) (10.48)

The second terms of Eqs. (10.47) and (10.48) are zero, for horizontal infiltration. The term S on the right side of Eqs. (10.47) and (10.48) was named sorptivity by Philip (1957). Sorptivity is a quantitative expression of those properties of soils that enable absorption of water by capillary forces. Sorptivity characterizes quantitatively not only soil properties, but also the influence of initial and boundary conditions on infiltration. It is an integral characteristic of the process, not of soil only. It is estimated from infiltration tests results. To estimate sorptivity, a horizontal infiltration test is suitable, and the second terms on the right side of Eqs. (10.47) and (10.48) can be neglected. For a relatively long time of infiltration (vertical), the first term of Eq. (10.48) can be neglected, and the second term (on the right side of the equation) becomes dominant and equals approximately the hydraulic conductivity k(θ ) of the transit (transmission) area of infiltration as a function of its soil-water content (Fig. 10.15). Then, Eqs. (10.47) and (10.48) can be written in the form: 1

i  St 2 + k(θ )t

(10.49)

10.9 Analytical Expression of Unsteady Infiltration into Homogeneous Soil

vi 

1 −1 St 2 + k(θ ) 2

161

(10.50)

Saturated soil hydraulic conductivity K in the second term on the right side of the equations will appear if there exists a ponding infiltration. If the soil-water content of the transit area of infiltration into homogeneous soil is smaller than the saturated soil-water content θ < θ s , then K > k. By managing the infiltration rates, the needed soil-water contents of a transition zone can be reached, and water will infiltrate by the rate corresponding to the soil hydraulic conductivity of a particular soil-water content. This method of hydraulic conductivity of soil evaluation was proposed independently (without knowing the Philip’s results which were published later) by Budagovskij (1955) and Youngs (1964).

10.10 Infiltration into Nonhomogeneous Soil Most soils are not homogeneous and isotropic (equally conductive in all directions). Therefore, fluxes of water and energy vary in space and are dependent on the direction of flow. Nonhomogeneity means variability of soil hydraulic conductivities and soil-water potentials in different directions. But, the basic calculation methods for mass and energy transport assume soil homogeneity. The main reason is the relative simplicity of soil-water and energy-flux quantification, which leads to acceptable errors when assuming soil homogeneity. But this is not universally valid; in the case of significant nonhomogeneities, it is necessary account for it. The heterogeneity of soil properties in the Darcian scale (i.e. above the level of pores) characterization and description is one of the basic problems of soil hydrology. It is extremely difficult (probably impossible) to quantify the spatial distribution of soil properties and apply them in calculations of water and energy transport in such porous medium. Variability of soil properties (in horizontal and vertical directions) seems to be accidental and can only be characterized by statistical distributions of soil hydraulic characteristics (hydraulic conductivity and soil-water retention curves). In principle, such distributions can be used in modelling soil-water movement in soils. Result of such an approach is a probability function of the occurrence of soil-water content (or soil-water potential) at a given time and location. The spatial variability of relatively homogeneous soil (e.g. sandy soil, because ideally homogeneous soil does not exist) is demonstrated explicitly during infiltration in such a way that the infiltration front does not advance continuously, but creates visible “fingers”, i.e. the infiltration front is “saw-like”. By averaging soil-water content at the infiltration front, the continuous infiltration front is created (Fig. 10.10). This is a typical case for sandy soil. The seemingly accidentally distributed properties in a relatively homogeneous porous media are typical of soils; but for the majority of soils, there are typically also macropores of various kinds (cracks, shear areas, biopores or macropores formed by soil fauna). Another reason for nonhomogeneities is the nonideal hydrophilicity

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of soils in space. The angle of contact between solid phases of soil and water is often much higher than the ideal “zero” angle, and those differences are distributed in the soil space nonuniformly. The non-zero contact angle is most often due to organic compounds on the surfaces of solid particles. In particular cases, soils are classified as hydrophobic soils, which can significantly influence soil-water retention and movement (Lichner 2003). Water in macropores can flow much faster than in micropores. This type of flow is denoted as preferential flow.

10.10.1 Infiltration into Layered-Soil Profiles Nonhomogeneous soil profiles in the vertical direction are often represented as soil profiles composed of discrete horizontal layers; discrete layers have varied hydraulic properties. Such a representation can be acceptable in some special cases like a soil profile divided into a ploughing layer and a much denser sub-ploughing layer. In this case, their properties change step-like. In the majority of cases, soil properties in the vertical direction change continuously, so identification of their vertical distribution is difficult. To do that, it is necessary to estimate soil properties of many soil samples to create a realistic picture of the distribution of soil hydraulic properties. The knowledge of spatial distribution of soil hydraulic characteristics (hydraulic conductivities and soil-water retention curves) enable calculation of infiltration into this porous medium (soil) by simulation models. The infiltration rate of water into ponded soil composed of discrete soil layers with various hydraulic conductivities can be estimated by infiltration tests. The steady infiltration into layered soil can be quantified as the flow through the homogeneous soil profile characterized by the effective hydraulic conductivities of the whole soil profile (Chap. 8). Two typical cases of water infiltration into soil composed of two layers of different hydraulic conductivities are exceptionally interesting and will be described. A soil layer with higher hydraulic conductivity (coarse soil) is above the soil layer with lower saturated hydraulic conductivity (fine soil) The infiltration rate during the initial stage is governed by the hydraulic conductivity of the upper (coarse) soil layer. After the infiltration front has reached the interface between both soil layers, the infiltration rate into this less conductive soil layer will be determined by the hydraulic conductivity of this fine soil layer. If the infiltration rate through the upper soil layer is higher than saturated hydraulic conductivity of the fine (bottom) layer, the saturated soil layer with free water table will form above the interface. As water infiltrates, the pressure head above the soil interface increases. This water pressure increase is followed by the increasing infiltration rate into the bottom soil layer. In the case of long infiltration durations, the soil profile can become saturated and a continuous water table on the soil surface can be formed—the soil is ponded (Miller and Gardner 1962).

10.10 Infiltration into Nonhomogeneous Soil

163

A soil layer with lower hydraulic conductivity (fine soil) is above the soil layer with higher saturated hydraulic conductivity (coarse soil) The infiltration rate during the initial stage of infiltration is governed by the hydraulic conductivity of the upper (fine) soil layer. After the infiltration front reaches the interface between both soil layers, the infiltration rate into the soil layer with higher saturated hydraulic conductivity can decrease significantly. This paradox is determined by the difference in pores dimensions of both layers. The movement of infiltration front is determined not by gravity only, but capillary forces are of importance too. The driving forces of infiltration are composed of the gravitational component and matric (capillary) component; when the infiltration front reaches the interface with relatively bigger pores, capillary forces become smaller, and the infiltration rate will decrease, because capillary forces decrease with an increase of pore radius. Miller and Gardner (1962) have shown (experimentally) the stopping of infiltration at the interface between both soil layers, until a water layer of critical thickness above the interface forms to overcome the difference between capillary forces (holding the soil water in fine soil) and pressure above the soil interface. When this barrier has been overcome, then water will infiltrate downward, and its rate will be determined by fine soil hydraulic properties. At the interface between the soils, interesting phenomena could be observed. The most important discontinuous phenomena are “fingering” and “funnelling”. The water movement at the interface called “fingering” is the flow of water having the form of fingers. It occurs due to different soil properties on a small scale; there are different contact angles followed by different wetting of the solid phase of soil and different densities of soil, as well as different distributions of small-scale hydraulic conductivities. All these non-regularities are the reason for such “discontinuous” phenomena (Fig. 18.2). After formation of fingers, their hydraulic conductivities are higher than in the surrounding dry soil; this provokes an increased infiltration rate and prolongation of fingers. This type of water movement is called “preferential” flow. Water moving in fingers is also infiltrating horizontally; in such a way, the infiltration front becomes relatively homogeneous. “Funnelling” is another phenomenon originating on declined interfaces between fine and coarse soil layers. As was mentioned, above such an interface, the continuous water table can form and can flow down slope. This type of flow is frequently illustrated by the image of an umbrella, made of coarse sand with water flowing along the interface and fingering down, after reaching the umbrella boundary. This phenomenon is important to know when planning playgrounds and draining of other horizontal areas (e.g. parking lots). The idea that coarse and highly conductive gravel is the best sub-layer below playgrounds to drain can be false. Application of this idea leads usually to a two-layer construction of a playground area, consisting of upper fine soil layer with grass canopy above the coarse (with high saturated hydraulic conductivity) material, as a drainage layer. It is not rare case that, after heavy rainfall, the surface of such a designed soil structure floods.

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However, the funnelling phenomenon is widely used in the design of capillary barriers to divert soil water flow around hazardous underground waste (Radcliffe and Šim˚unek 2010).

References Angulo-Jaramillo R, Bagarello V, Iovino M, Lassabatere L (2016) Infiltration measurements for soil hydraulic characterization. Springer International Publishing, Switzerland, p 386 Budagovskij AI (1955) Water infiltration into soils. Nauka, Moscow (In Russian) Císlerová M, Šim˚unek J, Vogel T (1988) Changes of steady state infiltration rates in recurrent ponding infiltration experiments. J Hydrol 104:1–16 Green WH, Ampt GA (1911) Studies on soil physics: I. Flow of air and water through soils. J Agric Sci 4:1–24 Hillel D (1973) Soil and water, physical principles and processes. Academic Press, New York, London Horton RE (1940) An approach towards a physical interpretation of infiltration capacity. Soil Sci Soc Am Proc 5:399–417 Jury WA, Horton R (2004) Soil physics. Wiley, Hoboken, NJ Kostjakov AN (1932) On the dynamics of the coefficients of water percolation in soils and on the necessity of studying it from a dynamic point of view for purpose of amelioration. In: Transactions of 6th committee international society of soil science, Russia, Part A, pp 17–21 Kutílek M (1978) Hydropedology. SNTL–SVTL, Prague (In Czech) Lichner L (2003) Soil water repellency. Part 1. Definition and characteristics of water repellent soils. J Hydrol Hydromech 51:309–320 (In Slovak with English abstract) Mezencev VJ (1948) Theory of surface runoff formation. Meteorologija i gidrologija 3:33–40 (In Russian) Miller EE, Gardner WH (1962) Water infiltration into stratified soil. Soil Sci Soc Am Proc 26:115–118 Morel-Seytoux HJ, Khanji J (1974) Derivation of an equation of infiltration. Water Resour Res 10:795–800 Morel-Seytoux HJ, Meyer PD, Nachabe M, Tourna J, van Genuchten MT, Lenhard RJ (1996) Parameter equivalence for the Brooks-Corey and van Genuchten soil characteristics: preserving the effective capillary drive. Water Resour Res 32:1231–1238 Morel-Seytoux HJ (2008) Infiltration. In: Chesworth W (ed) Encyclopedia of soil science. Springer, Dordrecht, Berlin, Heidelberg, New York, pp 350–361 Philip JR (1957) The theory of infiltration. 1. The infiltration equation and its solution. Soil Sci 83:345–358 Philip JR (1969) Theory of infiltration. Adv Hydrosci 5:215–296 Radcliffe DE, Šim˚unek J (2010) Soil physics with HYDRUS. Modeling and applications. CRC Press, Taylor & Francis Group, Boca Raton, USA Rubin J (1966) Theory of rainfall uptake by soils initially drier than their field capacity and their applications. Water Resour Res 2:739–749 Swiecicki C (1967) Soil water. SGGW, Warszawa (In Polish) Šim˚unek J, Hopmans JW, Nielsen DR, van Genuchten MT (2000) Horizontal infiltration revisited using parameter estimation. Soil Sci 165:708–717 van Genuchten MT (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Youngs EG (1964) An infiltration method for measuring the hydraulic conductivity of unsaturated porous materials. Soil Sci 97:307–331

Chapter 11

Redistribution of Water in Homogeneous Soil

Abstract The post-infiltration process of water movement in the soil root zone is known as soil-water redistribution. Understanding the redistribution process is important to prevent infiltrated water to penetrate below the soil root zone and thus decrease the amount of soil-water available for plants and increase the risk of groundwater pollution. The process of soil-water redistribution is described, as well as the influence of soil properties and boundary/initial conditions on the rate of redistribution. The quantification of the redistribution process by the solution of the Richards equation assuming constant hydraulic conductivity of soil demonstrates the logarithmic decrease of soil-water movement during redistribution, thus preserving soil water for plants.

11.1 Basic Characteristics of Water Redistribution in Soil Infiltration of water from rain, irrigation or ponds stops once all applied water has entered the soil. Infiltration is defined as a process of water entry into the soil through its surface. After infiltration, water in the soil will move further. The post-infiltration movement of water in the soil is known as redistribution. As will be shown later, the terms of the two following processes “infiltration” and “redistribution” are not only formal, but rather they are processes that differ by their dynamics and significance. The soil-water content increases in the soil space where water enters by infiltration. During the redistribution process following infiltration, soil-water content decreases in the space, where it had increased during infiltration. The gradient of total soil-water potential—gravitational and matric components—is directed upward and water flows downward; the soil-water content in the soil space below increases. The soil profile wetted during infiltration loses water; the matric potential of the drying segment of soil changes (decreases) according to the scanning drying branch of the soil-water retention curve (SWRC). The soil-water matric potential in the wetting part of the soil profile increases following the (scanning) wetting branch of the SWRC. The type of SWRC branch (scanning or main wetting branch) depends on the history

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of previous forms of the soil-water content profile; i.e. how the soil-water content profile of the wetting part of the soil profile was formed. The result of the redistribution process decreases the difference of soil-water matric potentials between the drying (upper) and wetting (lower) soil layers. It means that the rate of downward water movement (redistribution) will slow. The relatively fast infiltration movement is followed by redistribution; the movement of water during redistribution is therefore much smaller in comparison to the infiltration rate. By this way, the downward rate of water movement is relatively slow, which stabilizes the water content of the root zone of plants. The slow water movement downward during redistribution creates conditions for a relatively stable supply for plants of water even in inter-rainy periods. This knowledge inspired Veihmeyer and Hendrickson (1931) to introduce the term “field capacity” as the soil-water content of the soil root zone which changes slowly following infiltration and can be called the upper limit of soil-water content when water is available to plants. Because soil water in the upper part of the soil profile, previously wetted by infiltration, is drained during redistribution and the bottom soil layer is wetted, to analyse redistribution quantitatively hysteresis of the soil-water retention curve should be considered. Then, both main branches of the SWRC have to be known. Understanding of redistribution process and its quantification as it depend on soil properties and the rate of irrigation makes possible more effective irrigation and preventing of soil-water redistribution below the rooting zone of the soil.

11.2 Water Movement in Soil During Redistribution Soil-water content profiles during redistribution of soil water in silty (chernozem) and sandy soils are presented in Fig. 11.1. Soil-water content profiles were measured when evaporation from soil surface was negligible. Therefore, water quantity flowing downward from the upper, wetter part of the soil profile equals the soil water wetting the bottom part of the soil profile, as is indicated by the marked area in Fig. 11.1. Infiltration stopped at time t o ; the soil-water content profiles at time t 1 and t 2 are the results of redistribution. Soil-water content profiles during redistribution in sandy soil are of shape close to the “box”-like one in comparison to the soil-water contents profiles of silty loam soil. It is mostly the result of hydraulic conductivity function since it depends on soil-water matric potential. The hydraulic conductivity of sandy soil decreases strongly with a decreasing soil-water matric potential; the hydraulic conductivity function of the heavier silty loam soil decreases more fluently (Fig. 9.3). Those differences in hydraulic conductivity functions resulting in soil-water content profiles during redistribution. Soil-water content profiles formation depends on the difference between initial soil-water content and soil-water content profiles at the time of the start of redistribution; the lower the initial soil-water content is, the faster the redistribution. It is due to a relatively large gradient of soil-water matric potential between the infiltration area and the initial soil-water content.

11.2 Water Movement in Soil During Redistribution

167

Fig. 11.1 Soil-water content profiles during redistribution of soil water in silt and sand. Water infiltration stopped at the time t o ; redistribution of soil water is represented by the profiles at times t 1 and t 2 . Volume of soil water that flowed from the upper (wet) soil layer equals the volume of soil water that flowed in at the bottom (dry) soil layer. The two hatched areas are of equal size Fig. 11.2 Average soil root-zone water-content changes during redistribution; soil profile was initially wetted by infiltration. Estimated field capacities are also shown. (According to Hillel 1971)

Figure 11.2 shows changes in an average soil-water content of the drying part of the soil profile during redistribution for silty loam and sandy soil during the week time after irrigation. In sandy soil, the relatively stable soil-water content is reached in two days; in silty loam soil, this process takes longer. The soil’s constant (hydrolimit) field capacity (θ fc ) is indicated by the dashed line. This value can be estimated approximately only; therefore it depends on experience. As was mentioned before, this qualitative parameter is a useful tool to plan and manage irrigation. It can be concluded that: – Rate of redistribution (transport of soil-water from the infiltrated zone of soil to the dryer soil, not wetted by infiltration) decreases with time. – Soil-water redistribution is an hysteretic process; the upper part of the soil profile is drying, the bottom part of the soil is wetted. Therefore, to quantify the redistribution process, the hysteresis of the soil hydraulic characteristics needs to be considered.

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– Soil parameter (hydrolimit) field capacity is an actual estimation of soil water in the soil profile, available to plants. It is usually lower than saturated soil-water content; the main reason for this is assumed to be the presence of macropores (preferential ways) and entrapped air; both decrease the soil-water capacity.

11.3 Quantitative Analysis of the Redistribution Process Results of field and laboratory measurement were considered to formulate basic information about infiltration and the ensuing redistribution of water in the soil profile. The results of measurement provide information that can be generalised only qualitatively. Quantification of the redistribution process in homogeneous soil profile without measurement can be obtained by solution of the partial differential equation (PDE) of Richards. Equation (9.12) without the last term (which represents the root extraction pattern) can be applied to the redistribution of water in homogeneous soil in the vertical direction:   ∂ ∂k(h w ) ∂h w ∂θ  k(h w ) + (11.1) ∂t ∂z ∂z ∂z Redistribution is a process in which the hysteresis phenomenon is of importance, therefore hysteresis of the relationship hw  f (θ ) (soil water retention curve) and k  f (hw ) (hydraulic conductivity of soil as a function of soil-water matric potential) have to be considered. Equation (11.1) can be written in the form:     ∂θ ∂k(h w ) ∂ ∂h w kh (h w ) + (11.2)  ∂t h ∂z ∂z ∂z Index h denotes functions with hysteresis. Soil-water content distribution as a function of vertical coordinate z and time t (SWC profiles) can be obtained by the solution of Eq. (11.2), and initial and boundary conditions are described next. The soil-surface boundary condition is usually formulated with an assumption of zero water-flow rate through the soil surface; it means infiltration and evaporation can be neglected: v  0, z  0, t ≥ 0

(11.3)

Soil-water content distribution at the beginning of redistribution is known and expressed by the equation: θ  f (z), z ≥ 0, t  0

(11.4)

11.3 Quantitative Analysis of the Redistribution Process

169

Fig. 11.3 Soil root-zone water storage (W ) of the chernozem on silt as a function of time during redistribution. Evaporation from the soil surface was negligibly small, therefore it was neglected (According to Gardner et al. 1970)

The thickness of the soil profile is large enough that the redistributed soil water does not reach the bottom of the soil profile: θ  θi , z → ∞, t ≥ 0

(11.5)

where θ i is initial soil-water content. Equation (11.2) can be solved numerically; analytic methods can be applied only for constant hydraulic conductivity of soil k, or if the relationship k  f (hw ) is linear. The simplified solution of Eq. (11.2) leads to the function expressing the soil-water content of defined soil layer W and time t (Gardner et al. 1970): W  a(b + t)−c

(11.6)

W is the soil-water content of the defined soil layer; it is expressed usually by the thickness of the water layer (soil-water storage), a, b, c are empirically estimated parameters and t is time from the redistribution beginning. For relatively long time intervals (usually longer than one day), the parameter b can be neglected (Hillel 1982). To rewrite Eq. (11.6) in logarithmic scale, the following equation will be obtained: log W  log a − c log t

(11.7)

Figure 11.3 illustrates the results of soil-water content measurement in drained soil expressed for various soil layers. The calculated data are presented in logarithmic scale, and then the relationships W  f (t) are of linear form. In reality, they are logarithmic functions, like those in Fig. 11.2. From Fig. 11.3, it follows that the redistribution rate is the highest at the initial stage of the redistribution process (during the first days) and then, it slows significantly. Information about the logarithmic course of soil-water content redistribution can be utilized to make a prognosis of the process, knowing soil-water content at two time intervals after the redistribution beginning. Then, Eq. (11.7) can be used to interpolate (or extrapolate) the course of the process and estimate W or θ for other time intervals after the redistribution onset.

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References Gardner WR, Hillel D, Benyamini Y (1970) Post infiltration movement of soil water: I. Redistribution. Water Resour Res 6:851–861 Hillel D (1971) Soil and water. Physical principles and processes. Academic Press, New York Hillel D (1982) Introduction to soil physics. Academic Press, New York Veihmeyer FJ, Hendrickson AH (1931) The moisture equivalent as a measure of the field capacity of soils. Soil Sci 68:75–94

Chapter 12

Interaction of Groundwater and Soil Water

Abstract Groundwater and soil water can interact if the groundwater is relatively close to the soil root zone. Transport of water between groundwater and soil water can significantly improve the supply to plants of water and nutrients. A groundwater depth of about two metres below the soil surface is assumed to be the critical depth that contributes markedly to the supply of water to the plant canopy. In this chapter, the vertical transport of water from groundwater to the root zone of the soil is quantified. It depends on the soil hydraulic properties and on the depth of the groundwater table below the hypothetical level of water extraction by plant roots and on the soil-water matric potential at the same level. Internal drainage of the soil profile as a result of a lowering of the groundwater table is quantitatively expressed. The drainage equations are derived to calculate the parameters of the systematic drainage system: the depth, horizontal distance and diameter of pipe drains in a homogeneous soil under conditions of steady and unsteady drainage.

12.1 Types of Groundwater and Soil-Water Interactions Transport of water between groundwater and unsaturated zone of soil occurs if the groundwater table is close to the soil surface, i.e. it is in hydraulic contact with the soil layer. Water can flow vertically in both directions depending on the soilwater potential distribution. The depth of significant interactions between soil water and groundwater is usually in the range 1.0–2.5 m, while a lesser depth is typical for sands and greater for heavier soils. A groundwater-table depth influencing soil-water content can be found usually in lowlands and rivers alluvia. Characteristic processes of soil and groundwater interactions are the following: 1. Groundwater recharge. (a) Groundwater (GW) recharge by rain infiltration if the groundwater table is close to the soil surface (Fig. 12.1).

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Fig. 12.1 Groundwater recharge by infiltration through the soil profile; groundwater table is increasing

Fig. 12.2 Stream water and groundwater interaction. Groundwater-table level near the stream is higher than stream-water table (GWT1); groundwater infiltrates into the stream. If groundwater table near the stream is below the water table of the stream (GWT2), water infiltrates from the stream, recharging groundwater

(b) Groundwater can be recharged by infiltration from rivers and channels, if water tables in water bodies are higher than groundwater table. An example is the infiltration of water from great rivers (Danube, Rhine) into surrounding geological structures; groundwater recharged in this way is important source of fresh water for communal and industrial consumption (Fig. 12.2). 2. Groundwater table decreases by the process of evapotranspiration which consumes soil water transported to the soil from groundwater; the groundwater table is in close contact with the soil root zone. 3. Groundwater water table decreases due to river (channel) drainage (Fig. 12.2).

12.2 Water-Flow Direction Between Groundwater and an Unsaturated Zone of Soil Equilibrium soil-water content distributions above the groundwater table are shaped like one of the two main branches of soil-water retention curves (SWRC). The drying main branch of the SWRC is the result of the draining process (Fig. 12.3a, 1d); the wetting branch of the SWRC is the result of a continuous wetting process

12.2 Water-Flow Direction Between Groundwater and an Unsaturated …

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Fig. 12.3 Typical equilibrium soil-water content distributions above water table during drainage (a, curve 1d), and wetting (a, curve 1a). (b, curve 1) shows equilibrium soil-water content distribution after draining (corresponding to the SWRC on a, curve 1d). Soil-water content distribution (b, curve 2) represents the wetting stage of soil; soil water moves upward. Soil-water content distribution expressed by curve 3 represents the drainage stage of the process and water moves downward

(Fig. 12.3a, 1a). To make the analysis more instructive, the drying branch of the SWRC will be analyzed (Fig. 12.3b). In field conditions, equilibrium distribution of the soil-water content above the groundwater table is rare; water-transport processes naturally seek equilibrium, but this is not reached, mostly due to the non-regularity of boundary conditions at the soil surface and of groundwater, too. The simplified (non-hysteretic) approach to water flow in soil in contact with groundwater will be used to better understand of the process. Equilibrium distribution of soil-water content in the soil is reached when the total soil-water potential H is zero and the gradient of the total soil-water potential is also zero. Then, the gravitational component of the total soil-water potential hg equals the soil-water matric potential hw : H  hg + hw  0 Figure 12.3b shows curve (1) representing equilibrium soil-water content distribution above the GW table. The soil-water content distribution represented by curve (2) is below the equilibrium soil-water content distribution, or SWRC. But, soil-water content distribution represented by curve (3) is above the equilibrium soilwater content distribution. The equilibrium soil-water content distribution θ  f (z) is thermodynamically optimal. Soil-water transport processes lead to equilibrium distribution, therefore, if the SWC distribution θ  f (z) is expressed by curve (2), water will move upward to reach an equilibrium state; for the distribution θ  f (z) expressed by curve (3), water will move downward (to reach equilibrium), and the soil profile will be drained. The above discussion is valid for homogeneous soils and non-hysteretic processes. To consider hysteresis, the distribution of the soil-water matric potentials matches the scheme presented in Fig. 7.1.

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Fig. 12.4 Total soil-water potential distribution (H) and its components: gravitational (hg ) and matric (hw ) potentials, corresponding to the equilibrium distribution of soil-water content above the water table (curve 1, Fig. 12.3b)

Distribution of total soil-water potentials (H) and its components (gravitational (hg ) and matric (hw ) potentials) —during the equilibrium soil-water content distribution above the groundwater (Fig. 12.4)—corresponds to the drying branch of the soil-water retention curve (1) in Fig. 12.3b. The gravitational component of the total soil-water potential hg (expressed in pressure heads) equals the height above the reference level. Because the soil-water content distribution above this level is in equilibrium, the soil-water matric potential hw (with a negative sign) decreases proportionally with the height. Total soil-water potential (H) is the sum of both its components and therefore is zero along the soil profile. Figure 12.5a illustrates the distribution of the total soil-water potential (H) and its components (gravitational (hg ) and matric potential (hw )), corresponding to curve (3) in Fig. 12.3b. The soil-water content distribution θ  f (z) lies above the equilibrium distribution of soil-water content above the groundwater table level, even if soil-water matric potentials corresponding to this distribution (curve 3) are higher (less negative values), corresponding to the SWRC (1) in Fig. 12.3b. Because the gravitational component of the total soil-water potential does not change, the resulting total soilwater potential distribution (H) (Fig. 12.5a) moves soil water downward, in the direction of the arrow. Figure 12.5b depicts the total soil-water potential (H) and its component gravitational (hg ) and matric (hw ) potential distributions corresponding to curve (2) in Fig. 12.3b. In this case, soil-water contents are lower than the equilibrium ones, and the distribution of soil-water matric potentials hw  f (z) is shifted to the left of the SWRC, and matric potentials are lower than those corresponding to the SWRC. This total soil-water potentials distribution results in upward soil water movement. It should be noted, that the above analysis is valid for non-hysteretic flow.

12.3 Uptake of Water to the Soil from Groundwater

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Fig. 12.5 Total soil-water potential distributions (H) and its components: gravitational (hg ) and matric (hw ) potentials for downward water flow (a), corresponding to curve (3) in Fig. 12.3b, and for upward water flow (b), corresponding to curve (2) in Fig. 12.3b

12.3 Uptake of Water to the Soil from Groundwater If the nonequilibrium soil-water content distribution of homogeneous soil above the groundwater table is below the equilibrium soil-water content distribution (main wetting branch of SWRC), as it is shown in Fig. 12.3b (curve 2), then the soil water will move in the vertical direction upward. Vertical transport of water from the groundwater close to the soil surface is an often observed process, and it can significantly improve the water supply of plants during the vegetation period. This process is usually unsteady. Steady-state processes in the field are rare, but steady-state processes are easier to quantify and often can simulate the real processes with acceptable accuracy. Numerical methods make calculation possible even the unsteady soil-water fluxes between groundwater and soil (Chap. 21). A simple procedure to calculate steady-state flows of water to the soil from groundwater was proposed by Gardner and Fireman (1958). The Darcy–Buckingham equation describing the vertical flow of water in differential form can be written as: v  −k(h w )

dH dz

(12.1)

where v is water flow rate (LT−1 , m s−1 ); k(hw ) is hydraulic conductivity of the unsaturated soil function (LT−1 , m s−1 ); z is the vertical coordinate, positive upward, with the zero value at the GWT level (L, m); H  hw + z is total soil-water potential H, composed of matric hw and gravitational z components, both of which are expressed in pressure heads (L, m). To substitute H  hw + z into Eq. (12.1) and separating the variables, one can derive:

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h wz z− 0

dh w +1

v k(h w )

(12.2)

where hwz is matric potential at the level z above the groundwater table (L, m). The level z above the GWT corresponding to the vertical flow rate v, for matric potential hwz , can be calculated from Eq. (12.2). Císler (1969) substituted the function expressed by Eq. (12.3) into Eq. (12.2): k  K exp (c h w )

(12.3)

where K is saturated-soil hydraulic conductivity (LT−1 , m s−1 ) and c is an empirical coefficient (L, m). Hydraulic conductivity of unsaturated soil is always smaller than of the saturated one, so the empirical coefficient c is positive. The matric potential expressed as soil-water pressure height hw is negative, so the exponent (c hw ) is always negative. After integration, the resulting equation is:   K 1 − exp(z + h wz )c v (12.4) exp(c z) − 1 Equation (12.4) can be used to calculate the rate of water uptake from groundwater to the soil at level z above the groundwater table, when the soil-water matric potential at this level is hwz and the function k(hw ) is expressed by Eq. (12.3). The maximum possible rate of upward flow to the level z above the groundwater table level can be reached theoretically for hwz → −∞. Then, the second term of the numerator of Eq. (12.4) will be close to zero, it can be neglected and the equation can be written in the form: vmax 

K exp(c z) − 1

(12.5)

The matric potential of soil water hw is a negative value, and the coefficient c is positive. To calculate the rate of water flow using Eq. (12.4), it is necessary to know the SWRC (hw  f (θ )), the function k(hw ) and the soil-water matric potential hwz at the level where the water inflow is calculated. Parameter c can be calculated from Eq. (12.3). Coefficient c and values of K of some types of soils according Rijtema (1965) are in Table 12.1. The proper estimation of the level in the soil where water is to be extracted by plant roots is important. This should be positioned at the centre of the roots extraction pattern. Results of measurement have shown (Novák 2012) that this centre of roots extraction level is approximately at 1/3 of the rooting depth, usually at the bottom of ploughing soil layer. Up to 50% of total soil water extracted by crops is extracted from a depth of 0–25 cm, and from soil layer 0–30 cm was extracted up to 65% of the

12.3 Uptake of Water to the Soil from Groundwater Table 12.1 Saturated soil hydraulic conductivity K and coefficient c of various types of soils (Rijtema 1965)

177

Soil type

K (cm h−1 )

c (cm−1 )

Loamy sand

1.0

0.0269

Sandy loam

0.36

0.0378

Peat Clay

5.3 0.22

0.1045 0.0380

Fig. 12.6 Calculated rate of capillary rise vv from groundwater table (GWT) up to two levels above the groundwater table (z  70 and 100 cm) and various soil-water matric potentials hw at the water-extraction level (sandy loam)

soil water by winter wheat. Therefore, it is logical to localize the effective level of soil-water extraction by crop roots at a depth of 25 cm below the soil surface, which is the centre of gravity of the root extraction patterns of crops roots. Figure 12.6 depicts the calculated rates of soil-water uptake from the groundwater level up to two levels in the soil above the groundwater (z  70, 100 cm) and different soil-water matric potentials hwz at extraction level of sandy loam soil. This figure also shows the significant influence of the soil-water matric potential at the water extraction level hwz on the rate of water extraction. The high sensitivity of the extraction rate on hwz and on the position of extraction level above the groundwater is also shown. For a groundwater table at a depth of 70 cm below the water extraction depth, the maximum inflow rate is 7 mm d−1 (this is high rate, higher than the maximum evapotranspiration rate expected on a hot summer day in Central Europe); but, at a depth of 100 cm, the inflow rate decreases to 2 mm d−1 . Even this rate can contribute significantly to cover crop needs. Depths of groundwater table below 100 cm cannot contribute significantly to fulfil crop needs in sandy loam soil. What is interesting is the narrow interval of soil-water matric potentials at the extraction level hwz , under which the water flow rate to the defined level can significantly contribute to plant needs. This interval hwz is in the narrow range of 0 < hwz ≤ −200 cm. Those data are valid for particular sandy loam soil; limiting values for other soils vary, but the relatively narrow intervals of hwz values in which the inflow rate can be influenced can be found for any soil type. The proffered results well illustrate the extreme sensitivity of the rate of water flow from the groundwater to the extraction level to the values hwz and z.

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12 Interaction of Groundwater and Soil Water

Fig. 12.7 Calculated rate of capillary rise rate vv from groundwater table (GWT) at depth z below the extraction level and two soil-water matric potentials at the water extraction level hw  −100 and −1.000 cm (sandy loam)

Figure 12.7 shows the interesting relationship between the rate of soil water flow (v) from groundwater to the extraction level and the distance between the groundwater table and extraction level (z), calculated for two values of soil-water matric potentials at the extraction level. It confirms conclusions from Fig. 12.6: the water flow rate into the soil root zone is strongly influenced in a narrow interval of soil-water matric potential at the extraction level only. For sandy loam soil, hwz ,crit  −200 cm and the “critical” distance between the groundwater table and extraction level is z  120 cm. This means that, for lower values of soil-water matric potential than the “critical” one, the rate of water flow will not change significantly. Even for distances between the groundwater table and water extraction levels (the distance between extraction pattern centre and GWT level) higher than approximately 100 cm, the water-flow rate will be very small. Regarding various soil textures, the effective interval of depths and soil-water matric potential will be even narrower for sandy soils and a little broader for heavy soils.

12.4 Empirical Equations to Calculate Flow of Water from Groundwater to the Soil Empirical equations can be used to approximately calculate the water-flow rate from groundwater to the soil profile when the hydraulic characteristics of the soil profile are not known. The simple equation proposed by Averjanov (1950) incorporates generalisations of numerous field measurements:   z n (12.6) vv  vvm 1 − zk where vv is the rate of water flow to the level z above the groundwater table (mm d−1 ); vvm is the maximum rate of water flow to the level z, equal to the potential evapotranspiration rate (mm d−1 ); zk is the critical depth below the soil surface (i.e.

12.4 Empirical Equations to Calculate Flow of Water from Groundwater …

179

Fig. 12.8 Typical soil-water content profiles above groundwater table during infiltration into soil and water movement following infiltration. Infiltration front before the water table is reached (a), infiltration front reached the GWT, the transit-water flow follows (b), internal drainage following infiltration (c), infiltration, redistribution and internal drainage of soil profile (d), transit flow of water into groundwater following rain infiltration (θ < θ s ) (e). Numbers at the soil profiles denote the time sequence of the profiles

the depth at which the water flow from the groundwater to the soil is negligible (m)); and n is an exponent dependent on the soil properties, 1 ≤ n ≤ 3. The critical depth zk is in the range 1.4–3 m, while shallower depths are valid for sandy soils and the deeper for less conductive, heavy soils.

12.5 Flow of Soil Water to Groundwater—Internal Drainage of Soil Transport of water from soil to groundwater after infiltration is known as soil internal drainage. Transit transport of water through the soil to the groundwater occurs when the groundwater table is reached by the infiltration front and yet continues (Fig. 12.8b). When the infiltration front reaches the groundwater table and infiltration stops, then the process of internal drainage of soil occurs (Fig. 12.8c). Internal drainage continuously decreases the saturated soil-water content following ponding infiltration (Fig. 12.8c) or redistribution (Fig. 12.8d). The soil-water content of the soil profile can be lower than the saturated soil water content if the rain rate is lower than the ponding infiltration rate (Fig. 12.8e). The rate of soil-water movement to the groundwater will decrease when the infiltration front reaches the region of increased soil-water content above the water table (capillary fringe). The infiltration front then changes or disappears, i.e. the soil-water matric potential gradient at the infiltration front will decrease or will be zero, so the driving force at the infiltration front will be small or zero. The soil-water content during the internal drainage above the groundwater lowers, and the soil hydraulic conductivity decreases too. Then, the rate of soil-water movement to the groundwater will decrease; internal drainage slows down and can finally stop as the soil-water

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12 Interaction of Groundwater and Soil Water

content distribution above the groundwater approaches the equilibrium distribution, i.e. the main drainage branch of the SWRC. Hillel (1982) proposed a simple method for internal-drainage calculation. It is based on the following assumptions: the soil-water content profile is close to saturation and uniformly distributed; the soil profile is homogeneous and only the gravitational component of the soil-water potential is the driving force. Then, the rate of downward water flow equals the soil hydraulic conductivity; the rate of internal drainage of the soil profile decreases, too. The Darcy–Buckingham equation can be written as: v  −k(θ )

d (h w + z) dz

(12.7)

It is assumed that the internal drainage is not influenced by the soil-water matric potential gradient, so it can be neglected: dh w 0 dz

(12.8)

The gradient of the gravitational component of the total soil-water potential equals one; therefore, the rate of internal drainage can be expressed as: v  k(θ )

(12.9)

Knowing the function k(θ ), the rate of internal drainage at some depth z can be expressed by the exponential function of the soil-water content: k(θ )  a exp(b θ )

(12.10)

where a and b are empirical parameters. By substitution of Eqs. (12.9) and (12.10), the following equation is obtained: v  a exp(b θ )

(12.11)

Expressing Eq. (12.11) in logarithmic scale and substituting A  ln a, the following equation is obtained: ln v  A + b θ

(12.12)

Equation (12.12) shows that a small change in soil-water content during the internal drainage can be followed by a logarithmic change of internal drainage rate; the small soil-water content change means a fast change of the internal drainage rate. The internal drainage rate can be expressed by the soil-water content change of the upper soil layer above the level zv up to the soil surface. The soil-water content in this layer can be expressed as W  zv θ . Then, it can be written:

12.5 Flow of Soil Water to Groundwater—Internal Drainage of Soil

181

Fig. 12.9 Scheme used in derivation of the Hooghoudt (1937) drainage-design equation

v  k(θ )  −

dθ dW  −z v dt dt

(12.13)

Equation (12.13) demonstrates that the internal drainage rate increases with depth below the soil surface and decreases with time because the soil hydraulic conductivity decreases and is proportional to the soil-water content of the upper soil layer. Only knowing the function k(θ ) can help to quantify internal drainage rate (Eq. 12.13).

12.6 Groundwater Drainage Decreasing the groundwater table (GWT) located close to the soil surface by artificial drainage is an important technical step in managing the soil aeration necessary for root respiration and crop ontogenesis. Some crops (like rice) are able to utilize oxygen even from liquid water; the majority of plants need some minimum soil aeration (Table 19.1). The minimum groundwater table depth depends on soil and plant properties, but the critical depth is approximately 0.8 m below the soil surface. Therefore, GWT is often decreased (especially in river alluvia) to the required level by a systematic drainage system. Drains are usually ditches, pipes or so-called “mole channels” into which groundwater from the space above the drain level can flow (Fig. 12.9). To determine the desirable depths and spacing of drains, many drainage-design equations have been proposed. These “drainage equations” can be used to calculate the spacing of the systematic drainage system because it depends on drain depth, soil hydraulic properties and the specific outflow of drains resulting in the needed GWT level decrease during the prescribed time interval. Drainage-design equations make possible calculating the parameters of the systematic pipe drainage. Among the most important parameters are: drain spacing (L); depth of the drains (pipes) below soil surface (it depends mostly on crops needs) (s); and the radius of the drainage pipes (r) (Fig. 12.10). The next input parameters for the drainage-design equations are soil properties and the time needed to decrease the GWT to the desired depth at the centre of the drain spacing.

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12 Interaction of Groundwater and Soil Water

Fig. 12.10 Scheme to derive the drainage-design equation to the parallel drains (ditches) under unsteady conditions according to van Schilfgaarde (1957)

Drainage-design equations could be divided into two groups: equations assuming steady-state groundwater flow into the drainage system (the Hooghoudt equation, described later) and equations that quantitative describe unsteady drainage and enable quantifying also the kinetics of the water table between drains (Eq. 12.19; van Schilfgaarde 1957).

12.6.1 Drainage-Design Equation Describing Steady Groundwater Flow to Drains One of the oldest and simplest drainage-design equations was proposed in 1937 by Hooghoudt (Luthin 1966). This equation has been modified many times for various conditions. Here we will develop its simplest and easy understood form: steady drainage of homogeneous soil. Assumptions for the equation development are the following: (1) Homogeneous soil, hydraulic conductivity of saturated soil is constant; (2) drains are parallel and equidistant; (3) the hydraulic gradient in the vertical direction below the groundwater table is constant and equals the GWT slope (Dupuit–Forchheimer assumption); (4) the Darcy equation is valid; (5) an impermeable layer is below the drain depth; (6) a steady-state recharge of groundwater is assumed; (7) the GWT level is above the depth of drains because, due to systematic drainage system, the groundwater table decreases, followed by the soil-water content above it. Figure 12.9 is a scheme to develop the drainage-design equation. Let us assume that the flow field between drains is symmetric; then half of the flow field can be analysed. The water discharge through the vertical level at coordinate x is Q (L3 T−1 , m3 s−1 ); this is the sum of the specific discharges (discharges per unit horizontal length) q (L3 T−1 , m3 s−1 ), for the length (L/2 − x):   1 L−x (12.14) Q  −q 2 Discharge Q can be expressed also by the Darcy equation:

12.6 Groundwater Drainage

183

Table 12.2 Spacing and depth of drains of various soil types (US Bureau of Reclamation, according to Luthin 1966) Soil type

K (cm d−1 )

Clay

0.15

10–20

1–1.5

Loam Sandy loam

0.5–2.0 6.5–12.5

20–25 30–70

1–1.5 1–1.5

Sand

100

50–100

1–2

Spacing of drains (m)

Q  −K h

dh dx

Depth of drains (m)

(12.15)

where K is saturated hydraulic conductivity of soil (LT−1 , m s−1 ); and h is the height of GWT level above the impermeable layer (L, m). Combining these two equations:   dh 1 L−x  Kh (12.16) q 2 dx to separate the variables, it can be obtained that: 1 q L d x − q x d x  K h dh 2

(12.17)

To integrate Eq. (12.17) under conditions x  L/2, then h  H + d where d is the depth of the impermeable layer below the level of the drains, and the final equation for spacing of drains can be expressed: L2 

4K (H + d)2 q

(12.18)

This is the famous equation of Hooghoudt (1937), which often applied to calculate the design parameters of systematic pipe drainage systems, under a condition of steady recharge of the groundwater; the GWT level during drainage is steady, and its position depends on soil properties and drainage-system parameters. The spacing of the drainage system (Eq. 12.18) is proportional to the second root of the saturated hydraulic conductivity K and indirectly proportional to the second root of the specific recharge of the groundwater q. In other words, the more conductive is the soil, the higher is the spacing of the drains; drains spacing increases with decreasing specific recharge of groundwater. Drain spacing and depths in various soils are presented in Table 12.2 (U.S. Bureau of Reclamation, cit. acc. Luthin 1966).

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12 Interaction of Groundwater and Soil Water

12.6.2 Drainage-Design Equation of Unsteady Water Flow to Drains Van Schilfgaarde (1957) proposed the equation to calculate the design parameters of a systematic drainage system; this equation describes unsteady drainage flow of water between two drains with an impermeable layer at the depth d below the level of the drains. The additional information that was not needed in the equation describing steady-state water flow to the drains (Hooghoudt equation) is the drainable porosity (Pd ). The scheme of the drainage flow and terms used in the van Schilfgaarde equation are shown in Fig. 12.10. The Van Schilfgaarde equation can be written in the form:   m 0 (m + 2de ) −1 9 K de (t − t0 ) ln L  Pd m(m 0 + 2de ) 2

(12.19)

where L is the horizontal parallel-drain spacing (m) in vertical distance d above the impermeable soil layer (m); m0 is the initial GWT level above the level of the drains (m) in time t 0 (s); m is the GWT level above the level of drains (m) in time t (s); d e is the effective depth of the impermeable layer below the drains axis (m); K is hydraulic conductivity of saturated soil layer below GWT (m s−1 ); Pd is soil drainable porosity (specific yield) (−); and r is the drain pipe radius (m). Parameter d e characterizes the influence of the impermeable layer depth on drainage flow dynamics (d e ≤ d); it can be calculated from the equation: de   8 ln

πL  − 1.15

L r

(12.20)

Equation (12.20) is valid for d/L > 0.3, i.e. for relatively deep impermeable layers.

12.7 Drainable and Wettable Porosity A systematic drainage system can be used also for irrigation, after some technical arrangements enabling the pumping of water into the drains and thus inducing a water flow into the soil. Such a dual-purpose system is called “regulation drainage”, and it makes possible maintaining a certain level of the groundwater table that is suitable to supply plants with water. To calculate the optimum GWT level, it is necessary to know the water quantities that have to be pumped into the system, during the irrigation regimen or to flow from it when the GWT decreases during the drainage mode. The water quantity per unit soil surface which flows out to or in from the soil profile when the groundwater table level changes by a unit height is called the average drainable porosity for the drainage process. When the groundwater table increases (the irrigation mode of the regulation drainage system), then analogically to the “drainable” porosity, one can introduce the term “wettable porosity”, which

12.7 Drainable and Wettable Porosity

185

Fig. 12.11 Equilibrium distribution of soil-water content profiles corresponding to the groundwater-table level z1 (θ  f (z, t 1 )) and z2 (θ  f (z, t 2 )), following the drop of groundwater table from the level z1 to z2 . Marked area represents the water volume that flowed out of the soil profile

means the water quantity per unit soil surface that flows into the soil profile when the groundwater table increases by a unit height. They should be known to calculate groundwater table dynamics during the drainage or irrigation modes of the regulated drainage system. The steady groundwater table level (GWT1) with an equilibrium soil-water content distribution above it θ (z, t 1 ) will decrease by the depth interval z  z1 − z2 , to reach the groundwater table level (GWT2); after a period of time, the new, equilibrium soil-water content distribution θ (z, t 2 ) above it will be established, corresponding to the drainage main branch of the soil-water retention curve. The equilibrium soilwater content distribution established by the groundwater table, decreasing from the groundwater level (GWT1) to (GWT2), is pictured in Fig. 12.11. The marked area represents the water quantity drained from the soil profile when the groundwater table decreased by the vertical distance z  z1 − z2 . Zero evaporation from the soil is assumed. Knowing the volume of water V wd that flowed from the volume of soil V  A(zs − z2 ), where A  1 (area of horizontal cross-section), and zs is the distance of the soil surface to the reference level z  0, then the equation: Pd 

Vwd V

(12.21)

expresses the drainable porosity of soil Pd (specific yield). It is an average value of Pd corresponding to the decrease of GWT by z  z1 − z2 . Drainable porosities of various soils are found in Table 12.3. The coarser the soil texture is, the higher is the Pd . This implies that sandy soil is easily drainable by the systematic drainage system, but heavy soils are practically impossible to drain because their soil watercontent distribution above the GWT is nearly unchanged. The unit decrease of the GWT results in a decrease in the soil-water content above the new GWT by 0.02–0.03 (2–3% of volumetric soil-water content). The drop in groundwater due to the system-

186 Table 12.3 Drainable porosity (specific yield) Pd of various soil types (US Bureau of Reclamation according to Luthin 1966)

12 Interaction of Groundwater and Soil Water Soil type

Drainable porosity Pd

Sand Loamy sand

0.15–0.20 0.05–0.15

Sandy loam

0.04–0.1

Clay loam

0.03–0.06

Clay

0.02–0.03

atic pipe drainage in heavy soils does not mean a significant decrease in soil-water content above it. This problem (impossibility to drain heavy soils) is serious in the amelioration of heavy soils in alluvial lowlands around great rivers (Danube). Drainable porosity can be estimated by measurement with lysimeters, some examples of which are shown in Table 12.3. The U.S. Bureau of Reclamation (cit. acc. Luthin 1966) proposed this empirical equation to calculate Pd : Pd  16.5 K 1/2 Y 1/3

(12.22)

where Y is the thickness of the drained soil layer (m) with saturated hydraulic conductivity K (m s−1 ). To increase the GWT by the height z  z1 − z2 , the equilibrium soil-water content distribution must correspond to the main wetting branch of the soil-water retention curve; the soil-water content will be lower than this, corresponding to the decreased GWT, resulting from SWRC hysteresis. To apply the procedure just described and knowing the water volume that flowed into the soil from groundwater (V wn ) after GWT increase by z  z1 − z2 , and then dividing it by the soil volume V  A (zs − z2 ), the coefficient characterizing the relative volume of water needed to increase the soil-water content when GWT is increased by a unit height is denoted as Pn : Pn 

Vwn V

(12.23)

This coefficient is analogical to the drainable porosity, which is known as the coefficient of recharge or “wettable” porosity. Generally, Pn  Pd ; the drainable porosity is higher than the wettable porosity. For practical purposes (the design of regulation drainage systems), it is reasonable to use the drainable porosity coefficient.

12.8 Risk of Soil Salinization by Groundwater and Surface Irrigation Groundwater and other water used for irrigation contain dissolved mineral compounds; during evaporation and transpiration, dissolved compounds accumulate in the surface soil layer. The groundwater in the upper part of the Žitný ostrov (Rye

12.8 Risk of Soil Salinization by Groundwater and Surface Irrigation

187

Island, Western Slovakia), which is ideal for drinking purposes, contains 300–450 mg L−1 of dissolved mineral compounds. The purest water in nature is rain water containing approximately ten times less dissolved compounds than the surface water or groundwater. Groundwater in the eastern part of the Žitný ostrov lowland area (Rye Island) contains in average 600 mg L−1 of dissolved mineral compounds, but the maximum values at some sites are above 2000 mg L−1 , therefore this water is not suitable for irrigation (Kováˇcová and Velísková 2012). A real risk of salinization is usually under conditions of a high water-table level (close to the soil surface), with relatively high concentrations of dissolved compounds and high annual evapotranspiration totals (above the 600 mm per year). Under the conditions of Central Europe, the risk of salinization is low, but in arid regions this risk is real. Water used for irrigation has to be regularly verified that it is suitable for irrigation. In Central Europe, the maximum daily evapotranspiration total is approximately 7 mm d−1 (meaning seven litres per meter2 ), i.e. the mass of dissolved compounds can increase up to 8 g per day per meter2 . Part of this quantity is absorbed by plants during transpiration. The risk of salinization is usually minimized by natural leaching during the winter–spring periods, and thus the dissolved compounds concentration of the soil decreases; this type of soil-water regimen (periodically percolating regimen) is characteristic of Central Europe. In arid regions, where irrigation is the main source of water for plants, drainage systems have to be installed to drain artificially leached soil solute from the soil to prevent artificial salinization (Imperial Valley, California). The main risk of soil salinization comes from dissolved chemical elements like natrium, toxic chlorine and borum. It is assumed that the main reason for the crises in ancient civilizations (like Mesopotamia) was probably artificial salinization of arable lands by irrigation and the resultant increase in the saline groundwater-table level. Decreased plant production then probably forced people to move to other regions.

References Averjanov SF (1950) About the permeability of variable saturated soils. Inž Sbornik In-ta mech AN SSSR Moscow–Leningrad, Izd AN SSSR, t7 (in Russian) Císler J (1969) Theoretical solution of the steady capillary rise from groundwater. Vodohospodársky cˇ asopis SAV 17:181–190 (in Czech with English summary) Gardner WR, Fireman M (1958) Laboratory studies of evaporation from soil columns in the presence of a water table. Soil Sci 8:244–249 Hillel D (1982) Introduction to soil physics. Acad Press, New York Hooghoudt SB (1937) Bijdragen tot de kennis van eenige natuurkundige grootheden van de grond. deel 6, Versl Landbouwk Onderz 43(13) B:461–676 (in Dutch) Kováˇcová V, Velísková Y (2012) The risk of the soil salinization of the Eastern part of Žitný ostrov. J Hydrol Hydromech 60:57–63

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Luthin JN (1966) Drainage engineering. Willey, New York Novák V (2012) Evapotranspiration in the soil-plant-atmosphere system. Springer Science + Business Media, Dordrecht Rijtema PE (1965) An analysis of actual evapotranspiration. PUDOC, Wageningen van Schilfgaarde J (1957) Approximate solution to drainage flow problems. In: Drainage of Agric Lands, Am Soc Agron Monograph, No 7, pp 79–112

Chapter 13

Evaporation

Abstract Evapotranspiration is the most important term among the “sink” terms of the soil-water balance equation. In this chapter are described contemporary methods for evapotranspiration estimation, as well as the calculation methods to estimate its structural components: evaporation and transpiration. The basic equations of potential evapotranspiration calculation (the Penman equation) and actual evapotranspiration calculation (the Penman–Monteith equation) are presented, and their applications are described. The FAO modification of the Penman–Monteith equation to evaluate daily and hourly reference evapotranspiration is also presented. This chapter also contains the most popular empirical equations for the calculation of the potential evapotranspiration of various evaporating surfaces. Results of measurements of rootextraction patterns and proposed calculation methods for root water-uptake functions needed in mathematical models are briefly discussed.

13.1 Transport of Water in the Soil–Plant–Atmosphere Continuum (SPAC) Rain or irrigation water enters the soil through its surface, by infiltration or by capillary rise from the groundwater. In the opposite direction, from the soil to the atmosphere water is transported by evapotranspiration, which is the term comprising evaporation from wet surfaces—also soil or plants (evaporation)—and by water evaporation from substomatal cavities of plants (transpiration). The specific case of evaporation from a wet plants canopy surface is called evaporation of intercepted water. Part of rain water is intercepted by leaves and then evaporated. Transpiration is a much more complicated process than evaporation. The term “transpiration” contains the words “trans” which means the water is transported through the plant and then evaporated. Transpiration is a part of the water movement through the soil–plant–atmosphere continuum (SPAC). Formally, it is composed of water transport from soil to plant roots, then transport of water through the plant to the

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_13

189

190

13 Evaporation

substomatal cells and from substomatal cells to the atmosphere and finally the water transport (in the water-vapour phase) into the atmosphere. Therefore, transpiration is often denoted as the “catenary” process. Transport of water during transpiration can be divided into two stages: from soil to the evaporation horizon, water moves in the liquid phase, and from the evaporation level to the atmosphere water moves in the gaseous phase (water vapour). The evaporation horizon is situated in substomatal cavities (transpiration) or at the soil or leaf surfaces (evaporation); but usually, the evaporation horizon occurs below the soil surface in the process of soil-surface drying. There can be identified a special type of transpiration, so-called “cuticular” transpiration, which is evaporation of water from leaves through the leaf cuticula; this type of transpiration usually comprises less than 10% of overall transpiration. The enormous consumption of energy needed to change the liquid-water phase (liquid water–water vapour) is important; this process significantly influences the balance of energy of the Earth and secures stable conditions for life. Quantitatively, evapotranspiration can be expressed as the sum of two processes: evaporation and transpiration. Both processes are complicated; therefore, the simplified approach to evapotranspiration by van Honert (1948) led to a better understanding of the evapotranspiration process and its quantification. Analogically to Ohm’s law, van Honert proposed expressing the transpiration flow in the SPAC as being divided into three subsystems: Et 

Ψsr Ψrl Ψla   rsr rrl rla

(13.1)

where Ψ sr , Ψ rl , Ψ la are soil-water potentials differences between soil and roots (sr), roots and leaves (rl) and leaves and atmosphere (la), usually at standard height above the soil surface z  2 m (Pa); r sr , r rl , r la are partial resistances to the flow of water through the subsystems sr, rl and la (L−1 T, m−1 s); and E t is the transpiration rate (ML−2 T−1 , kg m−2 s−1 ). Evaporation E e is the process of water transport between two subsystems: soil–atmosphere (sa), and it can be expressed by the equation: Ee 

Ψsa rsa

(13.2)

Figure 13.1 displays the scheme of the water-potential distribution across the SPAC system. It reveals typical soil-water potential distributions for dry and wet soils. The interface between soil and plant is the root surface. The soil-water matric potential of soil covered by a plant canopy varies usually in the range –1.5 MPa up to zero (from the matric potential corresponding to the wilting point of plants up to saturated soil). In the field, soil-water matric potentials of −3 MPa have been measured, nearly twice as low as the soil-water matric potential corresponding to the wilting point. Yet, no signs of wilting were observed. This is not an exceptional phenomenon because plants can adapt to dry conditions.

13.1 Transport of Water in the Soil–Plant–Atmosphere Continuum (SPAC)

191

Fig. 13.1 Total water potential distribution in the Soil–Plant–Atmosphere continuum (SPAC) under dry and wet conditions

Soil is the critical section of the water transport in the SPAC system. Soil conductivity decreases exponentially with a decrease in soil-water matric potential; this limits the water flow to the roots, and plant resistance is usually not decisive. The high resistance to water flow is located in the leaf in the section between the surface of the substomatal cavities and the atmosphere, just above the leaf surface: it can be 10–50-times higher than the resistance in soil or plants. To compensate this high resistance, (and low hydraulic conductivity) there are high differences between potentials in substomatal cavities and on leaf surfaces. The resistance r la depends on leaves’ hydration, and it changes (increases) during hot days when the water transport from soil to leaves is lower than potential transpiration (maximal transpiration rate under actual meteorological conditions).

13.2 Evaporation as a Physical Process Transport of water to the atmosphere from wet soil or other surfaces containing water is called evaporation. Physically, evaporation is the process of the phase change of a mass (water) to the gaseous phase (water vapour). Evaporation from the solid phase (ice) is called sublimation; it has been shown that, even in the process called sublimation, molecules of the solid phase are transformed into the liquid phase at first, and then to the gaseous phase. Molecules of liquid are always in chaotic motion, their velocity proportional to the liquid (water) temperature. Those molecules, moving in the direction perpendicular to the liquid surface and their velocity is higher than the average velocity of liquid molecules that can penetrate the liquid surface and enter the atmosphere. Energy per molecule of water needed to overcome the energy of attraction among water molecules forming the thin, cohesive layer of molecules on the liquid surface (quantitatively characterized by the surface tension), equals the latent heat of evaporation (Budagovskij 1964). The evaporating liquid (water) loses the energy consumed by evaporation (latent heat of evaporation) which decreases the temperature and evaporation rate as well, if there is not a delivery of energy from other

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sources. Water-vapour molecules in the atmosphere just above the evaporating surface are moving too, and a portion of them returns to the liquid (evaporating surface), thus increasing its energy and temperature. Some basic conclusions from the kinetic theory of evaporation: 1. Energy consumption to evaporate a unit mass of liquid is a constant quantity at constant temperature; the latent heat of evaporation is the energy consumed to evaporate (to change the liquid to vapour phase) a unit mass of liquid. Steady evaporation needs continuous delivery of energy to the evaporation area at the rate equal to the latent heat of evaporation. If this is not so, the liquid temperature decreases, followed by a decrease in the evaporation rate. 2. Evaporating flux (in the vapour phase) is directed into the atmosphere; the direction of vapour flow is reversed during water vapour condensation.

13.2.1 Water Evaporation In the previous section, evaporation of liquids was analysed. But evaporation of water has some specific features that are related to water as the particular medium. Evaporation as a process is of basic importance for living organisms because it is not only a component of the biomass production, but it is also part of biomass. Evaporation is the process consuming the major portion of available energy and thus forming the structure of the energy balance equation of the Earth. Evaporation can be compared to a business transaction (Monteith 1965). The wet surface is trading water vapour with its environment. Every gram of evaporated water at 20 °C consumes 2450 J of energy; payment can be realised in various ways. The most frequent means of payment is in the form of Sun radiation, but it could be delivered even from any nearby area of higher temperature by advection. The energy can be delivered also from the evaporating surface; then its temperature, as well as its evaporation rate, is decreasing, up to the equilibrium state. Important is the extremely high “price” of energy (energy consumption needed to evaporate one gram of water—the latent heat of water evaporation). Compare the energy needed to evaporate one gram of water (2450 J g−1 ) to the specific energy needed to heat 1 g of water from 0 °C up to the boiling temperature of water (100 °C), “only” 418 J is consumed, which is 5.86-times less than to evaporate 1 g of water. The evaporation rate is not influenced only by the temperature, but also wind velocity, air humidity, atmospheric pressure and other meteorological parameters are important.

13.2.2 The Basic Characteristics of Water Evaporation Water can evaporate from all surfaces containing water. The surface, from which water evaporate, is called the evaporating surface, the most important of which in

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193

nature is water evaporation from the water table, plant canopies and soil. The most important component of evaporation from plant canopies is evaporation of water transported from soil, through plants and evaporated through stomata into the atmosphere. This evapotranspiration component is transpiration. Approximately 2/3 of the annual precipitation evaporates from the territory of Slovakia; approximately the same proportion of precipitation evaporates from other Central Europe countries. Up to 80% of all evaporated water of Slovak territory transpires during the growing period, while only about 20% of water evaporates from the soil surface. Transpiration is an indispensable component of the biomass production process and also creates the condition for biomass production (keeps turgor of plant tissues, transporting nutrients to plants and is a part of the plant itself). Plant water is also a key component of photosynthesis, which can be expressed by the equation: 6CO2 + 6H2 O + solar energy → C6 H12 O6 + 6O2 As can be seen, product of photosynthesis is organic matter and oxygen. Absorption of carbon dioxide in the photosynthesis keeps its concentration in the atmosphere relatively stable; the observed increase is due to fossil-fuel burning. Important is the consumption of energy by evaporation process which stabilizes the Earth’s temperature. Transpiration and evaporation are simultaneous processes. In the temperate climatic zone, a significant part of precipitation is intercepted by the plant canopy surface (interception) and then evaporated (evaporation of intercepted water). Interception and evaporation from interception can be a significant part of the water-balance equation on the Earth’s surface. Coniferous trees intercept (and evaporate) an average 38% of annual precipitation in Slovakia. The average of forest-leaves’ interception is less (28% of total annual precipitation) than the interception by crops and grass, which is usually less than 12% of total precipitation during the growing period (Miklánek and Pekárová 2006). Evaporation from ice and snow is less significant, and their daily totals are usually in the range 0.1–1 mm d−1 . Evaporation from wet surfaces (potential evaporation) is influenced mostly by meteorological conditions: solar radiation, temperature, wind velocity, air humidity and therefore their daily and annual courses resemble daily and annual courses of potential evaporation. Water can evaporate if there is a difference in water-vapour pressure just above the evaporating surface and at some level above it; the water-vapour pressure just above the wet evaporating surface is maximum (saturated water-vapour pressure) and becomes lower in the vertical upward direction. To express it mathematically, water evaporates when the gradient of air humidity is directed toward the evaporating surface. If it is in opposite direction, condensation will occur. Dalton (1802) expressed the evaporation rate quantitatively by the equation: E  C(eo − e)

(13.3)

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13 Evaporation

where E is evaporation rate, C is the coefficient dependent on the wind velocity, eo , e is the saturated water-vapour pressure just above the evaporating surface and water vapour pressure of the atmosphere above the evaporating surface. Dalton was the first to understand the evaporation process and identified the characteristic (water-vapour pressure) in which the difference, between the evaporating surface and at some level above, is the condition for evaporation. The evaporation rate depends on the following conditions: – Wet evaporating surface. – Delivery of energy to the evaporating surface needed for the phase change of liquid–water–water-vapour. – Transport of water vapour from the evaporating surface to the atmosphere, as will be shown later; the air movement above the evaporating surface is a key factor.

13.2.3 Evaporation from the Water Table and Bare Soil Evaporation from the water table is influenced by meteorological conditions and by the temperature of the surface layer of water. At constant meteorological conditions, the evaporation rate of water is constant, too. An example of evaporation from the water table is evaporation from the measuring device called an evaporimeter, which is a container with water. Evaporation of water is evaluated by measuring the water-table decrease or the water flow to the evaporimeter that compensates for the evaporation flux. Measuring the evaporation rate from the soil saturated with water (following a heavy rain) and at a steady state of the atmosphere (air temperature, incoming radiation, wind velocity, air humidity), water evaporate at a steady rate, like water from the water table. But, after some time, the evaporation rate decreases (if water in the soil is not transported from outside), and finally the evaporation rate approaches zero (Fig. 13.2). Results of measurements (Fig. 13.2) record a constant evaporation rate (E) at the range of relatively high soil-water content, and it does not depend on soil-water content; this evaporation rate resembles evaporation from the water table. Evaporation continues until it reaches some average soil-water content (θ ci ), where evaporation rate starts to decrease. Let us denotes this average soil-water content as the critical SWC (θ ci ). Figure 13.2 demonstrates the critical SWC (θ ci ) for various evaporation rates; the higher is the evaporation rate, the higher is the critical SWC. In other words, the lower the evaporation rate from the soil is, the longer is the SWC interval of a constant rate of evaporation. Relationships E  f (θ ) presented in Fig. 13.2 can be generalised by transformation of the relationship E  f (θ ) into the relationship E/E p  f (θ ). Relative evaporation E/E p is the ratio of actual evaporation rate E and potential evaporation rate E p . The potential evaporation rate E p is the maximum rate of evaporation under existing meteorological conditions (evaporation from the water table is always the potential one) . The calculated relationship E/E p  f (θ ) represented by one curve is presented in Fig. 13.3.

13.2 Evaporation as a Physical Process

195

Fig. 13.2 Evaporation rate (E) as a function of the average soil-water content (θ) for three evaporation rates corresponding to different meteorological conditions and to different critical soil-water contents (θ ci )

Fig. 13.3 The relative evaporation rate (E/E p ) and the soil-layer average water content (θ) with critical soil-water contents (SWC) and evaporation stages. Critical SWC θ c1 indicates the end of the first stage of (potential) evaporation; critical SWC θ c2 indicates the SWC range with minimum rate of evaporation (third stage of evaporation)

Fig. 13.4 Evapotranspiration rate of maize canopy E (potential evapotranspiration E p  3.2 mm d−1 ) and the average soil-water matric potentials hw ; ha , hc1 and hc2 are critical soil-water matric potentials corresponding to the anaerobiosis point and to critical soil-water contents θ c1 and θ c2 (Trnava site, the Western Slovakia)

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13 Evaporation

As is shown in Fig. 13.3, the evaporation process can be divided into three stages (Keen 1914; Budagovskij 1964; Novák 1995, 2012). 1. Constant-rate stage of evaporation. The maximum possible rate of evaporation E p is not limited by water content. The evaporation rate is controlled only by characteristics of atmosphere. This is called potential evaporation. This stage of evaporation is observed during wet periods of the year; and even the rate of water evaporating from the water table belongs to this stage. This stage of evaporation is rare in arid or semiarid regions. 2. Falling-rate stage of evaporation. As the soil-water content of an isolated soil volume decreases, its evaporation rate decreases, too; the soil hydraulic conductivity decreases, followed by a decreasing transport rate of water to the evaporating surface. The evaporating surface of the soil during this stage of evaporation is usually the topographic soil surface. When the rate of water transport to the topographic soil surface becomes smaller than the rate of potential evaporation E p , then the rate of evaporation starts to decrease, even if the atmospheric conditions remain the same. Average soil-water content of an analysed soil layer (θ ) at which evaporation rate starts to decrease is denoted as the critical soil-water content (θ c1 ); therefore, it depends even on the thickness of the soil layer under consideration. The thinner is the soil layer thickness, the higher is the critical soil-water content (θ c1 ). To calculate evaporation, the thickness of the soil layer equals usually the thickness of the plant canopy’s roots zone—often a one-meter-tick soil layer is considered. To quantify evaporation stages, it is important to have in mind the soil-layer thickness. The shape of the relationship E/E p  f (θ ) is the same for various soil layer thicknesses under consideration. 3. Slow-rate stage of evaporation. As water evaporates from an isolated soil volume, the soil-water content decreases, as well as water delivery to the evaporation horizon. The dry-soil layer forms at the soil surface, and the evaporation rate is limited also by this dry-soil layer because water vapour is transported through this dry layer by molecular diffusion. The molecular diffusion rate is less about a few orders of the convective transport of liquid water. The beginning of this stage is formally characterized by the critical soil-water content (θ c2 ); this approximation is proposed for practical purposes. In reality, the three stages of evaporation are not separated by distinct limits, rather they are separated fluently, as is obvious in Fig. 13.3. To estimate the critical soil-water content, empirical equations are used. The slow-rate stage of evaporation depends on the air humidity above the evaporating surface. When the soil-water content decreases and air humidity just above the evaporating surface is in equilibrium with the air humidity in the atmosphere, evaporation stops. The relationship between air humidity and soilwater content is characterized by the adsorption (desorption) isotherm (Chap. 4). The dry-surface soil layer prevents the loss of soil water by evaporation; the preserved water can be used for transpiration, which can help to avoid wilting. Relative evaporation (E/E p ) can be expressed also as a function of the soilwater matric potential of the soil layer (hw ). Figure 13.4 presents the relationship between daily evapotranspiration total of maize (E) and the average value of

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197

soil-water matric potential (hw ) of the root zone of loamy soil from the Trnava area in Western Slovakia. This relationship is the result of a field-measurement generalisation; critical soil-water matric potentials (hc1 , hc2 ), are presented, as well as the soil-water matric potential corresponding to the anaerobic threshold (ha ), i.e. to the soil-water content when plant roots respiration is limited by a lack of oxygen. In the calculation of evaporation, relationships of the type E/E p  f (θ ) are preferred in comparison to the relationships of the type E/E p  f (hw ). The only reason is the easier measurement of the relationship E/E p  f (θ ) as compared to E/E p  f (hw ).

13.3 Water Transport in Bare Soil During Evaporation Evaporation of water from bare soil and soil-water transport to the evaporating surface are simultaneous processes. Soil evaporation is maximal (potential evaporation) if the water vapour just above the evaporating (soil) surface is saturated, i.e. the maximum at the actual soil-surface temperature. Saturated air humidity above the wet evaporating surface is observed in the wide interval of water contents of the soil-surface layer, from water-saturated soil to the soil-water content corresponding to the relatively low soil-water content, hygroscopicity coefficient (θ h ), (Mitscherlich 1901). This coefficient corresponds to the relative air humidity of 0.98. From it follows that the evaporation rate can be at maximum (potential) over the wide range of soil-water contents of the upper soil layer and then is controlled by the atmosphere only. The problem is to deliver soil water to the evaporation surface at the rate covering the evaporative demand of the atmosphere, or more concisely, to satisfy the potential evaporation rate. Soil-water content distributions θ  f (z) in homogeneous, bare, sandy soil during evaporation are shown in Fig. 13.5; the initial soil-water content is (θ i ). The values at particular profiles of soil-water content show the time sequence of profiles. The soil-water content distribution (curve 3) corresponds to the soilwater content of the soil surface (θ h ), above which the air humidity just above the evaporating surface is a maximum at the given temperature, corresponding to the maximum rate of evaporation (the first, or steady-rate stage of evaporation); soil profiles (1–3) correspond to the first stage of the evaporation interval. When the soil-water content of the upper-soil evaporating layer decreases below (θ h ) and air humidity above it is less than the saturated one, the evaporation rate decreases below the potential evaporation rate and the second or falling-rate stage of evaporation takes place—represented by profiles (3–5). Later, the dry-soil layer forms, decreasing the evaporation rate continuously, proportionally to the decreasing hydraulic conductivity of the dry-soil layer. The evaporation rate decreases (the third, slow-rate stage of evaporation) until the soil-water content of the surface-soil layer (θ e ) is in equilibrium with the air humidity just above the evaporating surface. Then, the rate of water flow to the soil surface will be very slow (curves 6 and 7), determined by the gradient of soil-water potential in

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13 Evaporation

Fig. 13.5 Profiles θ  f (z) during evaporation from the homogeneous bare sandy soil. The initial SWC was (θ i ); θ h is SWC of soil-surface layer (with saturated water-vapour pressure above it); θ e is the equilibrium SWC corresponding to the water-vapour pressure in the atmosphere. Numbers denote the time sequence of the θ  f (z) development

the soil–atmosphere system. Hypothetical, equilibrium soil-water content distribution is represented by the line (8), which represents the equilibrium state between water-vapour pressure of the evaporating surface and the atmosphere.

13.3.1 Calculation of Water Movement in Bare Soil During Evaporation Water evaporates from soil surfaces or from the evaporating surfaces below the soil surface. The evaporating surface moves downward when the rate of soil-water flow to the soil surface does not meet the potential evaporation (or evaporative demand of the atmosphere). The evaporating surface level decreases below the soil surface and evaporation rate decreases too. Evaporation consumes a great amount of energy (latent heat of evaporation), therefore the area around the evaporating surface cools and thus the process of evaporation is non-isothermal. Solar radiation increases the soil-surface temperature, especially of bare soil. Soil-water transport to the evaporating surface is non-isothermal process and can be quantified by a system of partial differential equations describing the transport of water and energy in soil (Novák 1995, 2012). The transport of soil water can be calculated assuming isothermal process at an average temperature of soil layer under consideration. It is possible to use Eq. (9.29) without the root extraction term. In this case (evaporation), it is recommended to use diffusivity (D), involving also the transport of water vapour in soil, which can be significant in evaporation at low soil-water content:

13.3 Water Transport in Bare Soil During Evaporation

  ∂ ∂θ ∂k(h w ) ∂θ  D(θ ) + ∂t ∂z ∂z ∂z

199

(13.4)

An initial condition is the constant soil-water content along the soil profile at the start of evaporation: θ  θi , z ≥ 0, t  0

(13.5)

The evaporation rate is constant (first-rate stage of evaporation): v  const., z  0, t > 0

(13.6)

The soil-water rate through bottom boundary (z  L) is zero: v  0, z  L , t ≥ 0

(13.7)

A constant soil-water content at the evaporating surface is assumed: θ  θe , z  0, t > 0

(13.8)

Equation (13.4) can be solved numerically because both functions D(θ ) and k(hw ) are highly nonlinear.

13.4 Transpiration Transpiration in the temperate climatic zone is the most important component of the water-balance equation. Approximately two thirds of the annual precipitation total on the territory of Slovakia (768 mm) evaporates (497 mm), only one third forms runoff. Evaporation in the Danube lowland occurs for about 90% of the annual precipitation total; only the remainder forms runoff. But, in mountainous regions of Slovakia, only one third of annual precipitation total evaporates. The variability of the ratio of evapotranspiration to precipitation is great over the Slovakian mountainous territory and depends on the environmental conditions. The observed increase in annual precipitation totals (in Europe) is assumed to be the result of ongoing climate change; but the runoff is generally decreasing because evapotranspiration is increasing over the southern part of Slovakian territory. About 80% of evaporated water is transported to the atmosphere by transpiration. Those quantitative estimations are valid approximately also for the rest of the temperate zone of Europe. What is transpiration? Transpiration is transport of water through the plant to the substomatal cells; then, the liquid phase changes to the gaseous phase (water vapour) and is transported from substomatal cells to the atmosphere. Evaporation of water from the plants surface and evaporation from the soil surface are not part of the transpiration process. As will be

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13 Evaporation

shown later, biomass production is proportional to transpiration rate, and therefore transpiration is the basic transport phenomenon in the biomass production process. Not all water transported through the plant to the atmosphere is evaporated from substomatal cavities; a small amount of water (approximately 5–10%) evaporates through the cuticle, which is the surface of plant leaves not containing stomata. It usually comprises the upper surface of leaves and is covered by a thin layer of wax (Novák 2012). This part of transpiration is named cuticle transpiration, and it is usually higher in young than in older leaves. The differences between cuticle transpiration of various plants are also significant. Only a small part of transpired water is involved in the process of photosynthesis (usually less than 2% of absorbed water), the rest moving into the atmosphere. As was mentioned before, water is necessary to maintain the internal pressure of plant tissues (turgor) and thus secure the proper function of plant growth. Water transports nutrients (mineral compounds) dissolved in soil water at low concentrations to the plants, and therefore great fluxes of soil solution are necessary to supply nutrients to plants. A relatively high water content in plant tissues is necessary to maintain high values of the leaf-water potential necessary for stomata opening and to move carbon dioxide to the substomatal cavities. At the same time, but in the opposite direction, water vapour moved through the stomata. Cooling of plant tissues is another role of transpiration that secures the proper conditions for photosynthesis. Regulation of those processes is autonomous in the sense that the stomata’s opening/closing depend on illumination and plant tissues’ turgor; in this way, the plant controls the fluxes of soil solution (water), carbon dioxide (CO2 ) and also the energy balance of the plant to optimize the biomass production and protect plants from dehydration, overheating and finally wilting.

13.4.1 Soil-Water Movement During Transpiration Water for transpiration is extracted from soil by plant roots. The interface roots–soil is assumed to be the most important hydrological interface it channels the highest portion of water flow as a part of the water cycle of the continents (Budagovskij 1981). The transpiration rate depends on the properties of soil–plant–atmosphere continuum (SPAC) system, and roots are a critical part of this. Soil-water movement below the plant canopy can be calculated by Eq. (9.29), containing the term that quantitatively describes water extraction by roots from soil. A homogeneous plant canopy and also homogeneous horizontal (not vertical!) distribution of roots below the soil surface is assumed. The extraction term S(z, t) is always negative. The adapted equation can be written as:   ∂k(h w ) ∂h w ∂ ∂θ k(h w ) +  − S(z, t) (13.9) ∂t ∂z ∂z ∂z

13.4 Transpiration

201

where S(z, t) is the water extraction rate by plant roots; this is the volume of water extracted by roots from a unit volume of soil and in a unit time (L3 L−3 T−1 , cm3 cm−3 s−1 ). The volume of water extracted by roots in a time interval is the transpiration rate E t (retention by the plant canopy is small, and therefore it can be neglected): zr

E t  ∫ S(z)dz

(13.10)

0

where zr is the depth of the rooting zone, i.e. the depth to where plant roots penetrate (L, m). The solution of Eq. (13.9) (under particular initial and boundary conditions) is the distribution of soil-water contents below the soil surface θ  f (z, t). The input data needed to solve Eq. (13.9) are the soil-water retention curve, hydraulic conductivity and the root-extraction function S(z, t). Methods for its evaluation will be described later.

13.4.2 Soil-Water Uptake Patterns by Plant Roots Soil-water uptake rate by plant roots S(z, t) usually changes with the depth below the soil surface and over time. The function expressing the extraction-rate S(z, t) depends on environmental conditions. The rate of soil-water extraction by the roots of a particular plant canopy, sufficiently supplied with water, depends on properties of the atmosphere only. The distribution of the soil-water extraction rates with depth—under conditions of potential transpiration, i.e. the maximum possible under given meteorological conditions—depends on the distribution of the root-system properties only. Therefore, it is important to know the properties of the roots and their distribution below the soil surface. Assuming equal and stable properties of all the roots related to soil-water extraction, the roots-extraction rate can be expressed as proportional to the particular roots property. The following roots characteristics can be measured in the field: – Roots density; the mass of dry roots in a unit volume of soil as a function of depth below the soil surface and time – Specific roots length; the length of roots in a unit volume of soil as a function of depth below the soil surface and time – Specific roots surface; the surface of roots in a unit volume of soil as a function of depth below the soil surface and time. From analysis, it follows that the specific length of roots, or specific surface of the roots, should be the most influential characteristic that describes the role of roots in soil-water extraction. The problem is the relatively complicated method for their estimation. Tests were conducted to evaluate if the mass of dry roots (roots density)

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13 Evaporation

can be used as the key characteristic of the root system, instead of specific length of the roots or specific surface of the roots. A linear relationship was found between the roots dry mass, specific length of roots and specific surface of roots representing distribution of particular roots properties (Himmelbauer et al. 2008). Because the density of roots mass distribution is easy to measure, it can be recommended as the one to use as the key characteristic of the root system. Results of numerous field measurements have shown that the relative root mass density distribution, in homogeneous and well-irrigated soil, can be expressed by the exponential function (Novák 1995): n r d (z)  exp(− p(z/zr ))

(13.11)

n r d (z)  m r (z)/Mr

(13.12)

where mr (z) is dry roots mass density in the soil layer of thickness Δz at the depth z below the soil surface and a unit area of soil surface (kg m−2 ); M r is total mass of dry roots up to the maximum depth of the root system zr and a unit area of soil surface (kg m−2 ); z is the vertical coordinate (positive downward), measured from the soil surface (m). The maximum transpiration rate (potential transpiration) E tp equals the sum of soil-water extraction rates by a root system under potential (optimal) conditions S p (z): zr

E t p  ∫ S p (z)dz

(13.13)

0

where S p (z) is the function representing the potential (maximum) rate of soil-water extraction by plant roots as a function of vertical coordinate z (positive downward); it can be expressed by the equation: S p  Sm exp(− p(z/zr ))

(13.14)

where S m is the maximum rate of soil-water extraction by plant roots close to the soil surface (cm3 cm−3 s−1 ), and p is an empirical parameter; p  3.6 is recommended. Substituting Eq. (13.14) into Eq. (13.13): zr

E t p  Sm ∫ exp(− p(z/zr ))dz

(13.15)

0

By integrating Eq. (13.15) the following equation for S m can be written: Sm  E t p

p   zr 1 − exp(− p)

(13.16)

13.4 Transpiration

203

Fig. 13.6 Dimensionless function P(hw ) and absolute value of soil-water matric potential |hw |. Soil water matric potential |ha | denotes the anaerobic range beginning. Meanings of terms ha , hc1 and hc2 are shown in Fig. 13.4

Substituting Eq. (13.16) into Eq. (13.14), the function for the soil-water extraction rate by plant roots during potential transpiration S m becomes: S p (z)  E t p

p exp[− p(z/zr )]   zr 1 − exp(− p)

(13.17)

Equation (13.17) demonstrates the exponential nature of the distribution of the soil-water extraction rate if the soil water is not a limiting factor (wet soil); it is the same type as the distribution of particular properties of root system in the vertical direction; the maximum is situated just below the soil surface. If the soil-water content is limiting the transpiration rate, the transpiration rate is below the potential rate. Feddes et al. (1978) and Novák (1987, 1995, 2012) have shown that the soil-water root extraction function S(z, t) can be expressed as the product of functions S p (z, t) and P(hw ), which depends on the soil-water matric potential hw : S(z, t)  S p (z, t)P(h w )

(13.18)

Values of function P(hw ) are in the range 0 ≤ P(hw ) ≤ 1; the minimum of the function (P(hw )  0) is typical for dry soils with transpiration close to the zero rate; the maximum (P(hw )  1) is typical for potential, i.e. the maximum transpiration rate. Then, the soil-water potential does not limit transpiration rate. Figure 13.6 is a graph of the function P(hw ). Critical soil-water matric potentials are the same as they are shown in Fig. 13.4; they correspond to the critical soil-water contents θ c1 and θ c2 .

13.4.3 Distribution of Root-Extraction Patterns Evaluated from Field Measurements The distribution of the average soil-water extraction rates by roots of maize in the vertical direction S(z) during rain-free periods of the season 1982 in loamy soil near the Trnava site (Western Slovakia) are shown in Fig. (13.7). These root-extraction

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Fig. 13.7 Vertical distribution of root-water uptake rates of maize S(z) calculated from the results of field measurements during the season 1982. (1) May 25–June 7; (2) June 7–June 15; (3) June 15–June 25; (4) July 13–July 16; (5) August 10–August 24; (6) August 24–September 3 (Trnava site, the Western Slovakia)

rate distributions were calculated by analysis of the measured soil-water content profiles at the beginning and at the end of the rain-free intervals and represent the average rates of water extraction in the particular time intervals. Distributions S(z) represent the combined effect of roots parameters, as well as the effect of soil-water content (or better soil-water matric potential distributions). Relatively high soil-water content was measured in the time interval July 13–16; in the time interval August 24–September 3, the soil-water content was low, and the transpiration was low too, so the majority of leaves wilted. Distribution of the relative amounts (volumes) of soil-water extraction by roots (nrc  S(z)/S), during the growing period of plant canopies in the vertical direction (expressed by the relative depth z/zr , where zr is the depth of the root system) is depicted in Fig. (13.8). Integration of the extracted volumes of soil water starts at the soil surface (z/zr  0) and finishes at the maximum roots’ depth (z/zr  1), where nrc  1. Figure 13.8 shows the exponential distribution of soil-water extraction rates in the root zone, as it was assumed in the proposed function of soil-water extraction by roots (Eq. 13.17). The differences between various plant canopies are not significant. From Eq. (13.8), it is possible to determine the relative quantities (volumes) of extracted soil water from various soil layers during the growth period. As can be seen, about 50% of transpired water is extracted from the upper 0–20 cm of soil; the upper soil layer of 0–50 cm is the source of about 90% of transpired water. The development of soil layers’ thickness from which 50, 90 and 100% of all transpired water was extracted is shown in Fig. 13.9. The development of rootsystems extraction depths of sugar beet, winter wheat, maize, spring barley, potatoes and alfalfa canopies are shown. The rooting depth of crops during the vegetation period of non-irrigated crops increases approximately linearly. It is important to know that about 50% of the soil water transpired by non-irrigated crops is extracted from the ploughing soil layer.

13.5 Methods for Evapotranspiration Estimation

205

Fig. 13.8 Relative root-water uptake rates (nrc  S(z)/S) depending on the relative root depth below soil surface z/zr ; zr is the root-system depth. Sugar beet (1), winter wheat (2), maize (3), spring barley (4). (1, 2, 4—Strebel and Renger (1979); 3—Novák (1995))

Fig. 13.9 Seasonal course of soil layer’s depth from which was extracted 50, 90 and 100% of transpired water during the growth period of sugar beets (1, 7), winter wheat (2), maize (3), spring barley (4), potatoes (5) and alfalfa (6). (1, 2, 4, 5, 6, 7—Strebel and Renger (1979); 3—Novák (1995))

13.5 Methods for Evapotranspiration Estimation Methods for evapotranspiration estimation can be divided into methods for measurement and calculation. Methods for measurement are those methods that measure evapotranspiration directly; the frequently used instruments are lysimeters and evaporimeters. Lysimeters are containers filled with soil, eventually with soil and plants canopy. Evapotranspiration is measured usually by weighing with the lysimeter and evaporimeter. The difference between their masses in the precipitation free (or irrigation free) periods is assumed to be due to evapotranspiration. The influence of groundwater on evapotranspiration can be measured by so-called “compensation lysimeters” in which the constant water-table level below the soil surface is maintained. Then, evaporation rate equals the rate of water delivery to the lysimeter. Lysimeters should be installed in environments with the same properties as lysimeters; this means that a soil monolith of the local soil profile is installed in lysimeters, and the plant canopy in the lysimeter should be surrounded by a canopy with the same properties (height, LAI). This is often problematic; even monoliths taken from the field are difficult to sample without partially destroying them.

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13 Evaporation

An evaporimeter is a container with water wherein the indicated change of watertable level is the thickness of the water layer that has evaporated. Evaporimeters are among the standard equipment in meteorological stations. Evapotranspiration and its components (transpiration and evaporation) can be estimated by calculation, using parameters of the SPAC system which influence evapotranspiration or from the energy or water balance of the evaporating surface. – Methods for water and energy balances in the SPAC system can be used to calculate evapotranspiration. Water and energy balance equations are usually written for the evaporating surface, and evapotranspiration is calculated as the last term of those balance equations. – Combination methods for evapotranspiration calculation are based on energy and water balance equations, combined with equations describing movements of heat and water vapour above the evaporating surface (Penman 1948), Penman–Monteith equation (Monteith 1965), or their modifications (Allen et al. 1998; Budagovskij and Novák 2011). – Eddy-correlation method (Campbell and Norman 1998). – Empirical equations of potential evapotranspiration calculation; they generalise the measured relationships between the evapotranspiration and meteorological conditions of the environment. The list of methods for evapotranspiration calculation and measurement just described is not complete, but the most important methods are mentioned. Detailed information about methods for evapotranspiration measurement and calculation can be found in monographs by Brutsaert (1982), Burman and Pochop (1994), Abtew and Melesse (2013) and Novák (1995, 2012). In the next part, only the most popular methods will be described.

13.5.1 Method of Evapotranspiration Estimation from the Energy-Balance Equation of the Evaporating Surface Nearly all processes on the Earth are driven by the energy supplied by the Sun. The upper boundary of the atmosphere is irradiated by the Sun at the rate 1.4 kW m−2 . This rate of short-wave radiation is known as the solar constant. Part of the Sun’s radiation penetrating atmosphere is reflected back, part of it is heated the atmosphere, and then it is radiated back in all directions as long-wave radiation. The part of solar radiation reaching the Earth’s surface drives numerous processes in nature. The maximum average value of short-wave radiation reaching the Earth’s surface in the region near the Equator is 250 W m−2 . The rest of it is reflected or absorbed by the atmosphere. The energy reaching the Earth’s surface is used by plants to drive photosynthesis and to heat the soil and the atmosphere; part of it is reflected back to the atmosphere, heating it and then radiating the heat in all directions. But, the major portion of the

13.5 Methods for Evapotranspiration Estimation

207

energy reaching the Earth’s surface is consumed by evapotranspiration. The flux of solar energy at the surface in Central Europe averages 125 W m−2 , which is less than one tenth of solar constant. Approximately 45% of solar radiation in Central Europe is consumed by evapotranspiration, which is more than 100 times all the energy transformed in power-plant stations in this area. Therefore, evapotranspiration is the key process stabilizing the Earth’s climate. The climate (and especially the microclimate) can be modified by management of evaporating surfaces and the soilwater regimen. The simplified energy-balance equation at the evaporation surface level can be expressed (energy consumed by photosynthesis and the plant canopy’s heat capacity are small and were neglected) as: Rn  L E + H + G

(13.19)

where Rn is the net radiation at the evaporating surface or just above the plant-canopy level; it is the sum of all energy fluxes at the evaporation level (MT−3 , W m−2 ), E is evapotranspiration rate (ML−2 T −1 , kg m−2 s−1 ); L is latent heat of evaporation (L2 T−2 , J kg−1 ) and H is sensible heat rate; this is the turbulent flux of heat from the evaporating surface to the atmosphere (MT−3 , W m−2 ); G is the heat-flux rate into the soil through its surface (MT−3 , W m−2 ). The evaporation rate E can be calculated if three terms of Eq. (13.19) are known; the equation is applicable at the evaporating-surface level. Net radiation Rn is the energy (expressed by the rate of energy flux, i.e. by the energy income per unit time and unit surface area) that is consumed by processes involved in the energy-balance equation. Radiation reaching the Earth’s surface can be denoted as short-wave (s) and long-wave radiation (l). The wavelength of radiation is indirectly proportional to the radiating body’s temperature. The effective temperature of the Sun is approximately 6000 °C, while the Earth’s surface temperature is within the limits (−50) °C ≤ T ≤ 70 °C; the wavelength of solar radiation is shorter than the wave length of Earth´s radiation. Therefore, we denote solar radiation as short-wave radiation and the radiation of the Earth as long-wave radiation. Short-wave radiation balance at the evaporating-surface level Rs (MT−3 , W m−2 ) can be expressed by the equation: Rs  Rsd − Rsu  Rsd (1 − α)

(13.20)

where Rsd is the short-wave radiation reaching the evaporating surface (MT−3 , W m−2 ); Rsu is the short-wave radiation reflected by the evaporation surface (MT−3 , W m−2 ); and α is the reflection coefficient of the evaporating surface (albedo): α  Rsu /Rsd . Typical values of albedo of different surfaces are in Table 13.1. Radiation balance of the long-wave radiation at the evaporating-surface level is the sum of incoming and outgoing long-wave radiation. Net radiation of the long-wave radiation Rl can be expressed as:

208 Table 13.1 Reflection coefficient (albedo) of various evaporating surfaces

13 Evaporation Surface

Albedo

Snow Meadow Coniferous forest Broadleaf forest Crops

0.8 0.19 0.14 0.17 0.15–0.25

Water table

0.1

Rl  Rld − Rlu

(13.21)

where Rld , Rlu are long-wave energy fluxes from the atmosphere to the evaporating surface (d) and outgoing long-wave radiation of the evaporating surface directed upward (u) (W m−2 ). Net radiation at the evaporating-surface level can be expressed as the sum (with the sign characterizing their directions) of net short-wave and net long-wave radiation: Rn  Rs + Rl

(13.22)

As was mentioned, Eq. (13.19) can be used to calculate evapotranspiration E, knowing three terms of the equation: Rn , H and G. Terms H and G are calculated or measured independently (Novák 2012). The daily totals of evapotranspiration can be calculated easily from Eq. (13.19) by neglecting the terms H and G, and evapotranspiration can be calculated knowing the net radiation only. This is possible because the daily energy fluxes H and G are approximately zero (the directions of the daily fluxes of both H and G are opposite to the night-time fluxes). Figure 13.10 shows the daily courses of all fluxes of the energy-balance equation for a field covered by grass during a clear summer day at Pavˇcina Lehota (Central Slovakia). It can be seen that energy consumption by evapotranspiration is dominant.

13.5.2 Evaporation Calculation from a Wet Surface: The Penman Equation The evaporation rate from the wet surface can be expressed by an equation resembling those presented by Dalton. To express water-vapour movement from the evaporating surface (subscript s) to the standard level where meteorological characteristics are measured (z  z2 ), it is possible to write: E  ρa D(qs − q2 )

(13.23)

where E is the evaporation rate, (ML−2 T−1 , kg m2 s−1 ), D is the coefficient of water-vapour turbulent transport with the dimension of rate (LT−1 , m s−1 ); ρ a is air

13.5 Methods for Evapotranspiration Estimation

209

Fig. 13.10 Daily courses of the energy-balance equation components over grass canopy during clear day. Rn —net radiation, LE—latent heat of evapotranspiration, H—convective (sensible) heat flux to the atmosphere, G—soil heat flux (P. Lehota site, Slovakia, July 24, 1977)

density (ML−3 , kg m−3 ); qs is specific humidity just above the evaporating surface (kg kg−1 ); q2 is specific humidity at the reference level of the atmosphere (at the level of measurement of q2 ) (kg kg−1 ). The turbulent (convective) heat transport from the evaporating surface to the atmosphere is: H  ρa cD(Ts − T2 )

(13.24)

where H is the rate of convective heat transport from the evaporating surface to the atmosphere (MT−3 , W m−2 ); c is mass heat capacity of soil (L2 T−2 K−1 , J kg−1 K−1 ); T s is the air temperature just above the evaporating surface (K, °C); and T 2 is the air temperature at the measurement level (K, °C). In this development, the coefficient of turbulent transport D is assumed to be the same for the transport of water vapour and heat. The evaporation surface created by a horizontal, homogeneous plant canopy is replaced by the virtual evaporating level’s so-called “effective canopy height” and by the “effective air temperature” at this level. In this approach, one can calculate evaporation from the wet, homogeneous horizontal surface (or wet leaf) at the effective height above the soil surface. The structure of the energy-balance equation depends on plant-canopy properties. Net radiation and the soil-heat flux and turbulent heat flux (sensible heat, convective heat flux) depend on the evaporating surface properties. Evaporation rate can be calculated using Eqs. (13.19), (13.23) and (13.24) if all the input data are known. Air temperatures and air humidity above the evaporating surface (T s , qs ) and at some reference level above the evaporating level (T 2 , q2 ) have

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13 Evaporation

to be known. The problem is measuring the needed data just above the evaporating surface (level) (T s , qs ), which is technically difficult, sometimes impossible. The problem can be solved for wet evaporating surfaces when the air humidity just above the evaporating surface is saturated (qs  qso ). Then, it is not necessary to measure air humidity just above the evaporating surface qs , because it is function of the surface temperature T s , which is measurable. The relationship qso  f (T s ) can be expressed by the Magnus equation:   17.1Ts −2 (13.25) qso  0.38 × 10 exp 235 + Ts where qso is the specific humidity of air just above an evaporating surface saturated by water vapour (MM−1 , kg kg−1 ), at the temperature T s . To calculate qso more easily, Eq. (13.25) can be expressed by a Taylor series. Lozinskaja (1979) showed that the first two terms of the series can be used: qso  q2,0 + (Ts − T2 )ϕ2

(13.26)

where ϕ 2 is the derivation of the relationship qo  f (T ) at temperature T 2 (K−1 ); q2,0 is specific humidity of air saturated with water vapour at the temperature T 2 . System of Eqs. (13.23)–(13.26) contains four unknowns: E, H, D, qs ; assuming (qs  qso ). To substituting Eqs. (13.26) into (13.23), the result can be written:   E  ρa D d  + (Ts − T2 )ϕ2

(13.27)

where d  is the saturation deficit of the air, d   q2,0 − q2 . By combining Eqs. (13.19), (13.24) and (13.27) and substituting the term (T s − T 2 ) into Eq. (13.24) and then into Eq. (13.27), the final form of the equation for the calculation of potential evaporation E p from the horizontal, wet evaporation surface is: Ep 

ϕ2(Rn −G) + ρa cDd  c + Lϕ2

(13.28)

To calculate E p , the water-vapour pressure deficit d  can be replaced by the watervapour deficit expressed by the equation d  e2,o − e2 , where d   6.22 × 10−4 d; d is expressed in units hPa (hectoPascal). The structure of Eq. (13.28) resembles the equations published by Penman (1948) and Budagovskij (1964, 1981). The development was done according to Budagovskij (1964), which is much simpler than that published by Penman (1948). Penman’s approach was complicated and difficult to follow; the resulting equation published by Penman (1948) is:  Rn −G + Ea γ L (13.29) Ep  1+ γ

13.5 Methods for Evapotranspiration Estimation

211

where Δ is derivation of the relationship e2,0  f (T ) (Pa K−1 ); e2,0 , e2 are saturated water-vapour pressure and water-vapour pressure, both at height z2 (Pa); γ is the psychrometric constant (Pa K−1 ); E a is the empirical, so-called “wind function” (kg m−2 s−1 ). The empirical function E a has an integral-like nature; it characterizes the influence of the evaporating surface on evaporation because it depends on wind velocity u and saturation deficit d  e2,0 − e2 . Equation (13.28) contains physical characteristics and clearly defined parameters only. To compare Eqs. (13.28) and (13.29), function E a can be expressed by the parameters used in Eq. (13.24). The relationship between water-vapour pressure e and specific air humidity q is: eq

pa ε

(13.30)

where pa is air pressure (atmospheric pressure) (Pa); ε is the ratio of molecular masses of water vapour and air (ε  0.622). Reordering Eq. (13.30) according to temperature T , we get: pa dq de  dT ε dT After some rearrangement, we obtain:

p  a  ϕ ε

(13.31)

(13.32)

The psychrometric constant can be expressed as: γ 

c p pa L ε

Then, Eq. (13.28) can be rearranged in the form:  Rn −G + ρa pεa D d γ L Ep  1+ γ

(13.33)

(13.34)

where d is the saturation deficit at the height z  z2 , d  e2,0 − e2 (Pa). Compare Eqs. (13.28) and (13.29), function E a can be expressed as: E a  ρa

ε Dd pa

(13.35)

Penman (1948) expressed E a by the product of the aerodynamic function dependent on wind velocity (wind function) f (u) and saturation deficit d: E a  f (u)d From this, it follows that:

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13 Evaporation

f (u)  ρa

ε D pa

(13.36)

Using standard values of air parameters, ρ a  1.2 kg m−3 , pa  105 Pa, function f (u) can be written in the form: f (u)  7.46 × 10−6 D

(13.37)

The rate of evaporation can be expressed also in the form (van Honert 1948):   ε eo − e (13.38) E p  ρa pa ra where eo , e are saturated vapour pressures just above the evaporating surface and in the other (reference) level above this surface (hPa); r a is aerodynamic resistance to water-vapour transport between levels of measurement eo and e (s m−1 ). Combining Eqs. (13.23) and (13.38) (assuming qso  qo ), one obtains: ρa D(qo − q) 

ερa eo − e pa ra

(13.39)

Because q − qo  (eo − e)

ε pa

(13.40)

it leads to the known result: D

1 ra

(13.41)

Penman’s equation and the procedure for its derivation are basics of the evaporation theory, therefore this has been described in detail.

13.6 Calculation of Plant-Canopy Potential Evapotranspiration Equation (13.28) can be presented in the form: Ep 

ϕ(Rn − G) + ρa c(qso − qa )/ra c + Lϕ

(13.42)

where r a is aerodynamic resistance (L−1 T, s m−1 ); qa , qso are specific air humidity, measured at standard height and specific saturated-air humidity at the temperature of the evaporating surface (MM−1 , kg kg−1 ).

13.6 Calculation of Plant-Canopy Potential Evapotranspiration

213

Aerodynamic resistance r a , involves the resistances of the two sublayers of the boundary layers of an atmosphere (0, zo ) and (zo , z), enabling calculating of the potential evapotranspiration of soil and the plant canopy without knowing the plant canopy resistance r s (Monin and Obukhov 1954; Novák and Hurtalová 1987). The equation expressing r a can be written in the form:

p 

zo u ∗ e + ln z2 z−d νa o ra  (13.43) κu ∗ where u* is air-friction velocity (LT−1 , ms−1 ); zo is the roughness coefficient which changes during the growing season (L, m); z is the height of the measurement of meteorological characteristics (L, m); υ a is the air’s kinematic viscosity (L2 T−1 , m2 s−1 ); d e is the plant canopy’s effective height (L, m); κ is von Kármán constant; and p is an empirical exponent (−); usually p  0.5 is used; this value is typical for most plant canopies, but it can be different for particular ones. The method for calculating the potential evapotranspiration presented is applicable to all types of homogeneous plant canopies.

13.6.1 Evapotranspiration Calculation: The Penman–Monteith Equation Equation (13.42) is useful to calculate potential evaporation from horizontally homogeneous, wet surface where the only resistance between evaporating surface and the reference level in the atmosphere is aerodynamic resistance. Evaporation (transpiration in fact) from the plant canopy with dry-leaf surface can be calculated by the equation known as Penman–Monteith equation. Such a plant canopy is called “big leaf”. Leaves are dry on their surfaces, and water evaporates from substomatal cavities through the stomata, so it is necessary to consider, in the calculation, the resistance of the stomata r s in the path of water vapour from substomatal cavities–stomata–atmosphere. Water from leaves evaporates also through the cuticle; but the rate of this evaporation flux is approximately an order less than for transpiration; therefore it is usually neglected. Water-vapour transport from mesophyl cells through stomata into the atmosphere can be once more time expressed by the equation (van Honert 1948): LE 

ρa c eo − e γ ra + rs

(13.44)

where eo , e are water-vapour pressures just above the evaporating surface (here evaporating surface is represented by mesophyl cells of temperature T s ) and watervapour pressure at the defined (reference) level of an atmosphere at the temperature T (L−1 MT−2 , Pa), r s is stomata resistance (L−1 T, m−1 s), r a is aerodynamic resistance

214

13 Evaporation

between the leaves’ surface and the reference level of the atmosphere (L−1 T, m−1 s), the equivalency of resistances for water vapour and heat is assumed. The equation describing a steady, convective flux of heat from the evaporating surface at temperature T s to the atmosphere can be written as: H  ρa c

Ts − T ra

(13.45)

The next equation needed for is evaporating surface-energy balance (13.19). The water-vapour pressure change (eo − e) corresponding to the small temperature change can be expressed by the equation (Monteith 1965): 

eo − e Ts − T

(13.46)

where eo is saturated water-vapour pressure at temperature T (L−1 MT−2 , Pa). Comparing Eqs. (13.45) and (13.44), taking into account the energy-balance Eq. (13.19), it is possible to eliminate unknowns H, T s and eo . Then, leaves’ transpiration can be expressed by the equation: E

(Rn − G) + ρa c(eo − e)/ra L{ + γ [1 + (rs /ra )]}

(13.47)

Equation (13.47) is known as the Penman–Monteith equation (Monteith 1965). It is the basic tool to calculate transpiration (actual transpiration) from the evaporating surface known as “big leaf”. But making it harder, the main problem of this application is the estimation of the stomata resistance r s . In reality, the plant canopy is not “big leaf” but a spatially distributed surface. To characterize the spatial variation of leaves’ distribution, the leaf-area index (LAI) of the illuminated leaves can be introduced, the stomata resistance r s in Eq. (13.47) can be replaced by the canopy resistance r c (L−1 T, m−1 s), and then the plant canopy resistance can be approximately expressed by the equation (Szeicz and Long 1969): rc 

rs L AI

(13.48)

where r c is the resistance of the evaporating surface (plant canopy) involving the water-vapour transport through the canopy and from canopy surface, (L−1 T, s m−1 ); r s is leaves’ stomata resistance (L−1 T, s m−1 ); and LAI is leaf-area index of illuminated leaves (LAI of sunlit leaves), (L−2 L2 , m2 m−2 ). The elegant Penman–Monteith equation (13.47) is suitable for calculating plantcanopy evapotranspiration, knowing the parameters on the right side of the equation. The key problem is the estimation of r c ; its distinctive daily and seasonal course is specific for a particular plant and environment. This is the reason why less accurate, but much simpler methods for calculating evapotranspiration are used.

13.7 Reference Evapotranspiration and Plant-Canopy …

215

13.7 Reference Evapotranspiration and Plant-Canopy Evapotranspiration (FAO Method) 13.7.1 Reference Evapotranspiration The so-called “reference evapotranspiration” calculation method (Allen et al. 1998; Pereira et al. 2015) is a modification of the Penman–Monteith method and is denoted as the FAO Penman–Monteith method. Reference evapotranspiration (the frequently used term ET o ) is evaporation from the reference evaporating surface, which is characterized as a dense, green grass canopy of height zp  0.12 m, with canopy resistance to water vapour r c  70 s m−1 and albedo a  0.23. To use this method, the net radiation Rn , air temperature T , air humidity e and wind velocity u—all characteristics measured at the reference height 2.0 m—need to be known. To calculate reference evapotranspiration, the aerodynamic resistance is expressed by the equation:



 z m −de z h −de ln zm z oh (13.49) ra  2 κ u where r a is aerodynamic resistance (s m−1 ); zm , zh are the heights of wind velocity and air-humidity measurement (m); d e is the effective canopy height (m); zom , zoh are roughness coefficients of a evaporating surface for the transport quantity of motion and heat (m); u is wind velocity at the height z (m s−1 ); and κ is von Kármán constant. The FAO Penman–Monteith equation of reference-surface evapotranspiration with the just defined properties can be expressed in the form: E To 

0.408 (Rn − G) + γ T 900 u − e) +273 2 (eo + γ [1 + 0.34u 2 ]

(13.50)

where ET o is the reference evapotranspiration rate (mm d−1 ); Rn is net radiation at the plant-canopy surface (MJ m−2 d−1 ); G is heat flux to the soil (soil heat flux) (MJ m−2 d−1 ); T is an average daily air temperature at standard height 2 m (°C); u2 is wind velocity at standard height 2 m (m s−1 ); eo , e is saturated water-vapour pressure and measured water-vapour pressure at the air temperature T (kPa); Δ is the derivation of the relationship eo  f(T ) (kPa K−1 ); γ is the psychrometric constant (kPa K−1 ). Equation (13.50) can be used also to calculate daily course of reference evapotranspiration, in hourly time steps. Then, all the inputs should be expressed in appropriate units; energy fluxes in (MJ m−2 h−1 ) and the hourly average values of T , u2 , eo , e have to be used. Instead of constant C n  900 in the numerator of Eq. (13.50), the constant C n  37 needs to be used (Allen et al. 2006).

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13 Evaporation

13.7.2 Plant-Canopy Evapotranspiration This procedure (Allen et al. 1998) makes possible calculating potential (and actual) evapotranspiration of plant canopies other than the grass canopy (ETc ). Properties of a particular crop are considered in the calculation procedure based on the dimensionless crop coefficient (K c ): E Tc  K c E To

(13.51)

where ET c is the potential evapotranspiration of a particular plant canopy (mm d−1 ). The crop coefficient characterises the particular plant-canopy influence on evapotranspiration. This approach can be employed to calculate the potential evapotranspiration for one week (as a minimum) and for longer time intervals. To calculate evaporation and transpiration separately, the so-called “dual” crop coefficient was proposed: K c  K cb + K e

(13.52)

where K cb is “basal” crop coefficient characterising transpiration, and K e is the evaporation coefficient. Potential evapotranspiration of a particular plant canopy can be expressed by the equation: E Tc  (K cb + K e )E To

(13.53)

The influence of soil-water stress on transpiration can be expressed by the coefficient of water stress K s . Then: E Tca  (K s K c )E To

(13.54)

where ET ca is plant-canopy evapotranspiration (mm d−1 ); K s is the water-stress coefficient (dimensionless); K s  1 for a canopy well supplied with water; and K s < 1 for the canopy under water stress. The information necessary to calculate evapotranspiration can be found in FAO publications (Allen et al. 1998). The advantage of this method is its simplicity and its applicability to standard meteorological data. It is frequently used to manage the soil-water regimen.

13.8 Calculation of Actual Evapotranspiration …

217

13.8 Calculation of Actual Evapotranspiration from Potential Evapotranspiration The Penman–Monteith equation makes possible the calculation of (actual) evapotranspiration if the plant-canopy resistance r s is known. Other methods (like the equation of Penman) in which canopy resistance is not involved enable calculating the potential evapotranspiration only. Evapotranspiration (actual, or real) is usually calculated from potential evapotranspiration, using empirical relationships between the relative evapotranspiration (E/E p ) and the average soil-water content of the soil root zone (θ¯ ) or some average value of soil-water matric potential (hw ). Figure 13.3 shows the schematic course of the relationship E/E p  f (θ¯ ). It is important the evaluation of the so-called critical soil-water contents θ c1 and θ c2 . The relationship ¯ can be divided into three linear sections, which can be expressed by E/E p  f (θ) the equations: E  1 f or θ ≥ θc1 Ep E  a(θ − θc2 ) f or θc2 < θ < θc1 Ep E  0 f or θ ≤ θc2 Ep

(13.55) (13.56) (13.57)

The coefficient a can be expressed as: a

1 (θc1 − θc2 )

(13.58)

The critical soil-water content θ c1 indicates the soil-water content at which the evapotranspiration rate starts to decline below the rate of potential evapotranspiration. For the rates E p ≤ 5 mm d−1 , (it is typical for moderate climate zone; higher rates of evapotranspiration are rare) coefficient a can be expressed by the empirical equation (Novák 2012): a  −2.27E p + 17.5

(13.59)

Then, θ c1 can be calculated from Eqs. (13.58) to (13.59): θc1 

1 + aθc2 a

(13.60)

Approximately, both critical soil-water contents can be expressed by the equations θ c2  θ wp , (θ c2 corresponds to the soil-water content of wilting point θ wp ) and θ c1  2 × θ wp .

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13 Evaporation

13.9 Structure of Potential Evapotranspiration The term “evaporation structure” usually denotes the ratio of the evaporation and transpiration components of evapotranspiration. The structure of potential evapotranspiration means the ratio of potential evaporation and potential transpiration. Evapotranspiration structure is usually calculated from the components of potential evapotranspiration structure, and then, the components of evapotranspiration are calculated, using the data points of the soil-water content or soil-water matric potential of the soil–root zone. The empirical relationship between daily relative potential transpiration E tp /E p and leaf-area index (LAI) of winter wheat (1), cotton (2) and buckwheat (3) are in Fig. 13.11. It can be seen that, for LAI  1.0, the rates of evaporation and transpiration are approximately the same, but, for LAI  3.0 the transpiration share on evapotranspiration is approximately 80%. Results of numerous measurements (reviews can be found in the book by Novák (2012)) demonstrate the universality of the relationship E tp /E p  f (LAI) for low crops; it can be expressed by the equation:   E t p  E p 1 − exp(−β L AI )

(13.61)

where E p is potential evapotranspiration (ML−2 T−1 , kg m−2 d−1 ); E tp is potential transpiration (ML−2 T−1 , kg m−2 d−1 ); LAI is canopy-leaf area index (–) and β is the empirical coefficient. Results of measurements have shown that the value of coefficient β is close to the value β  0.5; this value is recommended for use when calculating the potential transpiration of low crops. The sum of potential transpiration E tp and potential evaporation E ep is potential evapotranspiration E p (in a particular time interval): E p  E t p + E ep

Fig. 13.11 Relative daily transpiration—the ratio of potential transpiration and potential evapotranspiration (E tp /E p ) and leaf-area index (LAI) of winter wheat (1), cotton (2) and buckwheat (3). Curve for winter wheat approximated by using Eq. (13.61)

(13.62)

13.9 Structure of Potential Evapotranspiration

219

If the potential evapotranspiration E p is known, potential transpiration E tp can be calculated from Eq. (13.61), knowing LAI; the potential evaporation E ep is calculated as a complementary term of Eq. (13.62).

13.10 Daily and Annual Courses of Evapotranspiration and Its Components Daily and annual courses of potential evapotranspiration follow the daily and annual courses of meteorological conditions above the evaporating surface. The highest correlation coefficient was estimated between daily totals of potential evapotranspiration and the net radiation (Rn ), then the coefficients of correlation decreased in the following sequence: air temperature (T ), followed by the wind velocity (u) and air humidity (e) (Novák et al. 1997). Empirical equations to calculate potential evapotranspiration (mentioned later) are based on the generalisation of such relationships. Daily and annual courses of evapotranspiration (not potential evapotranspiration) are strongly influenced not only by meteorological conditions, but also by the soil-water content profile. If the potential evapotranspiration rate is not fulfilled by the soil-water flow to the roots and by its extraction by roots, the courses of evapotranspiration do not follow courses of meteorological conditions, but are strongly influenced by the hydration of soil and the plant canopy. The seasonal course of leaf-area index (LAI), as well as seasonal courses of transpiration (1), soil evaporation (2) and possible increase of transpiration by elimination of the evaporation of the soil, e.g. by mulching (3) are pictured in Fig. 13.12.

Fig. 13.12 Seasonal course of winter-wheat leaf-area index (LAI) (a); seasonal courses of transpiration (1), soil surface evaporation (2) and possible increase of transpiration when evaporation is zero (b). Cerhovice site, Central Bohemia, 1978

220

13 Evaporation

Fig. 13.13 Seasonal courses of ten-day transpiration totals (a) and sums of daily totals (b). Winter wheat (1), spring barley (2), and maize (3) (Trnava site, 1981)

Water vapour as a product of soil evaporation increases the air humidity in the canopy; therefore, the transpiration rate decreases correspondingly. Seasonal courses of the ten-day sums of transpiration of winter wheat (1), spring barley (2) and maize (3) are shown in Fig. 13.13a, and their sums are shown in Fig. 13.13b. The courses of transpiration rates of various plants during the growth period are similar, but are delayed in time according to the growth periods of particular plants. The vegetation period of winter wheat (1) starts the earliest; the last is the growth period of maize (3). In the conditions of Western Slovakia, the growth period of maize begins during the last ten days of April. Seasonal transpiration totals were 130 mm (winter wheat), 110 mm (maize) and 80 mm (spring barley). The average maximum daily transpiration totals were 1.8 mm (winter wheat and maize) and 1.5 mm (spring barley). The maximum daily evapotranspiration totals of canopies well supplied with water (potential evapotranspiration) in the conditions of Central Europe are less than 6 mm d−1 and are limited by the available energy needed to cover transpiration. The ratio of transpiration and evaporation to evapotranspiration of all the growth periods of low crops in the southern part of Slovakia were: Et  0.8; E

Ee  0.2 E

(13.63)

Figure 13.14 summarizes the daily courses of soil evaporation rates (1), transpiration (2) and transpiration increasing (3), under condition of zero soil evaporation for various leaf-area index (LAI) of winter wheat and the southern part the Czech

13.10 Daily and Annual Courses of Evapotranspiration and Its Components

221

Fig. 13.14 Daily courses of soil evaporation (1), transpiration (2), and transpiration when evaporation is zero (3), for various leaf-area indexes (LAI) (Winter wheat, Cerhovice site, Central Bohemia, 1978)

Republic (Cerhovice 1978). It can be seen the structure of evaporation changes during the growth period as it depends on LAI. At the beginning of the growth period, evaporation dominates, and later the transpiration rate prevails.

13.11 Evapotranspiration Estimation by the Eddy-Correlation Method This method can be used to evaluate the evapotranspiration rate (or the rate of heat and quantity of motion) by processing the measured vertical components of wind velocity pulsations w (m s−1 ), pulsations of specific humidity q (kg kg−1 ) and pulsations of air temperature T  . Air density ρ a (kg m−3 ) is calculated as a function of air humidity and temperature. Horizontal lines above the terms w and q of Eq. (13.64) mean averaged pulsations. Based on the basic equation of evapotranspiration E (kg m−2 s−1 ), estimation by the eddy-correlation method yields: E  ρa q¯  w¯ 

(13.64)

222

13 Evaporation

The average pulsation frequency is about 20 Hz, therefore the average values of pulsations during some tenths of minutes are calculated. The method itself is easy in principle; but the measuring and processing instrument is complicated, but commercially available. Details about the method and its application can be found in the publications of Shuttleworth (1993), Sogard (1993), Campbell and Norman (1998).

13.12 Calculation of Potential Evapotranspiration by Empirical Equations Empirical equations can be used to calculate potential evapotranspiration or potential evaporation only. Empirical equations to calculate potential evapotranspiration, transpiration and evaporation are generalised relationships between them and one or a few meteorological conditions during particular time intervals. Potential evapotranspiration of a specific plant canopy and its components is controlled by meteorological conditions of the atmosphere and not by the soil-water state. The longer the time interval (period of time) during which potential evapotranspiration is calculated, the closer are the calculated data to the real (measured) data. As the minimum period of application of the empirical equations, one day is recommended; but most authors recommend a five-day period as the shortest acceptable time interval to apply empirical equations. Regarding parameters of the atmosphere, the closest correlation was found to be between potential evapotranspiration calculated by Penman’s equation and the net radiation at the evaporating surface level (E p  f (Rn )), characterised by the correlation coefficient k 1  0.92; the next highest correlation was air temperature (E p  f (T )); k 2  0.53. Correlations between potential evapotranspiration and other meteorological conditions at the standard level were low (Novák et al. 1997). Net radiation is not among the characteristics measured at any meteorological station; therefore, the empirical equations are oriented to the more easily available meteorological characteristics, like air temperature and air humidity, which led to the low accuracy of calculated results. Some proven empirical equations will be briefly described; results of their application can be accepted as approximations, not as exact data.

13.12.1 Linacre Equation The Linacre equation (1977) is suitable to calculate potential evapotranspiration of dense plant canopy for periods of time not less than five days: Ep 

500 Tu /(100 − ϕ) + 15(T − Td ) T − 80

(13.65)

13.12 Calculation of Potential Evapotranspiration …

Tu  T + 0.006 h

223

(13.66)

where E p is the average potential evapotranspiration during the period (mm d−1 ); T is the average air temperature for the calculation period (°C); ϕ is the geographical latitude in degrees; T d is the average dew-point temperature (°C); h is the height above the sea level (m). Good hydration of plant canopy is assumed; this assumption is valid for all empirical methods of potential evapotranspiration calculation.

13.12.2 Ivanov Equation Ivanov (1954) generalised his numerous lysimetric measurements and proposed the equation to calculate monthly totals of grass evapotranspiration. The input data are the average monthly air temperature and air humidity: E p  0.0018(25 + Tm )2 (100−r )

(13.67)

where E p is the potential evapotranspiration total per month (mm); T m is the average monthly air temperature (°C); r is the monthly average relative air humidity (%). This method has proved a good approximation of potential evapotranspiration not for grass canopies only, but also for other green, dense canopies.

13.12.3 Tichomirov Equation The Tichomirov equation has proven to be applicable to calculate the daily total evaporation of water reservoirs (Chrgijan 1986): E p  0.375 d2 (1 + 0.2 u 2 )

(13.68)

where E p is the daily evaporation total from the water body (mm d−1 ); u2 is the average wind velocity at the standard height 2 m above the water-table level (m s−1 ); d 2 is the daily average air deficit at the height 2 m above the water-table level (hPa).

13.13 Evaporation from Various Evaporating Surfaces Evapotranspiration is the most important term of the dry-land, energy-balance equation; in the Earth’s water-balance equation, it is the second most important term after precipitations. Potential evapotranspiration rates (and totals) depend on the energy supply only; therefore, the highest evaporation rates are expected (and measured)

224

13 Evaporation

Table 13.2 The average annual values of Earth’s water-balance equation components (Denmead 1973) Continent Water layer, mm/year Ratio O

E

E/P

O/P

Europe

P 789

305

489

0.62

0.38

Asia Africa North America South America Australia

742 742 755

332 151 339

410 591 417

0.55 0.80 0.55

0.45 0.20 0.45

1600

650

940

0.59

0.41

455

40

415

0.91

0.09

Precipitation (P), outflow (O), and evapotranspiration (E) of continents

in regions of high incoming solar energy and heavy rains or from the oceans in the bands between the Tropics of Cancer and Capricorn. The annual evapotranspiration total of the continents and islands is only a factor of 0.124 of the Earth’s total evapotranspiration. The annual evaporation from the Atlantic Ocean in the area between the tropics is 2400 mm (eastern part of the Atlantic Ocean) up to 3200 mm in its western part (Gulf of Mexico). It means about 10 mm per day. Evaporation from the eastern Pacific reaches about 1000 mm per year. The annual evaporation from the oceans of the temperate zone is in the range 400–1000 mm; in the northern part of the Pacific Ocean, it is about 200 mm per year. The average annual evaporation of the Pacific Ocean is 1400 mm. Evapotranspiration from the dry land depends mostly on the soil-water regimen, which is the limiting factor. Evapotranspiration rates of the continents depend on properties of the soil–plant–atmosphere system. Any part of the subsystem SPAC can be the limiting factor. Generally, annual evapotranspiration totals for polar areas have been estimated as less than 200 mm; for the temperate zone, it is 250–500 mm, and for subtropical and tropical regions over 1000 mm per year (Babkin 1984). The average annual values of Earth’s water balance are listed Table 13.2 (Denmead 1973). The structure of the water-balance equation of the continents is similar; the differing water-balance equation structure of Africa and Australia should be mentioned. The variation of water-balance structure can be illustrated by two contiguous countries of Central Europe: Hungary and Slovakia. The annual ratio of evapotranspiration to precipitation of Slovakia is 0.647, but for Hungary it is 0.91. The main difference is in topography; Slovakia is a mostly hilly country, but lowlands prevail in Hungary. Therefore, even in relatively small continental Europe, there are significant variations of the water-balance (and evapotranspiration) structure.

13.13 Evaporation from Various Evaporating Surfaces

225

Table 13.3 Grass-canopy annual evapotranspiration totals (E), potential evapotranspiration (E p ), the ratio of annual evapotranspiration totals (mm) and annual precipitation (E/P), as well as the ratio of annual totals (E/E p ) of some Slovak sites; the height above the sea level is indicated (Tomlain 1990). Site h E Ep E/P E/Ep m a.s.l.

mm year−1

mm year−1

–

–

Bratislava, airport

133

465

701

0.79

0.66

Brezno Hurbanovo Kuchyˇna

487 115 206

491 451 507

519 720 668

0.65 0.83 0.76

0.95 0.62 0.76

Lipt. Mikuláš

569

492

563

0.71

0.87

Michalovce Poprad

110 694

454 456

641 569

0.77 0.77

0.71 0.8

St. Smokovec

800

428

442

0.49

0.97

Bold numbers denote minimum and maximum in the field of data.

13.13.1 Evaporation in Slovak Territory The average monthly and annual evapotranspiration totals for grass in Central Europe (Slovakia) over a 30-year period were calculated by Tomlain (1990) using the combined methods of Zubenok (1976). The results of the calculations for some typical sites are summarised in Table 13.3; it shows the narrow range of annual evapotranspiration totals across the territory of Slovakia: from 428 mm (Starý Smokovec, High Tatras; 900 m a.s.l.) up to 507 mm (Kuchyˇna, Záhorská lowland, 150 m a.s.l.). The range of annual totals of grass-canopy potential evapotranspiration is more pronounced: from 442 mm (Starý Smokovec, 900 m a.s.l.) up to 720 mm (Hurbanovo, 130 m a.s.l.). The relative evapotranspiration of grass (E/E p ) ranges from 0.62 (Hurbanovo) up to 0.97 (Starý Smokovec). The maximum totals of average monthly evapotranspiration were observed in the hottest regions of Slovakia in July (more than 100 mm per month; the average rate was 3.3 mm/day). Average monthly potential evapotranspiration totals can be higher than 120 mm, i.e. more than four mm/d. During winter months (December and January), monthly evaporation totals are usually less than five mm. Characteristic daily evaporation totals from snow and ice are 0.1 mm d−1 , but, during snow thawing, the evaporation rate can increase up to 1 mm d−1 .

226

13 Evaporation

References Abtew W, Melesse A (2013) Evaporation and evapotranspiration, measurements and estimation. Springer Science + Business Media, Dordrecht Allen RG et al (2006) A recommendation on standardized surface resistance for hourly calculation of reference ETo by the FAO56 Penman-Monteith method. Agric Water Manag 81:1–22 Allen RG, Pereira LS, Raes D, Smith M (1998) Crop evapotranspiration—guidelines for computing crop water requirements. FAO Irrigation and drainage paper 56, FAO, Rome Babkin VI (1984) Evaporation from water surface. Gidrometeoizdat, Leningrad (in Russian) Brutsaert W (1982) Evaporation into the atmosphere. D Reidel Publishing Company, Dordrecht Budagovskij AI (1964) Evaporation of soil water. Nauka, Moscow (in Russian) Budagovskij AI (1981) Soil water evaporation. In: Fizika poˇcvennych vod, Nauka, Moscow, pp 13–95 (in Russian) Budagovskij AI, Novák V (2011) Theory of evapotranspiration: 1. Transpiration and its quantitative description. J Hydrol Hydromech 59:3–23 Burman R, Pochop LO (1994) Evaporation, evapotranspiration and climatic data. Developments in atmospheric science 22, Elsevier Science, BV Amsterdam Campbell GS, Norman JM (1998) An introduction to environmental biophysics. Springer, New York Dalton J (1802) On evaporation. Essay III. In: Experimental essays on the constitution of mixed gases; on the force of steam or vapour from water and other liquids in different temperatures, both in Torricellian vacuum and in air; on evaporation and on the expansion of gases by heat. Mem Proc Manchester Lit Phil Soc 5:574–594 Denmead OT (1973) Relative significance of the plant and water evaporation in estimating evapotranspiration. In: Proceedings of the Uppsala symposium 1970, UNESCO, Paris, pp 505–511 Feddes RA, Kowalik PJ, Zaradny H (1978) Simulation of field water use and crop yield. Wageningen, PUDOC Himmelbauer M, Novák V, Majerˇcák J (2008) Sensitivity of soil water content profiles in the root zone to extraction functions based on different root morphological parameters. J Hydrol Hydromech 56:1–11 Chrgijan ACh (1986) Physics of atmosphere. Izdat Moskovsk Universiteta, Moscow (in Russian) Ivanov NN (1954) About potential evapotranspiration estimation. Izv VGO, t 86 2:189–196 (in Russian) Keen BA (1914) The evaporation of water from soil. J Agric Sci 6:457–475 Linacre ET (1977) A simple formula for estimating evaporation rates in various climates using temperature data alone. Agric Meteorol 18:409–424 Lozinskaja EA (1979) Evaluation of the Magnus formula linearization on the accuracy of evaporation from irrigated soil and transpiration. In: Symp KAPG, Isparenie poˇcvennoj vlagi, Institut gidrologii SAN, Bratislava (in Russian) Miklánek P, Pekárová P (2006) Interception estimation of hornbeam and pine forest of the experimental microcatchments of the Institute of Hydrology SAS. J Hydrol Hydromech 54:123–136 (in Slovak with English summary) Mitscherlich EA (1901) Untersuchungen über die physikalischen Bodeneingeschaften. Landw Jahrb 30(B):360–445 Monin AJ, Obukhov AH (1954) Basic laws of turbulent mixing in boundary layer of atmosphere. Trudy Geofiz. In – ta 24, (151):163–186 (in Russian with English summary) Monteith JL (1965) Evaporation and environment. Symp Soc Exp Biol 29:205–234 Novák V (1987) Estimation of soil–water extraction patterns by roots. Agric Water Manage 12:271–278 Novák V (1995) Evaporation of water in nature and methods its estimation. Veda, Bratislava (in Slovak) Novák V (2012) Evapotranspiration in the soil-plant-atmosphere system. Springer Science + Business Media, Dordrecht

References

227

Novák V, Hurtalová T (1987) Coefficient of water vapor turbulent transport to calculate potential ˇ 35:3–21 (in Slovak with English abstract) evapotranspiration. Vodohosp Cas Novák V, Hurtalová T, Matejka F (1997) Sensitivity analysis of the Penman type equation for calculation of potential evapotranspiration. J Hydrol Hydromech 45:173–186 Penman HL (1948) Natural evaporation from open water, bare soil and grass. Proc Roy Soc Lond, Ser A, Math Phys Sci 193:120–145 Pereira LS, Allen RG, Smith M, Raes D (2015) Crop evapotranspiration estimation with FAO56: past and future. Agric Water Manage 147:4–20 Shuttleworth WJ (1993) Evaporation. In: Maidmend DR (ed) Handbook of hydrology. Mc GrawHill, New York Sogard H (1993) Advances in field measurements of evapotranspiration from a plant canopy. In: Becker A, Sevruk B, Lapin M (eds) Proceedings of symposium of precipitation and evaporation, Bratislava, Slovakia. Slovak Hydrometeorological Institute, Bratislava Strebel O, Renger M (1979) Geländeuntersuchungen zum wassernutzungdurch die wurzeln in abhängigkeit von klima, boden und kulturart. Mitteilng Dtsch Bodenkundl Gesselllsch 29:89–98 Szeicz G, Long IF (1969) Surface resistance of crop canopies. Water Resour Res 5:622–633 Tomlain J (1990) Coefficients of evapotranspiration across Slovakia. Acta Met Univ Comeniane, Meteorológia XIX, SPN, Bratislava, pp 15–32 (in Slovak with English abstract) van Honert TH (1948) Water transport in plants as a catenary process. Discus Faraday Soc 3:146–153 Zubenok II (1976) Evaporation from continents. Gidrometizdat, Leningrad (in Russian)

Chapter 14

Transport of Solutes in Soils

Abstract Soil water is in fact a solution; but in the majority of cases, the concentrations of soil solutes are small, and therefore soil-water physical characteristics are treated as the characteristics of pure water. Pollution transport, transport of dissolved fertilizers in infiltration water or transport of solutes in salt-affected soils should be treated as transport of solutions. Concentration of a solute is defined, and transport mechanisms of dissolved compounds in soil are described qualitatively and quantitatively in this chapter. Diffusion, convection and hydrodynamic dispersion are involved together with the continuity equation to derive the convective–diffusion transport equation. The Péclet number is used to evaluate the transport mechanisms of dissolved compounds. Outflow and breakthrough curves are defined, described and explained to identify the character and significance of the solute transport.

14.1 Basic Processes of Solute Transport in Soils Transport of solutes in soils as a process deals with transport of water and dissolved matter, but also involves chemical reactions and microbial transformations; therefore, its quantitative description is complicated. The presence of dissolved compounds and their transport in soil determines water quality. Understanding the transport of dissolved compounds in soils and their transformation is studied by hydrologists, chemists, physicists, biologists and soil physicists (Radcliffe and Šim˚unek 2010). The liquid phase of soil never only equal to chemically pure water, but always contains dissolved mineral and organic compounds. Soil solutions can contain nutrients important to plants, but also compounds decreasing their ability to produce biomass. The positive or negative influence of the dissolved compounds often depends on their concentration in soil solutions. Even high concentration of plant nutrients (N, P, K) can be toxic for plants. The transport of liquids can be divided into miscible and immiscible displacement. Immiscible displacement is the flow of two immiscible liquids, like oil and water or water and air. This type of transport is often observed especially during ecological disasters. © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_14

229

230

14 Transport of Solutes in Soils

Fig. 14.1 Dispersion of dissolved compounds applied on the soil surface at a single point. Dispersion in vertical (a) and horizontal directions (b) can be observed. Isolines of solution concentrations and concentrations distribution in both cross sections are demonstrated

The miscible displacement is often observed; it is a mixture of various liquids or a dissolution of solid compounds. An example of miscible displacement can be the infiltration of nutrients or herbicides into soil; solid compounds dissolve and infiltrate in form of the solution into soil. Another example is the leaching of salts from soil during heavy rain or irrigation. Figure 14.1 depicts an example of the dispersion of dissolved compounds in soil applied at one point of the soil surface. Two forms of dispersion should be specified: vertical (in the downward direction) and horizontal (in the horizontal direction). In the case of solution infiltration over an extensive soil-surface area (e.g. ponding infiltration), the solute displacement will performing in the vertical direction only. Solutes can be transported in soil that is saturated or unsaturated with liquid. The solutes displacement in the unsaturated zone of soil is much more complicated to quantify.

14.2 Concentration of Dissolved Compounds in Solution A soil solution is a liquid composed of dissolved mineral and organic compounds in soil water. When evaporating the water from a soil solution, the mineral and organic compounds, which can be identified as a solid phase, are finally deposited in the evaporating pot. The three phases of soil can be identified in Fig. 14.2; the dissolved compounds are particularly visible. The quantity of dissolved compounds can be expressed in various ways: 1. Concentration of the i-th component of dissolved compound C i expressed in a mass unit of this compound mi in a unit volume of soil V : Ci  ρi 

dm i dV

(14.1)

14.2 Concentration of Dissolved Compounds in Solution

231

Fig. 14.2 Dissolved compounds as a part of soil

where C i is concentration of i-th component of the dissolved compound (solute) expressed in mass units of this compound in a unit volume of soil (ML−3 , kg m−3 ); ρ i is bulk density of the dissolved compound (ML−3 , kg m−3 ); mi is mass of the i-th component of the dissolved compound (M, kg) in volume of soil V (L3 , m3 ). 2. Concentration of the i-th component of the dissolved compound ci expressed in mass units of this compound in a unit volume of soil solution (liquid) V w ; this term is usually denoted as mass concentration of solution: ci 

dm i d Vw

(14.2)

where ci is concentration of i-th component of the dissolved compound expressed in mass units of this compound (ML−3 , kg m−3 ); mi is expressed in mass unit of this compound (M, kg); V w is the volume of the soil solution (L3 , m−3 ). Between both expressions of soil-solution concentrations can be identified this simple relationship: Ci  ρi 

dm i d Vw dm i   ci θ dV d Vw d V

(14.3)

The volumetric soil-water content is expressed by the equation: θ

d Vw dV

(14.4)

14.3 Transport of Dissolved Compounds in Soils Dissolved compounds (solutes) are transported by various mechanisms and can be denoted formally as convection (J lc ), diffusion (J ld ) and hydrodynamic dispersion (J lh ). The solution transport rate (J s ) is the sum of transport rates by three particular mechanisms:

232

14 Transport of Solutes in Soils

Js  Jlc + Jld + Jlh

(14.5)

Particular processes involved in Eq. (14.5) will be described later.

14.3.1 Convection Convection is the transport of dissolved compounds together with solvent (water). It can be expressed by the equation: Jlc  Jw c

(14.6)

where J lc is the rate of solution flow (ML−2 T−1 , kg m−2 s−1 ); J w is the water-flow rate (L3 L−2 T−1 , m3 m−2 s−1 ); c is the dissolved-compounds concentration of the solution (ML−3 , kg m−3 ).

14.3.2 Diffusion Diffusion is the dissolved-compounds transport acting due to the difference of their concentrations. Impulses from the area of higher concentration of particular ions are higher than impulses from the area of lower ions concentration of particular ions. Ions of dissolved compounds move in the direction of decreasing concentrations of a particular ion. The rate of dissolved-ions diffusion (J ld ) in the direction of the z coordinate is quantitatively described by Fick’s law: Jld  −Dlw

dc dz

(14.7)

where J ld is the rate of dissolved-compound diffusion (ML−2 T−1 , kg m−2 s−1 ); 2 Dw l is the coefficient of diffusion of a particular ion in chemically clean water (L −1 2 −1 T , m s ); the coefficients of diffusion of various ions of dissolved compounds vary. A typical value of diffusion coefficient of the most frequently occurring ions is −9 m2 s−1 . approximately Dw l  1.0 × 10 Fick’s law formally resembles other relationships expressing mass and energy transport like the Fourier law that describes heat transport and the Darcy and Ohm’s law. The diffusion of ions in soil differs from diffusion in water; part of the space is occupied by the solid phase of soil, and pores occupy only a part of the soil volume. Therefore, the streamlines of the solution transport are not direct, but curved, which elongates their trajectories. To take into account the curvature of streamlines, the coefficient of tortuosity was introduced. The coefficient of tortuosity is a number less than one; usually the value τ  0.66 is applied. In a porous medium (soil)

14.3 Transport of Dissolved Compounds in Soils

233

unsaturated with water, the volume of the water in which the diffusion occurs is less than soil porosity, so the volumetric soil-water content θ must be involved in the calculation. The diffusion flux of the dissolved ions in soil can be expressed by a modification of Eq. (14.7): Jld  −θ τ Dlw

dc dc  −θ Dls dz dz

(14.8)

2 −1 Dw l is the effective coefficient of particular ion diffusion in soil (L T , m s ); it can be seen that the effective coefficient of diffusion in soil is always less than the coefficient of diffusion in water.

Dsl  τ 2 −1

14.3.3 Hydrodynamic Dispersion The phenomenon known as hydrodynamic dispersion is responsible for the “dispersion” of the advancing front of a solution in soil. Dispersion means that the front of the advancing solution is not sharp, but penetrates at locally unsteady velocities. Hydrodynamic phenomena are the main reason for this; therefore, this phenomenon is called hydrodynamic dispersion. Hydrodynamic dispersion is the result of these three phenomena: 1. The nonlinear distribution of the flow rates in the cross section of soil pores. To simplify the problem, pores of circular cross-section can be assumed; then, from the application of Poiseuille’s equation (Eq. 8.4) follows the parabolic distribution of water-flow velocities in the cross section of circular pores during laminar flow. This means the solution in the centre of the pore will be ahead of the solution closer to the walls of the pores. Thus, the average concentration of the solution in the direction of flow decreases (see Fig. 14.3a). 2. The non-regularity of the porous space and its influence on the velocity of water flow means that, in various pores, the soil solution is flowing at various velocities, depending on the properties of porous spaces. Therefore, the non-regular distribution of the soil-solution concentration at the penetrating front of the soil solution is observed (Fig. 14.3b). 3. The soil pores of various dimensions imply various velocities of the flowing soil solution; those differences are described by Poiseuille’s law; the average waterflow velocities are proportional to the second power of the pores’ radiuses. The solution velocity in wider pores is higher than the velocities of the solution in narrower pores, which increases the dispersion of the solution (Fig. 14.3c). Quantitatively, the hydrodynamic dispersion can be described by the Fick’s equation: Jlh  −θ Dlh

dc dz

(14.9)

234

14 Transport of Solutes in Soils

Fig. 14.3 Reasons of hydrodynamic dispersion; a non-uniform flow-rate distribution in the pores cross sections, b different flow trajectories in the porous system, c different average velocities of liquid flow in pores of various diameters

where Dlh is the coefficient of hydrodynamic dispersion (L2 T−1 , m2 s−1 ). The transport of solution in the soil can be expressed by all three mechanisms (Eq. 14.5): Js  Jw c − θ Dls

dc dc − θ Dlh dz dz

(14.10)

The coefficient of diffusion and the coefficient of hydrodynamic dispersion can be expressed by one coefficient, denoted as the effective coefficient of hydrodynamic dispersion De (L2 T−1 , m2 s−1 ): De  Dls + Dlh

(14.11)

Then, using Eqs. (14.11), (14.10) can be written as: Js  Jw c − θ De

dc dz

(14.12)

14.4 Equation of Soil-Solution Transport

235

14.4 Equation of Soil-Solution Transport The equation of soil solution transport can be developed by a standard procedure, using the continuity equation and solution-transport equation. The continuity equation of one-dimensional transport of soil solutions can be written in the form (Radcliffe and Šim˚unek 2010): d Js dCt − − S(z) dt dz

(14.13)

where C t is the total concentration of the solution expressed in units of mass in a unit volume of soil (ML−3 , kg m−3 ); J s is the rate of solution transport (ML−2 T−1 , kg m−2 s−1 ); S(z) is the rate of ion input or extraction (ML−3 T−1 , kg m−3 s−1 ); t is time (T, s); z is vertical coordinate (L, m). The total concentration of solution C t can be expressed as the sum of ions’ concentration in soil a solution and in the solid phase of soil: Ct  θ c + ρb s

(14.14)

where c is ions’ concentration in the liquid phase of soil expressed as mass of solute in a unit volume of soil (ML−3 , kg m−3 ); s is ions’ concentration in the solid-soil phase and adsorbed by a soil surface expressed in mass units (MM−1 , kg kg−1 ); ρ b is volumetric soil density (ML−3 , kg m−3 ). By substituting Eqs. (14.13), (14.14) and (14.12), the following equation is derived:   d dc d(θ c + ρb s) − Jw c − θ De − S(z) (14.15) dt dz dz When the extraction and source term can be neglected and adsorption (desorption) can be neglected too, Eq. (14.15) can be written as:   d d(Jw c) dc d(θ c)  θ De − (14.16) dt dz dz dz This equation is usually denoted as convective dispersion equation—CDE. The water-transport rate J w and the volumetric soil-water content θ are constant during steady-state transport. For a constant effective coefficient of hydrodynamic dispersion De , Eq. (14.16) can be written: θ

d 2c dc dc  θ De 2 − Jw dt dz dz

(14.17)

To divide Eq. (14.17) by the volumetric soil-water content θ and substituting vp  J w /θ , Eq. (14.17) can be written as:

236

14 Transport of Solutes in Soils

dc d 2c dc  De 2 − v p dt dz dz

(14.18)

where vp  J w /θ is the average water-flow rate in soil pores (LT−1 , m s−1 ). The solution of Eq. (14.18) are the profiles of concentration distribution in the vertical direction as a function of z and t; c  f (z, t). Equation (14.18) without the last term of the right side (which characterises convection) has the same shape as the equation describing the transport of heat. To solve Eq. (14.18), input data De and vp are needed, as well as the initial and boundary conditions, i.e. concentrations or flow rates at the boundary of the system (the soil surface and bottom soil-layer boundary in which solution concentrations are calculated). Also needed is the concentration distribution (concentration profile) at the beginning of the solution-transport process. The simplest conditions for analytical solution of Eq. (14.18) are: Initial condition: c(z, t  0)  0, means the zero concentration along the soil profile at the beginning of the infiltration; the soil profile contains water only. Boundary conditions: J w (0, t)  J w (co ), the rate of solution flow of constant concentration across the soil surface is J w (co ). The bottom boundary condition expresses the constant concentration of a particular ion during the process under study: dc(∞, t) 0 dz The analytical solutions of Eq. (14.18) for simple initial and boundary conditions were presented by Skaggs and Leij (2002). Equation (14.18) can be solved numerically; its solution is an example of modern simulation models; even more complicated equations with various initial and boundary conditions can be solved numerically (Radcliffe and Šim˚unek 2010).

14.5 Péclet Number and Identification of Transport Mechanisms of Dissolved Compounds Transport of diluted compounds in soil is usually the result of all three of the just described mechanisms. The importance of individual mechanisms can be identified according to the parameter called the Péclet number (Pe), defined by the ratio of the convective transport rate and diffusion transport rate. It can be expressed by the formula: Pe 

vp d vd w  P Dl Dlw

(14.19)

where v is the macroscopic (Darcian) rate of water flow in soil (LT−1 , m s−1 ); vp is the actual (in pores) rate of water flow (LT−1 , m s−1 ); P is porosity; d is the average

14.5 Péclet Number and Identification of Transport Mechanisms …

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Fig. 14.4 The ratio of the coefficient of hydrodynamic dispersion of ions in the soil (Dlh ), the coefficient of molecular diffusion of ions in water (Dw l ) and the Péclet number (Pe)

diameter of the primary soil material (grain) (L, m); Dwl is the coefficient of molecular diffusion of the diluted compound in water (L2 T−1 , m2 s−1 ). Pfannkuch (1962) showed that the ratio of the hydrodynamic dispersion coefficient of ions in soil Dlh and the coefficient of molecular diffusion of ions in water Dwl are a function of the Péclet number Pe. Figure 14.4 demonstrates the ratio of the hydrodynamic-dispersion coefficient of ions in soil Dlh and the coefficient of molecular diffusion of ions in water Dwl as a function of Péclet number: Dlh  Dlw τ + k1 (Pe)n

(14.20)

where k 1 is an empirical parameter 0.5 ≤ k 1 ≤ 2; n is an empirical parameter 1.0 ≤ n ≤ 1.2. It follows from Eq. (14.20) that molecular diffusion is dominant at small rates of transport; with increasing flow rates (v), convection becomes more important. To illustrate the problem, the Péclet number can be calculated corresponding to the flow rate 1.0 m d−1 (it is high rate of water flow in porous medium) and to the diameter of soil particles 0.1 mm (fine sand); the standard value of coefficient of molecular diffusion of dissolved compound (NaCl) is Dwl  1.1 × 10−9 m2 s−1 . The calculated Péclet number Pe  3.0 illustrates the approximately equal importance of both convective- and molecular-diffusion mechanisms in this case. Transport of dissolved compounds can be strongly influenced by adsorption or desorption of ions, by the solution concentration and by properties of the diluted compound.

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14.6 Outflow and Breakthrough Curves 14.6.1 Outflow Curves An outflow curve is the relationship between solution concentration that is forced to flow out of the soil (sample) by water (or by the solution of lower concentration) and time. The most common way to present an outflow curve is the relationship between the solution’s relative concentration c/co , the flowed out of the soil layer (soil sample in laboratory conditions) and the relative volume of solution n  V i /V o that flowed out of the soil with initial concentration of solution co . V i is the volume of water that flowed into the soil sample at time t; V o is the volume of liquid in soil sample. The outflow curve of solution from the soil is expressed by the ratio of the relative solution concentration and the relative volume of outflowed water (Fig. 14.5). Sometimes, so-called breakthrough curves are measured. The difference between these relationships is the liquid concentrations initially contained in the soil layer (or soil sample). To measure outflow curve, the solution initially contained in soil pores is pressed out by entering water (or solute with lower concentration). To measure a breakthrough curve, the soil is initially containing water (or the solution of low concentration) and is pressed out of the soil by the solution. A common example of an outflow curve is the infiltration of rain (or irrigation) water into soil and its subsequent transport across the soil (which always contains diluted compounds) and its flow from the soil profile by infiltrating rain water. This process resembles the outflow-curve measurement process; it can be characterised as a “pressing” of solution from the upper soil layer by rain water. This type of process is artificially realised in arid and semiarid zones to leach salts accumulated in the upper soil layer during evapotranspiration. A necessary component of the leaching procedure is the drainage of the leached solution by a drainage system and its transport out of the field. This process is simulated in laboratory conditions to study the process and to quantify it. The quantity and concentration of the solute

Fig. 14.5 Outflow curve of soil solution from soil; the soil solution is forced to flow out of the soil by water

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outflowing the soil sample is measured, as well as its dependence on time. Results of such experiments enable calculation of the coefficient of hydrodynamic dispersion. Typical for outflow curves is c/co  1 at the time t  0, (at the outflow onset), and then the relative concentration of the solute decreases.

14.6.2 Breakthrough Curves A breakthrough curve is the relationship between a liquid concentrate flowing off the soil, saturated or unsaturated with water, when soil water is forced to flow out by the solution versus elapsed time. Most frequently, it is expressed by the solution’s relative concentration c/co , the soil that flowed out (soil sample) and the relative volume of outflowed solution n  V i /V o (Fig. 14.6). Term c is the mass concentration of the solution at the outflow cross section, co is mass concentration of solution entering the soil (sample), V i is solution volume entered the soil during the time t, and V o is the initial water volume in the soil sample. The ratio n  V i /V o  1 means that the soil entered the solution at a volume equal to the volume of liquid (water) in the soil (sample) before the measurement. Breakthrough curves are used to characterise quantitatively solution inflow and transport into the soil by infiltration through the soil surface. This type of process is typical for infiltration of dissolved nutrients into soil saturated or unsaturated with water. Figure 14.7 diagrams such an experiment. It shows the simplest, so- called “piston” transport of solution through the cylinder filled with water, which is forced to flow out by the solution entering the soil. The interface between solution and water is sharp. The relative concentration of outflowed liquid from the early stage after the beginning of solution inflow is c/co  0, until full volume of liquid is displaced; then the relative concentration of outflowed liquid immediately increases to c/co  1 (line (a), Fig. 14.6). In reality, such a situation can be observed at high-infiltration velocity of the solution without notable dispersion. Solution dispersion in porous medium (soil) is demonstrated in Fig. 14.8, and the corresponding breakthrough curve in

Fig. 14.6 Breakthrough curve of soil solution displaced by “piston”-flow mechanism (a) (Fig. 14.7) and by dispersion in soil (b) (Fig. 14.8)

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Fig. 14.7 Schematic diagram of solution outflow through the soil sample initially saturated with water by “piston”-flow mechanism. The breakthrough curve of such displacement is presented in Fig. 14.6a

Fig. 14.6b. Comparing both breakthrough curves, it can be seen that dispersion can be significant factor. The increase of solution concentration at the outflow profile appears sooner than in the case of the “piston” mechanism of the solution transport because of the nonhomogeneity of porous spaces, as was mentioned before. Solution concentration at the outflow profile changes fluently. Figure 14.9 shows breakthrough curves of chlorine (Cl) in water-saturated clay loam (1) and sand (2). The shape of the breakthrough curve of sand is closer to the “piston” mechanism than it is for clay loam. Sand-pore diameters are concentrated in a narrow interval. Clay loam contains a wider range of pore diameters; among these are relatively wide pores that contribute to the “fast” preferential flow. The influence of initial soil-water content (saturation) on breakthrough curves is demonstrated in Fig. 14.10. Tritium displacement in the loamy sand soil with lower initial soil-water content (θ i  0.37 cm3 cm−3 ) is faster than in the same soil with higher soil-water content (θ i  0.39 cm3 cm−3 ). The solution uses the empty pores to achieve a faster breakthrough in the soil. Concentration profiles of chlorine ions during ponding infiltration of the chlorine solution into the soil and its subsequent redistribution are presented in Fig. 14.11. Distributions resemble water infiltration and resultant redistribution. When this happens, the dominant transport mechanism is convective transport of chlorine ions.

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Fig. 14.8 Schematic diagram of the solution dispersion in soil. Solution concentration at the inflow cross section is co; concentration of outflowed solution increases from zero up to co (Fig. 14.6b)

Fig. 14.9 Breakthrough curves of chlorine ions in soil saturated with water; clay loam (1) and sandy soil (2). According to Biggar and Nielsen (1976)

Breakthrough and outflow curves are important information about dispersion processes of solutes in soils. By the analysis of those curves, it is possible to calculate dispersion coefficients of solutes in soils and then to use them as input data for simulation models. Detailed information can be found in the publications by Kutílek and Nielsen (1994), Jury and Horton (2004) and Radcliffe and Šim˚unek (2010).

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Fig. 14.10 Breakthrough curves of tritium in unsaturated loamy sand with various initial soil-water contents (θ i  0.37 and θ i  0.39 cm3 cm−3 ). According to Biggar and Nielsen (1976)

Fig. 14.11 Concentration of chlorine ions as a function of depth for various times after ten hours of ponding infiltration, followed by redistribution. Numbers at particular profiles mean hours from the onset of infiltration

References Biggar JW, Nielsen DR (1976) Spatial variability of the leaching characteristics of a field soil. Water Resour Res 12:78–84 Jury WA, Horton R (2004) Soil physics, 6th edn. Willey, New York, p 384 Kutílek M, Nielsen DR (1994) Soil hydrology. Catena, Cremlingen, Destedt Pfannkuch HO (1962) Contribution à l’étude des déplacements de fluides miscibles dans le milieux poreux. Revue de l’Institut Français du Pétrole 18:215–270 Radcliffe DE, Šim˚unek J (2010) Soil physics with HYDRUS. Modeling and applications. CRC Press, Taylor & Francis Group, Boca Raton Skaggs TH, Leij FJ (2002) Solute transport: theoretical background. In: Dane JH, Topp GC (eds) Methods of soil analysis; Part 4, Physical methods (Chap. 6.3), 3rd edn. SSSA, Madison, pp 1353–1380

Chapter 15

Water and Energy Balance in the Field and Soil-Water Regimen

Abstract Management of soil water to optimize biomass production is the primary aim of amelioration activities. Soil-water management, i.e. the design and operation of irrigation and drainage systems in the field have to be based on diagnoses of soilwater regimen. The soil-water regimen is a statistical characteristic of individual (mostly annual) cyclic courses of soil water (or soil-water matric potential); it is the generalisation of the individual annual courses by processing long-time (many years) average values of regimen characteristics. Then, the type of soil-water regimen and ways to manage it can be proposed and an expected yield can be calculated. Based on analysis of soil-water balance equations, one can propose the principles of soil-water management. Hydrological classification of the soil-water regimen and the principles of soil-water regimen diagnostics are presented.

15.1 Soil-Water Balance The Sun is the source of energy driving the majority of processes on the Earth. Only a small part of available energy has its source in the deep layers of the Earth, or from nuclear energy. A substantial part of energy emitted by the Sun drives the water cycle of the Earth, and evaporation consumes the majority of the incoming energy as latent heat of evaporation. To control the quality and quantity of water in the landscape (or catchment), it is necessary to know the structure of the water-balance equation, i.e. the values of the individual components making up water balance. The water-balance equation is an application of the law of the conservation of matter to some aspect of the landscape. The water-balance equation of the defined part of the hydrosphere (it can be a catchment or a certain volume of soil profile) counts the incomes (inflows) and expenditures (outflows) of water in the analyzed body. The water-balance equation is in analogy to the financial statement because it compares the incomes and expenditures during a given period of time. The difference between inflows to the defined volume of hydrological structure and outflows from it results in water accumulation (incomes are higher than expenditures) or in drying (expenditures

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_15

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are higher than incomes). The water-balance equation makes possible calculating the water storage of the hydrological structure at a given time. Through water balance, we calculate those components of the water-balance equation of the soil volume that are not possible to measure or which are measured only with difficulty. The soil-water balance equation expresses changes in soil-water content of the defined soil volume as the difference between components of the soil-water balance equation that express incomes (inflows) and outcomes (outflows) of water fluxes. As was mentioned, the soil-water balance equation is a modified form of the mass-balance equation (or continuity equation). The water-balance equation can be written for a defined level (like the soil surface or bottom boundary of the soil profile), then upward and downward water fluxes crossing the defined level are considered. The soil-water balance equation can be written also for a defined volume of soil or catchment. The water balance is then limited to the chosen time interval: day, month or year. The soil-water balance equation is usually written for soil volume V  A.z, where A is the area of soil horizontal cross section; (usually A  1 m2 ; vertical water movement is assumed); z is the thickness of the soil layer; usually it is the difference between soil surface and depth of the root system. The soil- water balance equation of the soil volume V can be written in the form (see Fig. 1.1):   Vw  (Vi + Pd ) − I p + E e + E t

(15.1)

where V w is the soil-water content change in the soil volume V (L3 , m3 ); V i is the quantity (volume) of infiltrated water (from rain) to the soil volume V (L3 , m3 ); Pd is the volume of water that flowed into the soil volume from nearby soil or groundwater (L3 , m3 ); I p is the volume of soil-water outflow from the defined soil volume V downward (L3 , m3 ); and E e , E t are volumes of evaporated/transpired water from the defined soil volume V (L3 , m3 ). All the components of the water balance equation are attributed to the chosen time interval t. In principle, the water quantities in Eq. (15.1) can be expressed in units of volume, or in thicknesses of the water layer transported into/from the soil volume, or stored in the defined soil volume. The volume of infiltrated water into soil V i during the time interval t can be calculated from the equation: Vi  (Z + Z z ) − (O + I )

(15.2)

where Z is quantity of precipitation (volume or water-layer thickness) on the unit soil surface in time interval t; Z z is quantity of irrigation water applied to the soil surface of a unit area in time t; O is water of quantity that flowed from the unit soil surface (surface runoff); I is the quantity of water intercepted by the plant canopy corresponding to the unit soil-surface area. The rate of change of soil-water content in the balanced volume of soil can be expressed deriving it from Eq. (15.1):

15.1 Soil-Water Balance

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d Vw  (vi + pd ) − (i + ve + vt ) dt

(15.3)

Terms of the right side of Eq. (15.3) are expressed in the units of rates (L3 T−1 , m s ). The water-balance equation for the catchment can be written: 3 −1

Vw  Z − E + Pp − O p

(15.4)

where V w is the change of water quantity of the catchment in time interval t (L3 , m3 ); Z is the precipitation total during the time interval t (L3 , m3 ); E is evapotranspiration total (including evaporation of intercepted water) in time interval t (L3 , m3 ); Pp are the water fluxes of surface and subsurface water entering the catchment in time interval t (L3 , m3 ); Op is the outflow of surface and subsurface water from the catchment in time interval t (L3 , m3 ). The water-balance equation of a catchment is often used to calculate any one term of the equation that is not known; other terms should be known.

15.2 Water and Energy Balance of the Land The energy of the Sun radiated to the Earth keeps in motion Earth’s water cycle (hydrological cycle). Equation (15.5) contains the term (LE), representing the energy consumption by evapotranspiration (E), as the sum of evaporation (E e ) and transpiration (E t ). Evapotranspiration is the term of the water-balance equation that connects the equations of the water and energy balance of the Earth. The simplified equation of the energy balance of the soil surface is: Rn  L E + H + G

(15.5)

Under conditions typical of Central Europe, more than half of the energy reaching the soil surface is used for the phase change by evapotranspiration (LE); the rest is divided into turbulent (sensible) heat flux (H) and soil-heat flux (G). The structure of the components of the energy-balance equation on the right side of Eq. (15.5) depends on properties of the evaporating surface; for bare soil, the terms H and G are dominant. Desert regions without plant cover overheat due to the terms H and G, which makes those regions unsuitable for life. Areas covered by green-plant canopies and water surfaces (oceans) are dominated by term LE, and thus help to stabilize the Earth’s temperature. Only a small part of the incoming energy is consumed by mechanical processes like destruction of soil aggregates and the ensuing erosion. Knowledge of the components of the water- and energy-balance equations structure enhance understanding and evaluating generalised courses of the cited components during the year. Significant variations of the structure of water- and energybalance equations between individual years cover the typical, characteristic courses

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of the terms of the water- and energy-balance equation that are needed to manage and ameliorate the landscape. To understand and quantify the typical structure of the soil-water and energy-balance equations, it is necessary to know the courses of both equations’ components over many years, then to generalise them and to find the typical courses (water and energy regimen) needed to classify and later to manage the landscape of a particular site.

15.3 Soil-Water Regimen Hydrology is the science dealing with water on the Earth in the atmosphere, rivers, and lakes and below the land surface in interaction with the environment. Soil hydrology is a part of hydrology; its role is to study the movement of soil water and its quality as a part of the soil–plant–atmosphere continuum. Results of research are generalised to be utilized in water management. Hydrology as a science is divided accordingly to the subsystems (hydrology of surface water, hydrology of subsurface water); subsurface-water hydrology is divided into the hydrology of groundwater and soil-water hydrology. This division was natural process due to specific methods of subsurface water research in different conditions. Knowledge of the laws of water movement in an environment, as well as water-quality changes, are necessary to identify and forecast the development of the qualitative and quantitative parameters of water resources and thus create the scientific basis of their regulation. The income of solar energy to the Earth changes regularly in cycles; the same character is typical of the water dynamics of the Earth. Annual and daily cycles of the energy income are known; they form the annual and daily cycles of meteorological elements in the soil–plant–atmosphere system. The heterogeneity of the Earth’s surface and non-regular distribution of the energy across the Earth’s surface are responsible for the non-regularities of water and energy fluxes of the hydrosphere. The variability of weather is a typical example of such non-regularities. Those non-regularities are responsible for quite diverse annual courses of the SPAC system characteristics in consecutive years, even when the energy income to the upper boundary of an atmosphere is steady and regular. Diagrams of annual courses of meteorological and hydrological characteristics (discharges in rivers, precipitation, soil-water content, air temperature) reveal various courses in observed years, but there are observable some regularities, typical of a given site, which are different from other sites. A typical course (variability) of water-balance components in time and space is denoted as the hydrological regimen. A typical hydrological regimen (rivers, groundwater, soil water) can be determined by the generalisation of the annual (seasonal) courses of so-called “regimen characteristics” during the long-term interval (usually not less than twenty years). Then, the typical features of such generalised courses can be identified; the nonregularities (fluctuations) due to various courses of meteorological characteristics of individual years are suppressed.

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The knowledge of regimen characteristics (rivers, springs, groundwater table levels, soil-water content) is needed to rationally manage water resources. It is important to secure some minimum river discharge to supply enough water for communal and industrial consumption. To do that, water reservoirs need to be built, and they are designed based on knowledge of the particular river’s water regimen. Water reservoirs thus enable the accumulation of water (during high-water discharge) and an increase of outflows during dry spells. The same approach is used to design irrigation and drainage structures; they should be designed to secure optimal soil-water regimens during the vegetation period by regulation of irrigation and drainage operations. Knowledge of the water-regimen characteristics is the basis of successful design and management of amelioration systems. The soil-water regimen is the only one of several soil regimens. Biomass production is strongly influenced by the regimen of other soil characteristics, like the soil-temperature regimen, soil-air regimen, and plant-nutrient and dissolved-mineral regimens (Bedrna et al. 1989). All can be influenced by appropriate amelioration activities. But, the soil-water regimen is of key importance because all other’s soil regimens mentioned can be strongly influenced by its regulation.

15.4 Soil-Water Regimen and Its Diagnostics Soil water—particularly water in a variably saturated soil—is the source of water for plants, and plants are the first element of the trophic chain of animals. In principle, soil water can be used also as a source of water for industrial and communal consumption; but the extraction of water from soil is technically complicated (extraction, cleaning). Soil water usually contains a relatively high quantity of dissolved compounds and usually is not suitable for immediate communal use. A relatively high content of minerals (nutrients) is suitable for plants. Soil-water regimen is a statistical characteristic of individual (mostly annual) cyclic courses of regimen characteristics; it is the generalisation of the average values of long-term (many years) regimen characteristics. The term “regimen characteristics” is understood as the average value of soil-water content of the soil-root zone or the average value of soil-water matric potential during a chosen time interval; one day is usually used.

15.4.1 Soil-Water Regimen Classification The type of soil-water regimen (SWR) is possible to express by the generalised course of soil-water regimen characteristics. Usually, courses of the average soilwater content during the year (or season) expressed by the volumetric soil-water content (θ  f (t)), or by the course of soil-water content of the root zone expressed in

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mm of water layer contained in the soil-root zone as a function of time (V w  f (t)) are used as soil-water regimen characteristics. The use of the soil-water matric potential as regimen characteristics is rare because of difficulties with their measurement and the ambiguity of the relationship between the soil-water matric potential and soilwater content. The evaluation of the soil-water regimen depends on the aim of this procedure. The most important is soil-water regimen evaluation that leads to regulations to reach optimal yield of crops. The basic classification of soil-water regimen can be differentiated into two types: the natural soil-water regimen (not artificially influenced) and the artificial (or influenced) SWR. Another classification of the SWR is tripartite: – unacceptable (does not enable acceptable yield), – acceptable (delivers acceptable yield), – optimum (suitable to reach maximum yield).

15.4.2 Hydrological Classification of the Soil-Water Regimen Rode (1952) proposed a soil-water regimen classification denoted as the hydrological classification. It is based on the identification of dominant hydrological processes in soil. This classification is important during the primary decision stage of amelioration-activity design. According to Rode (1952), soil-water regimens can be divided into following classes: (a) Regimen of permafrost characterises soil that is permanently frozen; it occurs in the Arctic and Antarctic regions. (b) Percolation regimen is the type of soil-water regimen typified by the percolation of rain water every year (or many times during the year); it is characterised by the ratio Z/E > 1. Z is the total of average annual precipitation; E is the total of average annual evapotranspiration. This means the resulting water flux is directed downward; soil salinization is not assumed. This type of soil-water regimen is typical for the moderate climatic zone. (c) Periodically percolated soil-water regimen is typified by a dominant percolation process, but this type of process is not observed every year. (d) Nonpercolation type of soil-water regimen is typified by water transport restricted to the upper layer of soil; below this “active” soil layer, the soilwater content is low and relatively constant; the connection to the groundwater does not exist. This type of SWR is characterized by the equation Z/E  1 and is typical of soils without contact to groundwater; the GWT is behind the infiltration depth. (e) Perspiration type of soil-water regimen is characterized by higher totals of average annual evapotranspiration than the totals of average annual precipitation. Z/E < 1. The primary source of water is precipitation, but additional sources of water are usually groundwater and irrigation. This type of SWR can potentially

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lead to soil salinization, especially when the groundwater is mineralized, i.e. contains relatively high contents of dissolved minerals. This type of SWR is found in alluvia with high groundwater-table level. (f) Stagnant type of soil water regimen is due to a high groundwater-table level; the capillary fringe usually reaches the soil surface. (g) Irrigation type of soil-water regimen is an artificially created SWR; high evapotranspiration totals are typical of this type of SWR. Z/E < 1. In Central Europe nearly all types of these soil-water regimens can be found, with the exception of permafrost. In northern Europe, the percolation SWR is dominant, but, in southern part of Europe, the perspiration (evaporation) regimen is the most frequent.

15.5 Soil-Water Regimen Diagnostics and Biomass Production Soil constitutes a water reservoir (or better reservoir of soil solution) for plant canopies. Optimum biomass production can be achieved with an optimum soil-water regimen. There are also other conditions to support optimum biomass production, namely, optimal tillage, optimal-plant nutrition and elimination of other negative factors. The above mentioned risks can be eliminated relatively easy. But what is “optimum soil-water regimen”? Results of numerous measurements indicate that there is a linear relationship between biomass production Y (yield) and transpiration totals during the growth period E t (Hanks and Hill 1980; Vidoviˇc and Novák 1987). Figure 15.1 displays examples of published relationships Y  f (E t ); the conclusion can be drawn that the maximum biomass production under certain climatic conditions can be reached if the transpiration total is at maximum, i.e. it is not limited by lack of soil water. This assumption is valid when other conditions do not limit crop ontogenesis (tillage, nutrition and extreme dry and wet spells periods during crops ontogenesis). The optimum soil-water regimen does not limit plant-canopy transpiration during the growth period. To manage soil hydration to preserve potential transpiration, various types of irrigation systems are used. Chapter 13 analyses methods for calculating potential transpiration, as well as for the estimation of critical soil-water content above which transpiration will be maximum (potential). To design an irrigation system to optimize the soil-water regimen, it is necessary to know seasonal transpiration courses (and totals) of particular crops and sites at least during 20 years to diagnose the soil-water regimen (Novák and van Genuchten 2008). The only realistic possibility to do so is the application of so-called retrospective modelling of soil-water content. As a result, annual courses evapotranspiration and transpiration of plant canopies can be obtained. There are many commercially available simulation models, which are briefly described in Chap. 21. Stable physical and hydrophysical soil properties during the modelled period, as well as stable crops

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Fig. 15.1 Plots of the mass of maize grain yield versus transpiration total Y  f (E t ) during the growing season at the Trnava site, Southern Slovakia (1). Other relationships (Hanks and Hill 1980) represent relationships between the mass of maize grain yield and seasonal evapotranspiration. 2—Logan, USA (1975); 3a, 3b, 3c—Gilat, Israel (1968, 1969, 1970); 4—Cherson, Ukraine (1974–1978); 5—Greenville, USA (1978); 6—Farmington, USA (1978); 7—Evans, USA (1978)

Fig. 15.2 Exceedance curves of the grain corn yield (Y ), the calculated potential (maximum) yield (Y p ) and the difference (Y ) for 1971–2000, 2003 growing seasons at Most pri Bratislave, Slovakia

parameters, are assumed. Then, only meteorological conditions of the particular site were responsible for the variability of soil-water regimen characteristics over time. Variable meteorological characteristics, together with stable soil and plant data, are used as inputs to the mathematical simulation models. Seasonal courses and totals of transpiration (E t ) and potential transpiration (E tp ) of particular crops are the results of simulation. Then, using the known empirical relationship Y  f (E t ), for particular crops and sites, and values E t and E tp , the actual (Y ) and potential (optimum)

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(Y p ) yields for any season can be calculated, as well as the expected differences between actual and potential yields (Y ). Then, using the simulation results, curves of exceedance of calculated yields can be drawn. Figure 15.2 contains curves of exceedance of actual yields of corn grains (Y ), potential (optimum) yields (Y p ) and the differences between the yields of optimal and actual soil-water regimen (Y ). To analyse the results of simulation, one can estimate the effectiveness of investment for the optimization of the soil-water regimen for a particular crop. In the illustrative example (maize grains, site Most pri Bratislave, the Southern Slovakia), the average maize grain yield was 7.64 t ha−1 and the maximum (optimum) calculated average yield was 9.03 t ha−1 . The average difference between the yields was estimated as Y  1.4 t ha−1 . The question is: Is it reasonable to invest into an irrigation system to meet a targeted increase of average yield? It depends on the situation on the market, too. This type of analysis can be performed for any crop and site using the method of retrospective mathematical modelling. The weak point is the necessity to know the relationship Y  f (E t ), which has to be estimated from field data from a particular site.

References Bedrna Z, Hraško J (1989) Soil regimen. VEDA, Bratislava (In Slovak with English abstract) Hanks RJ, Hill RW (1980) Modelling crop responses to irrigation in relation to soils, climate and salinity (IIIC publication). Publication No. 6, Bet Dagan, Israel Novák V, van Genuchten MTh (2008) Using the transpiration regime to estimate biomass production. Soil Sci 173:401–407 Rode AA (1952) Soil water. Izd AN SSSR, Moscow (In Russian) Vidoviˇc J, Novák V (1987) The relationship of maize yield and maize canopy evapotranspiration. Rostlinní výroba 33(6):663–670 (In Slovak with English abstract)

Chapter 16

Swelling and Shrinking Soils

Abstract Soils containing clay minerals (illite, montmorillonite) changing their volumes with soil-water content variation. Soil-water content decreases are followed by soil shrinkage, and increases in soil-water content is followed by swelling. These phenomena are expressed by the creation or closing of soil cracks, and the soil surface decreases/increases simultaneously. Soil cracks can be characterized by crack porosity, by their specific volume and by the specific surface; nets of cracks on the soil surface can be characterized by their specific length. An equilibrium relationship of crack porosity to soil-water content is typical of any particular soil. Soil cracks characterised by their volume and surface play an important role in water infiltration into dry, heavy soils; they significantly increase the retention volume of such soils and the infiltration surface of water-filling soil cracks. The importance of water infiltration into the soil matrix from cracks surface is demonstrated.

16.1 Cracks Porosity and Soil-Water Content Soils containing clay minerals or organic compounds change their volumes when their water content is altered. The reason for the deformation of porous materials associated with changes in soil-water content is the specific structure of clay minerals as a part of soils. Clay minerals (illite, montmorillonite) are composed of parallel plates; water molecules are attracted by short-range van der Waals electrostatic forces and form layers along the negatively charged surfaces of the plate-like structures. Electrostatic bonds between polar water molecules and negatively charged plate surfaces increase the internal distance between parallel plates, and the soil increases its volume—it swells (Kutílek and Nielsen 1994). Organic compounds in organic soils swell too. The opposite process to swelling is shrinkage; during shrinkage the soil volume decreases with decreasing soil-water content. Soil shrinkage is associated with the creation of soil cracks and by soil-surface level decrease. Increasing soil-water content is followed by soil swelling as soil cracks close and the soil-surface level increases. © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_16

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Fig. 16.1 Crack porosity Pc and mass soil-water content w; (Pc  f (w)) of the heavy soil of the East Slovak Lowland. Pairs of data (Pc , w) represent measured data; ws is saturated soil-water content in mass units

A volume of soils containing clay minerals depends on soil-water content. It is convenient to express soil deformation in relative units; so-called cracks porosity Pc can be expressed as a function of mass soil-water content w. Cracks porosity is the ratio of cracks volume V c and volume of initially dry (non-swelled) soil volume V : Pc 

Vc V

(16.1)

An example of the relationship Pc  f (w) of heavy soil of the East Slovak Lowland is shown in Fig. 16.1. Porosity of soil cracks (cracks porosity) is estimated by the measurement of the relative area of cracks and area of the soil’s horizontal cross section of the soil sample in a standard sampling cylinder. An equilibrium state between soil-water content and cracks porosity is assumed. The continuous curve Pc  f (w) approximates the measured data. The relationship Pc  f (w) can be divided into three stages, depending on the rate of swelling/shrinking during the soil-water content change. Zero range. The soil deformation of soil saturated with water during initial soilwater content decrease is not observable. It is probably due to water outflow from macropores, as outflow water from such pores is replaced by air. Normal range. The volume of soil decreases proportionally with the water volume outflow (or evaporation); the relationship Pc  f (w) is linear. The volume of water held between the clay minerals plates decreases; all micropores are filled with water during this stage. Residual range. The soil-water content decreases, but the volume of soil decreases only slightly. Water is drained from micropores, but soil deformations are restricted by the structural stability of the soil matrix.

16.1 Cracks Porosity and Soil-Water Content

255

To calculate soil deformations, it is reasonable to approximate the curve Pc  f (w) by a linear relationship. This step can simplify the calculation of soil deformation in the normal range, where the basic part of it occurs. This approximation (thin line) can be seen in Fig. 16.1. The relationship Pc  f (w) in Fig. 16.1 can be used to calculate both shrinkage and swelling. However, ambiguity is expected in the relationship Pc  f (w) because of hysteretic behaviour. The relationship Pc  f (w) will be located below the function Pc  f (w) which was estimated for drying process. The increase (decrease) of the soil-surface level during soil swelling (shrinking) can be measured too. From a practical point of view, vertical changes of the soilsurface level are not important; in temperate climatic regions, the significant drying of soil occurs in the relatively thin upper- soil layer, and the corresponding vertical soil deformations are relatively small and are observed as an increase in the density of the upper dry-soil layer (Šutor et al. 2002).

16.2 Specific Volume and Specific Surface of Soil Cracks Specific volume of soil cracks (V cv ) is the volume of cracks in a unit soil volume. Specific surface of soil cracks (S cs ) is the area of soil-cracks surface in a unit soil volume. Results of measurement show the exponential change of specific volume of soil cracks (V cv ) with depth below the soil surface (z); it can be expressed (Kutílek and Novák 1976) as: ∞ Vcv 

Aco exp(−αz)dz

(16.2)

0

where Aco is the area of soil cracks on a unit soil-surface area (1 m2 ) (L2 L−2 , m2 m−2 ); α is an empirical coefficient. Integrating Eq. (16.2) within the boundaries (0, ∞), the equation is reduced to V cv  Aco /α. Equation (16.2) expresses the potential volume of soil cracks that can be filled by rain water. The soil-cracks volume can be as great as some tenth of millimetres of the water layer (see Table 16.1).

Table 16.1 The final relative changes of soil volume V r with various contents of physical clay (soil particles less than 0.002 mm) for heavy soils of the East Slovak Lowland (Šútor et al. 2002) Soil 1 2 3 4 5 Relative mass 0.098 of clay in soil

0.20

0.28

0.49

0.72

Relative maximum volume change

0.09

0.19

0.29

0.39

0.046

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Fig. 16.2 System of cracks on the 1-m2 area of chernozem soil surface. Trnava site, June 6, 1995. The specific length of cracks is 11.1 m m−2

Accordingly, the specific surface of soil cracks (S cs ) changes with soil depth below the soil surface exponentially too; it can be expressed by the equation: ∞ Scs 

2L co exp(−αz)dz

(16.3)

0

where L co is the length of soil cracks in the unit area of soil surface (specific soil cracks length) (LL−2 , m m−2 ). After integration, Eq. (16.3) reduces to the equation S cs  2 L co /α. Exponential functions after the integrals can be replaced by other types of functions, depending on the estimated distribution functions V cv  f (z) and S cs  f (z); the linear function can sometimes be acceptable. Those distributions can be easily measured by the area of soil cracks in the evaluation of horizontal cross sections at various depths below the soil surface (Kutílek and Novák 1976). It is a tedious, but simple, process and can give useful information regarding the additional retention of water in heavy soils. The area of soil-cracks surface is an important characteristic of swelling soils because it potentially indicates the additional surface through which water could infiltrate. The system of soil cracks in Fig. 16.2 is for specific cracks of length L co  11.1 m m−2 . If the depth of soil cracks is assumed to be one m, then the soil-cracks surface is S s  22.2 m2 . This is the maximum (potential) additional soil surface that could be used to infiltrate water into soil. It is 22.2-times the topographic soil surface. Therefore, soil cracks are an important feature of heavy soils allowing them to conserve rain water by increased infiltration during heavy rains and thus to form specific soil-water regimens of heavy soils.

16.3 Formation and Kinetics of Soil Cracks

257

Fig. 16.3 The relative changes of soil samples of volume V r with various contents of physical clay (particles less than 0.002 mm) during drying. Volumes of soil samples in metal cylinders (volume 100 cm3 ) are given as a function of time t (h). Contents of clay in a particular sample and final soil—volume changes correspond to data in Table 16.1 (According to Šútor et al. 2002)

16.3 Formation and Kinetics of Soil Cracks In the formation of water regimens of heavy soils, their horizontal deformations are especially important. The upper, dry-soil layer shrinks extremely, and soil cracks expressed by their specific volume (and specific surface) of cracks are maximum at the soil surface. The system of soil cracks (created by drying) on the chernozem (silt loam) soil of a soil-surface area of 1 m2 at the Trnava site (the Western Slovakia) is pictured in Fig. 16.2. Soil-cracks porosity at the soil surface is 0.046 m2 m−2 ; the specific soil-cracks length is 11.1 m m−2 . The relative changes of soil volume V r of soils from the East Slovak Lowland with various contents of clay particles (with diameter less than 0.002 mm) and time t are shown in Fig. 16.3; the data are collected in Table (16.1). The diameters and lengths of water saturated cylindrical soil samples as it was shrinking, as well as corresponding mass soil-water content, were measured (Šútor et al. 2002). Knowing the dimensions of soil samples, soil volumes were calculated and corresponding mass soil water content was evaluated by gravimetric method. The relationships V r  f (t) illustrate the influence of clay minerals (montmorillonite and illite) on the relative soil-volume changes and the kinetics of soil-cracks formation during drying of the cylindrical soil sample of volume 100 cm3 . In the field, soil cracks regularly appear along previously existing nets of soil cracks because those are the areas of lower mechanical firmness. Soil-cracks kinetics depends on initial soil-water content, as well as on the rate of infiltration and evaporation. The relationship between the relative maximum soil-cracks volume V r and the clay-fraction content p in the previously mentioned heavy soil is graphed in Fig. 16.4 (Šútor et al. 2002). This relationship is close to a linear one. The higher the clay

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Fig. 16.4 The final relative volume change of drying soil V r and the relative content of clay particles in heavy soil (p); see Table 16.1 and Fig. 16.3

fraction that the soil contains, the more the soil will swell or shrink. The relationship between the clay-fraction content and final (or maximum) soil deformation is the soil’s characteristics and can be different for various soils. For soils containing illite and montmorillonite, the linear relationship V r = f (p) can be used. Shrinkage was relatively slow (Fig. 16.3) even if the soil samples were relatively small (100 cm3 ). In the field, the daily evapotranspiration total is usually below 5 mm, so the soilwater content changes and soil-cracks formation rate are relatively small. Let’s saturate the heavy soil with water; then the saturated soil-water content is approximately w  0.5 (50%); the upper 0.5-m-thick soil layer has its soil-water content change in mass units per day by about 0.02 (or 2%), (assuming a daily evaporation total of 5 mm d−1 ). Then, the process of soil-cracks formation is relatively slow, and the first visible soil cracks appear on some days, depending on weather conditions. Accordingly, the process of soil-cracks closing is slow because of the low hydraulic conductivity of dry soil matrix. A few days were needed to create or close the soil cracks during rain and the post-rain drying of extremely heavy soils in the East Slovak Lowland (Novák 1986). Therefore, the volume of soil cracks is not influenced strongly even during the heavy rain in the first hours and days of precipitation. Several days are needed to close the soil cracks and thus to minimize the cracks infiltration rate. The volume of soil cracks is the important retention volume of rain water and minimizes surface runoff and ponding. Let´s assume the dry soil (maximum shrinkage) corresponding to the relative change in maximum volumes near the soil surface, which is 0.39 (Table 16.1). Assuming the soil-water content below the soil surface decreases linearly with depth, then the volume of soil cracks in the depth interval 0–1.0 m represents a water layer of 200 mm. This means up to 200 mm of rain water can be stored by soil cracks of maximum volume. Additionally, water can infiltrate into the soil matrix through the cracks surface. In reality, the soil-water content of heavy soils is usually much higher than previously assumed, and retention volumes are much lower.

16.4 Soil Characteristics Influenced by Soil Swelling and Shrinking

259

16.4 Soil Characteristics Influenced by Soil Swelling and Shrinking Soil-volume changes by the soil-water content are associated with the changes of the soil’s physical (and hydrophysical) properties. Soil density, porosity, hydraulic conductivity and soil-matrix retention change, too. When deformable soils dry, the soil-matrix density increases while the hydraulic conductivity, retention capacity and porosity decrease. The retention capacity of the soil matrix decreases with the soil-cracks volume. The measurement of the physical characteristics of swelling (or shrinking) soils is difficult because they change with the change of soil volume; soil pores’ dimensions change, too. The appropriate procedure to calculate water transport in deformable soils requires the application of various functions of soil-water retention curves (SWRC) hw  f (θ ), and hydraulic conductivities k  f (hw ) at any point of the soil space to be solved. Usually, the stability of the functions k  f (hw ) and hw  f (θ ) is assumed; swelling/shrinking processes are neglected.

16.5 Infiltration of Water into Soils with Cracks Water infiltration into homogeneous soil and its quantification were described in Chap. 10. To quantify the infiltration process into soil with cracks is much more complicated because water infiltrates not through the soil geographical surface only, but also through the cracks’ walls’ surface, when soil cracks are partially filled with water. To calculate infiltration curves and curves of cumulative infiltration to the cracked soil, the diagram in Fig. 16.5 can be used (Novák et al. 2000).

Fig. 16.5 The schematic of a submodel to calculate water infiltration into a soil with cracks. Vertical flow of water (rain, irrigation) qo (t) is divided into the infiltration flux across soil surface q(t) and the flux into soil cracks qf (t). S f (z, t) is the rate of horizontal infiltration from soil cracks into soil matrix; S r (z, t) is the rate of water extraction by roots

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Assume that the rain rate qo (t) is variable in time, and water starts to infiltrate through the soil surface at the rate q(t). When the rain rate is lower than the saturated hydraulic conductivity of the soil, then soil water will infiltrate through the soil surface without ponding. In the case of heavy soils, ponding is a frequent phenomenon because hydraulic conductivities of such soils are usually very low, less than 0.01 m d−1 . Ponding occurs usually following heavy rain during the spring, when the soilwater content is relatively high; soil cracks are not observed. After some time, small ponds form in topographic mini-depressions on the soil surface, and later ponding infiltration occurs. The period of time from the onset of infiltration to the formation of ponds covering the soil surface is the ponding time t p . Then, if soil cracks open (summer or autumn periods), water starts to flow into the soil cracks and infiltrates into the soil matrix through the cracks walls, too. During the spring, soil cracks usually close in heavy soils because of high soilwater content; in this case, ponding is often observed. This phenomenon limits soil tillage and delays ploughing and the sewing of crops. Infiltration into soil with cracks can be divided into following stages: 1. Infiltration of water into unsaturated soil, qo (t)  q(t), t < t p . 2. Rain rate is higher than infiltration rate (qo (t) > q(t)); water layer on the soil surface is formed (t  t p ). 3. After reaching some critical soil layer height hs and qo (t) > q(t), water from the ponded soil surface starts to flow into soil cracks to fill the soil cracks at the rate qf (t). 4. Surface runoff starts when soil cracks are full of water or soil cracks are closed. 5. Water infiltrates through the soil surface and through the soil-cracks surface. Infiltration into soil with cracks can be calculated by mathematical model, the governing equation is the Richards equation with an additional term characterising water infiltration from soil cracks (Novák et al. 2000):    ∂h w ∂ ∂θ  k(h w ) + 1 − Sr (z, t) + S f (z, t) (16.4) ∂t ∂z ∂z where S r (z, t) is the rate of soil-water extraction by plant-canopy roots (L3 L−3 T−1 , m3 m−3 s−1 ) and S f (z, t) is the rate of infiltration from the part of the soil cracks filled with water through the cracks surface (L3 L−3 T−1 , m3 m−3 s−1 ). The stable dimensions of soil cracks are assumed; this assumption is acceptable because soil cracks change their dimensions very slowly; it takes some days, but the heavy rain duration is in hours. The dynamic of swelling depends on the soil’s hydraulic conductivity, and it is usually very small for heavy, swelling soils. Significant changes in soil-cracks dimensions were not observed in the field during heavy rain lasting even some hours. Figure 16.6 contains cumulative water-infiltration curves into cracked soil at Trnava site (the Western Slovakia) calculated by the simulation model. The simulation model HYDRUS-ET (Šim˚unek et al. 1997) used, making possible the calculation of water infiltration into cracked soil. A system of soil cracks (depicted in Fig. 16.2)

16.5 Infiltration of Water into Soils with Cracks

261

Fig. 16.6 Cumulative infiltration of water into a soil with cracks i as a function of time t. Cumulative rain (irrigation) (1), infiltration through the soil surface (2), infiltration from cracks to the soil matrix (3) and cumulative infiltration through the soil surface and soil cracks (4). Trnava site, the Western Slovakia

was used; the specific length of cracks on the soil surface was L co  11.1 m m−2 , depth of cracks was 40 cm, cracks porosity at zero soil-water content was Pco  0.046 m3 m−3 and hydraulic conductivity of a soil-matrix saturated with water was K  5 cm d−1 . This illustrative (but close to the real situation) case shows the significant contribution of the infiltration from cracks to the cumulative infiltration (curve 3, Fig. 16.6); it is 1.5-times higher than water infiltration through the soil surface (curve 2, Fig. 16.6). The maximum retention capacity of the soil cracks was estimated to be 46 litres of water in a 1-m3 volume of soil. This is a hypothetical (maximum) retention capacity of cracks because it was estimated for a volume of cracks of zero soil-water content, which is not realistic. But approximately one tenth of the maximum retention capacity was often observed (4.6-mm water layer); this is also a significant contribution to the soil-water capacity of heavy, swelling soils.

References Kutílek M, Novák V (1976) The influence of soil cracks upon infiltration and ponding time. In Kutilek M, Šútor J (eds) Proceedings of the Symposium on Water in Heavy Soils, CSVTS Publisher, Bratislava, pp 126–134 Kutílek M, Nielsen DR (1994) Soil hydrology. Catena, Cremlingen—Destedt Novák V (1986) Regulation of the soil water regimen of the East Slovakia Lowland. In: Novák V (ed) Ekologická optimalizácia využívania Východoslovenskej nížiny II. Zborník z vedeckého sympózia CPZV 614, Bratislava. (In Slovak with English summary) Novák V, Šim˚unek J, van Genuchten MTh (2000) Infiltration of water into soil with cracks. J Irrig Drain Eng 126(1):41–47

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Šim˚unek J, Huang S, Šejna J, van Genuchten MTh, Majerˇcák J, Novák V, Šútor J (1997) The HYDRUS—ET software package for simulating the one-dimensional movement of water, heat and multiple solutes in variably saturated media. Version 1.1, Institute of Hydrology, Slovak Academy of Sciences, Bratislava Šútor J, Gomboš M, Mati R, Ivanˇco J (2002) Characteristics of the East Slovak Lowland heavy soils aeration zone. ÚH—SAV Bratislava, OVUA Michalovce. (In Slovak with English summary)

Chapter 17

Stony Soils

Abstract This chapter provides a basic overview of the properties of stony soils: their main differences compared to non-stony soils, their occurrence, classification, basic physical and hydrophysical properties, as well as the methodology for their quantification, sampling methods, measurement of the hydrophysical properties of stony soils and the definition of so-called effective parameters needed for water-flow modelling in stony soils. The chapter also contains links to the latest literature in this area. An illustrative example of the evaluation of effective stony-soil parameters for particular stony soil from the Western High Tatra Mts. (Slovakia) and results of the illustrative water-flow modelling in the stony soil (a case study) are presented and discussed.

17.1 Specific Features of Stony Soils Stony soils differ from non-stony soils by a significant content of rock fragments (RF, called also skeleton), i.e. soil mineral particles larger than 2 mm in diameter; they include gravels, cobbles, stones and boulders. The size, shape, degree of weathering and geological origin, position and the spatial distribution of RF in a soil profile can strongly influence the stony soils’ properties (mainly the soil’s water retention and hydraulic conductivity) and can affect soil-water movement, infiltration and the runoff formation. Despite that stony soils occur in many forested, mountainous and even agricultural areas, the influence of the presence of RF on hydrological processes, water balance or water-storage estimations are still often neglected. In the past, only the hydraulic properties of fine earth were used to determine input parameters to water-flow modelling in stony soils, which could lead to substantial errors. Why is it so problematic to quantify hydraulic processes in stony soils? There are several reasons for that. One of them is the sample size. Rock fragments in stony soils are often several times larger than fine-soil mineral particles. Therefore, measurement of the characteristics of bulk stony soils have to be based on an adequately representative elementary volume (REV), the size of which may be 1 m3 or larger, © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_17

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Fig. 17.1 Typical distribution of the relative values of fine earth (Rf ), stoniness (Rv ) and total porosity (Pb ) of the stony soil in the soil profile. Site FIRE, High Tatra Mts., Slovakia

which is often impossible to do. The presence of RF causes also other practical problems connected with field measurements like the difficulties in inserting probes into stony soils or installing lysimeters. Furthermore, field measurements encounter a large variability of stony-soil characteristics even on a small plot, difficult soil sampling and infiltration measurements. The use of tracers is complicated in stony soils as well. Comparing with fine earth, rock fragments’ retention capacity is usually significantly lower than retention capacity of a fine earth, as well as hydraulic conductivity, which is the case of RF often so small that it is negligible. However, some particular types of rock fragments, especially of volcanic origin (tuffs) and carbonates, or others with a high degree of weathering, can possess a relatively high retention capacity and specific porous media can be formed (Cousin et al. 2003; Rouxel et al. 2012). The presence of rock fragments in soil significantly influences also soil porosity. The volume of the fine-earth fraction is reduced by the volume of rock fragments. The chemical and biological processes are concentrated in fine earth (Gömöryová et al. 2006). The stone content (stoniness) and the depth of a stony soil horizon are the results of particular pedogenetic processes. Mountainous and forest stony soils created on deluvial and glacial sediments are usually shallow soils with depths of 0.6–1 m. The stoniness of such soils usually increases with the soil depth, as is shown in Figs. 17.1 and 17.5. Figure 17.1 illustrates the decrease of a stony soil’s total porosity with soil depth (with increasing stoniness).

17.2 Stony Soils Classification There are various systems of stony-soil classification. Morphogenetic classification system of Slovak soils (Societas pedologica slovaca 2014) classifies stony soils according to the volumetric participation of the rock fragments. This classification

17.2 Stony Soils Classification

265

system is based on the relative content of fine-earth particles (fine earth) , rock fragments and organic compounds. According to the degree of stoniness, soil can be divided into a slightly stony (rock fragments relative volume is 5–10%), a moderately stony (10–25% of rock fragments) and a highly stony soil (25–50% of rock fragments). Within the category of psefitic soils belong soils with more than 50% of rock fragments by volume and less than 30% of organic material. The IUSS Working Group WRB (2015) classifies skeletic soil as a soil with 40% or more (by volume) of coarse fragments averaged over a depth of 100 cm from the soil surface or to continuous rock, technically hard material or a cemented or indurate layer, whichever is shallower. It can be further classified as akroskeletic, orthoskeletic and technoskeletic, according to the presence of coarse fragments either on the soil surface or in the entire soil profile. According to the IUSS Working Group WRB (2015), hyperskeletic soils are soils having less than 20% (by volume) fine earth, averaged over a depth of 75 cm from the soil surface or to continuous rock, technically hard material or a cemented or indurate layer starting at least 25 cm below the soil surface, whichever is shallower. USDA classification makes possible classifying soils according to the volume of RF as: (1) non-gravelly, non-cobbly, non-stony soils with less than 15% of RF; (2) gravelly, cobbly, or stony soils with 15–35% of RF; and (3) extremely gravelly, extremely cobbly, or extremely stony soils with 35–60% of RF (USDA 2017). Rock fragments are described by size (diameter) as gravels (2–75 mm), cobbles (75–250 mm), stones (250–600 mm) and boulders (>600 mm) (Soil Survey Division Staff 1993, cit. acc. to Garcia-Gaines and Frankenstein 2015).

17.3 Stony Soils Occurrence About 30% of soils in Western Europe and about 60% of soils in the Mediterranean region are referred to as stony soils (Poesen and Lavee 1994). National surveys and a large-scale inventory in Sweden estimated that, on average, 43.4% of volume of stones and boulders in Sweden’s forest soils, the variation was spatially correlated and related to bedrock composition and Quaternary development (Stendahl et al. 2009). Šály (1978) estimated that up to 80% of Slovak forest soils contain rock fragments. The systematic analysis of agricultural soils across the Slovak territory has shown that 47.2% of Slovak arable soils are classified as stony soils. According to the European Soil Database, 41% of European soils contain a stone volume of at least 10% (Stendahl et al. 2009). Soils in arid and semiarid parts of Chile are known for their moderate to high stoniness near the soil surface. They are the result of slow soilforming processes and of the soil degradation by deforestation and increased erosion, which are serious hazards in that country (Verbist et al. 2009; Baetens 2007).

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17.4 Representative Elementary Volume of Stony Soils The presence of two qualitatively different components of a stony soil (fine earth and rock fragments) creates the heterogeneous soil structure. The integral characteristics of stony soils have to be measured using a large enough volume of soil that depends on rock fragments’ dimensions. The appropriate soil volume is estimated on the basis of the representative elementary volume theory (REV). According to this theory, heterogeneous porous medium can be characterized by some average properties of a specific volume, denoted as the representative elementary volume (REV) . This volume has to be large enough to average out the discontinuities of the soil’s structural elements (spatial arrangement of pores and particles), but it should be small enough compared to the total volume of porous medium in which the transport processes are performed (Buchter et al. 1994). The appropriate one is the REV if the porous medium property is invariant to the REV dimensions, i.e. in some range of soil volumes, V min –V max this property is constant. The REV size depends on the soil property we are looking for, but also on the scale at which the particular soil property is studied. Therefore, the REV dimension can fluctuate for the evaluation of various properties of the same soil. The heterogeneous porous medium (soil) is replaced by the “hypothetically homogeneous” porous medium, and it is characterized by the effective characteristics. The effective characteristics of stony soils characterize a stony soil as a whole and are denoted as bulk characteristics, later indicated by the subscript „b“. A standardised methodology for REV estimation did not exist until now. An REV of 100 cm3 (standard sampling cylinder) has proved to be large enough for relatively homogeneous (fine-earth) soils. A standard cylinder can be used for the fine-earth fraction only in the case of stony soils. Baker and Bouma (1976), Kutílek and Nielsen (1994) and Lichner (1994) recommend to define the representative elementary volume of a soil as the soil-sample volume that should contain not less than 20 basic elements of the soil structure (in this case of rock fragments) in the soil-sample cross section. Buchter et al. (1994) recommended that the dry weight of the stony soil REV to be at least 100 times the weight of the largest solid-rock fragment in the stony soil. If the maximum diameter of rock fragments is d  10 cm, its weight is about 1.5 kg, so the minimum REV weight should be 150 kg.

17.5 Sampling of Undisturbed Stony Soils An enormous dimension of REV which is needed to be sampled to measure stonysoil characteristics is associated with technical problems. As has been mentioned, standard sampling cylinders (100 cm3 ) can be used to sample fine earth only. Soil samples of volume approximately 8000–10,000 cm3 (a cube with an edge of approximately 20 cm) can be used to measure hydraulic conductivity of stony soils with rock fragments if the dimensions of gravel. Special technique should be used to sample

17.5 Sampling of Undisturbed Stony Soils

267

soil blocks of volume 1 m3 and larger. This approach enables performing a detailed study of a monolith structure. Fixing, impregnating and cutting off particular soil sections enable the measurement of properties of the sample in the laboratory, as was suggested and performed by Buchter et al. (1987). However, this approach is expensive and tedious, and therefore it is not suitable as a standard procedure. In a case when it is not possible to sample a stony soil of REV dimensions, the samples of fine-earth fractions and rock fragments from various soil horizons are recommended to be sampled separately in at least five replicates (Cools and de Vos 2013). Disturbed fine-earth soil samples are needed to estimate soil texture, content of organic matter or other chemical components. Samples of fine-earth fractions are needed for laboratory measurement of saturated hydraulic conductivity and soilwater retention curves. Rock-fragment properties (porosity and retention capacity) are possible to evaluate using laboratory procedures too. According to European regulation EN ISO 11274:2014, measurements performed with soil samples of less than 20% rock fragments by volume do not have to be modified to account for the stoniness, and are considered as measurements of fine-earth soil. If undisturbed samples of fine earth are not available, disturbed samples have to be prepared.

17.6 Physical Characteristics of Stony Soils The basic physical characteristics of stony soils are stoniness, bulk density and porosity. As was already mentioned, the integral (effective) characteristics of stony soils can be measured with soil samples of REV dimension in specific situations only. Therefore, the alternative procedure of separately measured properties of rock fragments and fine-earth fractions of stony soil is recommended.

17.6.1 Stoniness of Stony Soils Stoniness is an important physical characteristic of a stony soil. Stoniness can be expressed in volumetric units Rv (L3 L−3 ) or in mass units Rm (MM−1 ). The volume of rock fragments divided by the total volume of a stony soil is the volumetric stoniness: Rv 

Vrf Vb

(17.1)

The volume of the stony soil (total) (V b ) is the sum of rock fragments volume (V ) and the volume of fine earth (V f ) (all the three phases): rf

Vb  Vrf + V

f

(17.2)

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Stoniness, expressed in mass units Rm (MM−1 ), is the ratio of dry rock fragments rf mass (md ) and mass of dry stony soil (together with rock fragments) (mbd ): rf

Rm 

md m bd

(17.3)

The relationship between Rv and Rm is (Ravina and Magier 1984): Rv  Rm

rf ρb

ρbb rf

ρb

(17.4)

Where ρ bb is the bulk density of the stony soil (rock fragments and fine earth), and is the rock fragments bulk density (ML−3 ). Equation (17.2) can be written also in the form: Vb  f

m bd Rm m bd (1 − Rm )m bd  + b rf f ρb ρb ρb

(17.5)

where ρ b is the fine-earth bulk density. Soil stoniness can be measured directly in the field by measurement of rock fragments volume in a particular soil volume. This method is quite laborious, but the most accurate. Detailed texture analysis of smaller gravels may be performed in the laboratory on disturbed stony soil samples. Soil stoniness can be estimated also by visual estimation of the stones and boulders in the soil-profile section, but this method is suitable only for large rock fragments as it tends to underestimate the content of small gravels. An alternative non-destructive method was presented by Viro (1952) (called the “rod penetration method” or “Finnish method”) for determining stone and boulder content in the topsoil (Stendahl et al. 2009). Contemporary indirect methods of stoniness estimation use the relationship between the electrical resistivity of a soil and the soil stoniness. The resistivity of fine earth is a few orders smaller than the resistivity of rock fragments. The soil resistivity increases with stoniness increase. The application of this method depends on actual soil-water content and the particular type of rock fragments. Rock fragments containing iron are of very low resistivity (Tetegan et al. 2011), therefore the calibration of measurement method is necessary. New methods of stoniness measurement are less time consuming, like classical methods based on digging a hole of REV dimensions, separating rock fragments and fine earth and measurement of their volumes, but they are not so reliable. Therefore, the results of measurements need to be calibrated. The classical method of stoniness estimation is still recommended for use.

17.6 Physical Characteristics of Stony Soils

269

17.6.2 Stony Soils’ Bulk Density The bulk density of a stony soil can be expressed as the sum of the bulk densities of fine earth and rock fragments: f

rf

ρbb  (1 − Rv )ρb + Rv ρb

(17.6)

The fine-earth bulk density of some stony soils can be lower than it would be in a non-stony soil. The reason is the larger porosity of fine earth as a part of stony soil (treated later).

17.6.3 Stony Soils’ Porosity Stony-soil porosity is the sum of fine earth porosity and porosity of rock fragments, according to their partial volumes in the stony soil: P b  (1 − Rv )P f + Rv P r f

(17.7)

where Pb is the stony-soil porosity, and Pf is the fine earth porosity; Prf is the rock fragments porosity. Stony soils can eventually contain specific type of pores, called lacunar pores, voids along the soil/stone interface. Fiés et al. (2002) proposed their definition. The term “lacunar pores” is rooted in the Latin word “lacuna” which means “free space, or lack of something”. Lacunar pores are located at the interfaces of rock fragments and fine earth of (mostly) heavy soils with high content of clay; they develop by shrinking in the process of soil drying. They also can be found if the space between rock fragments and fine earth is not filled with fine earth completely (Fiés et al. 2002; Poesen and Lavee 1994). The prediction of the presence of lacunar pores in a stony soil therefore depends to a large extent on the soil-matrix texture and on the process of soil formation. They are effective in relatively dry, heavy soils and can substantially influence stony soil properties like hydraulic conductivity and water retention. Concerning porosity, not only overall porosity is important information, but also the distribution of pores of various sizes, tortuosity of pores and connectivity of pores. In the case of lacunar pores, at smaller stoniness they can occur in a stony soil, but do not have to be active if they are not connected. Another situation is when lacunar pores are connected and they can create the continuous water pathways (Fig. 17.2) enhancing preferential flow rate (Zhou et al. 2009). An example of visualization of lacunar pores by the dye tracers’ technique can be found in Capuliak et al. (2010). Visualization and quantification of lacunar pores in a particular stony soil and their impact on hydraulic conductivity and water retention of stony soils are still not well understood, hence this area of research is still challenging.

270

17 Stony Soils

Fig. 17.2 Lacunar pores and their possible interconnection that can enhance water flow

17.7 Hydrophysical Characteristics of Stony Soils Basic hydrophysical characteristics of stony soils include: soil-water content, soilwater retention curve and hydraulic conductivity.

17.7.1 Stony Soils’ Volumetric Water Content The volumetric soil-water content of a stony soil is the ratio of the bulk-water volume in the total volume of the stony soil: f

θb  f

rf

Vwb Vw + Vw  b V Vb

(17.8)

rf

where V w , V w , V bw are volumes of water in the fine earth fraction, in the rock fragments and in the whole stony soil; V b is the total stony soil volume. rf The volume of water in rock fragments V w is relatively small compared to the rf water capacity of fine earth. Therefore V w is usually neglected.

17.7.2 Stony Soils’ Water Retention Water retention of stony soil (the ability to hold water) is the sum of the water retention of rock fragments and the water retention of fine earth. The water retention of rock fragments depends mostly on their geological origins and on the degree of skeleton weathering (Poesen and Lavee 1994; Brouwer and Anderson 2000). Rock fragments’ water retention is usually small compared to that of fine-earth fractions.

17.7 Hydrophysical Characteristics of Stony Soils

271

The amount of rock fragments present in a stony soil usually determines its overall water retention capacity of the soil. Some types of rock fragments are of high porosity, like, e.g. weathered granite, whose porosity can reach 36% (Rouxel et al. 2012) or volcanic tuff with a porosity of 27% (Šajgalík et al. 1986). The retention curve of rock fragments can be measured similarly as is routinely done for fine earth. A rock fragment saturated with water is placed on a conductive porous material (fine sand) to establish good hydraulic contact between the rock fragment and a pressure-chamber apparatus (Parajuli et al. 2017; Cousin et al. 2003; Novák and Šurda 2010). The effective volumetric stony-soil water content can be calculated knowing the soil-water contents of fine earth (Bouwer and Rice 1984): θ b  (1 − Rv )θ f

(17.9)

Equation (17.9) can be applied to a stony soil with very low or zero water retention of rock fragments. The effective soil-water content of a stony soil θ b represents the relative volume of water held in the fine earth at a particular soil-water matric potential of the unit volume of the stony soil. In the case of significant water-retention capacity of rock fragments in the stony soil, it is necessary to estimate the rock fragments’ soil-water content corresponding to particular soil-water matric potentials (soil-water retention curve) and then, the soil-water content of the stony soil can be expressed as: θ b  (1 − Rv )θ f + Rv θ r f

(17.10)

where θ rf is the volumetric soil-water content of rock fragments corresponding to the particular soil-water matric potential. Retention curves of rock fragments differ from the retention curves of fineearth fractions. Rock fragments’ retention capacity is usually small and with small change due to change of matric potential. Measurement methodologies (pressure plate extractor, pressure membrane cell, sand tank) of soil-water retention curves of fine earth and stony soils are available in regulation EN ISO 11274:2014. New devices have appeared enabling the measurement of soil-water retention curves of stony soils. The example of the application of simplified evaporation method (modified former Wind (1968) evaporation method) to stony soils in conjunction with a pressure plate and the dew point potentiometer method were published by Parajuli et al. 2017. The authors studied the influence of rock fragments´ water retentions (limestone, sandstone, and pumice) on the whole stony soil mixtures. They suggested the Durner (1994) dual-porosity model for stony-soil water-retention curves in case of considerable rock fragments retention.

272

17 Stony Soils

17.7.3 Hydraulic Conductivity of Stony Soils There are only a few empirical equations for estimating the saturated hydraulic conductivity of stony soils and most are derived from laboratory experiments. The methods of calculation involve easily measureable characteristics of fine earth and stoniness. Ravina and Magier (1984) introduced the term relative saturated hydraulic conductivity of stony soil as the ratio K b /K f . Kr s 

Kb  1 − Rv Kf

(17.11)

where K rs is the relative saturated hydraulic conductivity of the stony soil (–); K b is the effective saturated hydraulic conductivity of the stony soil (LT−1 ); K f is the saturated hydraulic conductivity of the fine earth in the stony soil (LT−1 ); and Rv is the relative volume of rock fragments (stoniness) (L3 L−3 ). Equation (17.11) assumes zero (or negligible) water-retention capacity of rock fragments. The authors recommend applying it to sandy and non-structural soils (for soils not susceptible to deformation). Brakensiek et al. (1986) expressed K rs by the relative mass of rock fragments: Kr s 

Kb  1 − Rm Kf

(17.12)

where Rm is the relative mass of rock fragments (MM−1 ). Bouwer and Rice (1984) expressed K rs as: Kr s 

Kb θb  sf f K θs

(17.13)

where θ bs is the effective saturated soil-water content of a stony soil (equals to the f stony-soil total porosity) and θ s is the saturated volumetric soil-water content of the stony soil’s fine earth (equal to the fine-earth fraction porosity). All equations demonstrate the linear decrease of saturated hydraulic conductivity of stony soil with the stoniness increase (Fig. 17.3). The function K b (Rv ) was calculated using Eq. (17.11) for sandy-loam soil with spherical rock fragments of diameter d  1 cm. The presence of rock fragments affects the hydraulic conductivity of a soil in two main ways. Firstly, rock fragments reduce the effective cross-sectional area through which water flows, and codetermines that an increase in stoniness results in greater curvatures of flow paths (larger tortuosity). These phenomena result in lower hydraulic conductivities of stony soils as is shown in Eqs. 17.11–17.13. Secondly, the shrink-swell phenomena may create, as was mentioned previously, lacunar pores, which can counterbalance the effect of increasing tortuosity. This effect is neglected in Eqs. 17.11–17.13. Which effect prevails, it depends on many factors, among them

17.7 Hydrophysical Characteristics of Stony Soils

273

Fig. 17.3 Effective saturated hydraulic conductivity of the stony soil K b and the relative rock-fragments’ content (stoniness) Rv . (K f is the saturated hydraulic conductivity of the stony-soil fine-earth fraction)

on the amount and dimensions of rock fragments and on the texture and structure of the fine earth. The effective saturated hydraulic conductivity of the stony soil can be estimated from results of field measurements of cumulative infiltration, using single- or doublering infiltrometer of appropriate diameter, corresponding to the rock fragments’ dimension. This method can be applied to soils of low or medium stoniness. The installation of an infiltrometer into stony soils of high stoniness is problematic because the structure of soil in the infiltrating area is changed because of ring installation. Another possibility is to measure saturated hydraulic conductivity of fine earth on undisturbed or disturbed soil samples under laboratory conditions.

17.7.4 Effective Hydrophysical Characteristics of Stony Soils The effective hydrophysical functions (hydraulic conductivity and soil-water retention curves) and their parameters are the main inputs in numerical waterflow modelling expressing soil properties in the vadose zone. As was mentioned previously, their measurements on large REVs are often not possible. Therefore, alternative ways are needed. One way is to measure the characteristics of both components separately (a fine-earth fraction and rock fragments) and then to combine them or, in a case of small or zero retention capacity of rock fragments, to measure the parameters of the fine earth and stoniness only. This approach was adopted for the estimation of stony-soil water retention by Bouwer and Rice (1984), for the estimation of saturated hydraulic conductivity by Brakensiek et al. (1986), Peck and Watson (1979) and Ravina and Magier (1984) (mentioned in previous subchapters), and for the unsaturated hydraulic conductivity by Hlaváˇciková and Novák (2014). Another approach is to estimate the hydrophysical characteristics of stony soils by numerical modelling. Novák et al. (2011) were the first who used the HYDRUS2D model in the numerical Darcy experiment. The same approach was later applied

274

17 Stony Soils

Table 17.1 Van Genuchten’s parameters of the effective SWRCs and saturated hydraulic conductivities of stony soils K b of various stoniness Rv , calculated from the fine-earth SWRC (Rv  0 cm3 ˇ cm−3 , representative soil sample from depth horizon 30–35 cm) and known Rv . Site Cervenec, the Western Tatra Mts. (1420 m a.s.l.) Slovakia Rv (cm3 cm−3 )

0

0.1

0.2

0.3

0.4

0.5

θ br

(cm3

cm−3 )

0.05

0.045

0.04

0.035

0.03

0.025

θ bs

(cm3

cm−3 )

0.630*

0.567

0.504

0.441

0.378

0.315

α

(cm−1 )

0.211

0.211

0.211

0.211

0.211

0.211

n (–)

1.147

1.147

1.147

1.147

1.147

1.147

K b (cm h−1 )

36.00*

32.40

28.80

25.20

21.60

18.00

* Measured

value; upper index “b” is valid for effective characteristics

by Hlaváˇciková et al. (2016) and Beckers et al. (2016). These studies examined the role of stoniness, the size of RF (Novák et al. 2011) and the shape, distribution and position of RF in the stony soil on saturated (Hlaváˇciková et al. 2016) and unsaturated hydraulic conductivities (Beckers et al. 2016). Another prospective method for the estimation of the hydrophysical properties of stony soils is the evaporation method used by Beckers et al. (2016) and Parajuli et al. (2017). However, this method is limited to relatively small stony-soil samples with only small gravels (not applicable to forest and mountain soils with large stones and boulders) and to a small range of soil-water potential. It is still challenging to identify other easy applicable methods. An illustration of the modification of stony-soil hydrophysical functions caused by the stoniness Figure 17.4 is a soil-water retention curve (SWRC) of fine earth (Rv = 0 cm3 cm−3 ) ˇ from the mountainous, forested site Cervenec (1420 m a.s.l.) in the Western Tatra Mountains, Slovakia. Also shown are hypothetical SWRCs of stony soils of various stoniness. SWRCs parameters, like residual volumetric soil-water content and saturated soil-water content (θ br , θ bs ) and saturated hydraulic conductivity (K b ) were calculated using Eqs. (17.9) and (17.11). They are collected in Table (17.1) together with van Genuchten (1980) SWRC parameters α and n. Figure 17.4 demonstrates the significant decrease of retention capacity of the stony soil with an increase in rock fragments content. If there is a hypothetical stony soil with the soil depth of 1 m containing 50% rock fragments by volume regularly distributed in the soil with tight contact between rock fragments and the fine earth fraction, the former values of its water retention and saturated hydraulic conductivity is decreased by half. This certainly influences hydraulic processes like infiltration, runoff formation, water movement and evaporation. This example is hypothetical and very simplified, but it indicates the importance of RFs’ effect on stony soils’ properties. The estimation of the hydrophysical functions of stony soils is very complex problem; several aspects are still not well-understood and cannot be well-assessed. Therefore, estimation of the hydrophysical functions

17.7 Hydrophysical Characteristics of Stony Soils

275

Fig. 17.4 The soil-water retention curve of the fine earth, forest Cambisol, ˇ sampled at the site Cervenec (1420 m a.s.l., the Western Tatras, Slovakia) and calculated soil-water retention curves of stony soils of various stoniness Rv (cm3 cm−3 )

of stony soils and modelling of water flow in stony soils still requires multiple simplifications and approximations.

17.7.5 Soil-Water Content of Stony Soils Measurement Direct, destructive method of soil-water content measurement can be applied also for stony soils with low stoniness and small dimensions of rock fragments. Knowing the stoniness and the volume of water in a soil sample, the bulk-water content or water content of the soil matrix can be expressed (see Eq. (17.9)). Indirect methods of continual stony-soil water-content measurements are described in Chap. 5 (based on TDR or FDR technique). The problems associated with their use are: stony-soil heterogeneity and difficulties of measuring-device installation. Measured data should be interpreted accounting for the rock fragments’ distribution around the device. Calibration curves have to be corrected depending on the soil stoniness and dielectric properties of rock fragments (Coppola et al. 2013).

17.8 Water Flow in Stony Soils The shape and size of rock fragments, their distribution in a soil profile and rock fragments’ surface properties (smooth, rough) together with other soil characteristics are factors that influence water transport in stony soils.

276

17 Stony Soils

Rock fragments on the soil surface usually contribute to the increase of the infiltration rate by preventing peptization by rain drops’ impact followed by crusting. But, rock fragments firmly embedded in the surface layer of the fine earth act as an infiltration barrier that decreases the infiltration rate, which can lead to the increase of surface runoff (Cousin et al. 2003). The presence of rock fragments usually decreases the evaporation rate. The properties and configuration of rock fragments in the soil profile influence the rate of infiltration. Large rock fragments in a stony soil tend to increase soil tortuosity and thus decrease the soil’s hydraulic conductivity and the rate of water flow. The presence of RF reduces the volume of fine earth that would be available for water flow and thus reduces water retention and the effective cross section of the stony soil. A low-permeable compact layer of rock material below the stony-soil profile is typical for stony soils, and it limits deeper water percolation. The relatively impermeable bedrock below the stony-soil profile allows the formation of the temporary saturated zone of the soil at the soil/bedrock interface. Stony soils located on steep slopes increase the risk of the formation of rapid subsurface water flow, contributing to the overall runoff. They also increase the risk of landslides.

17.8.1 Modelling of Water Flow in Stony Soils The dynamics of the stony-soil water content (or water storage) can be estimated by measurement of the volumetric soil water content or by soil-water flow modelling at various soil profile depths. As was mentioned, the installation of measuring devices in the field is not a simple procedure; interpretation of the measured data is not easy too. For modelling of water flow in a relatively homogeneous soil, the necessary input data characterizing the soil profile at various depths are: saturated hydraulic conductivities, soil-water retention curves and appropriate initial and boundary conditions (see more about modelling in Chap. 21). For the modelling of water flow in stony soils, the input data are different: the stoniness, the saturated hydraulic conductivity of the fine earth and the soil-water retention curve of the fine earth. These data enable calculating the effective hydrophysical characteristics of stony soils which can be used in the deterministic simulation models of variably saturated soils using the governing Richards equation. A similar principle was used by, e.g. Wegehenkel et al. 2017, Coppola et al. 2013, and Novák and Kˇnava 2012. The soil profile can be divided into several relatively homogeneous soil layers with known characteristics (stoniness, saturated hydraulic conductivity and soil-water retention curve). This approach is suitable for calculating changes of stony-soil water storage in various soil horizons and water fluxes at the boundaries of the whole system.

17.8 Water Flow in Stony Soils

277

17.8.2 More Complex Modelling Approach to Water Flow in Stony Soils A two-domain approach can also be applied to the modelling of the soil-water flow in stony soils. In the dual permeability model (see Chap. 21), the faster flow of water (preferential flow) can be considered in the faster domain and slow water flow in the soil matrix can be considered in the slow domain. Flow parameters in the slow domain, which include the soil matrix together with rock fragments, can be estimated as in the one-domain model. To estimate parameters of flow in the fast domain (hydraulic conductivity of fast domain and parameters characterizing interaction between slow and fast domain), it is necessary to perform more complex field measurements, e.g. by indicators. Rock fragments of large porosity can represent the special domain, which is wetted and dried but is not involved in the soil-water flow. An example of the application of this approach is shown in Ma and Shao (2008) (see dual-porosity model in Chap. 21).

17.8.3 An Example of Measured Characteristics of the Stony Soil and Estimation of Input Data for a Deterministic Simulation Model (a Case Study) This case study aims to show the influence of the presence of RF in the stony-soil profile on the hydrophysical properties of the soil and the effect of RF on water-flow modelling in a stony soil with an emphasis on water storage and the modification of potential outflow formation. ˇ The chosen study site was Cervenec that is a part of the Jalovecký creek catchment, the westernmost catchment of the Western Tatra Mts. (Slovakia). This mountain range is part of the highest peaks of the Carpathians. The study site is located at the height of 1420 m a.s.l, with 100-year old spruce trees. The soil is shallow, and the forest is located on weathered crystalline bedrock. The soil type is Cambisol. The prevailing rock fragments are sharp gneisses and paragneisses with irregular shapes, dimensions of 5–10 cm. The soil is slightly stony at the depth of 0–40 cm, moderately stony at the depth of 40–60 and highly stony at 60–95 cm, according to Societas pedologica slovaca (2014) classification (Fig. 17.5). The stoniness varies widely, as was proved by geophysical measurements, and the soil structure is influenced not only by the presence of rock fragments, but also by the presence of spruce forest roots. The average slope angle at the site is 24–30%. The stoniness of the soil was evaluated by measurement of rock fragments volume directly in the field—by gradual digging and excavating of soil layers up to 95-cm deep in a soil pit (down to the soil/weathered bedrock interface). Samples of fine earth and stones were collected to measure the saturated hydraulic conductivity of fine earth, soil-water retention curves of fine earth and maximum retention

278

17 Stony Soils

Fig. 17.5 Stoniness of individual soil layers of forest Cambisol soil from the ˇ site Cervenec (1420 m a.s.l.), the Western Tatras, Slovakia

Table 17.2 Van Genuchten’s parameters of SWRC (θ br , θ bs , α, n), hydraulic conductivity (K b ) ˇ and stoniness (Rv ) of individual soil layers. Site Cervenec, the Western Tatra Mts. (1420 m a.s.l.) Slovakia Soil depth 0–10 cm 10–40 cm 40–60 cm 60–85 cm 85–95 cm (cm) θ br (cm3 cm−3 ) 0.05 θ bs (cm3 α

cm−3 )

(cm−1 )

0.05

0.05

0.05

0.05

0.600*

0.597*

0.487

0.323

0.241

0.04319

0.25982

0.25982

0.25982

0.25982

n (–)

1.5038

1.13859

1.13859

1.13859

1.13859

K b (cm h−1 )

210*

45*

45

45

45

(cm3

0.00

0.00

0.20

0.50

0.65

Rv cm−3 )

* Measured

value; upper index “b” is valid for effective characteristics

capacity of rock fragments. Single-ring infiltration measurements were performed to estimate the saturated hydraulic conductivity of the soil layer with low stoniness (down to 40 cm). The simulation model of unimodal porosity HYDRUS-1D (Šim˚unek et al. 2008, 2016) was used. To estimate stony-soil hydrophysical characteristics as basic input parameters in modelling, measured stoniness distributions in the soil profile were used (Fig. 17.5), as well as measured properties of the fine earth of the stony soil (Table 17.2, layers 0–10 cm and 10–40 cm). Soil-profile stoniness at the depth interval of 0–40 cm was very low (max. value 0.05), therefore, it was not involved in the modelling. The upper-forest floor soil horizon (L, Of , and Oh ) was quantitatively characterized by the analytical equation of the SWRC according to van Genuchten; data were taken from the average values of soil-water contents corresponding to particular soil-water matric potentials measured on soil samples taken from the soil layer of 0–10 cm. The SWRC of soil horizon at 10–40 cm was determined similarly, using undisturbed fine-earth soil samples.

17.8 Water Flow in Stony Soils

279

Fig. 17.6 Effective soil-water retention curves (SWRC) of the forest-soil individual layers from the ˇ site Cervenec (1420 m a.s.l.), the Western Tatras, Slovakia. SWRC of sigmoidal shape represents the upper, organic soil horizon

The saturated hydraulic conductivity of the forest-floor horizon (depth interval 0–10 cm) was estimated in the laboratory by measurement of undisturbed fine-earth soil samples; the average values are listed in Table (17.2). The saturated hydraulic conductivities of soil layers 10–40 cm and 40–60 cm were estimated from the results of cumulative infiltration measurements performed by a single-ring infiltrometer of diameter 28 cm. K b values of deeper soil horizons were not measured, and the same value as in the above horizon was assumed, or it can be corrected according to actual stoniness by the Eq. (17.11). The influence of such correction in this case was assumed to not be significant. The potential influence of lacunar pores may counteract against K b decreasing, therefore a constant value of K b was used for deeper soil layers. The influence of stoniness on water-retention reduction was assumed to be more significant. This effect was involved in the SWRC parameters for soil depths at 40–95 cm, which were estimated by Eq. (17.10), and the maximum retention capacity of rock fragments (0.05 of volumetric soil-water content) was involved. This method is approximate because the water dynamics in fine earth and in rock fragments are different. Soil-water retention curves of individual soil layers are in Fig. 17.6, and their parameters are gathered in Table 17.2. The proposed parameters in Table 17.2 can be used in a standard modelling procedure in stony soils using one of the soil-water flow deterministic models. Results of stony soil water flow modelling The results of soil-water flow modelling in the upper, approximately one-meter-thick soil layer, show that soil-water storage of the stony soil can be lower by 16–31% compared to non-stony soil. From the point of view of soil-water balance, this difference is significant. It was also shown that soil-water dynamics in the stony soil is different than that in a non-stony one. This difference was found especially during

280

17 Stony Soils

wet (rainy) periods of the vegetation season. The stony soil was wetted by water faster; the runoff from the soil-profile bottom was formed faster, and its maximum was a little higher compared to non-stony soil. Runoff formation in small catchments is usually determined by rain intensity and its duration, initial soil-water content, soil hydraulic conductivity, bottom rock layers properties (permeable/impermeable) and plant-canopy properties. Rock fragments are only one of the many landscape components that affect runoff formation. The important factors are the properties of individual soil horizons, the configuration of rock fragments and properties of the bottom, usually less permeable, layer. The presence of rock fragments in the upper-soil layer and stoniness above 25% of the soil volume can significantly influence surface-runoff formation (especially if it is firmly embedded in the soil surface). On the contrary, the larger stoniness of the bottom soil layers will certainly affect subsurface-runoff formation. Further research concerning structured stony soils and the verification of available methodology are the next tasks of soil hydrology. The effects of rock fragments on the hydrological cycle of forested or agricultural stony soils are crucial, but yet often neglected. An overview of up-to-date literature concerning the influence of rock fragments on hydrological processes can be found in Zhang et al. (2016).

References Baetens J (2007) The effect of rock fragments on hydrophysical properties of a small watershed in North Chile. Dissertation, Ghent University, Belgium Baker FG, Bouma J (1976) Variability of hydraulic conductivity in two subsurface horizons of two silt loam soils. Soil Sci Am J 40:219–222 Beckers E, Pichault M, Pansak W, Degré A, Garré S (2016) Characterization of stony soils’ hydraulic conductivity using laboratory and numerical experiments. Soil 2:421–431 Bouwer H, Rice RC (1984) Hydraulic properties of stony vadose zones. Ground Water 22:696–705 Brakensiek DL, Rawls WJ, Stephenson GR (1986) Determining the saturated hydraulic conductivity of a soil containing rock fragments. Soil Sci Soc Am J 50:834–835 Brouwer J, Anderson H (2000) Water holding capacity of ironstone gravel in typical Phlintoxeralf in Southeast Australia. Soil Sci Soc Am J 64:1603–1608 Buchter B, Hinz C, Flühler H (1994) Sample size for determination of coarse fragment content in a stony soil. Geoderma 63:265–275 Buchter B, Leuenberger J, Richard F, Flühler H, Selim HM (1987) Preparation of large cross sections from stony soils. Soil Sci Soc Am J 51:494–495 Capuliak J, Pichler V, Flühler H, Pichlerová M, Homolák M (2010) Beech forest density control on the dominant water flow types in andic soils. Vadose Zone J 9:747–756 Cools N, De Vos B (2013) Forest soil: characterization, sampling, physical, and chemical analyses (Chap 15). Dev Env Sci 12:267–300 Coppola A, Dragonetti G, Comegna A, Lamaddalena N, Caushi B, Haikal MA, Basile A (2013) Measuring and modeling water content in stony soils. Soil Tillage Res 128:9–22 Cousin I, Nicollaud B, Coutadeur C (2003) Influence of rock fragments on the water retention sand and water percolation in a calcareous soil. CATENA 53:97–114 Durner W (1994) Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resour Res 30:211–223

References

281

EN ISO 11274:2014 Soil quality—determination of the water-retention characteristic—laboratory methods (ISO 11274:1998 + Cor. 1:2009). European Committee for Standardization, CENCENELEC Management Centre, Brussels Fiés JC, De Louvigny N, Chanzy A (2002) The role of stones in soil water retention. Eur J Soil Sci 53:95–404 Garcia-Gaines RA, Frankenstein S (2015) USCS and the USDA soil classification system: development of a mapping scheme. Army Engineer Research and Development Center, Vicksburg, U.S., p 46 Gömöryová E, Gregor J, Pichler V, Gömöry D (2006) Spatial patterns of soil microbial characteristics and soil moisture in a natural beech forest. Biologia 61(suppl 19):S329–S333 Hlaváˇciková H, Novák V, Šim˚unek J (2016) The effects of rock fragment shapes and positions on modeled hydraulic conductivities of stony soils. Geoderma 281:39–48 Hlaváˇciková H, Novák V (2014) A relatively simple scaling method for describing the unsaturated hydraulic functions of stony soils. J Plant Nutr Soil Sci 177:560–565 IUSS Working Group WRB (2015) World reference base for soil resources 2014, update 2015 international soil classification system for naming soils and creating legends for soil maps. World Soil Resources Reports No. 106. FAO, Rome Kutílek M, Nielsen DR (1994) Soil hydrology. Catena Verlag, Cremlingen–Destedt, p 370 Lichner L (1994) Contribution to the saturated hydraulic conductivity of soils with macropores measurement. J Hydrol Hydromech 42:421–430 (In Slovak with English abstract) Ma DH, Shao M (2008) Simulating infiltration into stony soils with a dual-porosity model. Eur J Soil Sci 59:950–959 Novák V, Kˇnava K (2012) The influence of stoniness and canopy properties on soil water content distribution: simulation of water movement in forest stony soil. Eur J Forest Res 131:1727–1735 Novák V, Šurda P (2010) Water retention of granite rock fragments of stony soils of High Tatra Mts. J Hydrol Hydromech 58:181–187 (In Slovak with English abstract) Novák V, Kˇnava K, Šim˚unek J (2011) Determining the influence of stones on hydraulic conductivity of saturated soils using numerical method. Geoderma 161:177–181 Parajuli K, Sadeghi M, Jones SB (2017) A binary mixing model for characterizing stony-soil water retention. Agr For Met 244–245:1–8 Peck AJ, Watson JD (1979) Hydraulic conductivity and flow in non-uniform soil. Workshop on soil physics and soil heterogeneity. CSIRO Division of Environmental Mechanics, Canberra, Australia Poesen J, Lavee H (1994) Rock fragments in top soils: significance and processes. Catena 23:1–28 Ravina I, Magier J (1984) Hydraulic conductivity and water retention of clay soils containing coarse fragments. Soil Sci Soc Am J 48:736–740 Rouxel M, Ruiz L, Molénat J, Hamon Y, Chirié G, Michot D (2012) Experimental determination of hydrodynamic properties of weathered granite. Vadose Zone J. https://doi.org/10.2136/vzj2011. 0076 ˇ Šajgalík J, Cabalová D, Schütznerová V, Šamalíková M, Zeman O (1986) Geology. Alfa, Bratislava and SNTL, Prague, p 563 (In Slovak) Šály R (1978) Soil—the base of forest production. Príroda, Bratislava, p 235 (In Slovak) Šim˚unek J, van Genuchten MTh, Šejna M (2016) Recent developments and applications of the HYDRUS computer software packages. Vadose Zone J. https://doi.org/10.2136/vzj2016.04.0033 Šim˚unek J, Šejna M, Saito H, Sakai M, van Genuchten MTh (2008) The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media, version 4.0, Hydrus Series 3. Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA Societas pedologica slovaca (2014) Morphogenetic classification system of Slovak soils. Basal reference taxonomy. Second revised edition Bratislava: NPPC—VÚPOP Bratislava (In Slovak with English abstract) Stendahl J, Lundin L, Nilsson T (2009) The stone and boulder content of Swedish forest soils. Catena 77:285–291

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Tetegan M, Nicoullaud B, Baize D, Bouthier A, Cousin I (2011) The contribution of rock fragments to the available water content of stony soils: proposition of new pedotransfer functions. Geoderma 165:40–49 USDA (2017) Determination of grain size distribution. http://www.nrcs.usda.gov/wps/portal/nrcs/ detail/soils/survey/office/ssr10/tr/?cid=nrcs144p2_074845#item1b. Accessed 20 June 2017 van Genuchten MTh (1980) A closed form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44:892–898 Verbist K, Baetens J, Cornelis WM, Gabriels D, Torres C, Soto G (2009) Hydraulic conductivity as influenced by stoniness in degraded drylands of Chile. Soil Sci Soc Am J 73:471–484 Viro PJ (1952) On the determination of stoniness. Communicationes Instituti Forestalis Fenniae 40:23 Wegehenkel M, Wagner A, Amoriello T, Fleck S, Messenburg H (2017) Impact of stoniness correction of soil hydraulic parameters on water balance simulations of forest plots. J Plant Nutr Soil Sci 180:71–86 Wind GP (1968) Capillary conductivity data estimated by a simple method. In: Rijtema PE, Wassink H (eds) Water in the unsaturated zone, proceedings of the Wageningen symposium, June 1968. Int Assoc Sci Hydrol Publ (IASH), Gentbrugge, The Netherlands and UNESCO, Paris, pp 181–191 Zhang Y, Zhang M, Niu J, Li H, Xiao R, Zheng H, Bech J (2016) Rock fragments and soil hydrological processes: significance and progress. CATENA 147:153–166 Zhou B, Shao M, Shao H (2009) Effects of rock fragments on water movement and solute transport in a Loess Plateau soil. CR Geoscience 341:462–472

Chapter 18

Water Repellent Soils

Abstract Capillary forces in unsaturated soils depend on the surface tension of water, the dimensions of soil pores and on the contact angle of the interfaces of the solid phase of soil (soil matrix) and liquid water. The contact angle (angle of wetting) is assumed to be zero, but in real soils is higher, depending on the properties of the thin surface layer (organic) covering soil particles. There are some classes of soils with limited affinity for soil water. Soils are classified as wettable with contact angles ϕ < 90°; water-repellent soils are characterized by contact angles of ϕ ≥ 90°. The occurrence of water-repellent soils is rare and usually their repellency is temporary (dry soils). This chapter defines the characteristics of soil-water repellency: contact angle, severity of water repellency, persistence of soil water repellency and index of water repellency. It also presents the hydrophobic compounds in soils and the influence of soil-water repellency on hydrological processes in soil (infiltration, preferential flow, evaporation).

18.1 Water Repellency of Soils and Its Identification Soils are hydrophilic in general, and their contact angles are close to zero. Hydrophobic soils have a contact angle greater than 90°, water menisci in soil pores are of convex shape, and capillary depression is observed. Water does not infiltrate into hydrophobic soil because capillary forces act upward, and the only downward force is the gravitational force. Growth of plants in such soils is limited if not impossible. The majority of soils are not ideally hydrophilic, but subcritically water repellent. It means that the contact angle is higher than zero, but lower than 90°. Subcritical water repellency refers to soil in which water uptake appears to occur readily, yet is impeded to some extent by the presence of hydrophobic surface films (Hallett et al. 2001). Subcritical water repellency is a common feature of many soils and it can influence the soil-water dynamics. The significant influence of subcritical water repellency on soil-water movement is rare, but this phenomenon deserves to be studied.

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_18

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The type of water affinity to the solid phase of soil (hydrophilicity or water repellency) can be identified simply by pouring water onto the soil surface. The majority of soils are not ideally hydrophilic, the contact angle between soil and a drop of water is usually higher than zero, but a zero contact angle is usually a good approximation. To classify such soils from the view of the interactions of the solid phase of soil and water, it is necessary to introduce a classification scheme. The water repellency of soil (WR) is not the best term to identify interactions of the solid phase of soil and liquid water because no soil is completely water repellent, and the solid phase of the soil attracts water but with varying forces that are characterized by the varying wetting of the solid phase of soil by water. Adamson (1990) defined water repellency (or hydrophilicity) of solids by the contact angle of the solid phase of soil with liquid water. The contact angle of wettable soil is less than 90° (ϕ < 90°), and water-repellent soil is characterized by a contact angle ϕ ≥ 90°. Ideally, the wettable solid phase of soil is characterized by a contact angle ϕ  0°, and ideally water repellent soil is at ϕ  180° (Chap. 4.1). Analysing the capillary rise (or infiltration) of liquid water in soils, a contact angle ϕ  0° is usually assumed. This approximation is acceptable in the majority of cases. In reality, the ideal hydrophilic, nor the ideal water-repellent soil does not exist, but the soil contact angle of wet solid phase of soil is close to an ideally hydrophilic surface. Another definition of water repellency (or its opposite state of hydrophilicity) is based on the evaluation of the difference between the free energy of the solid phase of soil in comparison to the liquid-water surface tension (Doerr et al. 2000). According this classification, the solid phases (soils) are hydrophilic if the free energy of the solid phase of soil is greater than 72.75 mN m−1 , and the opposite case, if the free energy of solid phase of soil is lower than 72.75 mN m−1 , the solid surface is water repellent. This definition incorporates the reasoning: if the cohesion of liquid water molecules is greater than the adhesion between water and the solid phase of soil, water does not wet the solid phase, and the soil is water repellent.

18.2 Characteristics of Soil-Water Repellency 18.2.1 Severity of Soil-Water Repellency The severity of soil-water repellency quantitatively expresses the affinity between the liquid and solid phases; it can be expressed by the contact angle ϕ, the height of capillary rise in the soil hk and the so-called “bubbling pressure”—the pressure head of soil water corresponding to the pressure of the entrance of air into the soil hb . The measurement of the contact angle of the water drop on the soil surface is not an easy task since the soil surface is usually not flat. Therefore, to measure the contact angle, thin layers of dry soil suspension on the solid (usually glass) plates are used, and the contact angles are measured.

18.2 Characteristics of Soil-Water Repellency

285

The MED (molarity of an ethanol droplet) test is the measurement methods. The MED test is an indirect measure of the surface tension of the soil surface and indicates how strongly a water drop is repelled by a soil at the time of application (i.e. how strongly it will ‘ball up’). The MED test measures the molarity of an aqueous ethanol droplet required for soil infiltration within 10 s (Wallis and Horne 1992) or 3 s (Doerr 1998). The indirect method of contact-angle estimation is the measurement of the capillary height of water in soil or the estimation of bubbling pressure. Assuming the soil pores can be approximated by the bundle of capillary tubes of radius r, then the relationship between the height of capillary rise of water hk , capillary tube radius r and contact angle ϕ can be expressed by the equation: hk 

2σw cos ϕ ρw gr

(18.1)

where σ w , g, ϕ are surface tension of water, acceleration of gravity and contact angle of water. Water can enter the porous space (Eq. 18.1) when the contact angle of the system capillary tube/water ϕ is in the range 0° ≤ ϕ < 90°, and the cosine of the contact angle is positive. Knowing the height of capillary rise, the contact angle can be calculated by Eq. (18.1). A contact angle greater than ϕ  90° identifies soil as water repellent, and water cannot enter the porous space spontaneously; a pressure corresponding to the pressure head hb , (bubbling pressure head) cause the water to enter into the soil pores.

18.2.2 Persistence of Soil-Water Repellency The persistence of soil-water repellency (SWR) is usually/commonly estimated by the WDPT (water- drop penetration time) test. The WDPT test measures how long the hydrophobicity persists on a porous surface. It corresponds to the hydrological implications of hydrophobicity because the amount of surface runoff is affected by the time required for the infiltration of droplets (Doerr 1998). The WDPT test consists of simply placing water (or distilled water) on the soil surface and recording the time it takes for the water to penetrate the sample (Wallis and Horne 1992). A standard droplet-release height of approximately 10 mm above the soil surface was used to minimize the cratering effect on the soil surface. The following classes of the persistence of WR were distinguished: wettable or non-water-repellent soil (WDPT < 5 s); slightly (WDPT  5–60 s), strongly (WDPT  60–600 s), severely (WDPT  600–3600 s), and extremely persistent (WDPT > 3600 s) water-repellent soil (Bisdom et al. 1993). As a consequence of SWR, the infiltration rate is reduced at the initial stage of the process as shown by the plot of cumulative infiltration (i) versus time (t) that typically exhibits an upwardly concave shape indicative of an initial hydrophobicity

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18 Water Repellent Soils

Fig. 18.1 “Hockey- stick-like” cumulative infiltration of water (i) and the square root of time t (SQRT t) for the water-repellent soil. The water-repellency cessation time (WRCT) was estimated from the point of the intersection of two straight lines, representing the i = f (SQRT t) relationships for hydrophobic and nearly wettable states of the crust (modified from Lichner et al. 2013)

that disappears as infiltration proceeds (Beatty and Smith 2014). The occurrence of the phenomenon is even more evident when i data are plotted as a function of the square root of time. In this case, the “hockey-stick-like” relationship of the cumulative infiltration of water against the square root of time (SQRT t) (Fig. 18.1) makes possible evaluation of the subcritical SWR that is the condition in which infiltration is reduced, but not completely prevented, in the case of severely water-repellent soils. Lichner et al. (2013) used the water-repellency cessation time (WRCT) to assess the persistence of SWR. In their method, the water sorptivity S wh for the water-repellent state of soil was estimated from the slope of i = f (SQRT t) relationship for a short time of infiltration (a straight line, representing the less steep part of the hockey stick). The water sorptivity S ww for the nearly wettable state of soil was estimated from the slope of i = f (SQRT t) relationship for a longer time of infiltration (a straight line, representing the steeper part of the hockey stick). The WRCT was estimated from the point of intersection of two straight lines, representing the relationships i = f (SQRT t) for water-repellent and nearly wettable states of the soil. The WRCT depends on both the WDPT and thickness of the water repellent layer, which has to be penetrated by water. The WRCT relates to the hydrological implications of SWR because the amount of surface runoff depends on the amount of infiltration. The described cumulative infiltration curve is a special case of a water-repellent soil, occurring in specific circumstances only, and to identify such porous media is rare case.

18.2 Characteristics of Soil-Water Repellency

287

18.2.3 The Repellency Index The repellency index is another criterion to express the so-called extent of soilwater repellency. The repellency index (Tillman et al. 1989) can be expressed by the equation: R I  1.95 Se /Sw

(18.2)

The sorptivity to water S w (influenced by soil-water repellency) and sorptivity to 95% ethanol S e (not influenced by soil-water repellency) are commonly measured using the minidisk infiltrometer (Decagon Devices, Inc. Pullman, USA). Sorptivity is a well-defined soil parameter with physical significance and may be easily measured in the laboratory or in the field (Wallis and Horne 1992). The higher the RI is, the higher is the soil water repellency.

18.3 Water Repellent Compounds in Soils The source of compounds that contribute to water repellency can be vegetation, soil microorganisms and soil organic matter. Vegetation Soil-water repellency is frequently associated with the secretions of evergreen trees growing on coarse-textured soils. These trees secrete resins, waxes, and aromatic oils that can increase the contact angle between the interface of solid phase of soil and water. These, together with the litter of plants and secretions of roots bacteria and fungi, can change the soil-solid phase properties and increase the contact angle of the solid phase of soil with water. Microorganisms Soil-water repellency can be associated with some kinds of microscopic fungi, actinomycetes, algaes, mosses and lichens. Microorganisms can increase water repellency of soil, but also clog the pores and decrease the soil’s hydraulic conductivity. Organic compounds in soil The surface tension of humic acid is approximately one third less than the surface tension of water, and therefore the solid phase of soil covered by it can be water repellent. Organic matter usually covers the soil particles with a thin layer of water-repellent compound; or it can occur in soil as discrete fragments. It has been discovered that non-arable soils contain more hydrocarbons that have an increasing effect on water repellency.

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18 Water Repellent Soils

18.4 Water Repellency and Soil-Water Characteristics Soil-water content is the soil characteristic that strongly influences soil-water repellency. Soil-water repellency is usually temporary; it is not a stable soil property, but depends on soil-water content. Wet soil is not water repellent. Numerous measurements have shown that the greatest water repellency was measured in dry soil and that it decreases with increasing soil-water content, and at some critical soil-water content the soil becomes hydrophilic. Soil texture and clay content strongly influence the affinity of water to the solid phase of a soil. It is generally accepted that the persistence of soil-water repellency increases with an increasing an content of soil organic matter and with a decreasing content of clay fraction. Soil temperature. While the influence of current soil temperatures on soil-water repellency was not found to be significant in general, the effects of the high temperatures (mainly during fires) on soil properties of surface and subsurface soil layers is significant (Malkinson and Wittenberg 2011; Inbar et al. 2014; Cawson et al. 2016). Results of measurements showed that soils heated to temperatures below 175 °C did not change their water repellency, but soil becomes water repellent in the temperature range 175–200 °C (DeBano 2000). Soil temperatures above the 280 °C eliminate water repellency. The reason of this phenomenon is probably the burning of the organic matter on the soil surface.

18.5 The Effects of Soil-Water Repellency on Soil-Water Movement Infiltration of water into water-repellent soil Water does not infiltrate ideally water-repellent soil (angle of contact is 180°). Of course, the definition of such soil is hypothetical because plants cannot grow on such soil. Infiltration into water-repellent soil can occur once water accumulating on the soil surface reaches the pressure making possible its entry into the porous system. The required pressure is the positive pressure expressed by the pressure height of ponded water zo  hk (Eq. 18.1). This pressure height is an analogous to the bubbling pressure for air entry into soil pores full of water. Usually, soil contains macropores, and water can infiltrate via those pores. Water-repellent compounds in the soil decrease the rate of water infiltration into soil, increasing the occurrence of ponding and the risk of surface runoff and erosion. Fingered flow in water-repellent soil Preferential water movement in soil is fast movement through macropores, usually in the vertical direction. The rate of preferential flow is usually of an order higher than in the soil matrix. Preferential flow is a general phenomenon that is not limited

18.5 The Effects of Soil-Water Repellency on Soil-Water Movement

289

Fig. 18.2 Vertical cross section of soil profile during infiltration of dyed water into sandy soil at site Sekule, Záhorská Lowland (modified from Lichner et al. 2012)

to water-repellent soils; but in water-repellent soil, a specific type of flow, known as the “fingering”, can occur. The areas of wet soil during this type of infiltration have the shape of fingers. Water-repellent soil layers (or parts of them) can slow down the vertical movement of soil water, and eventually a water layer at the interface of layers can form and then, water can flow quasi horizontally along the sloped interface of two different soil layers of different textures. Fingering can occur as the result of differences in water-repellency distribution. Soil water in the areas of lower water repellency can infiltrate downward in the form of “fingers” (Fig. 18.2). Wet soil in “fingers” has higher hydraulic conductivity in comparison to the nearby dry areas. The majority of water will flow in fingers in all directions and thus homogenize the soil-water content in the infiltration area (Lichner et al. 2012). Evaporation through water-repellent soil Evaporation from a water-repellent soil can be decreased significantly. The waterrepellent layer covering the solid phase of the soil prevents the water flow in thin films and thus blocks the capillary rise of water to the soil surface. Therefore, the evaporating surface is located below the soil surface, and water vapour can be transported to the atmosphere by molecular diffusion only the rate of which is of an order smaller than that of the convective flow in thin water layers or the rate of upward water movement via capillary rise. This phenomenon can be utilized to decrease the evaporation of water from the soil surface and thus increase the share of water stored in the soil and then transported to the atmosphere by transpiration to increase biomass production.

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18 Water Repellent Soils

Fig. 18.3 Time evolution of evaporation, expressed as percentage of total evaporation, from wettable sandy soil (h = 0) and from soil covered by the water-repellent sandy soil layers of h thickness. Site Sekule, Slovakia

The location of the thin (a few centimetres) layer of water-repellent soil on the soil surface can be an effective factor to decrease “ineffective” evaporation. It has been found that the 1-cm-thick water-repellent sand layer (sand below a pine forest) covering sandy soil conserved about 52% more of soil water in comparison to the non-covered sandy soil. Figure 18.3 illustrates the influence of different soil covers on the evaporation kinetics.

References Adamson AW (1990) Physical chemistry of surfaces, 5th edn. Wiley, New York Beatty SM, Smith JE (2014) Infiltration of water and ethanol solutions in water repellent post wildfire soils. J Hydrol 514:233–248 Bisdom EBA, Dekker JW, Schoute JFT (1993) Water repellency of sieve fraction from sandy soils and relationships with organic material and soil structure. Geoderma 56:105–118 Cawson JG, Nyman P, Smith HG, Lane PNJ, Sheridan GJ (2016) How soil temperatures during prescribed burning affect soil water repellency, infiltration and erosion. Geoderma 278:12–22 DeBano LF (2000) The role of fire and soil heating on water repellency in wildland environments: a review. J Hydrol 231–232:195–206 Doerr SH (1998) On standardizing the “water drop penetration time” and the “molarity of an ethanol droplet” techniques to classify soil hydrophobicity: a case study using medium textured soils. Earth Surf Process Landforms 23:663–668 Doerr SH, Shakesby RA, Walsh RPD (2000) Soil water repellency, its causes, characteristics and geomorphological significance. Earth Sci Rev 51:33–65 Hallett PD, Baumgartl T, Young IM (2001) Subcritical water repellency of aggregates from a range of soil management practices. Soil Sci Soc Am J 65:184–190

References

291

Inbar A, Lado M, Sternberg M, Tenau H, Ben-Hur M (2014) Forest fire effects on soil chemical and physicochemical properties, infiltration, runoff, and erosion in a semiarid mediterranean region. Geoderma 221–222:131–138 Lichner L, Holko L, Zhukova N, Schacht K, Rajkai K, Fodor N, Sándor R (2012) Plants and biological soil crust influence the hydrophysical parameters and water flow in an aeolian sandy soil. J Hydrol Hydromech 60(4):309–318 Lichner L, Hallett PD, Drongová Z, Czachor H, Kovacik L, Mataix-SoleraJ Homolák M (2013) Algae influence hydrophysical parameters of a sandy soil. CATENA 108:58–68 Malkinson D, Wittenberg L (2011) Post fire induced soil water repellency–modeling short and long-term processes. Geomorphology 125:186–192 Tillman RW, Scotter DR, Wallis MG, Clothier BE (1989) Water-repellency and its measurement by using intrinsic sorptivity. Aust J Soil Res 27:637–644 Wallis MG, Horne DJ (1992) Soil water repellency. In: Stewart BA (ed) Advances in soil science. Springer, New York, pp 91–146

Chapter 19

Soil Air and Its Dynamics

Abstract Soil air as a part of the unsaturated zone of soil is a necessary soil constituent for the growth of the majority of plants. Oxidation of assimilates (respiration) is necessary for biomass production, and oxygen is also needed for the respiration of living organisms in soil. The composition of soil air is close to that of the atmosphere because small deficits of oxygen and a surplus of carbon dioxide are quickly equilibrated by air interchanges with an atmosphere. This chapter analyses the convection and diffusion of air in the soil and expresses them quantitatively. It also quantifies oxygen transport to the plant roots to cover respiration by solving the simplified transport equation. The influence of the oxygen diffusion rate on plant canopy growth is also presented.

19.1 Soil Aeration and Plant Respiration Soil air as a part of soil is a necessary ingredient for biomass production and the existence of microorganisms’ in the soil root zone. Respiration is the basic physiological process that transforms the chemical energy of organic matter on the cell level (albumen, sugar, fat) by biological oxidation into the forms of energy needed to drive all the processes of plant development (growth, active absorption of ions, transport of compounds and the division and growth of plant cells) (Masaroviˇcová et al. 2002). Oxygen is consumed by plants in the process of respiration of oxidizing assimilates; respiration produces carbon dioxide, water and energy. Transformed energy is used mostly in the growth process; part of energy dissipates into the plant environment as heat. Plants respire through their entire surface, even the surface of roots; therefore, roots have to be located in an environment containing oxygen. Organic matter is produced in the process of photosynthesis by the transformation of carbon dioxide and solar energy, with oxygen as a by-product. Photosynthesis occurs in the plant’s green parts during the day when the incoming solar energy is above the critical rate; but respiration is a continuous process. The rate of photosynthesis is 5–25-times higher than the respiration rate; therefore, the result of these contrasting processes is biomass production (plant growth). As a result of the different rates of the two © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_19

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19 Soil Air and Its Dynamics

Table 19.1 Recommended soil aeration of several plants (Kopecký 1928) Plant Soil aeration (% vol.) Grass (meadows)

Minimum

Optimal

5

10

Wheat, oat

10–15

15–20

Potatoes, barley

15–20

18–24

Table 19.2 Oxygen content in soil water and in soil air (Swiecicki 1967) Soil oxygen concentration Low Optimal Soil water (mg

l−1 )

Soil air (% vol.)

High

1–3

3–5

>5

2.4–7

7–12

>12

processes, there is a much higher production of oxygen in the process of photosynthesis than in its consumption by respiration. Through the process of photosynthesis, plants are net producers of oxygen and consumers of carbon dioxide. The relative volume of soil air can be expressed by the term a, called aeration. It is the difference between porosity P and volumetric soil-water content θ : a  P −θ

(19.1)

Soil aeration, like soil-water content, can be also expressed by the ratio of air volume V a and soil volume V : a

Va V

(19.2)

Maintaining soil aeration in the range suitable for plants is important for the proper functioning of soil microorganisms that mineralize organic compounds and create humus. At low levels of soil aeration in anaerobic conditions, reduction processes can occur and toxic gases (like mercaptans, methane and ammonium) could be produced. Those gases, which can enter the soil even from industrial emissions, are toxic for plants even in small concentrations and the result of their action is the wilting of plants. Anaerobic processes in soils are typical during soil flooding. Therefore, appropriate soil aeration is a necessary condition of plant growth (Table 19.1). The classification system of oxygen content in soil air and soil water presented by Swiecicki (1967) is shown in Table 19.2. A soil environment with an oxygen concentration in its soil water less than 1 mg l−1 (corresponding to 2.4% of the air volume in the soil) is classified as an anaerobic environment, which is not suitable for plants growth.

19.2 Composition of Soil Air

295

19.2 Composition of Soil Air The composition of soil air in properly aerated soil is close to the composition of the atmosphere; oxygen consumed by respiration is replaced by the atmospheric oxygen, and the carbon dioxide produced is transported into the atmosphere. In poorly aerated soils, the processes of gas exchange between the soil and atmosphere are slow; therefore, the concentration of carbon dioxide in the soil is higher than in the atmosphere, and concentration of oxygen in the soil is lower than in the atmosphere. The concentration of carbon dioxide in the atmosphere has been increasing during recent decades as the result of the combustion of the fossil fuels from 330 up to 400 × 10−6 cm3 cm−3 ; also, the concentration of carbon dioxide in the soil has been increasing correspondingly. The CO2 concentration increase should not influence plant growth negatively; on the contrary, the rate of photosynthesis should increase. Soil-air humidity is close to the saturated state, i.e. its water-vapour pressure is close to the maximum at a particular temperature. Table 19.3 reports the composition of the air in the atmosphere and in the soil.

19.3 Transport of Soil Air Air is fluid; therefore, its movement is governed by the same physical laws as those governing liquid movement (the Poiseuille equation). The ratio of the air’s density to the density of water is 1.19 kg m−3 /1000 kg m−3 (i.e. 0.00119), and the ratio of dynamic viscosities of air and water is 0.171 × 10−6 kg m−1 s−1 /1002 × 10−6 kg m−1 s−1 (i.e. 1.71 × 10−4 ) at 20 °C. The rate of liquid movement according to the Poiseuille (Eq. 8.4) is directly proportional to the liquid density and indirectly proportional to the dynamic viscosity. The ratio of water density and its dynamic viscosity is 1 × 106 s m−2 , and the ratio of air density and its dynamic viscosity is 6.95 × 106 s m−2 . According to the Poiseuille equation, the rate of air flow between two points will be approximately seven-times higher than the rate of water flow in a porous medium at the same pressure difference. This means that soil-air dynamics is much greater than soil-water dynamics; the gases exchange between soil and atmosphere is rapid when the minimum soil aeration criteria are fulfilled (see Table 19.1). Convection is the mechanism of fluid transport between two points due to a pressure difference. As can be seen in Eq. (8.4), the air

Table 19.3 Relative composition of an air in the atmosphere and in the soil Air component Atmosphere Soil air Nitrogen (N2 )

0.78

Oxygen (O2 )

0.21

>0.78 <0.21

Carbon dioxide (CO2 )

0.00038

>0.00038

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19 Soil Air and Its Dynamics

is flowing due to the difference between air pressures. The differences of air pressure between the soil and atmosphere are evident during the atmospheric-pressure changes, during changes in temperature, during the pressurising of soil air before the infiltration front and during groundwater-table fluctuation and soil deformation by heavy machinery. Molecular diffusion is another mechanism of soil-air movement. Air in the soil can be transported by molecular diffusion also when the air pressure in porous system is constant. The soil-air transport rate due to molecular diffusion is proportional to the gradient of partial pressure of the individual constituents of soil air. If the total (atmospheric) pressure of soil air in the porous system is constant and the convective transport rate of soil air is zero, but there are differences in partial pressures of the soil air constituents (like oxygen or carbon dioxide) between two points of the porous system, the molecules of individual constituents are transported from the points of greater to the points of lesser concentration of a particular constituent of soil air. This type of transport process is denoted as molecular diffusion, and its final state is uniform concentration of the molecules of a particular constituent of soil air in the porous space. The differences between soil-air constituents’ concentrations are the result of non-uniform distribution of the processes of respiration of plant roots and microorganisms in soil. The dominance of convective transport rate of soil air in comparison to the molecular diffusion of gases in comparable conditions can be demonstrated, but the importance of both processes can be comparable, depending on the actual conditions. A quantitative description of the transport processes of molecular diffusion and convection will be presented later.

19.3.1 Convective Flow of Soil Air A differential equation to describe soil-air convection can be derived by the standard procedure through a combination of the continuity equation and transport equation. The rate of convective transport of air in soil in the direction of coordinate x can be expressed by an equation resembling the equation of Darcy: qc  −

ρa K a dp ηa d x

(19.3)

where qc is the rate of convective mass air flow (ML−2 T−1 , kg m−2 s−1 ), ρ a is air density (ML−3 , kg m−3 ), ηa is dynamic viscosity of air (ML−1 T−1 , kg m−1 s−1 ), K a is permeability of the pores filled with air (L2 , m2 ), and p is the air pressure in the pores (ML−1 T−2 , kg m−1 s−2 ). The continuity equation for a deformable liquid in the direction x can be written in the form:

19.3 Transport of Soil Air

297

dqc dρa − dt dx

(19.4)

where t is time (T, s). The air density changes with air pressure (p) and its temperature (T ). If the temperature of the air is constant, then the left side of Eq. (19.4) can be written as: dρa dρa dp dp   ca dt dp dt dt ca  dρa /dp Combine the above equations, the resulting differential equation is:   d ρa K a dp dp  ca dt dx ηa d x

(19.5) (19.6)

(19.7)

The characteristics of the air ca , ρ a , ηa , K a can be assumed as constant values when convection of air occurs at small pressure differences; then, the simplified equation of air convection in the soil can be written: d2 p dp α 2 dt dx ρa K a α ηa ca

(19.8) (19.9)

Equation (19.8) can be solved by analytical method, and the results can be used as good quantitative approximations for air flow in the soil.

19.3.2 Diffusion of Soil Air The diffusion of gases, such as CO2 and O2 , occurs not only in the gaseous phase of the soil, but also in the liquid phase, if a concentration gradient of particular air constituent exists. The most important is the diffusion of CO2 and O2 in the soil and diffusion of oxygen through the thin water layers that cover the plant roots. Both processes can be expressed (in one dimensional form, i.e. diffusion performs in the direction of coordinate x) by the equation known as Fick’s law: qd  −Das

dc dx

(19.10)

where qd is the diffusion flux of an air (or its component) (ML−2 T−1 , kg m−2 s−1 ), Das is the diffusion coefficient of air (or its component) in the soil (L2 T−1 , m2 s−1 ), and c is the concentration (density) of air (or its components) in the soil (ML−3 , kg m−3 ).

298

19 Soil Air and Its Dynamics

The coefficient of air diffusion (or its components) in soil is smaller than the diffusion coefficient of an air component in the atmosphere. The reason for this difference is the limited effective cross section of soil pores filled with air and thus the limited diffusion in the soil. In soil, only part of the cross section of soil pores is represented by pores filled with air. Trajectories of gaseous molecules during the diffusion process are not direct, but they follow the curved diffusion path of the soil’s porous space. Some of the pores are filled with liquid and thus block the diffusion paths. The coefficient of air diffusion (or its components) in the soil (Das ) can be expressed by the equation: Das  Da τ (P − θ )

(19.11)

where Da is the coefficient of diffusion of a particular air component in the air (L2 T−1 , m2 s−1 ); (the coefficient of diffusion of water vapour in the air is Dva  1.89 × 10−5 m2 s−1 ); P is soil porosity; θ is volumetric soil water content; and τ is the coefficient characterising the influence of diffusion-path curvature on the diffusion coefficient (tortuosity); a generally accepted value is τ  0.66. As was done before, to develop equation of diffusion transport of the air components in the soil, the continuity equation is needed: dqd dc − +A dt dx

(19.12)

where A is the rate of extraction (or generation) of gas in a unit volume of soil (ML−3 T−1 , kg m−3 s−1 ). To substitute the transport equation (Eq. 19.10) into the continuity equation, the final differential equation describing unsteady diffusion of gas (or individual gas component) in soil becomes:   d dc dc  Das +A (19.13) dt dx dx The solutions of Eq. (19.13) under specified boundary and initial conditions are the distributions of gas concentration c (or its components O2 , CO2 , water vapour) as a function of coordinate x and time t; c  f (x, t).

19.4 Movement of Oxygen from Soil to Plant Roots Equation (19.13) describes quantitatively changes of the concentration of air components (and oxygen, too) as a function of time t and coordinate x. Van Bavel et al. (1968) published a solution for Eq. (19.13) for a constant extraction rate of oxygen by plant roots (A  qr ) and a constant coefficient of oxygen diffusion in a soil (Dos )

19.4 Movement of Oxygen from Soil to Plant Roots

299

when the depth of the active (respirating) soil root zone is L r . Then, Eq. (19.13) for vertical coordinate can be written in the form: dc d 2c  Dos 2 − qr dt dz

(19.14)

Boundary conditions are: qr  const., for 0 ≤ z ≤ L r , (constant extraction of oxygen in the soil root zone) , qr  0, at z ≥ L r , (no extraction of oxygen below the rooting depth L r ), c  co , for z ≥ L r , t > 0, (constant oxygen concentration below the soil root zone) and then dc/ dz  0, no oxygen transport below the depth L r ). Initial condition: c  0, for 0 ≤ z ≤ L r , t  0 (zero oxygen concentration at the beginning of the transport). The solution to Eq. (19.14) the steady oxygen concentration distribution along soil active depth of soil root zone is: c(z)  co −

 qr  2L r z − z 2 2Dos

(19.15)

The depth of aerobic zone L r can be also calculated from Eq. (19.15). Substituting into Eq. (19.15), the condition of zero concentration of oxygen c = 0, at the depth z ≥ L r , the depth of aerobic zone is: 

Dos L r  2c0 qr

0.5 (19.16)

The oxygen concentration at sea level is co  0.2972 kg m−3 , so Eq. (19.16) can be written as:   Dos 0.5 L r  0.597 qr

(19.17)

The depth of the soil aerobic zone (Eq. 19.17) is proportional to the coefficient of diffusion of oxygen in the soil (Dos ) and indirectly proportional to the rate of oxygen extraction by roots qr . The rate of oxygen extraction in soil is the key characteristic of biological activity in soil because it represents the rate of respiration by plants and microorganisms. In general, the higher the biological activity of the soil is, the higher is the biomass production. Results of measurements indicate that the rate of oxygen consumption in the soil is in the range 1 × 10−7 ≤ q ≤ 6 × 10−7 kg m−3 s−1 ; in soils containing a high content of organic matter, oxygen consumption can be an order higher than in other soils (Kowalik 1973). The coefficient of oxygen diffusion in the air at T  20 °C is Doa  1.89 × 10−5 2 −1 m s (Koškin and Širkeviˇc 1976). The coefficient of oxygen diffusion in soil Dos

300 Table 19.4 The influence of ODR (oxygen diffusion rate) in soil on plant growth (according to Kowalik 1973)

19 Soil Air and Its Dynamics ODR (g cm−2 min−1 )

ODR and plant growth

ODR < 20 × 10−8

Slow growth

20 × 10−8 < ODR < 50 × 10−8

Some processes are slowed down

ODR > 50 × 10−8

Normal growth

can be calculated using Eq. (19.11) (Dos  6 × 10−6 m2 s−1 ). Assuming the oxygen consumption rate is qr  5 × 10−7 kg m−3 s−1 (Kowalik 2001), then, substituting the just mentioned characteristics into Eq. (19.17), the depth of the aerobic zone is 1.66 m. The depth of the aerobic zone increases with a decreasing rate of oxygen consumption, and vice versa. This means that, in light (sandy) soil in comparison to heavy soils, a shallower depth of aerobic zone is needed. Oxygen transport to the roots happens via the soil air and then through the thin water layer that covers the roots’ surface. Diffusion of oxygen through the air-filled soil pores is relatively fast; it is proportional to the soil aeration because the diffusion coefficient of oxygen through water is about four orders smaller than the coefficient of oxygen diffusion in air. The diffusion coefficient of oxygen in water is Dow  2.5 × 10−9 m2 s−1 , and the diffusion coefficient of oxygen in air is Doa  1.89 × 10−5 m2 s−1 . Therefore, the thickness of the water layer covering the roots surface is decisive for the rate of oxygen transport to the roots. The transport of oxygen by diffusion to a root of radius r with a thickness of water layer on the root surface d can be expressed by Fick’s law: qdo  −Dow

Δc d

(19.18)

where c is the difference between oxygen concentrations on both sides of the thin water layer on the root surface (ML−3 , kg m−3 ); qdo is the oxygen diffusion rate to the root surface through the thickness of the water layer d (ML−2 T−1 , kg m−2 s−1 ); and Dow is diffusion coefficient of oxygen in water (L−2 T−1 , m−2 s−1 ). The oxygen diffusion rate in soil is often expressed by the term ODR (oxygen diffusion rate); its dimension is (g cm−2 s−1 ). The influence of ODR values on the plants photosynthesis rate is listed in Table 19.4 (Kowalik 1973). ODR is measured by an apparatus made of a thin platinum electrode with a constant electrical voltage on its surface, as a part of the polarographic circuit; the platinum electrode reduces the oxygen on the electrode surface. By this process, the constant difference in the oxygen concentrations on the electrode surface and above it establishes the steady transport of oxygen to the electrode surface. The oxygen-rate flux to the electrode is proportional to the electrical voltage on its surface and depends on soil properties too. The electrode simulates the function of the root. The result of the measurement is the relative rate of oxygen extraction (consumption) by the model root (electrode) and is proportional to the ODR because the relative rate of oxygen transport is influenced by the same soil parameters as for ODR. The most important

19.4 Movement of Oxygen from Soil to Plant Roots

301

soil characteristics are: soil-water content and the temperature and dimensions of pores. The result of measurement (Kowalik 1973) is the ODR distribution above the groundwater. ODR of the soil root zone was found to be proportional to the decreasing soil water content. The optimal conditions for oxygen diffusion to the roots are in properly aerated soils, which is in agreement with experience.

References Kopecký J (1928) Soil science. Prague (In Czech) Koškin NI, Širkeviˇc MG (1976) Handbook of physics. Nauka, Moscow (In Russian) Kowalik P (1973) Basic soil physics. Politechnika Gdaˇnska, Gdaˇnsk (In Polish) Kowalik P (2001) Soil conservation. Wydawnictwo naukowe PWN, Warszawa (In Polish) Masaroviˇcová E, Repˇcák M et al (2002) Plant physiology. Vydavateˇlstvo UK, Bratislava (In Slovak) Swiecicki C (1967) The basic knowledge about soil water. Wyd. SGGW, Warszawa (In Polish) van Bavel CHM, Stirk GB, Brust KJ (1968) Hydraulic properties of a clay loam soil in the field measurements of the water uptake by roots: I. Interpretation of water content and pressure profiles. Soil Sci Soc Am Proc 32:310–317

Chapter 20

Soil Temperature and Heat Transport in Soils

Abstract Soil temperature varies over a wide range, depending on annual and diurnal cycles of incoming solar energy and on the soil surface properties. Plants can grow when soil root zone temperatures are in the relatively narrow range 0–40 °C. Soil-water hydraulic conductivity and soil-water retention curves are functions of temperature because of the temperature dependence of viscosity, surface tension and the density of water. Characteristic annual and diurnal cycles of soil temperature (at various depths) characterise the soil-temperature regimen. This statistical characteristic is typical of particular soils and depends on climate. The basic soil-heat characteristics (specific soil heat capacity, thermal conductivity) and basic modes of heat transport, namely, conduction, convection and radiation, are defined. The chapter presents governing equations of simultaneous heat and water transport in non-isothermal soil conditions. The influence of soil temperature on basic soil-water transport and retention characteristics are addressed.

20.1 Soil Temperature Plants can survive in the full temperature range occurring on the Earth (−88 °C up to +58 °C). The seeds of plants are extremely resistant to the extreme temperatures. Plants can thrive in a limited range of temperatures, starting from the temperatures just above the freezing point, up to temperatures of about 40 °C. Crops in conditions of Europe grow at leaves’ temperature ranging 5–35 °C. Optimal growth occurs in the relatively narrow interval of air temperatures 25–35 °C, depending on the variety of the plant. Soil temperature depends on the temperature of the atmosphere; maximum soil temperatures can be higher than the temperature of the atmosphere. The behaviour of microorganisms and the dynamic of chemical reactions in the soil depend on soil temperature. Soil temperature and especially its vertical distribution can strongly influence the transport of heat and water in the soil. The basic source of energy consumed by the various processes on the Earth is the energy from solar radiation. Even energy stored in fossil fuels (coal, oil, gas) © Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_20

303

304

20 Soil Temperature and Heat Transport in Soils

is a product of plant photosynthesis, energetically supplied by the solar radiation. The energy of water and wind is transformed energy of the Sun, too. Daily and annual cycles of the rate of solar energy striking the Earth’s surface are explicitly transformed to daily and annual cycles of air, soil and ocean temperature. Heat transport in the soil is a relatively complicated phenomenon; soil is a threephase system with variable water content and with a relatively high content of organic compounds (up to 10%). A small portion of heat is generated by respiration and consumed by photosynthesis or stored by plants. But, the major portion of incoming energy is consumed by evapotranspiration.

20.1.1 Soil-Temperature Regimen The term soil-temperature regimen denotes the typical daily and annual cyclic changes of soil temperature at the defined depth below the soil surface. The soiltemperature regimen depends mostly on the solar-radiation regimen but also on the properties of the soil-plant-atmosphere system. Ideally (under a clear sky), those cyclic changes can be expressed by a sine function. In the case of overcast skies or when cloudiness varies irregularly, soil-temperature variations are non-regular too and follow the course of solar radiation reaching the soil surface. The amplitude of soil-temperature decreases with soil depth. Figure 20.1 graphs the daily courses of soil-surface temperature and temperatures at two depths below the soil surface with a grass canopy at the Pavˇcina Lehota site (Central Slovakia). Non-regularities are the result of variable cloudiness. It is known that the maximum soil temperatures are postponed to the later afternoon hours, proportionally with soil depth. Those differences are due to the time needed to transport heat (and to increase the soil temperature) to the deeper soil horizons. As will be demonstrated later, soil-water content significantly influences the heat transport: the higher the soil-water content is, the higher the rate of heat transport is expected. The annual courses of the average daily temperatures depend on the energy income (rate of solar radiation) and on soil-heat conductivity. Annual courses of the average

Fig. 20.1 Daily courses of temperatures at the soil surface covered with grass at depths 5 and 20 cm during three consecutive sunny days as a function of time. Pavˇcina Lehota site, Central Slovakia

20.1 Soil Temperature

305

Fig. 20.2 Annual courses of the average daily temperatures of soil with grass canopy at depths 0.5 and 2.0 m below the soil surface in years 1953–1967 in Bratislava, Slovakia (Climate and Bioclimate of Bratislava 1979) Fig. 20.3 Vertical distribution of soil temperatures below grass during sunny day; numbers at temperature profiles denote hours of temperature measurement. Pavˇcina Lehota, Central Slovakia, July 2, 1997

daily soil temperatures at two depths below a soil surface (0.5 and 2.0 m) covered by a grass canopy in Bratislava (Central Europe) are in Fig. 20.2 (Climate and Bioclimate of Bratislava 1979). The amplitude of annual changes at a depth of 0.5 m is 17 °C, but even at the depth 2 m, it is relatively high: 9 °C. It is interesting that maximum of soil temperatures at both depths are postponed about one month; the deeper the soil horizon is, the longer delay of maximum soil temperature is observed. Figure 20.3 shows the vertical distribution of soil temperature below a grass canopy during one sunny day at the Pavˇcina Lehota site, with numbers showing the hour of the day. The amplitude of soil-surface temperature was as high as 17 °C. The highest soil-temperature changes were measured in bare soil with a dark-coloured surface; the lowest soil temperatures were recorded below dense-plant canopy.

20.2 Soil-Heat Transport There are three principal modes of heat transport: radiation, convection and conduction.

306

20 Soil Temperature and Heat Transport in Soils

Radiation refers to the emission of energy by electromagnetic waves from bodies of temperatures above 0 °K. According to the Stephan–Boltzmann law, the total energy emitted per unit area of radiating body per unit of time J can be expressed as proportional to the fourth power of the absolute radiating-body temperature T : J  εσ T4

(20.1)

where σ is the Stephan–Boltzmann constant (σ  5.67 × 10−8 W m−2 K−4 ), ε is emissivity coefficient, usually in the range 0.95 ≤ ε ≤ 1, ε  1 is emissivity of perfect emitter, the so-called black body. Radiation as a mode of heat transport in the soil is usually not important; but the temperature differences between water menisci in soil pores can drive this process, and transport in the direction of decreasing temperature can occur. The convection mode of heat transport in soil can be important in the case when the flowing-water temperature significantly differs from the temperature of the solid phase of soil. As an example of such a mode is the infiltration of cold water into a soil of high temperature or the infiltration of water from melting snow. Conduction is usually the most important mode of soil-heat transport. This mode transports the heat by the energy exchange between textural elements of soil bodies in close contact, in the direction toward the colder side of the soil. The molecules of the solid phase of soil of higher temperature collide with the molecules of the solid body of lower temperature, and those share part of their energy, tending to equilibrate the temperature of the soil.

20.2.1 Transport of Heat in Soil by Conduction Heat is transported between two points of soil if there exists a temperature difference between those two points; the direction of transport toward the lower temperature (the first law of thermodynamics). In other words, the heat transport is directed against the temperature gradient. The heat conduction rate can be expressed by the Fourier’s law: qh  −λ(θ )grad T

(20.2)

where qh is the rate of heat flux (MT−3 , W m−2 ); λ(θ ) is the coefficient of thermal conductivity (specific heat conductivity) as a function of soil-water content (LMT−3 K−1 , W m−1 K−1 ); and T is the soil temperature (K, °C). Transport of heat in soils is governed mostly by solar radiation (during the day) and by the soil radiation to the atmosphere (by night). The direction of soil-heat flux is mostly vertical; therefore, Eq. (20.2) can be rewritten in one-dimensional form in direction z as follows:

20.2 Soil-Heat Transport

307

Fig. 20.4 Typical course of soil thermal conductivity λ and soil volumetric heat capacity C as a function of volumetric soil-water content θ at the temperature T  20 °C

qh  −λ(θ )

dT dz

(20.3)

The coefficient of thermal conductivity λ is constant for the bodies with constant composition like metals, completely dry soil or soil saturated with water. In variably saturated soil, the water content varies, and soil thermal properties vary too. Soilwater content is the most important characteristic influencing λ, as can be seen in Fig. 20.4. Equation (20.3) demonstrates also the definition of thermal conductivity, which equals the rate of heat flux under a unit of temperature gradient (the temperature change 1 °K per length unit). The heat flux Qh across the area A perpendicular to the direction of transport during the period t can be expressed by the equation: Q h  qh A t  −λ(θ )A t

dT dz

(20.4)

Equation (20.4) quantitatively describes the conductive transport of heat; it is known as the Fourier equation. The negative sign on the right side of the equation indicates the direction of heat flux against the temperature gradient, or against the direction of increasing temperature. It is the result of the definition of the term “gradient”. The temperature gradient is positive in the direction of temperature increase; but heat is conducted in the opposite direction, in the direction of lower temperature. The steady transport of heat in soil can be expressed by Eq. (20.4); this means that the soil temperature must be constant in time. Soil-heat transport is in most cases an unsteady process because solar radiations, as well as soil-water content, change during the day. The law of energy conservation in simplified form applied to soil expresses the heat capacity of the defined soil volume as the difference between the incoming and outgoing heat fluxes of the defined soil volume. The law of energy conservation can be then expressed by the equation:

308

20 Soil Temperature and Heat Transport in Soils

ρb c

dqh dE dT − −L ± H (z, t) dt dz dz

(20.5)

where ρ b is the soil bulk density (ML−3 , kg m−3 ); c is specific heat capacity of soil (L2 T−2 K−1 , J kg−1 K−1 ); L is the latent heat of evaporation (L2 T−2 , J kg−1 ); and H is the rate of heat sources or sinks of a given soil volume (ML−1 T−3 , J m−3 s−1 ). The left side of Eq. (20.5) represents the change of soil heat capacity; often the reason for it is the conductive heat flux in soil (the first term on the right side of the equation). A significant decrease in the heat capacity of the surface soil layers can be due to the energy consumption by evaporation, especially from bare soil. The consumption of energy by evaporation (the latent heat of evaporation) or LE results in cooling the evaporating surface of the soil; it is expressed by the second term on the right side of Eq. (20.5). Chapter 13 contains a detailed description of the energy consumption by evaporation. Other processes producing or consuming heat are represented by the term H(z, t). This term can contain the condensation of water vapour on, or in, the soil, microorganisms and plants’ respiration; these processes generate heat. Processes expressed by the term (H(z, t)) are usually not important and often are neglected. By combining the heat transport Eq. (20.2) and equation of continuity (Eq. 20.5), the equation of unsteady transport of heat in the soil can be developed:   d dT dE dT  λ(θ ) −L ± H (z, t) (20.6) ρb c dt dz dz dz The solutions of Eq. (20.6) by numerical methods under specific initial and boundary conditions are temperature distributions as a function of coordinate z and time t: T  f (z, t)

(20.7)

Equation (20.6) can be simply expressed in two-dimensional or three-dimensional form. To solve Eq. (20.6), the basic input characteristics of the soil need to be known: specific heat capacity of soil c and soil thermal conductivity λ.

20.3 Soil-Heat Capacity Soil-heat capacity (or heat capacity of other porous body) is defined as the change of heat content in a unit quantity of soil per unit change of temperature. Depending on the unit of soil quantity, soil-heat capacity can be expressed as the specific volumetric heat capacity C or the specific-mass heat capacity c. Volumetric-heat capacity of soil C characterises the heat-content change of the unit bulk volume of soil; the mass-heat capacity of soil c is the heat-content change of the unit mass of soil. Their relationship can be expressed by the equation:

20.3 Soil-Heat Capacity

309

Table 20.1 Thermal characteristics of soil components at temperature 10 °C Soil component Density (ρ) Specific mass heat Specific thermal capacity (c) conductivity (λ) (g cm−3 ) (J kg−1 K−1 ) × 103 (W m−1 K−1 ) Quartz

2.66

0.790

8.8

Other minerals Organic matter

2.65 1.3

0.800 2.500

2.9 0.25

Water (liquid)

1.0

4.200

0.57

Air

0.00125

1.010

0.025

C  ρb c

(20.8)

where C is the specific volumetric heat capacity of a soil (ML−1 T−2 K−1 , J m−3 K−1 ), c is the specific-mass heat capacity of soil (L2 T−2 K−1 , J kg−1 K−1 ) and ρ b is the bulk density of the soil (ML−3 , kg m−3 ). Soil is a three-phase system composed of the solid (s), liquid (w) and gaseous (a) phases; the soil-heat capacity consists of the heat capacities of all three soil components. It can be expressed by the equation: C  n s Cs + n w Cw + n a Ca

(20.9)

where ns , nw , na are volumetric relative portions of particular soil phases and C s , C w , C a are specific volumetric heat capacities of individual soil phases. The solid phase of soil is composed of various parts: the mineral (inorganic) (m) and organic component (o); their heat capacities are very different. Heat capacity of air can be neglected, but the heat capacity of soil water (nw . C w ) cannot. For practical purposes, the soil-heat capacity can be calculated by the equation: C  n m Cm + n o Co + n w Cw

(20.10)

The subscripts denote the mineral (m) and organic (o) component of the solidphase of soil; soil water is denoted by subscript (w). Substituting the standard values (at 20 °C) of heat capacities of individual components of soil (see Table 20.1) expressed in units (J m−3 K−1 ) into Eq. (20.10), the formula applicable in practice is: C  2.0 × 106 n m + 2.56 × 106 n o + 4.2 × 106 n w

(20.11)

The specific volumetric heat capacity of soil C calculated by Eq. (20.11) is then expressed in (J m−3 K−1 ). The components ratio of a particular soil is stable; only the soil-water content expressed by the ratio nw can change; nw equals the volumetric soil-water content (θ ). This means that the relationship C  f (θ ) is linear. An example of such a relationship is shown in Fig. 20.4.

310

20 Soil Temperature and Heat Transport in Soils

20.4 Thermal Diffusivity of Soil The thermal conductivity of soil is defined by Eq. (20.2). The coefficient is often incorrectly confused with another thermal characteristic, the thermal diffusivity a. Thermal diffusivity is defined by the equation: a

λ c ρb

(20.12)

where a is the thermal diffusivity (coefficient) (L2 T−1 , m2 s−1) ; λ is the thermal conductivity coefficient (LMT−3 K−1 , W m−1 K−1 ); c is the mass-heat capacity of soil (L2 T−2 K−1 , J kg−1 K−1 ); and ρ b is the soil-bulk density (ML−3 , kg m−3 ). The value of the thermal diffusivity of soil depends on the soil composition (mineralogical composition, organic-compounds content), but the soil-water content (the relative soil-water content) is the most important. The thermal conductivity of loamy soil and its dependence on the soil-water content is illustrated in Fig. 20.4. The dominant role of water as a soil component in the heat transport in soil is evident. The thermal conductivity of soil depends on the soil texture and the arrangement of solid particles in the soil. The method to measure thermal conductivity is based on the physical model of Fourier’s equation (Eq. 20.4). This realization is technically not a simple problem; the measurement of temperature distribution along the transport direction, heat flow measurement and elimination of the heat loses from the area of measurement are each complicated. Therefore, inversion methods to calculate thermal conductivity of soils λ are often applied. Knowing the initial and boundary conditions and the temperature distribution in the heat-transport direction, the effective value of λ can be calculated. The thermal characteristics of individual soil components of dry soils are listed in Table 20.1. The thermal characteristics of two soils (sandy soil and clay) with different soil-water contents are in Table 20.2, according to van Wijk and de Vries (1963). The soil-water content is the dominant component of the soil influencing the soil-heat transport.

Table 20.2 Thermal characteristics of sand and clay for two volumetric soil-water contents (SWC). According to van Wijk and de Vries (1963) Soil type Porosity (%) SWC Specific thermal Volumetric heat conductivity (λ) capacity (C) (cm3 cm−3 ) (W m−1 K−1 ) (J m−3 K−1 ) × 106 Sand

0.4

Clay

0.4

0.0 0.4 0.0

0.29 2.17 0.250

1.25 2.92 1.25

0.4

1.59

2.92

20.5 Soil-Water Transport Under Non-isothermal Conditions

311

20.5 Soil-Water Transport Under Non-isothermal Conditions Soil temperature changes in time and space, thus influencing the transport of heat, soil water and solute in soil. Isothermal conditions in soils are rare, but the transport of soil water even in non-isothermal conditions can be calculated by equations describing the isothermal-water flow. This approach is usually a good approximation also for standard, non-isothermal conditions. Extreme changes of air temperature create great temperature gradients in soil, so it is important to account for the influence of such extreme temperature gradients on soil-water transport. Extreme soil temperature differences are observed in bare soils, in the vertical direction. The differences can reach up to 50 °C, and even cyclic oscillations of the groundwater table close to the soil surface corresponding to the soil-temperatures oscillation have been observed. The probable reasons are: the changes of soil-water surface tension with soil temperature; and the increase of the surface tension of water during the night that provokes the capillary fringe upward movement, as a result of which the groundwater-table decreases. During the day, the capillary fringe drains and the groundwater table increases. These oscillations are some centimetres in amplitude, but this is one of the many examples of the influence of a non-isothermal regimen of soil and its influence on soil water. These changes of soil temperatures do not change the physical properties of the solid phase of soil and also the soil’s porous space is preserved. Soil temperature changes are associated with changes in soil-water properties (soil-water retention curves, hydraulic conductivity of soil). Water density, (ρ w  f (T )), the dynamic viscosity of water (ηw  f (T )) and the surface tension of water (σ w  f (T )) are the most important soil-water characteristics that change with temperature (T ). A non-isothermal state of soil means that the soil-water transport is accompanied by simultaneous heat transport in the soil. A mathematical description of the simultaneous transport of heat and water in soils is not so difficult, but it is difficult to determine the characteristics of water and heat transport in variably saturated soil and their dependence on temperature. The resulting governing differential equations describing the transport of water and heat in non-isothermal conditions of soil will be presented to enable understanding of the structure of such processes (Philip and de Vries 1957; Globus 1983; Novák 1975, 2012). The equation describing the non-isothermal transport of soil water in the direction of vertical variable z is:      ∂h w ∂ ∂ ∂q ∂(ρw θ )  ρw k(h w ) +1 + ρa Dvs ∓ S(z, t) (20.13) ∂t ∂z ∂z ∂z ∂z The equation of unsteady soil-heat transport can be written in the form:   ∂T ∂E ∂T ∂T ∂ λ −L ρb c  + cw vw + H (z, t) (20.14) ∂t ∂z ∂z ∂z ∂z

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where ρ w is the liquid-water density (ML−3 , kg m−3 ); ρ b is the soil-bulk density (ML−3 , kg m−3 ); ρ a is air density (it is the function of temperature) (ML−3 , kg m−3 ); θ is the volumetric soil-water content; c is the specific-mass heat capacity of the soil (a function of soil-water content) (L2 T−2 K−1 , J kg−1 K−1 ); cw is the specific-mass heat capacity of water (a function of temperature) (L2 T−2 K−1 , J kg−1 K−1 ); λ is the specific thermal conductivity of soil (LMT−3 K−1 , W m−1 K−1 ); vw is the rate of soil-water transport in soil (ML−2 T−1 , kg m−2 s−1 ); hw is the matric potential of soil water expressed by the pressure head (is a function of temperature) (L, m); and k(hw ) is the soil’s hydraulic conductivity; it is a function of soil-water matric potential (and soil-water matric potential is a function of temperature) (LT−1 , m s−1 ); q is the specific air humidity (MM−1 , kg kg−1 ); t is time (T, s); z is the vertical coordinate, positive upward (L, m); Dvs is the coefficient of molecular diffusion of water vapour in a soil as a function of soil-water content (L2 T−1 , m2 s−1 ); L is the latent heat of evaporation (L2 T−2 , J kg−1 ); T is the temperature (K, °C); E is the evaporation rate (ML−2 T−1 , kg m−2 s−1 ); S(z, t) is the extraction rate of soil water by roots (with negative sign), or the inflow of water from the system (ML−3 T−1 , kg m−3 s−1 ); and H(z, t) is the rate of internal sources or extraction of heat (ML−1 T−3 , J m−3 s−1 ). Equations (20.13) and (20.14) can be solved numerically, knowing the initial and boundary conditions. As was mentioned before, the basic problem is to know the necessary soil characteristics: soil hydraulic conductivity as a function of soil-water matric potential and temperature k(hw , T ); the soil-water retention curve as a function of temperature hw (θ , T ) and to know the rate of water flow in a soil vw . To overcome the difficulties related to the estimation of the input characteristics of non-isothermal soil, it is possible to use as a first approximation the system of differential equations governing the isothermal transport of water in soil. Equations (20.13) and (20.14) help to understand the complexity of the nonisothermal transport of water and energy in soil. The left side of Eq. (20.13) represents the change of soil-water content in time; it is the result of the liquid-water flow (first term on the right side of the equation). The second term on the right side of Eq. (20.13) represents water-vapour flow and the third term represents (mainly) the water extraction rate by plant roots. The soil-water retention curve and soil hydraulic conductivity are functions of soil-water content and temperature, as well. The equation of unsteady heat transport in soil (Eq. 20.14) quantitatively describes the heat capacity change of a unit soil of volume (left side of the equation). A soilheat capacity change occurs by heat conduction (first term on the right side of the equation), heat consumption by evapotranspiration (the second term), the convective transport of heat (third term) and the extraction of heat from or to the soil and its delivery from (or to) the system.

20.6 Temperature and Water Properties

313

20.6 Temperature and Water Properties 20.6.1 Water Density and Temperature Water density ρ w increases in the temperature range 0–3.9 °C where lies its maximum; then water density decreases because the thermal motion of water molecules intensifies. This phenomenon is of fundamental importance for the life on the Earth. In the range of temperatures 20–50 °C, water density changes from 998.2 to 998.07 kg m−3 , a difference of only 0.0125% (Table 1.1). Such small water density change with temperature can be neglected when calculating soil-water transport.

20.6.2 Water Viscosity and Temperature The dynamic viscosity of water ηw decreases as temperature increases significantly; therefore, this effect usually cannot be neglected. In the range of temperatures 20–50 °C, the dynamic viscosity of water decreases from a value 1004 × 10−6 to 551 × 10−6 Pa s−1 ; the ratio of these values is 1.822, far too much to be neglected. The dynamic viscosity of water can be found in Table 1.1, or it can be calculated using the empirical formula of Poiseuille: ηw 

0.00179 1 + 0.0368 T + 0.000224 T 2

(20.15)

where ηw is water dynamic viscosity, Pa s−1 (ML−1 T−1 , kg m−1 s−1 ), T is temperature (°C).

20.6.3 Surface Tension of Water and Temperature The surface tension of water σ w significantly influences capillary phenomena in soils. In the temperature range of 20–50 °C, the water surface tension decreases from 72.8 to 67.78 Nm−1 , a difference of 7.4%. This difference cannot be neglected in the majority of cases. The height of capillary rise in a capillary tube of diameter 0.1 mm—assuming ideal wetting—is hk  148 cm, at 20 °C; at the temperature 50 °C, hk  138 cm. Therefore, analysing the capillary phenomena in non-isothermal systems, the temperature dependence of the water-surface tension should be taken into account (Table 1.1). The water-surface tension in the temperature range 0–100 °C can be calculated using the empirical equation: σw  (75.6 − 0.157 T ) × 10−3

(20.16)

where σ w is the water-surface tension, N m−1 (MT−2 , kg s−2 ), at temperature T (°C).

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20.7 Hydrophysical Characteristics of Soil and Temperature 20.7.1 Hydraulic Conductivity of Saturated Soil and Temperature The hydraulic conductivity of soil saturated with water K can be expressed by Eq. (8.12) using the universal characteristic of soil, denoted as soil permeability K p and the physical characteristics of a flowing liquid expressed by the fraction on the right side of the equation: K  Kp

ρw g ηw

(20.17)

where K is hydraulic conductivity of soil saturated with water (LT−1 , m s−1 ); K p is soil permeability; this is the soil characteristic of the porous space only and can be understood as the effective area of pores through which liquid can flow; (L2 , m2 ), ηw is the dynamic viscosity of water, (ML−1 T−1 , Pa s−1 ); ρ w is water density (ML−3 , . kg m−3 ); and g is the acceleration of gravity, g  9.81 m s−2 . Permeability K p is characteristic of the solid phase (porous space) and does not depend on the flowing liquid or on temperature. The fraction on the right side of Eq. (20.17) contains the properties of the transporting liquid only. Knowing the permeability of the porous body (soil) K p and characteristics of the liquid, it is possible to calculate the saturated (hydraulic) conductivity of any fluid, even water and air. The temperature dependence of water density can be neglected, but what cannot be neglected is the dependence of the dynamic viscosity on temperature. Equation (20.17) for two different temperatures T 1 and T 2 can be written: K (T1 )  K p

ρw g , ηw (T1 )

K (T2 )  K p

ρw g ηw (T2 )

(20.18)

Compare them, the following equation is obtained: K (T1 ) ηw (T2 )  ) K (T2 ) ηw (T1 )

(20.19)

Equation (20.19) demonstrates the proportionality of the saturated hydraulic conductivity of soils to the temperature of soil. Equation (20.19) is often used to recalculate saturated conductivity of soil K measured at the temperature (T 2 ) to the temperature (T 1 ). The data of dynamic viscosity in Table 1.1 can be used. This simple method to calculate the temperature dependence of saturated hydraulic conductivity cannot be used for variably saturated soils. In such soils, viscosity is not the only characteristic determining the change in soil-water properties with temperature. In variably saturated soil, the surface tension strongly influ-

20.7 Hydrophysical Characteristics of Soil and Temperature

315

Fig. 20.5 Hydraulic conductivity (k) of silicon sand SIAL-3, measured by the outflow method at temperatures 20, 30, 40 and 50 °C as a function of volumetric soil-water content (θ)

ences the capillary forces in soil. Figure 20.5 contains the hydraulic conductivities of variably saturated silicon sand SIAL-3 at temperatures 20, 30, 40 and 50 °C and volumetric soil-water content measured by the outflow method (Novák 1975). As was expected, the highest values of hydraulic conductivities were measured for silicon sand at the highest temperature applied (50 °C) because water-surface tension, as well as its dynamic viscosity, is the lowest among the temperatures applied. The important differences between the measured hydraulic conductivities were estimated in the range of relatively small soil-water contents, where the phase interfaces are the greatest.

20.7.2 Soil-Water Retention Curves and Temperature The significant dependence of soil-water retention curves (SWRC) on temperature is expected because the surface tension of water is strongly dependent on the temperature. The SWRC of silicon sand SIAL-3 at temperatures 20, 30, 40 and 50 °C are shown in Fig. 20.6; the silicon sand is chemically inert, and 72% of sand particles are in the range 0.05–0.25 mm, so they are coarse and of uniform texture. This sand was chosen to illustrate such a type of porous material covering the broad range of water contents by the relatively narrow range of negative-pressure heights (0–300 cm). As was expected, the highest retention capacity of porous medium was measured at the lowest of the temperatures used (20 °C), and the lowest soil-water capacity was found at the highest soil temperature (50 °C). The differences between SWRCs at various temperatures are significant.

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Fig. 20.6 Main drying branches of the soil-water retention curves of silicon sand SIAL-3 at temperatures 20, 30, 40 and 50 °C

20.8 Soil-Water Movement Under Non-isothermal Conditions As was shown, the quantification of non-isothermal conditions influence on soilwater movement is complicated. One of the possibilities to quantitatively express non-isothermal transport processes is the solutions of Eqs. (20.13) and (20.14). In principle, the soil temperature significantly influences evaporation, which is the drying of the surface-soil layer. The influence of temperature gradients in the surface layer of soils of temperate climate regions on the soil-water transport is generally not significant. The main reason is that the significant changes in soil temperatures can be observed mostly in the upper, ploughing soil layer, which is usually relatively dry. Laboratory experiments have shown that the water flux in an unsaturated zone of soil due to a temperature gradient is directed to the colder side of the soil profile, i.e. it means that the soil water moves to the bottom part of the soil profile during the day, where the colder side of the soil body is located. Usually, the downward water flow is not significant. Figure 20.7 is an illustration of this phenomenon; here are shown soil-water content profiles that are the results of evaporation from loamy soil to the controlled atmosphere. The soil-temperature gradient was 1 °C cm−1 , and the maximum temperature was at the soil surface. Accumulation of soil water can be seen at the bottom of the soil sample. Generalising this result, it can be expected that during the day, with the higher soil temperature at the soil surface, part of the soil water will be delivered to the bottom of the soil profile. During the night, the colder side of the soil profile will be near the soil surface, so soil water will be delivered toward

20.8 Soil-Water Movement Under Non-isothermal Conditions

317

Fig. 20.7 Soil-water content θ profiles along the loess soil sample during the evaporation of water from its surface. Duration of experiment was 48 h, the relative humidity above the soil surface was 0.1 (1) and 0.968 (2). Temperature gradient 1 °C cm−1 was directed upward, soil surface temperature was 36 °C, and initial volumetric soil water content was θ i  0.175 cm3 cm−3

the soil-surface direction, thus improving conditions for plants’ evapotranspiration during the next day (Novák 2012). It can be expected that water fluxes in the soil due to variable temperatures in the soil profile are not significant and are measurable under controlled conditions in the laboratory only, but it can be of importance during the extreme meteorological conditions and can contribute to survival of plants. In the field, cyclic fluctuations of the groundwater-table level located close to the soil surface were measured; the amplitude of such fluctuations was a few centimetres, and they were in accordance with the cyclic soil-temperature change. It is expected that the higher are the daily temperatures of soils (and lower water-surface tension), a drop in the groundwater-table level can be expected, as it can be seen in Fig. 20.6. The described groundwater-table fluctuations are probably not significant, but they indicate the principles of temperature influence on the soil-water movement. Differences between the SWRC at various temperatures can change the soil-profile water retention at a particular soil temperature.

References Climate and Bioclimate of Bratislava (1979) Veda, Bratislava (in Slovak) Globus AM (1983) Physics of non-isothermal water transport in soils. Gidrometeoizdat, Leningrad (in Russian) Novák V (1975) Non-isothermal flow of water in unsaturated soils. J Hydrol Sci (Poland) 2:37–52 Novák V (2012) Evapotranspiration in the soil-plant-atmosphere system. Springer Science + Business Media, Dordrecht

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Philip JR, de Vries DA (1957) Moisture movement in porous materials under temperature gradients. Trans Am Geophys Union 38:222–232 van Wijk WR, de Vries DA (1963) Periodic temperature variations inhomogeneous soil. In: van Wijk WR (ed) Physics in plant environment. North Holland Publ., Amsterdam

Chapter 21

Modelling of Water Flow and Solute Transport in Soil

Abstract The chapter presents the basic state of the art of water flow and solute transport modelling in unsaturated zone of soil. It provides a brief overview of model development and categorisation according to various criteria. The governing equations of water flow and solute transport, the time and space discretization of the computing domain, the initial and boundary conditions, the necessary input data and model outputs are presented. The chapter includes a short description of the most popular models of water and energy transport in variably saturated porous media. The challenges regarding the modelling of water flow, solute transport and marginally of other soil processes in the soil, along with references, are also mentioned.

21.1 Modelling in Soil Hydrology The aim of modelling in soil hydrology is to quantify and predict soil hydrological processes taking place in the unsaturated zone of a soil on various temporal and spatial scales. During recent past decades, numerous models were developed to simulate water flow and solute transport in the unsaturated soil zone. This progress is very closely related to the increasing development and power of personal computers. It has opened new possibilities for the numerical solution of partial differential equations describing water flow and solute transport in porous media. Simulation models are now of greater complexity and ability to characterise the complicated computational domain of geometry of a transport area in 2D or 3D environment than previous models. Originally, interest in studying soil water flow was initiated by agricultural needs, primarily to optimize soil-moisture conditions in the root zone for crop production. Later this interest shifted to the environmental area to study transport of various contaminants, such as pesticides, nutrients, pathogens, and pharmaceuticals. Applications in the environmental area were intended to evaluate water recharge through the vadose zone, salt leaching in arid areas with drip irrigation (Šim˚unek and Bradford 2008) and soil organic carbon and nutrient dynamics (Vereecken et al. 2016).

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváˇciková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1_21

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21 Modelling of Water Flow and Solute Transport in Soil

Models can be divided into two main categories: physical models and mathematical conceptual models. Physical models reflect the physical substance of the modelled medium, contrary to conceptual mathematical models in which flow and transport processes are expressed by a system of mathematical equations. An example of the physical model is a soil cylinder, or a soil lysimeter, in which soil hydrological and chemical processes are studied. Conceptual models include deterministic, stochastic and black-box models. Deterministic models, in which causal relations are strictly quantitatively defined, are dominant for vadose-zone modelling. The application of deterministic-stochastic models is also possible, and some inputs into the deterministic models can be stochastic variables. Artificial neural-network models belong to the group of “black-box” models which have also been frequently used in recent years. They are an alternative to conceptual deterministic mathematical models. Neural nets are data-controlled models; their applications are to be found even in the area of water management and hydrology. They use existing (measured) inputs and outputs of a process with an unknown mechanism, and they are able to find quantitative relationships among them. This type of model generates unknown results using known inputs into the self-educated structure. These models resemble a biological neural-network structure, adapted in the process of self-education for the solution of new problems (Bowden et al. 2005). Models can be further divided into various categories according to the temporal and spatial scales involved, and according to the complexity of the modelled processes. It is very important to decide how large a spatial scale is appropriate for solving the problem. Spatial scale can be a microscopic scale (soil pores), macroscopic scale or map scale (Kodešová 2012). Models at the microscopic scale quantify processes in individual soil pores. These kinds of models have made significant progress thanks to analytical techniques, such as X-ray tomography, neutron tomography, magnetic resonance, and others that enable the study of soil-pore structure development, or water flow (liquid and vapour) and chemical transport in individual soil pores. The macroscopic approach assumes a porous medium (soil) to be a homogeneous continuum quantitatively described by soil parameters valid for a representative elementary volume (REV). Soil characteristics differ for individual soil horizons, or they can also differ in a soil horizon. The macroscopic approach is widely used in a pedon scale. Large map scales are suitable for studying soil properties as a function of spatial scale, which enables identification of longer-running soil-formation processes (Kodešová 2012). Different time scales are used for various purposes. With the increasing capabilities of computers, larger amounts of data can be processed. It enables solution of problems on a short-time scale, e.g. a rainfall event, or on the longer scale of several decades, e.g. in the case of climate-change impact studies. Both time scales can be further modelled in finer time-step resolution according to the modelling purpose. The complexity of models is rapidly increasing, as well. The role of soil structure in the formation of soil hydrophysical properties was conceptualised in 1990s by Gerke and van Genuchten (1993). They worked out the concept of dual-porosity model for preferential water-flow modelling (described later). It is also known that

21.1 Modelling in Soil Hydrology

321

soil properties, like soil hydraulic conductivity and soil-water retention, vary with time and space. This variation can be incorporated in models by scaling factors (Vogel et al. 1991). Schwen et al. (2011) used different hydraulic properties for various time periods, which thus improved the modelling results. According to the solution techniques for the governing equations, the models can be divided into analytical, semi-analytical and numerical types. Analytical and semianalytical models solve governing equations by relatively simple analytical methods. They can be used only for simple applications with simple and well-defined boundary conditions. Numerical models solve governing equations by numerical methods; they are now widely used to solve more complex nonlinear processes (Šim˚unek and Bradford 2008).

21.2 Governing Equations of Water and Solute Transport in Soil Analytical, semi-analytical and numerical models are usually based on the solution of the following governing equations describing water, solute, and heat movement (Šim˚unek 2005): The Richards equation of water flow is:    ∂ ∂h w ∂θ (h w )  k(h w ) + 1 − S(z) (21.1) ∂t ∂z ∂z The equation describing solute transport is:   ∂ ∂c ∂θ Rc  θD − qc − φ ∂t ∂z ∂z Heat movement is described by the equation:   ∂ ∂T ∂T  λ(θ ) − Cw qT C(θ ) ∂t ∂z ∂z

(21.2)

(21.3)

where z is the vertical coordinate positive upward (L), t is time (T), hw is the soil-water pressure head (or soil matric potential) (L), θ is the volumetric soil water content (L3 L−3 ), S(z) is a sink term representing root water uptake or some other sources or sinks (L3 L−3 T−1 ), k(hw ) is the unsaturated hydraulic conductivity function; the latter is a function of the soil-pressure head (LT−1 ). In Eq. (21.2), known as the convection–dispersion equation, c is the solution concentration (concentration of the dissolved mass) (ML−3 ), R is the retardation factor that accounts for adsorption (−), D is the dispersion coefficient (L2 T−1 ), quantitatively characterising both molecular diffusion and hydrodynamic dispersion, q is the volumetric fluid flux density (LT−1 ) and φ is the sink (source) of solutes (ML−3 T−1 ), expressing zero- and first-order or

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21 Modelling of Water Flow and Solute Transport in Soil

other reactions. In Eq. (21.3), T is the temperature (K), λ is the soil specific thermal conductivity (LMT−3 K−1 ), C and C w are the volumetric heat capacities (ML−1 T−2 K−1 ) of the solid and the liquid phases respectively.

21.3 Characteristics of the Soil-Plant-Atmosphere Continuum: Model Inputs Necessary input data for deterministic soil-water flow models are meteorological conditions, plant (phenological) characteristics and the hydrophysical characteristics of a soil. Meteorological conditions strongly influence evapotranspiration and the infiltration rate of water out of, and into, a soil, respectively. Meteorological conditions include rainfall rates, air temperature, wind velocity, air humidity, sunshine duration, cloudiness and rates of solar radiation. Phenological characteristics of plants influence the transpiration rate and interception of water. The most important phenological characteristic of plants is LAI (leaf-area index), roots depth, vertical and horizontal distribution of roots, albedo and roughness of plant canopy. Hydrophysical characteristics (hydraulic properties) of the soil are the key inputs into simulation models. Among the most important hydrophysical characteristics of a soil belong the soil-water retention curve and the soil hydraulic-conductivity function. Soil profile is often divided into layers or spatial elements with particular hydrophysical characteristics to model two or three dimensional flow.

21.4 Main Model Outputs Among the main outputs of soil-water flow models belong soil-water content, soil pressure head and water flux as a function of space and time. In the case of solutetransport modelling, an additional output is solute concentration as a function of space and time. Because the vertical soil-water movement is dominant, it is usually simulated in the vertical direction. Simulation models make possible calculating the components of water-balance equation, as well, which include: actual and cumulative water and solute fluxes across the upper and lower boundary of the unsaturated soil zone (infiltration, evaporation and transpiration, capillary rise or groundwater recharge in the case of the groundwater table) and soil-water storage and its variation in time.

21.5 Governing Equations and Their Solutions

323

21.5 Governing Equations and Their Solutions Equations (21.2) and (21.3) that quantify solute and heat transport in soil, under certain conditions (usually steady-state ones), can be expressed as linear equations. The Richards Eq. (21.1) is highly nonlinear because of the nonlinear character of hydraulic functions θ (hw ) and k(hw ). Therefore, in most cases, the Richards equation can be solved by numerical methods only (Šim˚unek 2005). The first step in modelling is to define the geometry of the water-flow space, the flow domain. The area of a nonhomogeneous soil has to be divided into relatively homogeneous areas. In the case of one-dimensional modelling, the soil profile should be divided into individual soil layers to which particular soil-hydrophysical characteristics are assigned. Hydrophysical characteristics are estimated by field or laboratory measurements. The Richards equation can be solved by different numerical methods, such as the method of finite elements, finite differences or finite volumes, respectively. The governing equation is transformed into a system of algebraic equations solved by iterative methods; the different solutions in two consecutive calculation time-steps are compared. If an acceptable difference is found, the result can be considered as a final approximate solution. Spatial discretization of the flow domain into small discrete elements connected with each other in nodal points is needed (Fig. 21.1). The calculation grid is denser in the space where the rapid change of modelled characteristics is expected, as well as where the extreme hydraulic gradients are expected, as in the case of water infiltration into a dry soil. The discretization of time steps is needed, as well, but it usually changes during the modelling. Therefore, at the beginning of modelling, the limits have to be introduced (maximum and minimum time steps) which will not be exceeded during modelling. The solution of the governing equation depends on the correct specification of the initial and boundary conditions.

Fig. 21.1 Example of discretization of the flow domain (finite-element mesh)

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21.5.1 Initial and Boundary Conditions The initial condition is expressed by the distribution of soil-water contents or soilwater pressure heads in the flow domain at the beginning of the simulation. Boundary conditions are prescribed at the boundaries of the modelled area, and they define the interaction between the flow domain and surrounding environment. Boundary conditions may be generally applied to all transport-domain boundaries. When modelling is performed in one dimension, usually in the vertical one, they are defined at the upper and lower boundary. Boundary conditions can be expressed in “flux” form (Neumann’s condition) or in “pressure-head” form (Dirichlet’s condition). Boundary conditions can be introduced as system-independent (Dirichlet and Neumann) or as system-dependent. An example of the system-independent boundary condition is the prescribed soil-water pressure head at the upper soil boundary: h w (0, t)  h w0

(21.4)

This condition is defined for ponding infiltration, when the soil layer on the soil surface hwo (L) is applied. In this case the water-pressure head at the soil surface is positive and corresponds to the hydrostatic pressure. The pressure head of dry soil below the soil surface can be very negative; therefore, a large hydraulic gradient is expected. This situation may occur at the beginning of ponding infiltration into dry soil. Accordingly, at the bottom boundary of the soil profile, a constant value of pressure head can be applied. The boundary between the unsaturated zone of soil and the groundwater table is characterised by the zero pressure head. The bottom boundary condition can be set as free drainage, corresponding to the steady-state of downward water flow at a unit hydraulic gradient; then the water table is situated far below the flow domain. In this case, the water flux is equal to the soil hydraulic conductivity (Radcliffe and Šim˚unek 2010). A constant-value boundary condition cannot be set in some cases. The soil surface, the soil–air interface, is an example of a system-dependent boundary condition. The potential water flux at this boundary is controlled by external conditions (precipitation, evaporation). However, the actual water flux depends on the transient moisture conditions. An example is rainfall infiltration where infiltration rate equals the rainfall rate until the infiltration capacity of the soil is exceeded. This state occurs when ponding starts. Then, the infiltration rate is no longer controlled by the rainfall rate, but by the infiltration capacity of the soil. A similar case occurs when the potential evaporation rate, which is controlled by the evaporative demand of the atmosphere, exceeds the capability of the soil to deliver enough water to the evaporating surface. In this case, the potential evaporation rate can be significantly reduced to the evaporation rate which is controlled by the soil (Radcliffe and Šim˚unek 2010).

21.6 Water Flow Modelling in Heterogeneous Soils

325

21.6 Water Flow Modelling in Heterogeneous Soils The traditional water-flow modelling approach explained in the previous section assumes the transport of water, solute and heat in a unimodal porous system. It assumes the presence of a one-type soil porous system in a soil resulting in a uniform water and solute transport across the whole soil matrix. This type of model can be applied only for relatively homogeneous soils without macropores or soil cracks, where the wetting front is regular and stable; flow instability is not observed. During recent decades, it has been recognized that the water flow and solute transport in structured heterogeneous soils cannot be accurately modelled by the application of the classical single-domain approach because of the nonequilibrium nature of the water flow. Beven and Germann (1982) pointed out the role of macropores on soil hydrological behaviour. The non-uniform character of the water flow is the effect of the heterogeneity of a soil’s porous system, soil texture and structure. Soil structure belongs to the most important soil properties that affect soil porosity and consequently the nature of water flow in a soil. Soil structure heterogeneities result from long-term, soil forming processes, biological activities of soil organisms and plants, and human activities, e.g. soil tillage and soil cultivation. Systems of soil pores, through which water and solutes are transported, are very dynamic in space and time. Actually, strong links were found between soil-structure heterogeneity and the occurrence of fast non-uniform preferential soil-water flow (Lin 2010). Preferential flow is a good example of a non-uniform water flow in a soil. The fast flowing water in the system of macropores or other preferential pathways does not have enough time to consolidate with the water flowing at much lower rates in a soil matrix and thus are creating non-uniform flow fields with widely varying velocities (Gerke and van Genuchten 1993). Preferential flow can very quickly transport the large portion of water and solutes into deep soil horizons, so eventually the solute front reaches the groundwater table, bypassing the soil matrix. The soil-profile retention capacity in such a case is used only partially (Weiler and Naef 2003). This type of flow is often observed in soils with a high content of clay minerals. Clayey soils are typified by their low saturated hydraulic conductivity. Soil cracks appear as a result of soil shrinkage during soil drying; their properties are quite different than the soil matrix pores. However, the presence of macropore flow was observed as well in agricultural, grassland and forest soils, respectively, by many field experiments (Scherrer and Naef 2003; Sander and Gerke 2007; Capuliak et al. 2010; Alaoui et al. 2011; Gerke et al. 2015; Laine-Kaulio et al. 2015). Advances and future perspectives concerning an up-to-date understanding of preferential flow phenomena can be found in Jarvis et al. (2016). The influence of soil structure on soil hydrophysical processes was conceptualised in “a dual-porosity approach” introduced by Gerke and van Genuchten (1993). The concept of dual-porosity modelling, which can quantitatively better describe the preferential water flow, and solute transport in heterogeneous soils was later elaborated. Models for the principle of dual-porosity and dual-permeability have been enhanced to the multi-porosity and multi-permeability concepts. They are based on the same

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concept as dual-porosity and dual-permeability models, but include additional interacting pore regions (Šim˚unek and van Genuchten 2008). Dual-porosity and dual-permeability models are being applied to both water flow and solute transport. The principles of dual porosity and dual permeability models applied primarily to water flow are briefly described.

21.6.1 Dual-Porosity Model A dual-porosity model assumes the water flow is restricted to the fast-flow domain (fractures, inter-aggregate pores or macropores). The matrix (the soil matrix, intraaggregate pores or the rock matrix) represents an immobile water domain that is only exchanging water with the fast-flow domain (drying or wetting) and not participating on the soil-water movement (Fig. 21.2). Water flow in the dual-porosity model is expressed by the combination of the Richards equation and the mass-balance equation (Šim˚unek and van Genuchten 2008):    ∂ ∂h mo ∂θmo (h mo )  k(h mo ) + 1 − Smo (h mo ) − Γw (21.5) ∂t ∂z ∂z ∂θim (h im )  −Sim (h im ) + Γw (21.6) ∂t where S mo and Sim are sink terms for the mobile and immobile domains, respectively (L3 L−3 T−1 ) and Γ w is the mass-transfer rate between the mobile and immobile domains (T−1 ). Various formulations for the evaluation of the mass-transfer rate Γw are discussed by Šim˚unek et al. (2003), Köhne et al. (2004), and Gerke (2012).

Fig. 21.2 Dual-porosity model; arrows show the flow direction. Adapted from Šim˚unek and van Genuchten, (2008)

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21.6.2 Dual-Permeability Model A dual-permeability model assumes the soil is composed of two porous domains both participating in the soil-water flow (Fig. 21.3). The first domain represents the fastflow domain (fractures, inter-aggregate pores or macropores). The second domain represents the slow-flow domain (the soil matrix, intra-aggregate pores and the rock matrix). Soil-water flow in the dual-permeability model is expressed by two Richards equations applied to each pore domain, separately (Šim˚unek and van Genuchten 2008):    ∂h f ∂θ f (h f ) ∂ Γw  k f (h f ) + 1 − S f (h f ) − (21.7) ∂t ∂z ∂z w    ∂θm (h m ) ∂ ∂h m Γw  km (h m ) + 1 − Sm (h m ) + (21.8) ∂t ∂z ∂z 1−w where w is the ratio of the macropores volume (inter-aggregate pores or cracks) and the volume of the soil porous system. Index f denotes the macropores domain; index m denotes the soil matrix. This approach is relatively complicated because it requires knowing the water retention and hydraulic conductivity function for both porous domains, as well as the hydraulic-conductivity function of the macropores-matrix interface (Šim˚unek and van Genuchten 2008). Dual-porosity and dual-permeability models differ mainly by how they implement the water flows in both domains, especially in the fast-flow domain. Ahuja and Hebson (1992) expressed the water flow in macropores by the Poiseuille’s equation; the Chézy-Manning equation describing turbulent flow was applied by Chen and Wagenet (1992); the kinematic wave approach was applied by Germann and Beven (1985) and Jarvis (1994); and Richards equation was used by Gerke and van Genuchten (1993). The main problem of multi-domain models is the large number of parameters characterising each pore domain and the interactions between them. The number of

Fig. 21.3 Dual-permeability model; arrows show the flow direction. Adapted from Šim˚unek and van Genuchten (2008)

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unknown parameters increases proportionally to the model complexity. For example, there are four unknown parameters in a single-porosity model, whereas in a dualpermeability model there are nine unknown parameters. An example of the parameter estimation of dual-permeability modelling by cumulative-infiltration measurements in combination with inverse modelling was demonstrated by Kodešová et al. (2010). Detailed information about the principles, measurements and modelling of the preferential nonequilibrium water flow and solute transport in heterogeneous soils can be found in Jarvis (2007), Allaire et al. (2009), Gerke (2006), Šim˚unek and van Genuchten (2008), Köhne et al. (2009), and Jarvis et al. (2016).

21.7 Calibration, Verification and Validation of Soil-Water Flow Models To quantify the water flow and the solute transport in soil, it is necessary to know all parameters in the governing equations that characterise the soil properties. The parameters characterising soil properties such as the soil-water retention and hydraulic-conductivity functions are estimated by field and laboratory measurement. This method is called the direct solution. The results of the modelling are water contents, pressure heads and water fluxes. However, such results differ from the measured soil properties (measured water contents, pressure heads, water fluxes or solute concentrations in a soil profile). Therefore, to estimate soil hydrophysical properties using the above measured data, the inverse solution is used. There are two main reasons to use this type of solution. First is the possibility to estimate the unknown soil parameters needed in governing equations using relatively easily measured water contents, pressure heads, water fluxes (e.g. infiltration rates) or solute concentrations during soil-water movement. This application of inverse solution is called parameter estimation or parameter optimization. This approach can be applied to dual-permeability models in structured soils where large number of parameters is unknown and it is not possible to measure them. Infiltration measurements are widely used to estimate unknown soil parameters. The second reason for using the inverse solution is to achieve better modelling results. This application of inverse solution is called model calibration. According to Šim˚unek and de Vos (1999), it is defined as the process of tuning a model for a particular problem by manipulating the input parameters (e.g. hydraulic conductivities) and initial or boundary conditions, in reasonable ranges until minimum differences between modelled and observed variables are achieved. The primary goal is to find “a best set of parameters” that produce minimum differences between measured and modelled data. The numerical solution to achieve model calibration or parameter estimation is called inverse modelling, or inverse numerical optimization. In the past, model calibration or parameter optimization were performed manually. Nowadays, it is a fully computational and numerical process with specific algorithms. The best choice of a

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parameter set is obtained by solving the governing equations repeatedly with various sets of parameters and by finding the minima of the objective function using a minimization technique. It is very important to define a suitable objective function for a particular problem. Various inverse parameter optimization algorithms have been developed and their improvement is ongoing (Vrugt et al. 2008). The main tasks are: Is it possible to find a best fit parameters set; Is this set unique; or What are the uncertainties connected with the application of such a set of parameters in a model? Verification of the model can be understood as a demonstration that the numerical code of the model (governing equations) accurately describes the simulated process. Validation of the model means determining if the model was correctly set up for a particular problem. The measured data set is divided into two groups: the first group of data are used to calibrate the model and to estimate all needed parameters; the second data group is used to validate the model. This involves comparing the modelled and measured data values using calibrated parameters gained from the first data group. Calibrated and validated model can then be used for various applications. Actually, the term “validation” has a more complicated meaning among model developers (Šim˚unek and de Vos 1999). Examples of inverse modelling and discussions of their applicability can be found in Šim˚unek and de Vos (1999), Šim˚unek and Hopmans (2002), Hopmans et al. (2002), Vrugt et al. (2008), Köhne et al. (2009) and Twarakavi et al. (2010).

21.8 Overview of Some Soil-Water Flow Models Soil-water flow modelling started with a simple bucket or reservoir models, which were referred to as water-balance models. These models represented the unsaturated zone as one or more storage reservoirs or soil layers. The reservoirs could fill and drain according to actual hydrological processes. The water content in each reservoir was calculated using the soil-water balance equation. The disadvantage of these models was in the discontinuous information about simulated variables, e.g. soilwater contents. They presented the state of soil water, soil-water content, in a given time interval, but not the mechanism of soil-water movement. However, improved versions of bucket models are still being used; they avoid the computational requirements that are needed for numerical solutions of the Richards equation. Their simple nature is used in some watershed models as an alternative way of estimating nearsubsurface water flow over large areas (Healy 2008). Finally, numerical models, based on the Richards equation, replaced bucket models; those numerical models make it possible to simulate water flow and multiphase flow in various geometries of transport domains, with various initial and boundary conditions. Model HYDRUS The model HYDRUS is a mathematical, deterministic model simulating the transport of water, heat and multiple solutes in variably saturated media, respectively (Šim˚unek et al. 2013). The model makes it possible to estimate infiltration, soil-moisture stor-

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age, evaporation, plant-water uptake, groundwater recharge and surface-runoff calculation. In the one-dimensional version, the model is publicly available, whereas the 2D/3D versions are distributed commercially. The model is typified by an interactive, graphics-based, user-friendly interface for the MS Windows environment. The Richards equation is solved numerically by the finite element method (Radcliffe and Šim˚unek 2010). In the 1D dimension, the Richards equation can be applied for one-dimensional water flow in various directions (vertical and horizontal, or in any other direction characterised by the angle α):    ∂h w ∂ ∂θ  k(h w ) + cos α − S(z) (21.9) ∂t ∂z ∂z where hw is the water-pressure head (L), θ is the volumetric soil water content (L3 L−3 ), t is time (T), z is the spatial coordinate (positive upward) (L), k(hw ) is the unsaturated soil hydraulic conductivity (LT−1 ), S(z) is the sink term (L3 L−3 T−1 ) and α is the angle between the flow direction and the vertical axis (i.e. α  0° for vertical flow, 90° for horizontal flow, and 0° < α < 90° for inclined flow). The sink term accounts for water uptake by plant roots as a function of both water and salinity stress. Model HYDRUS enables simulating ponding infiltration with a prescribed depth of soil-water layer on the soil surface or the rainfall infiltration by setting the rain rate of the precipitation event. The infiltration rate is calculated numerically as the water flux at the soil surface (z  0) described by the Darcy-Buckingham equation in the form (Radcliffe and Šim˚unek 2010):    ∂h w (z, t) +1 (21.10) vi (t)  −q(t)  k(h w ) ∂z where vi (t) is the infiltration rate (LT−1 ), q(t) is the flow rate (LT−1 ), k(hw ) is the unsaturated soil hydraulic conductivity (LT−1 ) and hw is the soil-water pressure head at the soil surface (L). Modelling rainfall infiltration, it is possible to set the criteria for the surface-runoff formation. This means it is possible to include a small critical water layer which can form on the soil surface, corresponding to soil-surface depressions, so that excess water is immediately removed and contributes to surface runoff. Model HYDRUS-1D provides: – coupled transport of water, vapour and energy (thermal and isothermal, in the liquid and gaseous phases), – dual-porosity and dual-permeability approaches to water flow and solute transport, – surface energy balance for bare soils, – potential evapotranspiration calculated by the Penman–Monteith equation or by the empirical Hargreaves equation, – the opportunity to achieve soil hydrophysical parameters from a small soil catalogue or by the Rosetta Lite program using soil textural data,

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– daily variations in evaporation, transpiration and precipitation rates generated by HYDRUS from daily values, – plant canopy interception, – uncompensated and compensated root-water and solute uptake, – snow accumulation, melting, and evaporation. Except for the standard module, HYDRUS 2D/3D provides modules for transport of particle-like substances (e.g. viruses, colloids, bacteria) and particle-facilitated transport (e.g. heavy metals, radionuclides, pharmaceuticals, pesticides etc.), a dualpermeability module for structured soils, the module for transport of major ions, the wetland module and the fumigant module. Nonstandard modules are available too. An overview of all available modules either for HYDRUS-1D or 2D/3D version, as well as their applications together with the review of developments in the recent decade, can be found in Šim˚unek et al. (2016). The conceptual submodel quantitatively describing the infiltration of water into fine-textured soils with cracks was part of the HYDRUS-ET model. This model arose as the modification of the classical HYDRUS model thanks to AmericanSlovak cooperation (US Salinity Laboratory, Riverside, CA, USA, and The Institute of Hydrology, Slovak Academy of Sciences, Bratislava) (Šim˚unek et al. 1997). Water infiltrating from cracks into the soil matrix was estimated using the Green–Ampt approach. It was then added as a source term to the Richards equation describing water flow in the soil matrix (Novák et al. 2000; Šim˚unek et al. 2003). This model required crack input parameters, like the specific length of cracks per unit soil-surface area and the crack-porosity profile (Novák et al. 2000). Model SWAP The model SWAP is the agro-ecohydrological (Soil-Water-Atmosphere-Plant) onedimensional (in the vertical direction) mathematical model. Model SWAP is the next version of the previous agrohydrological model SWATR and crop-growth routine CROPR, designed in Wageningen (Feddes et al. 1978; van Dam et al. 2008). After some modifications of the SWATR model, the model name was changed to SWATRE, and finally to SWAP in 1997. Model SWAP is primarily used for agricultural and water-management applications. It can evaluate the influence of various land uses on crop growth and crop production, or the impact of surface and groundwater strategies on crop development. Drainage to, or infiltration, from surface-water systems are calculated with two-dimensional drainage equations, which makes possible evaluation of drainage-system design. Actually, it can simulate coupled water flow, solute transport, heat flow, flow in macroporous and water-repellent soils and crop growth. Model SWAP calculates the water flow by the Richards equation and the solute transport by the convectiondispersion equation, similarly to the HYDRUS model. Specific features of the SWAP model are simulation of crop growth (using submodel WOFOST), optimal irrigation doses, evapotranspiration of partly covered soils by plants, flow in water-repellent and macroporous soils, interaction of soil moisture and surface-water management and

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possible coupling with pesticide and nutrient models (PEARL and ANIMO model, respectively). Applications of the model and detailed information can be found in the work of van Dam et al. (2008). Model COUP The COUP model was developed in the early 1980s in Sweden. It was primarily intended for forest soils and later modified for soils with other canopies. It is a onedimensional model (in the vertical direction) simulating water flow, solute and heat transport in various types of soils covered or uncovered with vegetation. The soil profile is divided into several soil layers with different soil properties. The model contains a database of Swedish forest and agricultural soils and their parameters. The use of the database enables estimating soil parameters from commonly available information such as soil texture and organic-matter content. Excepting water and heat module, the model provides tracer, chloride, nitrogen and carbon modules to simulate interactions in the soil-plant-atmosphere system. The snow module (snow accumulation and melting), evaluation of rapid-bypass flow in macropores and simulation of processes under frozen conditions are available as well. The model provides the possibility to change some dynamic parameters at specified dates during the simulation, and model output variables can be compared with measurements. The model performance is evaluated by statistical methods. Detailed information can be found in the paper by Jansson (2012), and Jansson and Karlberg (2004). Model MACRO The MACRO model is a widely used one-dimensional, dual-permeability model for water flow and reactive or non-reactive solute transport in a one-dimensional (vertical) soil environment. The model was primarily developed and is used to estimate the effects of macropore flow and contaminant transport in structured soils. Its main application is in risk assessment for groundwater and surface waters. The model calculates the water flow in the soil matrix by the Richards equation and the solute transport by convection-dispersion equation. The macropore flow is calculated as a gravity-driven flow (ver. 5.2) (Jarvis and Larsbo 2012). Main application areas include: simulations of water flow and pesticide leaching from drained agricultural fields; estimation of contaminant concentration in groundwater; and reactive and non-reactive solute transport in a macroporous soil. Further descriptions and application of the model are available in Stenemo and Jarvis (2010), Bachmair et al. (2010) and Jarvis and Larsbo (2012). Model DAISY DAISY was developed in Denmark as a one-dimensional agro-ecosystem model for simulating water and nitrogen dynamics and crop growth in a soil root zone. The model simulates water balance, nitrogen balance and losses, development in soil organic matter and crop growth and production in crop rotations under alternate management strategies (Abrahamsen and Hansen 2000). The model inputs are meteorological conditions (mainly precipitation, global radiation, air temperature), agronomic data (soil tillage, time of sewing and yield harvesting, quantities and time of irrigation application, as well as data about application of fertilizers and pesticides) and soil characteristics (soil texture and organic-matter content). The model

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has been rewritten and improved so that it can be coupled with other models. Details can be found in the papers by Hansen et al. (1991), Abrahamsen and Hansen (2000) and Hansen et al. (2012). Model TOUGH—“Transport of Unsaturated Groundwater and Heat” This is a model simulating coupled soil-water, vapour and heat flow in porous soil and fractured unsaturated media. The specific feature of the model is that it is focused on hydrogeological features, physical processes and thermodynamic properties that are very useful for studying and predicting more complex phenomena in near-surface or in deeper unsaturated zones. It was developed at the Lawrence Berkeley National Laboratory (LBNL), USA, in the early 1980s with the aim to evaluate the function of geothermal reservoirs. It is currently utilized by governmental organisations and universities and in the nuclear industry to process nuclear waste; or in geothermal energy transformation facilities. The model is continually updated and provides several simulation modules. Their description can be found at the website of the LBNL. The application areas of this model and its ability to be coupled with other models are described in Finsterle et al. (2008). Models Coupling The unsaturated zone is part of the complex natural environment. It is confined from above by the atmosphere and from below by the saturated zone in the case of an existing groundwater level (GWL). Naturally complicated boundary conditions in water-flow models in the unsaturated zone are usually simplified, which is often sufficient for many model applications. The same is valid for models of the water flow in saturated zone. A reasonable degree of simplification is useful because it simplifies the mathematical solution and speeds up the simulation time. On the other hand, it leads to incompleteness (and less accuracy) of the model (Furman 2008). Models simulating water flow in an unsaturated zone of soil often simplify spatial and temporal changes at their boundaries (e.g. changes in GWL). This may be caused by the lack of sufficient data. Similarly, regional groundwater-flow models simplify vadose-zone processes and require only that groundwater recharge from the vadose zone is calculated externally by other simplified procedures. Therefore, to solve the regional spatial interaction between both water zones, there is a need to create “largescale” models or coupled models. There are examples of such model couplings, and one of them is the, HYDRUS-MODFLOW. The MODFLOW is a three-dimensional, finite-difference, groundwater-flow model. The advantage of such integration is in the ability of the HYDRUS model to provide simulated recharge fluxes at the lower boundary (of the groundwater table), which serve as inputs (the upper-boundary condition) for the MODFLOW model. The MODFLOW model provides simulated positions of the groundwater table which serves as the bottom boundary condition in the next time step to the HYDRUS model. More details can be found in Twarakavi et al. (2008) and Furman (2008). Other examples of coupling models can be found in large watershed models and in models coupling water flow, solute transport and biological processes in Soil-Plant-Atmosphere systems. In the abovementioned models, no reference was made to their possibilities concerning time resolution and model calibration. Nowadays, almost all frequently used

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models also provide parameter estimation or calibration, which is incorporated in the model. Time scales may differ; usually the smallest time step is expressed in minutes (or seconds), and the largest time step may be in years. Several models are freely available with open-source codes, others not. Comparisons of the abovementioned and other models can be found in Köhne et al. (2009) and Moriasi et al. (2012).

21.9 Current Trends and Future Challenges Concerning Water-Flow Modelling in the Unsaturated Soil Zone Current trends in the area of mathematical modelling of energy and mass transport in the vadose zone are driven by the necessity mainly to solve environmental problems (the impact of pollution, climate change, water supply, crop production and many others). Current trends in simulation models development can be divided into three categories (Finsterle et al. 2008): 1. The necessity to solve problems related to the heterogeneity and complexity of hydrophysical and biogeochemical processes in the vadose zone; better understanding of such processes and their incorporation into models is needed. 2. The vadose zone is located at the interface between spheres (types of environments, like atmosphere, biosphere and hydrosphere) that interact between each other. Therefore, integration of particular submodels into “large-scale” models is needed. 3. Complex nonlinear processes solved on various spatial and time scales, together with model coupling, pose new computing challenges in the area of numerical codes development. Simulation-model development is enormous. The application of contemporary computers and parallel-computation methods enables the processing of great amounts of data in a short-time interval. There are many factors that influence the physical, chemical and hydraulic properties of soil in space and time. Among them are, e.g. soil formation, erosion, soil biological activities, nutrient cycle, climate change and anthropogenic influences. They are often modelled by separate models. Today’s need is to improve the exchange of knowledge and experience among the different disciplines involved in soil science. Soil physicists, hydrologists, chemists, soil biologists etc. should cooperate and contribute their knowledge in larger complex models. The latest review of not only water flow but also other soil-processes modelling in its maximum complexity, as well as key challenges and new perspectives, are found in Vereecken et al. (2016). Why are there still smaller or larger differences between simulated and measured data? There are more reasons for that. They can be generalized into three main groups of uncertainties: (1) the numerical code incorporated in the model does not include all processes occurring in the soil, (2) another group of uncertainties is connected with the difficulties of soil hydraulic functions and parameters estimation and their

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representativeness for a particular site and time and, (3) uncertainties connected with measured values of water content and pressure heads (accuracy and calibration of a measurement device, its representativeness, interpretation of measured values). Model developers are trying to develop more and more sophisticated, complex models. One way to improve the model is, for example, by sensitivity analysis. The results of sensitivity analysis can reveal the importance of individual model parameters. This can provide directions for further research (Šim˚unek and de Vos 1999). Model users must always clarify the purpose for which the model is intended to be used, the reasonable accuracy needed and, last but not least, a cost benefit analysis should be known, as well. It is very important to check in advance if all the needed data are available. For the simulation of simple processes on a relatively small area, a simpler model is sufficient. In the case of describing complicated processes with a high degree of expected accuracy or simulating processes on a regional scale, a more sophisticated model is needed. However, each model represents a simplified picture of natural processes in a certain way. Therefore, the modelling results have validity that is always bound with limitations for their use.

References Abrahamsen P, Hansen S (2000) Daisy: an open soil-crop-atmosphere system model. Environ Model Softw 15:313–330 Ahuja LR, Hebson C (1992) Water, chemical, and heat transport in soil matrix and macropores. In: Root zone water quality model, version 1.0 technical documentation. GPSR Tech. Report No. 2, USDA-ARS-GPSR, Colorado Alaoui A, Caduf U, Gerke HH, Weingartner R (2011) Preferential flow effects on infiltration and runoff in grassland and forest soils. Vadose Zone J 10:367–377 Allaire SE, Roulier S, Cessna AJ (2009) Quantifying preferential flow in soils: a review of different techniques. J Hydrol 378:179–204 Bachmair S, Weiler M, Nützmann G (2010) Benchmarking of two dual-permeability models under different land use and land cover. Vadose Zone J 9:226–237 Beven K, Germann P (1982) Macropores and water flow in soils. Water Resour Res 18:1311–1325 Bowden GJ, Dandy GC, Maier HR (2005) Input determination for neural network models in water resources applications. Part 1- background and methodology. J Hydrol 301:75–92 Capuliak J, Pichler V, Flühler H, Pichlerová M, Homolák M (2010) Beech forest density control on the dominant water flow types in andic soils. Vadose Zone J 9:747–756 Chen C, Wagenet RJ (1992) Simulation of water and chemicals in macropore soils. Part 1. Representation of the equivalent macropore influence and its effect on soil water flow. J Hydrol 130:105–126 Feddes RA, Kowalik PJ, Zaradny H (1978) Simulation of field water use and crop yield. Simulation Monographs, Pudoc for the Centre for Agricultural Publishing and Documentation, Wageningen, The Netherlands, p 189 Finsterle S, Doughty C, Kowalsky MB, Moridis GJ, Pan L, Xu T, Zhang Y, Pruess K (2008) Advanced vadose zone simulations using TOUGH. Vadose Zone J 7:601–609 Furman A (2008) Modeling coupled surface-subsurface flow processes: a review. Vadose Zone J 7:741–756

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Gerke HH (2006) Preferential flow descriptions for structured soils. J Plant Nutr Soil Sci 169:382–400 Gerke HH (2012) Macroscopic representation of the interface between flow domains in structured soil. Vadose Zone J 11. https://doi.org/10.2136/vzj2011.0125 Gerke HH, van Genuchten MT (1993) A dual-porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resour Res 29:305–319 Gerke KM, Sidle RC, Mallants D (2015) Preferential flow mechanisms identified from staining experiments in forested hillslopes. Hydrol Process 29:4562–4578 Germann PF, Beven K (1985) Kinematic wave approximation to infiltration into soils with sorbing macropores. Water Resour Res 21:990–996 Hansen S, Jensen HE, Nielsen NE, Svendsen H (1991) Simulation of nitrogen dynamics and biomass production in winter wheat using the Danish simulation model Daisy. Fertil Res 27:245–259 Hansen S, Abrahamsen P, Petersen CT, Styczen M (2012) Daisy: model use, calibration, and validation. Trans ASABE 55:1315–1333 Healy RW (2008) Simulating water, solute, and heat transport in the subsurface with the VS2DI software package. Vadose Zone J 7:632–639 Hopmans JW, Šim˚unek J, Romano N, Durner W (2002) 3.6.2 inverse methods. In: Dane JH, Topp GC (eds) Methods of soil analysis. Part 4. Physical methods. SSSA Book Series 5. SSSA, Madison, WI, pp 963–1008 Jansson PE (2012) Coup model: model use, calibration, and validation. Trans ASABE 55:1303–1313 Jansson PE, Karlberg L (2004) Coup manual. Coupled heat and mass transfer model for soilplant-atmosphere systems. Royal Institute of Technology, Department of Water and Resources Engineering, Stockholm. https://www.researchgate.net/publication/292875837 Jarvis N (1994) The MACRO model (version 3.1): technical description and sample simulations. Reports and Dissertations 19, Swedish Univ of Agric Sci, Dept Soil Sci, Uppsala Jarvis NJ (2007) A review of non-equilibrium water flow and solute transport in soil macropores: principles, controlling factors and consequences for water quality. Eur J Soil Sci 58:523–546 Jarvis N, Larsbo M (2012) MACRO (v5.2): model use, calibration, and validation. Trans ASABE 55:1413–1423 Jarvis N, Koestel J, Larsbo M (2016) Understanding preferential flow in the vadose zone: recent advances and future prospects. Vadose Zone J 15. https://doi.org/10.2136/vzj2016.09.0075 Kodešová R (2012) Modeling in pedology. Czech University of Life Sciences, Prague, pp 152 (in Czech) Kodešová R, Šim˚unek J, Nikodem A, Jirk˚u V (2010) Estimation of the dual-permeability model parameters using tension disk infiltrometer and Guelph permeameter. Vadose Zone J 9:213–225 Köhne JM, Mohanty B, Šim˚unek J, Gerke HH (2004) Numerical evaluation of a second-order water transfer term for variably saturated dual-permeability models. Water Resour Res 40:W07409. https://doi.org/10.1029/2004WR003285 Köhne JM, Köhne S, Šim˚unek J (2009) A review of model applications for structured soils: a) Water flow and tracer transport. J Contam Hydrol 104:4–35 Laine-Kaulio H, Backnäs S, Koivusalo H, Laurén A (2015) Dye tracer visualization of flow patterns and pathways in glacial sandy till at a boreal forest hillslope. Geoderma 259–260:23–34 Lin H (2010) Linking principles of soil formation and flow regimes. J Hydrol 393(1–2):3–19 Moriasi DN, Wilson BN, Douglas-Mankin KR, Arnold JG, Gowda PH (2012) Hydrologic and water quality models: use, calibration, and validation. Trans ASABE 55:1241–1247 Novák V, Šim˚unek J, van Genuchten MT (2000) Infiltration of water into soils with cracks. J Irrig Drain Eng 126:41–47 Radcliffe DE, Šim˚unek J (2010) Soil physics with HYDRUS. Modeling and applications. CRC Press, Taylor & Francis Group, Boca Raton, USA Sander T, Gerke HH (2007) Preferential flow patterns in paddy fields using a dye tracer. Vadose Zone J 6:105–115 Scherrer S, Naef F (2003) A decision scheme to indicate dominant hydrological flow processes on temperate grassland. Hydrol Process 17:391–401

References

337

Schwen A, Bodner G, Loiskandl W (2011) Time-variable soil hydraulic properties in near-surface soil water simulations for different tillage methods. Agric Water Management 99:42–50 Šim˚unek J (2005) 78 models of water flow and solute transport in the unsaturated zone. In: Anderson MG (ed) Encyclopedia of hydrological sciences, Part 6. Soils. Wiley, New York, pp 1171–1180 Šim˚unek J, Bradford SA (2008) Vadose zone modeling: introduction and importance. Vadose Zone J 7:581–586 Šim˚unek J, de Vos JA (1999) Inverse optimization, calibration and validation of simulation models at the field scale. In: Feyen J, Wyio K (eds) Modelling of transport processes in soils at various scales in time and space. International workshop of EurAgEng’s field of interest on soil and water, 24–26 Nov, Leuven (Belgium), Wageningen Pers., Wageningen, pp 431–445 Šim˚unek J, Hopmans JW (2002) 1.7 parameter optimization and nonlinear fitting. In: Dane JH, Topp GC (eds) Methods of soil analysis. Part 4. Physical methods. SSSA Book Series 5. SSSA, Madison, WI, pp 139–158 Šim˚unek J, van Genuchten MT (2008) Modeling nonequilibrium flow and transport processes using HYDRUS. Vadose Zone J 7:782–797 Šim˚unek J, Huang K, Šejna M, van Genuchten MT, Majerˇcák J, Novák V, Šútor J (1997) The HYDRUS-ET software package for simulating the one-dimensional movement of water, heat and multiple solutes in variably saturated media, version 1.1. Institute of Hydrology, Slovak Academy of Sciences, Bratislava, p 184 Šim˚unek J, Jarvis NJ, van Genuchten MT, Gärdenäs A (2003) Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J Hydrol 272:14–35 Šim˚unek J, Šejna M, Saito H, Sakai M, van Genuchten MT (2013) The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat, and multiple solutes in variably-saturated media, version 4.17. Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, p 308 Šim˚unek J, van Genuchten MT, Šejna M (2016) Recent developments and applications of the HYDRUS computer software packages. Vadose Zone J. https://doi.org/10.2136/vzj2016.04.0033 Stenemo F, Jarvis N (2010) Users guide to MACRO 5.2, a model of water flow and solute transport in macroporous soil. Technical descriptions. Swedish University of Agricultural Sciences, Department of Soil and environment, Division Biogeophysics and Water Quality. https://www. slu.se/globalassets/ew/org/centrb/ckb/modeller_dokument/macro-users-guide-2010.pdf Twarakavi NKC, Šim˚unek J, Seo S (2008) Evaluating interactions between groundwater and vadose zone using the HYDRUS-based flow package for MODFLOW. Vadose Zone J 7:757–768 Twarakavi NKC, Saito H, Šim˚unek J, van Genuchten MT (2010) Inverse modeling of vadose zone flow processes using squared ε-intensitivity loss function. J Hydrol Hydromech 58:188–200 van Dam JC, Groenendijk P, Hendriks RFA, Kroes JG (2008) Advances of modeling water flow in variably saturated soils with SWAP. Vadose Zone J 7:640–653 Vereecken et al (2016) Modeling soil processes: review, key challenges, and new perspectives. Vadose Zone J. https://doi.org/10.2136/vzj2015.09.0131 Vogel T, Cislerova M, Hopmans JW (1991) Porous media with linearly variable hydraulic properties. Water Resour Res 27:2735–2741 Vrugt JA, Stauffer PH, Wöhling Th, Robinson BA, Vesselinov VV (2008) Inverse modeling of subsurface flow and transport properties: a review with new developments. Vadose Zone J 7:843–864 Weiler M, Naef F (2003) An experimental tracer study of the role of macropores in infiltration in grassland soils. Hydrol Process 17:477–493

Index

A Air absolute humidity, 9 in soil, 3, 16 in water, 7, 32, 146 relative humidity, 10, 45, 92–94 specific humidity, 8, 210, 221 Atmosphere, 1–3, 6–11, 17, 32, 44, 85, 91, 92, 129, 145, 189–194, 196–201, 206–209, 213, 214, 222, 224, 246, 289, 293, 295, 298, 303, 306, 316, 324, 333, 334 B Biomass production, 2, 3, 6–8, 11, 12, 29, 86, 192, 193, 200, 243, 247, 249, 289, 293, 299 Breakthrough curve, 229, 238–242 Bubbling pressure, 74, 80, 84, 88, 89, 132, 145, 284, 285, 288 C Canopy density, 12 height, 12, 209, 213, 215 Capillary condensation, 37, 44 height, 26, 37–39, 42–44, 79, 80, 82, 284, 285, 313 pressure, 39–41, 43 rise, 26, 37–39, 42–44, 79, 80, 177, 178, 189, 284, 285, 289, 313, 322 tube, 4, 37–44, 79, 80, 82, 92, 99, 285, 313 Carbon dioxide, 1, 7, 8, 11, 30, 32, 193, 200, 293–296 Clausius–Clapeyron equation, 10

Contact angle, 38–43, 79, 162, 163, 283–285, 287 Continuity equation, 121, 122, 134, 229, 235, 244, 296, 298 Convection, 133, 229, 231, 232, 236, 237, 293, 295–297, 303, 305, 306 Convective Dispersion Equation (CDE), 235 Crop Growth Rate (CGR), 12 D Darcy-Buckingham equation, 65, 119–122, 124, 128, 130, 132, 175, 180, 330 Darcy equation, 5, 65, 97, 107, 182 Diffusion of ions, 232, 237 of soil air, 297 of water vapour, 133–135, 298, 312 Diffusivity of soil, 132, 310 Dirichlet’s condition, 324 Drainage groundwater, 2, 172, 181–184 internal, 171, 179–181 E Effective characteristics of stony soils, 266 Effective saturation, 50, 84 Energy-balance equation, 207–209, 223, 245, 246 Energy of soil water, 63, 64, 67 Evaporation basic characteristics, 1, 192 from bare soil, 197, 308 from water table, 193, 194, 196 from wet surface, 189, 193 of intercepted water, 189, 193, 245

© Springer Nature Switzerland AG 2019 V. Novák and H. Hlaváčiková, Applied Soil Hydrology, Theory and Applications of Transport in Porous Media 32, https://doi.org/10.1007/978-3-030-01806-1

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340 Evaporative demand of the atmosphere, 197, 198, 324 Evaporimeter, 194, 205, 206 Evapotranspiration actual, 189, 217 annual courses, 219 daily courses, 208, 220, 221 estimation, 205, 206, 221 plant-canopy, 212, 214–216, 249 potential, 178, 189, 206, 213, 216, 218, 219, 222 reference, 189, 215 structure, 218 F Field capacity, 51, 77, 85–87, 116, 166–168 Fine earth, 20, 21, 263–274, 276-279 G Green and Ampt method, 149, 154 Groundwater, 2, 3, 26, 37, 53, 72, 73, 97, 114, 119, 127, 147, 153, 165, 171–179, 181–187, 244, 246, 248, 301, 311, 322, 325, 330–333 Groundwater level (GWL), 185 H Hydraulic conductivity of soil saturated, 53, 97, 100, 101, 105–116, 121, 126, 127, 131, 140, 141, 143, 145, 150, 155–157, 161–163, 183, 186, 260, 267, 272–274, 276–279, 314, 325 unsaturated, 121, 123, 125, 127–131, 273, 321, 330, 333 Hydraulic gradient, 100, 102, 103, 106, 110, 111, 114, 141, 147, 182, 323, 324 Hydrodynamic dispersion, 229, 231, 233–235, 237, 239, 321 Hydrolimits, 77, 84–87 Hydrological regimen, 246 Hydrophobic soil, 39, 162, 283 Hydrophobicity, 285 Hygroscopicity, 45, 133, 197 I Infiltration calculation, 149 characteristics, 138, 153 curve, 139–147, 149, 153, 155–159, 286 from rain, 137, 139, 144, 146, 155, 165, 244 front, 70, 128, 141–156, 158, 159, 161–163, 179, 296

Index ponding, 70, 71, 112, 137, 141–143, 145–148, 150–155, 161, 179, 230, 240, 242, 260, 324, 330 steady, 115, 128, 129, 138, 140, 145, 154, 162, 164 unsteady, 138, 158 Interception, 12, 193, 322 Isotherm adsorption, 37, 45, 46, 53, 196 desorption, 37, 45, 53, 77 Isothermal conditions, 45, 63, 70, 123, 134, 135, 311 L Latent heat of evaporation, 9, 93, 191, 192, 198, 207, 243, 308, 312 Leaf Area Index (LAI), 12, 205, 214, 218–221, 322 Lysimeter, 186, 205, 264, 320 M Modelling of water flow direct solution, 328 in stony soil, 263, 275–277, 279 inverse solution, 328 Models black-box, 320 calibration, 328, 333 complexity, 320, 328, 334 coupling, 332, 333 deterministic, 276, 277, 279, 320, 322, 329 dual-permeability, 326–328, 332 dual-porosity, 271, 277, 320, 325–327 parameter optimization, 328 validation, 329 verification, 328, 329 N Neumann’s condition, 324 Non-isothermal conditions, 311, 316 O Outflow curve, 238, 239, 241 Oxygen, 7, 8, 10, 15, 27, 29, 30, 120, 135, 181, 193, 197, 293–295, 297–301 Oxygen Diffusion Rate (ODR), 293, 300, 301 P Pedogenesis, 7, 15, 16, 19, 20 Pedotransfer functions (PTFs), 97, 116 Penman equation, 189, 208 Penman–Monteith equation, 189, 206, 213–215, 217, 330

Index Piezometer, 63, 66, 67, 70, 72, 73, 103 Plant available water capacity, 85, 86 Plant canopy characteristics, 1, 12, 13 Poiseuille equation, 99, 295 Porosity cracks, 253, 254, 261 drainable, 184–186 wettable, 184, 186 Precipitation, 2, 3, 37, 87, 137, 139, 193, 199, 205, 223, 224, 244–246, 248, 258, 324, 330–332 Pressure head, 67–69, 71, 73, 77, 80, 81, 91, 94, 95, 102, 104, 106, 113, 115, 121, 123, 129, 154, 162, 174, 175, 284, 285, 312, 321, 322, 324, 328, 330, 335 R Radiation long-wave, 8, 206–208 net, 207–209, 215, 219, 222 short-wave, 206–208 solar, 193, 198, 206, 207, 303, 304, 306, 307, 322 Representative Elementary Volume (REV), 1, 5, 6, 17, 106, 263, 266–268, 320 Richards equation, 122, 123, 132, 133, 158, 159, 165, 260, 276, 321, 323, 326, 327, 329–332 Rock fragments relative mass, 272 relative volume, 272 Roots density, 12, 201, 202 specific length, 12, 201, 202 specific surface, 13, 201, 202 system depth, 12, 205 Runoff, 3, 137, 139, 140, 199, 244, 258, 260, 263, 274, 276, 280, 285, 286, 288, 330 S Saturated soil with water, 66, 85, 86, 120, 131, 146, 147, 150, 152, 161, 168, 179, 258, 274 Scale Darcian, 4, 5, 161 macroscopic, 4, 5, 320 megascopic, 5 microscopic, 4, 320 Sensible heat, 207 Soil aeration, 7, 8, 15, 18, 120, 155, 293–295, 300

341 aggregates, 21, 26, 106, 149, 155, 245 air, 3, 7, 11, 16, 18, 32, 111, 145, 284, 288, 293–296, 298 anisotropy, 110 bulk density, 5, 15, 52–54, 95, 267–269, 309, 310, 312 cracks, 24, 26, 110, 141, 253–261, 325, 331 liquid-phase, 3, 4, 16, 17, 31, 37, 67, 97, 229, 235, 238, 284 matrix, 15–17, 90, 97, 111, 253, 254, 258–261, 269, 275, 277, 283, 288, 325–327, 331, 332 particle density, 18, 22 pedogenesis, 7, 15, 20 pores, 3, 16–18, 44, 53, 67, 70, 81, 90, 97–100, 104, 110, 111, 119, 131, 145, 148, 233, 236, 238, 259, 283, 285, 288, 298, 300, 306, 320 porosity, 18, 19, 120, 134, 294 capillary, 4, 16, 34, 43, 44 lacunar, 269 macropores, 26 porous medium, 4, 7, 15, 16, 63–65, 80, 83, 99, 161, 162, 232, 237, 239, 266, 320 salinization, 1, 187, 248, 249 solid-phase, 19, 69, 309 specific surface, 15, 27, 46, 85, 99, 253, 255, 256 structure, 5, 20, 21, 26, 163, 266, 273, 277, 280, 320, 325, 328, 331, 332 temperature, 3, 112, 197, 198, 288, 296, 297, 303–306, 311, 314, 317 texture, 15, 20, 21, 23, 27, 58, 80, 83, 110, 142, 178, 185, 267, 288, 310, 325, 332 three-phase system, 3, 15, 17, 39, 56, 304, 309 type, 15, 16, 19, 20, 23, 59, 60, 80, 84, 91, 92, 94, 99, 141, 176, 177, 187, 243, 247–249, 277, 310, 332 water, 1–4, 6, 7, 12, 15, 26, 29, 37, 39, 41, 42, 44–46, 49–52, 54–60, 63, 65–72, 75, 77–99, 104, 108, 110, 111, 116, 119–133, 135, 137, 138, 140–154, 156, 158–163, 165–169, 171–182, 185, 186, 190, 191, 194–196, 203, 204, 207, 216–219, 222, 224, 229–231, 233, 235, 307, 310–317, 319, 321, 322, 324, 325, 327–330, 333 Soil classification, 20 stony-soil classification, 264 Soil heat capacity, 303, 308 Soil heat flux, 209, 215 Soil hydrology, 1, 3, 5, 65, 161, 246, 280, 319

342 Soil permeability, 100, 107, 314 Soil-Plant-Atmosphere Continuum (SPAC), 1, 3, 4, 6, 7, 12, 189–191, 200, 206, 224, 246 Soil shrinkage, 253, 325 Soil solution concentration, 49, 72, 235, 239, 294, 296, 321 Soil swelling, 253, 255, 259 Soil thermal conductivity, 307, 308 Soil-water balance equation, 189, 243, 244 Soil-water content mass, 19, 49–51, 54, 95, 254, 257 measurement, 45, 49, 51, 53, 54, 58, 78, 275 of stony soil, 270 residual, 50, 83, 131, 254, 274 volumetric, 18, 49–54, 57, 58, 77, 78, 87–89, 91, 94–96, 119–121, 126, 127, 130, 132–134, 144, 151, 185, 231, 233, 235, 247, 270–272, 274, 276, 279, 294, 298, 307, 309, 310, 312, 315, 317, 321, 330 Soil-water flow Darcian, 97, 98 laminar, 99 modelling, 87, 133, 161, 273, 276, 277, 279 non-Darcian, 111 non-uniform, 325 preferential, 162, 277, 288, 320, 325 turbulent, 102, 327 Soil-water potential gravitational, 68, 69, 71, 147, 148, 151, 174, 175, 180 matric, 63, 66, 68–72, 74, 75, 77–79, 81, 83–85, 90, 92–94, 108, 116, 119, 121, 122, 124–130, 132, 144, 146, 148, 151, 154, 165, 166, 168, 171, 173, 174, 176–180, 190, 191, 195–197, 203, 217, 218, 243, 247, 248, 271, 278, 312, 321 measurement, 63, 72, 74 pneumatic, 70, 71 total, 63, 67–72, 121, 122, 127, 129, 132, 142, 148, 151, 165, 173–175, 270 Soil-water redistribution, 165, 167 Soil-Water Regimen (SWR), 53, 119, 243, 247–251, 256 Soil-water retention curve analytical expression, 77, 83 definition, 78 hysteresis, 77–79, 166–168, 186 measurement, 78, 87, 88, 90, 92, 94, 96

Index Soil-water uptake, 177, 201 Solute transport, 229, 319, 321, 325, 326, 328, 330–333 Sorptivity, 160 Specific yield, 184–186 Sprinkling irrigation, 140, 144, 146 Stoke’s equation, 22, 100, 111 Stoniness, 264, 265, 267–269, 272–280 Stony soils, 5, 24, 59, 263–267, 269–277, 279, 280 Subsurface water, 1, 2, 103, 245, 246, 276, 329 Surface tension of water, 9, 33, 34, 40, 283, 285, 287, 311, 313, 315 Surface water, 2, 187, 246, 332 T Tensiometer, 63, 66, 67, 74, 75, 78, 130 Transpiration, 2, 3, 7, 85, 86, 186, 187, 189–191, 193, 196, 199–204, 206, 213, 214, 216, 218–222, 245, 249, 250, 289, 322, 331 Transport of heat in soil, 306, 307 U Unsaturated soil with water, 67 V Vadose zone, 3, 120, 273, 319, 333, 334 van Genuchten equation, 50, 83, 96, 154 Viscosity of water dynamic, 9, 22, 99, 112, 313, 314 kinematic, 35, 213 W Water density, 9, 22, 29, 33, 35, 42, 52, 68, 69, 99, 295, 311–314 Water-Drop Penetration Time (WDPT), 285, 286 Water meniscus, 38, 43 Water repellency, 283–285, 287, 288 Water repellent soils, 283 Water-vapour deficit, 10 flow, 132, 192, 312 pressure, 8–10, 37, 43–46, 65, 66, 193, 194, 198, 210, 211, 213–215 partial, 9 saturated, 9, 10, 43, 198, 215 Wetting angle, 37 Wilting point, 77, 85, 86, 190, 217