Conduction Phenomena.pdf

CHAPTER 9

9.1

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Conduction Phenomena

INTRODUCTION

Virtually all geotechnical problems involve soil or rock deformations and stability and/or the flow through earth materials of fluids, chemicals, and energy in various forms. Flows play a vital role in the deformation, volume change, and stability behavior itself, and they may control the rates at which the processes occur. Descriptions of these flows, predictions of flow quantities, their rates and changes with time, and associated changes in the properties and composition of both the permeated soil and the flowing material are the subjects of this chapter. Water flow through soil and rock has been most extensively studied because of its essential role in problems of seepage, consolidation, and stability, which form a major part of engineering analysis and design. As a result, much is known about the hydraulic conductivity and permeability of earth materials. Chemical, thermal, and electrical flows in soils are also important. Chemical transport through the ground is a major concern in groundwater pollution, waste disposal and storage, remediation of contaminated sites, corrosion, leaching phenomena, osmotic effects in clay layers, and soil stabilization. Heat flows are important relative to frost action, construction in permafrost areas, insulation, underground storage, thermal pollution, temporary ground stabilization by freezing, permanent ground stabilization by heating, underground transmission of electricity, and other problems. Electrical flows are important to the transport of water and ground stabilization by electroosmosis, insulation, corrosion, and subsurface investigations. In addition to the above four flow types, each driven by its own potential gradient, several types of coupled

flow are important under a variety of circumstances. A coupled flow is a flow of one type, such as hydraulic, driven by a potential gradient of another type, such as electrical. This chapter includes a review of the physics of direct and coupled flow processes through soils and their quantification in practical form, an evaluation of relevant parameters, their magnitudes, and factors influencing them, and some examples of applications.

9.2

FLOW LAWS AND INTERRELATIONSHIPS

Fluids, electricity, chemicals, and heat flow through soils. Provided the flow process does not change the state of the soil, each flow rate or flux Ji (as shown in Fig. 9.1) relates linearly to its corresponding driving force Xi according to Ji ⫽ Lii Xi

(9.1)

in which Lii is the conductivity coefficient for flow. When written specifically for a particular flow type and using familiar phenomenological coefficients, Eq. (9.1) becomes, for cross section area A Water flow

qh ⫽ khih A

Darcy’s law

(9.2)

Heat flow

qt ⫽ ktit A

Fourier’s law

(9.3)

Electrical flow

I ⫽ eie A

Ohm’s law

(9.4)

Chemical flow

JD ⫽ Dic A

Fick’s law

(9.5)

In Eqs. (9.2) to (9.5) qh, qt, I, and JD are the water, heat, electrical, and chemical flow rates, respectively. 251

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9.3

HYDRAULIC CONDUCTIVITY

Darcy’s law1 states that there is a direct proportionality between apparent water flow velocity vh or flow rate qh and hydraulic gradient ih, that is, vh ⫽ kh ih

(9.6)

qh ⫽ kh ih A

(9.7)

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where A is the cross-section area normal to the direction of flow. The constant kh is a property of the material. Steady-state and transient flow analyses in soils are based on Darcy’s law. In many instances, more attention is directed at the analysis than at the value of kh. This is unfortunate because no other property of importance in geotechnical problems is likely to exhibit such a great range of values, up to 10 orders of magnitude, from coarse to very fine grained soils, or show as much variability in a given deposit as does the hydraulic conductivity. Some soils exhibit 2 or 3 orders of magnitude variation in hydraulic conductivity as a result of changes in fabric, void ratio, and water content. These points are illustrated by Fig. 9.2 in which hydraulic conductivity values for a number of soils are shown. Different units for hydraulic conductivity are often used by different groups or agencies; for example, centimeters per second by geotechnical engineers, feet per year by groundwater hydrologists, and Darcys by petroleum technologists. Figure 9.3 can be used to convert from one system to another. The preferred unit in the SI system is meters/second.

Figure 9.1 Four types of direct flow through a soil porous

mass. A is the total cross-section area normal to flow; n is porosity.

Coefficients kh, kt, e, and D are the hydraulic, thermal, electrical conductivities, and the diffusion coefficient, respectively. Typical ranges of values for these properties are given later. The driving forces for flow are given by the respective hydraulic, thermal, electrical, and chemical gradients, ih, it, ie, and ic, respectively. The terms in Eqs. (9.2) through (9.5) are identified in Fig. 9.1 and in Table 9.1, which also shows analogs between the various flow types. As long as the flow rates and gradients are linearly related, the mathematical treatment of each flow type is the same, and the equations for flow of one type may be used to solve problems of another type provided the property values and boundary conditions are properly represented. Two well-known practical illustrations of this are the correspondence between the Terzaghi theory for clay consolidation and one-dimensional transient heat flow, and the use of electrical analogies for the study of seepage problems.

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Theoretical Equations for Hydraulic Conductivity

Fluid flow through soils finer than coarse gravel is laminar. Equations have been derived that relate hydraulic conductivity to properties of the soil and permeating fluid. A usual starting point for derivation of such equations is Poiseuille’s law for flow through a round capillary, which gives the average flow velocity, vave, according to vave ⫽

p R2 i 8 h

(9.8)

where  is viscosity, R is tube radius, and p is unit

1 This ‘‘law’’ was established empirically by Darcy based on the results of flow tests through sands. Its general validity for the description of hydraulic flow through most soil types has been verified by many subsequent studies. Historical accounts of the development of Darcy’s law are given by Brown et al. (2003).

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HYDRAULIC CONDUCTIVITY

Table 9.1

253

Conduction Analogies in Porous Media Fluid

Heat

Electrical

Chemical

Potential

Total head h (m)

Temperature T (C)

Voltage V (volts)

Storage

Fluid volume W (m3 /m3) Hydraulic conductivity kh (m/s) qh (m3 /s) qh /A (m3 /s/m2) h ih ⫽ ⫺ (m/m) x Darcy’s law h qh ⫽ ⫺kh A x Coefficient of volume change dW a M⫽ ⫽ w v ⫽ dh 1⫹e kh cv W q ⫹ h ⫽0 t A 2qh ⫽ 0 h k 2h ⫽ h 2 t M x

Thermal energy u (J/m3) Thermal conductivity kt (W/m/ C) qt (J/s) qt /A (J/s/m2) T it ⫽ ⫺ (C/m) x Fourier’s law T qt ⫽ ⫺kt A x Volumetric heat C(J/ C/m3) dQ C⫽ dT

Charge Q (Coulomb)

Flow Flux Gradient Conduction

Capacitance

Continuity Steady state Diffusion

冉冊

冉

冊

冉冊

u q ⫹ t ⫽0 t A 2qt ⫽ 0 T k 2T ⫽ t 2 t C x

冉 冊

weight of the flowing fluid. Because the flow channels in a soil are of various sizes, a characteristic dimension is needed to describe average size. The hydraulic radius RH flow channel cross-section area wetted perimeter

is useful. For a circular tube flowing full, RH ⫽

Current I (amp) I/A (amp/m2) V ie ⫽ ⫺ (v/m) x Ohm’s law V V I ⫽ ⫺e A⫽ x R Capacitance C (farads ⫽ coul/volt)

jD (mol/s) JD ⫽ jD /A (mol s⫺1 m⫺2) c ic ⫽ ⫺ (mol m⫺4) x Fick’s law c JD ⫽ ⫺D A x Retardation factor, Rd (dimensionless)

冉冊

Q I ⫹ ⫽0 t A 2I ⫽ 0 V  2V ⫽ t C x2

(m) ⫹ JD ⫽ 0 t 2JD ⫽ 0 c D* 2c ⫽ t RD x2

k ⫽a C

k ⫽ cv M

RH ⫽

Electrical conductivity  (siemens/m)

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Conductivity

Chemical potential  or concentration c (mol m⫺3) Total mass per unit total volume, m (mol/m3) Diffusion coeff. D (m2 /s)

R2 R ⫽ 2R 2

qcir ⫽

1 p 2 R ia 2  Hh

where a is the cross-sectional area of the tube. For other shapes of cross section, an equation of the same form will apply, differing only in the value of a shape coefficient Cs, so q ⫽ Cs

(9.9)

so Poiseuille’s equation becomes

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(9.10)

p RH2 ia  h

(9.11)

For a bundle of parallel tubes of constant but irregular cross section contributing to a total cross-sectional area A (solids plus voids), the area of flow passages Aƒ filled with water is

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Figure 9.2 Hydraulic conductivity values for several soils. Soil identification code: 1, compacted caliche; 2, compacted caliche; 3, silty sand; 4, sandy clay; 5, beach sand; 6, compacted Boston blue clay; 7, Vicksburg buckshot clay; 8, sandy clay; 9, silt—Boston; 10, Ottawa sand; 11, sand—Gaspee Point; 12, sand—Franklin Falls; 13, sand–Scituate; 14, sand–Plum Island; 15, sand–Fort Peck; 16, silt—Boston; 17, silt—Boston; 18, loess; 19, lean clay; 20, sand—Union Falls; 21, silt—North Carolina; 22, sand from dike; 23, sodium Boston blue clay; 24, calcium kaolinite; 25, sodium montmorillonite; 26–30, sand (dam filter) (From Lambe and Whitman (1969). Copyright  1969 by John Wiley & Sons. Reprinted with permission from John Wiley & Sons.

Figure 9.3 Hydraulic conductivity and permeability conversion chart.

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HYDRAULIC CONDUCTIVITY

Aƒ ⫽ SnA

(9.12)

where S is the degree of saturation and n is the porosity. For this condition the hydraulic radius is given by

⫽

Aƒ Aƒ L volume available for flow ⫽ ⫽ P PL wetted area Vw Vs S0

(9.13)

where P is the wetted perimeter, L is the length of flow channel in the direction of flow, Vs is the volume of solids and S0 is the wetted surface area per unit volume of particles. The wetted surface area depends on the particle sizes and the soil fabric and may be considered as an effective surface area per unit volume of solids. It is less than the total specific surface area of the soil since flow will not occur adjacent to all particle surfaces. For void ratio e and volume of solids Vs, the volume of water Vw is Vw ⫽ eVs S

(9.14)

Equation (9.11) becomes q ⫽ Cs

冉冊

(LT⫺1), and the absolute or intrinsic permeability K has units of area (L2). The effects of permeant properties are accounted for by the  / p term, provided the fabric of the soil is the same in the presence of different fluids. The pore shape factor k0 has a value of about 2.5 and the tortuosity factor has a value of about 兹2 in porous media containing approximately uniform pore sizes. For equal size spheres, S0 becomes 6/D (⫽surface area/volume of a sphere), where D is the diameter. If a soil is considered to consist of spheres of different sizes, an effective diameter Deff can be computed from the particle size distribution (Carrier, 2003) according to

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RH ⫽

冉冊 冉 冊

p e RH2 Snih A ⫽ Cs p RH2 S i A   1⫹e h

(9.15)

and substitution for RH using Eqs. (9.13) and (9.14) gives

冉 冊冉 冊 冉 冊

q ⫽ Cs

p 3 e3 S i A  1⫹e h

1 S 02

(9.16)

By analogy with Darcy’s law,

冉冊 冉 冊

kh ⫽ Cs

p 1 e3 S3 2  S0 1 ⫹ e

(9.17)

For full saturation, S ⫽ 1, and denoting Cs by 1/ (k0T 2), where k0 is a pore shape factor and T is a tortuosity factor, Eq. (9.17) becomes K ⫽ kh

冉冊

冉 冊

 1 e3 ⫽ 2 2 p k0 T S 0 1 ⫹ e

(9.18)

This is the Kozeny–Carman equation for the permeability of porous media (Kozeny, 1927; Carman, 1956). The hydraulic conductivity kh has units of velocity

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255

Deff ⫽

100%

兺(ƒi /Dave,i)

(9.19)

where fi is the fraction of particles between two sizes (Dli and Dsi) and Dave,i is the average particle size be0.5 tween two sizes (⫽D0.5 li Dsi ); S0 can also be estimated from the specific surface area. Methods for nonplastic soils and clayey soils are given in Chapter 3 and also are summarized by Chapuis and Aubertin (2003). Various modifications for S0 are available to take irregular particle shapes (Loudon, 1952; Carrier, 2003) into account. The Kozeny–Carman equation accounts well for the dependency of permeability on void ratio in uniformly graded sands and some silts; however, serious discrepancies are often found when it is applied to clays. The main reasons for these discrepancies are that most clay soils do not contain uniform pore sizes and changes in pore fluid type are often accompanied by changes in the clay fabric. Particles in clays are grouped in clusters or aggregates that have large intercluster pores and small intracluster pores. The influences of fabric and nonuniform pore sizes on the hydraulic conductivity of fine-grained soils are discussed further later in this section. If comparisons are made using materials having the same fabric, the influence of permeant on hydraulic conductivity is quite well accounted for by the p /  term. If, however, a fine-grained soil is molded or compacted in different permeants, then the fabrics may be quite different, and the hydraulic conductivities for samples at the same void ratio can differ greatly. If Cs in Eq. (9.17) is taken as a composite shape factor, and noting that total surface area per unit volume is inversely proportional to particle size, then kh ⫽ CD2s

冉冊

w e3 S3  1⫹e

where Ds is a characteristic grain size.

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(9.20)

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Validity of Darcy’s Law

A basic premise of Darcy’s law is that flow is laminar and steady through saturated porous media. If particle and pore sizes and flow rates are sufficiently great, then flow is turbulent, and Darcy’s law no longer applies. Turbulent flow conditions are likely in flows through gravel and rockfill (Ahmed and Sunada, 1969; Arbhabhirama and Dinoy, 1973; George and Hansen, 1992; Hansen et al., 1995; Li et al., 1998).2 Some modification of Darcy’s law is needed also to account for nonsteady and wave-induced flows through sands, silts,

2

and clays (Khalifa et al., 2002). These nonsteady and turbulent flow conditions are not treated herein. As early as 1898, instances were cited in which hydraulic flow velocity in fine-grained materials in which laminar flow can be expected increased more than proportionally with increases in gradient (King, 1898). The absence of water flow at finite hydraulic gradients in ceramic filters of 0.1-m average pore diameter was reported by Derjaguin and Krylov (1944). Oakes (1960) found no detectable flow through a 30-cm-long suspension of 6 percent Wyoming bentonite subjected to a 50-cm head of water. Experiments by Miller and Low (1963) led to the conclusion that there was a threshold gradient for flow through sodium montmorillonite. Flow rates through clay-bearing sandstones were found to increase more than directly with gradient up to gradients of 170 by von Englehardt and Tunn (1955). Deviations from Darcy’s law in pure and natural clays up to gradients of 900 were measured by Lutz and Kemper (1959). Apparent deviations from Darcy’s law for flow in undisturbed soft clay are shown in Fig. 9.4. The reported deviations from linearity between flow rate and hydraulic gradient are most significant in the lower range of gradients. Hydraulic gradients in the field are seldom much greater than one. Thus, deviations from Darcy’s law, if real, could have very important implications for the applicability of steady-state and transient flow analyses, including consolidation, that are based on it. Furthermore, gradients typically used in laboratory testing are high, commonly more than 10, and often up to several hundred. This brings the suitability of laboratory test results as indicators of field behavior into question. Three hypotheses have been proposed to account for nonlinearity between flow velocity and gradient: (1) non-Newtonian water flow properties, (2) particle migrations that cause blocking and unblocking of flow passages, and (3) local consolidation and swelling that is inevitable when hydraulic gradients are applied across a compressible soil. The apparent existence of a threshold gradient below which flow was not detected was attributed to a quasi-crystalline water structure. It is now known, however, that many of the effects interpreted as resulting from unusual water properties can be ascribed to undetected experimental errors arising from contamination of measuring systems (Olsen, 1965), local consolidation and swelling, and bacterial growth (Gupta and Swartzendruber, 1962). Additional careful measurements by a number of investigators (e.g., Olsen, 1969; Gray and Mitchell, 1967; Mitchell and Younger, 1967; Miller et al., 1969; Chan and Kenney, 1973) failed to confirm the existence of a threshold gradient in clays. Darcy’s law was

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Like the Kozeny–Carman equation, Eq. (9.20) describes the behavior of cohesionless soils reasonably well, but it is inadequate for clays. For a uniform sand with bulky particles and a given permeant, Eqs. (9.17) and (9.20) indicate that kh should vary directly with e3 /(1 ⫹ e) and D2s , and experimental observations support this. Despite the inability of the theoretical equations to predict the hydraulic conductivity accurately in many cases, they do reflect the overwhelming importance of pore size. Flow velocity depends on the square of pore radius, and hence the flow rate depends on radius to the fourth power. The specific surface in the Kozeny– Carman equation and the representative grain size term in Eq. (9.20) are both measures of pore size. All other factors equal, the hydraulic conductivity depends far more on the fine particles than on the large. A small percentage of fines can clog the pores of an otherwise coarse material and result in a manyfold lower hydraulic conductivity. On the other hand, the presence of fissures, cracks, root holes, and the like can result in enormous increases in the rate of water flow through an otherwise compact soil layer. Equation (9.20) predicts that the hydraulic conductivity should vary with the cube of the degree of saturation, and some, but not all, experimental data support this, even in the case of fine-grained soils. Consideration of flow through unsaturated soils is given in Section 9.4.

Flow transitions from laminar to turbulent flow when the Reynolds number Re, defined as the ratio of inertial to viscous forces, exceeds a critical value. For flow through soils the critical value of interstitial flow Re is in the range of 1 to 10, with Re defined as (Khalifa et al., 2002) Re ⫽

4 v (1 ⫺ n)Avd

in which  is fluid density,  is tortuosity (ratio of flow path mean length to thickness), v is flow velocity, n is porosity, and Avd is the ratio of pore surface area exposed to flow to the volume of solid.

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Figure 9.4 Dependence of flow velocity on hydraulic gradient. Undisturbed soft clay from Ska˚ Edeby, Sweden (from Hansbo, 1973).

obeyed exactly in several of these studies. Thus it is unlikely that unusual water properties are responsible for non-Darcy flow behavior. On the other hand, particle migrations leading to void plugging and unplugging, electrokinetic effects, and chemical concentration gradients can cause apparent deviations from Darcy’s law. Analysis of interparticle bond strengths in relation to the magnitude of seepage forces shows that particles that are not participating in the load-carrying skeleton of a soil mass can be moved under moderate values of hydraulic gradient. Soils with open, flocculated fabrics and granular soils with a relatively low content of fines appear particularly susceptible to the movement of fine particles during permeation. Internal swelling and dispersion of clay particles during permeation can cause changes in flow rate and apparent non-Darcy behavior. Tests on illite–silt mixtures showed that the hydraulic conductivity depends on clay content, sedimentation procedure, compression rate, and electrolyte concentration. Subsequent behavior was quite sensitive to the type and concentration of electrolyte used for permeation and the total throughput volume of permeant. Changes in relative hydraulic conductivity that occurred while the

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electrolyte concentration was changed from 0.6 to 0.1 N NaCl are shown in Fig. 9.5. The cumulative throughput is the ratio of the total flow volume at any time to the sample pore volume. The hydraulic conductivities for these materials ranged from more than 1 ⫻ 10⫺7 to less than 1 ⫻ 10⫺9 m/s. Practical Implications Evidence indicates that Darcy’s law is valid, provided that all system variables are held constant. However, unless fabric changes, particle migrations, and internal void ratio redistributions caused by effective stress and chemical changes can be shown to be negligible, hydraulic conductivity measurements in the laboratory should be made under conditions of temperature, pressure, hydraulic gradient, and pore fluid chemistry as closely approximating those in the field as possible. This is particularly important in connection with the testing of clays as potential waste containment barriers, such as slurry walls and liners for landfills and impoundments (Daniel, 1994). Microbial activities may be important as well, as they can lead to formation of biofilms, pore clogging, and large reductions in hydraulic conductivity as shown, for example, by Dennis and Turner (1998). Unfortunately, duplication of field conditions is not always possible, especially as regards the hydraulic

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Figure 9.5 Reduction in hydraulic conductivity as a result of internal swelling (from Hardcastle and Mitchell, 1974).

gradient. If hydraulic gradients are low enough to duplicate those in most field situations, then the laboratory testing time usually becomes unacceptably long. In such cases, tests over a range of gradients are desirable in order to assess the stability of the soil structure against changes due to seepage forces. Similarly, the gradients that are developed in laboratory consolidation tests on thin samples are many times greater than exist in thick layers of the same clay in the field. The variation of hydraulic gradient i with time factor T during one-dimensional consolidation according to the Terzaghi theory is shown in Fig. 9.6. The solution of the Terzaghi equation gives excess pore pressure u as a function of position (z/H) and time factor

冘 2uM 冉sin MzH 冊e ⬁

u⫽

⫺M2T

0

(9.21)

m⫽0

where M ⫽  (2m ⫹ 1)/2. Thus, the hydraulic gradient is  i⫽ z

冉冊

u 2u0 ⫽ w wH

冘 ⬁

m⫽0

冉 冊

Mz ⫺M2T cos e H

冘 cos冉MzH 冊e ⬁

⫺M2T

u0 p wH

(9.24)

The real gradient for any layer thickness or loading intensity can be obtained by using actual values of u0 and H and the appropriate value of p from Fig. 9.6. For small values of u0 / w H, as is the case in the field, for example, for u0 ⫽ 50 kPa, H ⫽ 5m, then u0 / w H ⫽ 1, and the field gradients are low throughout most of the layer thickness during the entire consolidation process. On the other hand, for a laboratory sample of 10 mm thickness and the same stress increase, u0 / w H is 500, and the hydraulic gradients are very large. In this case a gradient-dependent hydraulic conductivity could be the cause of significant differences between the laboratory-measured and field values of coefficient of consolidation. Constant rate of strain or constant gradient consolidation testing of such soils is preferable to the use of load increments because lower gradients minimize particle migration effects. Anisotropy

(9.22)

If a parameter p is defined by p⫽2

i⫽

(9.23)

m⫽0

Eq. (9.22) becomes

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Anisotropic hydraulic conductivity results from both preferred orientation of elongated or platy particles and stratification of soil deposits. Ratios of horizontal-tovertical hydraulic conductivity from less than 1 to more than 7 were measured for undisturbed samples of several different clays (Mitchell, 1956). These ratios correlated reasonably well with preferred orientation of the clay particles, as observed in thin section. Ratios of 1.3 to 1.7 were measured for kaolinite consolidated one dimensionally from 4 to 256 atm, and 0.9 to 4.0

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Figure 9.6 Hydraulic gradients during consolidation according to the Terzaghi theory.

were measured for illite and Boston blue clay consolidated over a pressure range up to more than 200 atm (Olsen, 1962). A ratio of approximately 2 was measured for kaolinite over a range of void ratios corresponding to consolidation pressures up to 4 atm (Morgenstern and Tchalenko, 1967b). Thus, an average hydraulic conductivity ratio of about 2 as a result of microfabric anisotropy may be typical for many clays. Large anisotropy in hydraulic conductivity as a result of stratification of natural soil deposits or in earthwork compacted in layers is common. Varved clays have substantially greater hydraulic conductivity in the horizontal direction than in the vertical direction owing to the presence of thin silt layers between the thin clay layers. The ratio of horizontal values to vertical values determined in the laboratory, rk, is 10  5 for Connecticut Valley varved clay (Ladd and Wissa, 1970). Similar values were measured for the varved clay in the New Jersey meadows. Values less than 5 were measured for New Liskeard, Ontario, varved clay (Chan and Kenney, 1973). The practical importance of a high hydraulic conductivity in the horizontal direction depends on the distance to a drainage boundary and the type of flow. For example, the rate of groundwater flow will clearly be affected, as will the rate of consolidation when vertical

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drains are used. On the other hand, lateral drainage beneath a loaded area may not be greatly influenced by a high ratio of horizontal to vertical conductivity if the width of loaded area is large compared to the thickness of the drainage layer. Fabric and Hydraulic Conductivity

The theoretical relationships developed earlier in this section indicate that the flow velocity should depend on the square of the pore radius, and the flow rate is proportional to the fourth power of the radius. Thus, fabrics with a high proportion of large pores are much more pervious than those with small pores. For example, remolding several undisturbed soft clays reduced the hydraulic conductivity by as much as a factor of 4, with an average of about 2 (Mitchell, 1956). This reduction results from the breakdown of a flocculated open fabric and the destruction of large pores. An illustration of the profound influence of compaction water content on the hydraulic conductivity of fine-grained soil is shown in Fig. 9.7. All samples were compacted to the same density. For samples compacted using the same compactive effort, curves such as those in Fig. 9.8 are typical. For compaction dry of optimum,

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Figure 9.8 Influence of compaction method on the hydraulic

conductivity of silty clay. Constant compactive effort was used for all samples.

Figure 9.7 Hydraulic conductivity as a function of compac-

tion water content for samples of silty clay prepared to constant density by kneading compaction.

clay particles and aggregates are flocculated, the resistance to rearrangement during compaction is high, and a fabric with comparatively large pores is formed. For higher water contents, the particle groups are weaker, and fabrics with smaller average pore sizes are formed. Considerably lower values of hydraulic conductivity are obtained wet of optimum in the case of kneading compaction than by static compaction (Fig. 9.8) because the high shear strains induced by the kneading compaction method break down flocculated fabric units.

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Three levels of fabric are important when considering the hydraulic conductivity of finer-grained soils. The microfabric consists of the regular aggregations of particles and the very small pores, perhaps with sizes up to about 1 m, between them through which very little fluid will flow. The minifabric contains these aggregations and the interassemblage pores between them. The interassemblage pores may be up to several tens of micrometers in diameter. Flows through these pores will be much greater than through the intraaggregate pores. On a larger scale, there may be a macrofabric that contains cracks, fissures, laminations, or root holes through which the flow rate is so great as to totally obscure that through the other pore space types.

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aggregates or clusters as shown schematically in Fig. 9.10. These aggregates of N particles each have an intracluster void ratio ec. The spaces between the aggregates comprise the intercluster voids and are responsible for the intercluster void ratio ep. The total void ratio eT is equal to the sum of ec and ep. The clusters and intracluster voids comprise the microfabric, whereas the assemblage of clusters comprises the minifabric. Fluid flow in such a system is dominated by flow through the intercluster pores because of their larger size. The sizes of clusters depend on the mineralogical and pore fluid compositions and the formational process. Conditions that favor aggregation of individual clay plates produce larger clusters than deflocculating, dispersing environments. There is general consistency with the interparticle double-layer interactions described in Chapter 6. When a fine-grained soil is sedimented in or mixed with waters of different electrolyte concentration or type or with fluids of different dielectric constants, quite different fabrics result. This explains why the  / term in Eqs. (9.18) and (9.20) is inadequate to account for pore fluid differences, unless comparisons are made using samples having identical fabrics. This will only be the case when a pore fluid of one type replaces one of another type without disturbance to the soil. The cluster model developed by Olsen (1962) accounts for discrepancies between the predicted and measured variations in flow rates through different soils. The following equation can be derived for the ratio of estimated flow rate for a cluster model, qCM to the flow rate predicted by the Kozeny–Carman equation (9.18) qKC:

Figure 9.9 Contours of constant hydraulic conductivity for silty clay compacted using kneading compaction.

Figure 9.10 Cluster model for permeability prediction (after Olsen, 1962).

Co py rig hte dM ate ria l

These considerations are of particular importance in the hydraulic conductivity of compacted clays used as barriers for waste containment. The controlling units in these materials are the clods, which would correspond to minifabric units. Acceptably low hydraulic conductivity values are obtained only if clods and interclod pores are eliminated during compaction (Benson and Daniel, 1990). This requires that compaction be done wet of optimum using a high effort and a method that produces large shear strains, such as by sheepsfoot roller. The wide range of values of hydraulic conductivity of compacted fine-grained soils that results from the large differences in fabric associated with compaction to different water contents and densities is illustrated by Fig. 9.9. The grouping of contours means that selection of a representative value for use in a seepage analysis is difficult. In addition, if it is required that the hydraulic conductivity of earthwork not exceed a certain value, such as may be the case for a clay liner for a waste pond, then specifications must be carefully drawn. In so doing, it must be recognized also that other properties, such as strength, also vary with compaction water content and density and that the compaction conditions that are optimal for one property may not be suitable for the other. A procedure for the development of suitable specifications for compacted clay liners is given by Daniel and Benson (1990). The primary reason equations such as (9.18) and (9.20) fail to account quantitatively for the variation of the hydraulic conductivity of fine-grained soils with change in void ratio is unequal pore sizes (Olsen, 1962). A typical soil has a fabric composed of small

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9

CONDUCTION PHENOMENA

qCM (1 ⫺ ec /eT)3 ⫽ N2/3 qKC (1 ⫹ ec)4 / 3

(9.25)

9.4

vi ⫽ ⫺k(S)

冉

冊

 z ⫹ xi xi

FLOWS THROUGH UNSATURATED SOILS

Darcy’s law [Eq. (9.7)] also applies for flow through unsaturated soils such as those in the vadose zone above the water table where pore water pressures are negative. However, the hydraulic conductivity is not constant and depends on the amount and connectivity of water in the pores. For instance, Eq. (9.20) predicts that hydraulic conductivity should vary as the cube of the degree of saturation.3 This relationship has been 3

The hydraulic conductivity can also be a function of volumetric moisture content or matric suction . These variables are related to each other by the soil–water characteristic curve as described in Chapter 7.

Copyright © 2005 John Wiley & Sons

(9.26)

where k(S) is saturation-dependent hydraulic conductivity,  is the matric suction equivalent head (L), and z/ xi is the unit gravitational vector measured upward in direction z (1.0 if xi is the direction of gravity z). When percolating water infiltrates vertically into dry soil, the hydraulic gradient near the sharp wetting front can be very large because of a large value of the  / x term. However, the wetting front becomes less sharp as the infiltration proceeds and the gravity term then dominates. The hydraulic gradient then is close to one and the magnitude of flux is equal to the hydraulic conductivity k(S). Using Eq. (9.26), the equation of mass conservation becomes

Co py rig hte dM ate ria l

Application of Eq. (9.25) requires assumptions for the variations of ec with eT that accompany compression and rebound. Olsen (1962) considered the relative compressibility of individual clusters and cluster assemblages. The compressibility of individual clusters is small at high total void ratios, so compression is accompanied by reduction in the intercluster pore sizes, but with little change in intracluster void ratio. This assumption is supported by the microstructure studies of Champlain clay by Delage and Lefebvre (1984) Thus, the actual hydraulic conductivity decreases more rapidly with decreasing void ratio during compression than predicted by the Kozeny–Carman equation until the intercluster pore space is comparable to that in a system of closely packed spheres, when the clusters themselves begin to compress. Further decreases in porosity involve decreases in both ec and eT. As the intercluster void ratio now decreases less rapidly, the hydraulic conductivity decreases at a slower rate with decreasing porosity than predicted by the Kozeny–Carman equation. During rebound increase in porosity develops mainly by swelling of the clusters, whereas the flow rate continues to be controlled primarily by the intercluster voids. Recent attempts to quantify saturation and hydraulic conductivity of fine-grained soils containing a distribution of particle sizes and fabric elements in terms of pore-scale relationships have given promising results (Tuller and Or, 2003). Expressions for clay plate spacing in terms of surface properties and solution composition derived using DLVO theory (see Chapter 6), combined with assumed geometrical representations of clay aggregates and pore space in combination with silt and sand components, are used in the formulation.

found reasonable for compacted fine-grained soils and degrees of saturation greater than about 80 percent. Similarly to Eq. (9.7), the unsaturated flow equation in the direction i can be written as

(nS)  ⫽ t xi

冋 冉

冊册

 z ⫹ xi xi

k(S)

R w

⫹

(9.27)

where n is the porosity, w is the density of the water, and R is a source or sink mass transfer term such as water uptake by plant roots (ML⫺3). If the soil is assumed to be incompressible and there is no sink/sources (R ⫽ 0), Eq. (9.27) becomes n

S   ⫽  t xi

or C()

  ⫽ t xi

冋 冉 冋 冉

冊册 冊册

k()

 z ⫹ xi xi

k()

 z ⫹ xi xi

(9.28)

where C() ⫽ n(S/ ) and k(S) is converted to k() using the soil–water characteristic curve (S–  relationships). Equation (9.28) is called the Richards equation (Richards, 1931). For given S–  and k() relationships and initial/boundary conditions, the nonlinear governing equation can be solved for  (often numerically by the finite difference or finite element method). The hydraulic conductivity of unsaturated soils can be a function of saturation, water content, matric suction, or others. Measured hydraulic conductivities of well-graded sand and clayey sand as a function of (a) matric suction and (b) saturation ratio are shown in Fig. 9.11. Both figures are related to each other, as the matric suction is a function of saturation ratio by the soil moisture characteristic curve as described in Sec-

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FLOWS THROUGH UNSATURATED SOILS 1.E+01

1.E+01 Sand

Clayey Sand Clayey Sand > Sand

1.E-11 1.E-13 1.E-15 1.E-17

Sand Clayey Sand

1.E-01 1.E-03 1.E-05 1.E-07 1.E-09

Hydraulic Conductivity (m/s)

Hydraulic conductivity (m/s)

1.E-01 1.E-03 1.E-05 1.E-07 1.E-09

1.E-11 1.E-13 1.E-15 1.E-17

Sand > Clayey Sand 1.E-19 1.E-21 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Matric Suction (kPa)

20

40 60 Saturation (%)

80

Co py rig hte dM ate ria l

1.E-19 1.E-21 0

(a)

100

(b)

1.E+00

Sand

1.E-02

Clayey Sand

1.E-04 1.E-06 1.E-08 1.E-10 1.E-12

Relative Permeability kr

1.E+00 Relative Permeability kr

263

1.E-02 1.E-04 1.E-06 1.E-08 1.E-10 1.E-12

1.E-14

1.E-14

1.E-16 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 Matric Suction (kPa)

1.E-16

(c)

Sand Clayey Sand

0

20

40 60 Saturation (%)

80

100

(d)

Figure 9.11 Hydraulic conductivity of partially saturated sand and clayey sand as a function of matric suction and degree of saturation (from Stephens, 1996).

tion 7.12. Various methods to measure the hydraulic conductivity of unsaturated soils are available (Klute, 1986; Fredlund and Rahardjo, 1993). However, the measurement in unsaturated soils is more difficult to perform than in saturated soils because the hydraulic conductivity needs to be determined under controlled water saturation or matric suction conditions. A general expression for the hydraulic conductivity k of unsaturated soils can be written as k ⫽ krK

g ⫽ kr ks 

(9.29)

where ks is the saturated conductivity, K is the intrinsic permeability of the medium (L2) such as given by Eq. (9.18),  is the density of the permeating fluid (ML⫺3), g is the acceleration of gravity (LT⫺2),  is the dynamic viscosity of the permeating fluid (MT⫺1L⫺1), and ks is the conductivity under the condition that the pores are fully filled by the permeating fluid (i.e., full saturation). The dimensionless parameter kr is called the relative permeability, and the values range from 0 (⫽ zero per-

Copyright © 2005 John Wiley & Sons

meability, no interconnected path for the permeating fluid) to 1 (⫽ permeating fluid at full saturation). The equation can be used for a nonwetting fluid (e.g., air) by substituting the values of  and  of the nonwetting fluid. The data in Fig. 9.11a and 9.11b can be replotted as the relative permeability against matric suction in Fig. 9.11c and against saturation ratio in Fig. 9.11d. The two different curves in Fig. 9.11d clearly show that kr ⫽ S3 derived from Eq. (9.20) is not universally applicable. At very low water contents, the water in the pores becomes disconnected as described in Chapter 7. Careful experiments show that the movement of water exists even at moisture contents of a few percent, but vapor transport becomes more important at this dry state (Grismer et al., 1986). Therefore, Eq. (9.20) is not suitable for low saturations. One reason for this discrepancy is that soil contains pores of various sizes rather than the assumption of uniform pore sizes used to derive Eq. (9.20). Considering that the soil contains pores of random sizes, Marshall (1958) derived the following equation

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264

9

CONDUCTION PHENOMENA

for hydraulic conductivity as a function of pore sizes for an isotropic material: K⫽

n2 r 21 ⫹ 3r 22 ⫹ 5r 23 ⫹    ⫹ (2m ⫺ 1)r 2m m2 8 (9.30)

Co py rig hte dM ate ria l

in which K is the specific hydraulic conductivity (permeability) (L2), n is the porosity, m is the total number of pore classes, and ri is the mean radius of the pores in pore class i. Pore sizes can be measured from data on the amount of water withdrawn as the suction on the soil is progressively increased. Using the capillary equation, the radius of the largest water-filled pore under a suction of  (L) is given by 2 r⫽ wg

(9.31)

in which  is the surface tension of water, w is the density of water, and g is the acceleration of gravity. As it is usually more convenient to use moisture suction than pore radius, Eq. (9.29) can be rewritten as K⫽

 2 n2 ⫺2 [ ⫹ 32⫺2 ⫹ 5⫺2 3 22wg2 m2 1

(9.32)

The permeability K can be converted to the hydraulic conductivity k by multiplying the unit weight (wg) divided by the dynamic viscosity of water . This gives  2 n2 ⫺2 ⫺2 [ ⫹ 3⫺2 2 ⫹ 53 2wg m2 1

⫹    ⫹ (2m ⫺ 1)⫺2 m ]

(9.33)

Following Green and Corey (1971), the porosity n equals the volumetric water content of the saturated condition S, and m is the total number of pore classes between S and zero water content ⫽ 0. A matching factor is usually used in Eq. (9.33) to equate the calculated and measured hydraulic conductivities. Matching at full saturation is preferable to matching at a partial saturation point because it is simpler and gives better results. Rewriting Eq. (9.33) and introducing a matching factor gives k( i) ⫽

ks  2 2S ksc 2wg m2

m S ⫽ l S ⫺ L

冘 [(2j ⫹ 1 ⫺ 2i) l

⫺2 j

]

j⫽1

(i ⫽ 1, 2, . . . , l)

(9.34)

Copyright © 2005 John Wiley & Sons

(9.35)

A constant value of l is used at all water contents, and the value of l establishes the number of pore classes for which ⫺2 terms are included in the calculation at j saturation. Other pore size distribution models for unsaturated soils are available, and an excellent review of these models is given by Mualem (1986). Equation (9.34) can be written in an integration form as (after Fredlund et al., 1994)

冕



ks  2 Sp ksc 2wg

k( ) ⫽

⫹    ⫹ (2m ⫺ 1)⫺2 m ]

k⫽

in which k( i) is the calculated hydraulic conductivity for a specified water content i; is i the last water content class on the wet end, for example, i ⫽ 1 denotes the pore class corresponding to the saturated water content S, and i ⫽ l denotes the pore class corresponding to the lowest water content L for which hydraulic conductivity is calculated; ks /ksc is the matching factor, defined as the measured saturated hydraulic conductivity divided by the calculated saturated hydraulic conductivity; and l is the total number of pore classes (a pore class is a pore size range corresponding to a water content increment) between ⫽ L and S. Thus

L

⫺x dx 2(x)

(9.36)

where suction  is given as a function of volumetric water content , and x is a dummy variable. The hydraulic conductivity for fully saturated condition is calculated by assigning ⫽ S. For generality, the term 2S in Eq. (9.34) is replaced by ps , where p is a constant that accounts for the interaction of pores of various sizes (Fredlund et al., 1994). From Eq. (9.36), the relative permeability kr is a function of water content as follows: kr( ) ⫽

冕



r

冒冕

⫺x dx 2(x)

S

r

⫺x dx 2(x)

(9.37)

Herein, the lowest water content L is assumed to be the residual water content r. If the moisture content –suction  relationship (or the soil–water characteristic curve) is known, the relative permeability kr can be computed from Eq. (9.37) by performing a numerical integration. The hydraulic conductivity k is then estimated from Eq. (9.29) with the knowledge of saturated hydraulic conductivity ks. The use of the soil–water characteristic curve to estimate the hydraulic conductivity of unsaturated soils is attractive because it is easier to determine this curve

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THERMAL CONDUCTIVITY

in the laboratory than it is to measure the hydraulic conductivity directly. Apart from Eq. (9.37), the following relative permeability function proposed by Mualem (1976) is often used primarily because of its simplicity: kr( ) ⫽

冊 冉冕 d( )冒冕

冉

⫺ r s ⫺ r

q



s

r

r

d ( )

冊

2

(9.38)

kr( ) ⫽

冉

larger than the vertically infiltrating water flow. However, if the matric suction is reduced by large infiltration, the barrier breaks and water enters into the initially dry coarse layer. Solutions are available to evaluate the amount of water flowing laterally across the capillary barrier interface at the point of breakthrough for a given set of fine and coarse soil hydraulic properties and interface inclination (Ross, 1990; Steenhuis et al., 1991; Selkar, 1997; Webb, 1997). Capillary barriers have received increased attention as a means for isolating buried waste from groundwater flow and as part of landfill cover systems in dry climates (Morris and Stormont, 1997; Selkar, 1997; Khire et al., 2000). The barrier can be used to divert the flow laterally along an interface and/or to store infiltrating water temporarily in the fine layer so that it can be removed ultimately by evaporation and transpiration. Capillary barriers are constructed as simple two-layer systems of contrasting particle size or multiple layers of fine- and coarse-grained soils. If the thickness of the overlying fine layer is too small, capillary diversion is reduced because of the confining flow path in the fine layer. The minimum effective thickness is several times the air-entry head of the fine soil (Warrick et al., 1997; Smersrud and Selker, 2001). Khire et al. (2000) stress the importance of site-specific metrological and hydrological conditions in determining the storage capacity of the fine layer. The soil for the underlying coarse layer should have a very large particle size contrast with the fine soil, but fines migrations into the coarse sand should be avoided. Smesrud and Sekler (2001) suggest the d50 particle size ratio of 5 to be ideal. The thickness of the coarse sand layer does not need to be great, as the purpose of the layer is simply to impede the downward water migration.

Co py rig hte dM ate ria l

where q describes the degree of connectivity between the water-conducting pores. Mualem (1976) states that q ⫽ 0.5 is appropriate based on permeability measurements on 45 soils. van Genuchten et al. (1991) substituted the soil–water characteristic equation (7.52) into Eq. (9.38) and obtained the following closed-form solution4:

冊再 冋 冉

⫺ r S ⫺ r

p

1⫺ 1⫺

冊 册冎

⫺ r S ⫺ r

1/m

m

2

(9.39)

Both Eq. (9.39) as well as Eq. (9.37) using the soil– water characteristic curve by Fredlund and Xing (1994) give good predictions of measured data as shown in Fig. 9.12. The two hydraulic conductivity–matric suction curves shown in Fig. 9.11a cross each other at a matric suction value of approximately 50 kPa (or 5 m above the water table under hydrostatic condition). Below this value, the hydraulic conductivity of sand is larger than that of the clayey sand. However, as the matric suction increases, the water in the sand drains rapidly toward its residual value, giving a very low hydraulic conductivity. On the other hand, the clayey sand holds the pore water by the presence of fines and the hydraulic conductivity becomes larger than that of the sand at a given matric suction. If the sand is overlain by the clayey sand, then the matric suction at the interface is larger than 50 kPa, and the water infiltrating downward through the finer clayey sand cannot enter into the coarser sand layer because the underlying sand layer is less permeable than the overlying clayey sand. The water will instead move laterally along the bedding interface. This phenomenon is called a capillary barrier (e.g., Zaslavsky and Sinai, 1981; Yeh et al., 1985; Miyazaki, 1988). The barrier will be maintained as long as the lateral discharge along the interface (preferably inclined) is

4 m ⫽ 1 ⫺ 1 / n is assumed (van Genuchten et al., 1991). See Eq. (7.52).

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265

9.5

THERMAL CONDUCTIVITY

Heat flow through soil and rock is almost entirely by conduction, with radiation unimportant, except for surface soils, and convection important only if there is a high flow rate of water or air, as might possibly occur through a coarse sand or rockfill. The thermal conductivity controls heat flow rates. Conductive heat flow is primarily through the solid phase of a soil mass. Values of thermal conductivity for several materials are listed in Table 9.2. As the values for soil minerals are much higher than those for air and water, it is evident that the heat flow must be predominantly through the solids. Also included in Table 9.2 are values for the heat capacity, volumetric heat, heat of fusion, and heat of vaporization of water. The heat capacity can be used

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9

CONDUCTION PHENOMENA

Hydraulic Conductivity, k (cm/day)

1

0.1

0.01

Predicted coefficient of permeability (drying) Predicted coefficient of permeability (wetting)

Co py rig hte dM ate ria l

Hydraulic Conductivity, k × 10(m/s)

10

0.001

Measured coefficient of permeability (drying) Measured coefficient of permeability (wetting)

0.0001 20

30 40 50 Volumetric Water Content (a)

60

Figure 9.12 Comparisons of predicted and measured relationships between hydraulic con-

ductivity and volumetric water content for two soils. (a) By Eq. (9.37) with the measured data for Guelph loam (from Fredlund et al., 1994) and (b) by Eq. (9.39) with the measured data for crushed Bandelier Tuff (van Genuchten et al., 1991).

to compute the volumetric heat using the simple relationships for frozen and unfrozen soil given in the table. Volumetric heat is needed for the analysis of many types of transient heat flow problems. The heat of fusion is used for analysis of ground freezing and thawing, and the heat of vaporization applies to situations where there are liquid to vapor phase transitions. The denser a soil, the higher is its composite thermal conductivity, owing to the much higher thermal conductivity of the solids relative to the water and air. Furthermore, since water has a higher thermal conductivity than air, a wet soil has a higher thermal conductivity than a dry soil. The combined influences of soil unit weight and water content are shown in Fig. 9.13, which may be used for estimates of the thermal conductivity for many cases. If a more soil-specific value is needed, they may be measured in the laboratory using the thermal needle method (ASTM, 2000). More detailed treatment of methods for the measurement of the thermal conductivity of soils are given by Mitchell and Kao (1978) and Farouki (1981, 1982). The relationship between thermal resistivity (inverse of conductivity) and water content for a partly saturated soil undergoing drying is shown in Fig. 9.14. If drying causes the water content to fall below a certain value, the thermal resistivity increases significantly. This may be important in situations where soil is used as either a thermally conductive material, for example,

Copyright © 2005 John Wiley & Sons

to carry heat away from buried electrical transmission cables, or as an insulating material, for example, for underground storage of liquefied gases. The water content below which the thermal resistivity begins to rise with further drying is termed the critical water content, and below this point the system is said to have lost thermal stability (Brandon et al., 1989). The following factors influence the thermal resistivity of partly saturated soils (Brandon and Mitchell, 1989). Mineralogy All other things equal, quartz sands have higher thermal conductivity than sands containing a high percentage of mica. Dry Density The higher the dry density of a soil, the higher is the thermal conductivity. Gradation Well-graded soils conduct heat better than poorly graded soils because smaller grains can fit into the interstitial spaces between the larger grains, thus increasing the density and the mineral-to-mineral contact. Compaction Water Content Some sands that compacted wet and then dried to a lower water content have significantly higher thermal conductivity than when compacted initially at the lower water content. Time Sands containing high percentages of silica, carbonates, or other materials that can develop ce-

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ELECTRICAL CONDUCTIVITY

Table 9.2

Thermal Properties of Materials a

Material

Btu/h/ft2 / F/ft

W/m/K

Air Water Ice Snow (100 kg m⫺3) (500 kg m⫺3) Shale Granite Concrete Copper Soil Polystyrene

0.014 0.30 1.30

0.024 0.60 2.25

0.03 0.34 0.90 1.60 1.0 225 0.15–1.5 (⬇1.0) 0.015–0.035

0.06 0.59 1.56 2.76 1.8 389 0.25–2.5 (⬇1.7) 0.03–0.06

Material

Btu/lb/ F

kJ/kg/K

Co py rig hte dM ate ria l

Thermal Conductivity

Heat Capacity

Volumetric Heat

Heat of Fusion

Heat of Vaporization a

267

Water Ice Snow (100 kg m⫺3) (500 kg m⫺3) Minerals Rocks

1.0 0.5

4.186 2.093

0.05 0.25 0.17 0.20–0.55

0.21 1.05 0.710 0.80–2.20

Material

Btu/ft3 / F

kJ/m3 /K

Unfrozen Soil Soil Frozen soil Snow (100 kg m⫺3) (500 kg m⫺3) Water Soil Water Soil

d (0.17 ⫹ w/100)

d (72.4 ⫹ 427w/100)

d (0.17 ⫹ 0.5w/100)

d (72.4 ⫹ 213w/100)

3.13 15.66 143.4 Btu/lb 143.4(w/100) d Btu/ft3 970 Btu/lb 970(w/100) d Btu/ft3

210 1050 333 kJ/kg 3.40 ⫻ 104(w/100) d kJ/m3 2.26 MJ/kg 230(w /100) d MJ/m3

d ⫽ dry unit weight, in lb/ft3 for U.S. units and in kN/m3 for SI units; w ⫽ water content in percent.

mentation may exhibit an increased thermal conductivity with time. Temperature All crystalline minerals in soils have decreasing thermal conductivity with increasing temperature; however, the thermal conductivity of water increases slightly with increasing temperature, and the thermal conductivity of saturated pore air increases markedly with increasing temperature. The net effect is that the thermal conductivity of moist sand increases somewhat with increasing temperature.

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9.6

ELECTRICAL CONDUCTIVITY

Ohm’s law, Eq. (9.4), in which e is the electrical conductivity, applies to soil–water systems. The electrical conductivity equals the inverse of the electrical resistivity, or e ⫽

1 L (siemens/meter; S/m) RA

(9.40)

where R is the resistance ( ), L is length of sample

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9

CONDUCTION PHENOMENA

Co py rig hte dM ate ria l

fects particle size, shape, and surface conductance, soil structure, including fabric and cementation, and temperature. Electrical measurements found early applications in the fields of petroleum engineering, geophysical mapping and prospecting, and soil science, among others. The inherent complexity of soil–water systems and the difficulty in characterizing the wide ranges of particle size, shape, and composition have precluded development of generally applicable theoretical equations for electrical conductivity. However, a number of empirical equations and theoretical expressions based on simplified models may provide satisfactory results, depending on the particular soil and conditions. They differ in assumptions about the possible flow paths for electric current through a soil–water matrix, the path lengths and their relative importance, and whether charged particle surfaces contribute to the total current flow.

Figure 9.13 Thermal conductivity of soil (after Kersten,

1949).

Nonconductive Particle Models

Formation Factor The electrical conductivity of clean saturated sands and sandstones is directly proportional to the electrical conductivity of the pore water (Archie, 1942). The coefficient of proportionality depends on porosity and fabric. Archie (1942) defined the formation factor, F, as the resistivity of the saturated soil, T, divided by the resistivity of the saturating solution, W, that is, F⫽

T  ⫽ W W T

(9.41)

where W and T are the electrical conductivities of the pore water and saturated soil, respectively. An empirical correlation between formation factor and porosity for clean sands and sandstones is given by F ⫽ n⫺m

Figure 9.14 Typical relationship between thermal resistivity

and water content for a compacted sand.

(m), and A is its cross-sectional area (m2). The value of electrical conductivity for a saturated soil is usually in the approximate range of 0.01 to 1.0 S/m. The specific value depends on several properties of the soil, including porosity, degree of saturation, composition (conductivity) of the pore water, mineralogy as it af-

Copyright © 2005 John Wiley & Sons

(9.42)

where n is porosity, and m equals from 1.3 for loose sands to 2 for highly cemented sandstones. An empirical relation between formation factor at 100 percent water saturation and ‘‘apparent’’ formation factor at saturation less than 100 percent is FatSw⫽1 ⫽ (Sw)p

W T

(9.43)

where p is a constant determined experimentally. Archie suggested a value of p ⫽ 2; however, other published values of p range from 1.4 to 4.6, depending on the soil and

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ELECTRICAL CONDUCTIVITY

269

whether a given saturation is reached by wetting or by drainage. Capillary Model In this and the theoretical models

tance of clayey particles to the total current flow would be small.

that follow, direct current (DC) conductivity is assumed, although they may apply to low-frequency alternating current (AC) models as well. Consider a saturated soil sample of length L and cross-sectional area A. If the pores are assumed to be connected and can be represented by a bundle of tubes of equal radius and length Le and total area Ae, where Ae ⫽ porosity ⫻ A, and Le is the actual length of the flow path, then an equation for the formation factor as a function of the porosity n and the tortuosity T ⫽ Le /L is

Conductive Particle Models

Co py rig hte dM ate ria l T2 n

F⫽

(9.44)

T ⫽ X(W ⫹ s)

For S ⬍ 1, and assuming that the area available for electrical flow is nSA, then F ⫽ T 2 /nS. In principle, if F is measured for a given soil and n is known, a value of tortuosity can be calculated to use in the Kozeny– Carman equation for hydraulic conductivity. Cluster Model As discussed earlier in connection with hydraulic conductivity, the cluster model (Olsen, 1961, 1962) shown in Fig. 9.10 assumes unequal pore sizes. Three possible paths for electrical current flow can be considered: (1) through the intercluster pores, (2) through the intracluster pores, and (3) alternately through inter- and intracluster pores. On this basis the following equations for formation factor as a function of the cluster model parameters can be derived (Olsen, 1961): F ⫽ T2

冉

冊冉 冊

1 ⫹ eT eT ⫺ ec

1 1⫹X

X⫽Y⫹Z

Y⫽

In conductive particle models the contribution of the ions concentrated at the surface of negatively charged particles is taken into account. Two simple mixture models are presented below; other models can be found in Santamarina et al. (2001). Two-Parallel-Resistor Model A contribution of surface conductance is included, and the soil–water system is equivalent to two electrical resistors in parallel (Waxman and Smits, 1968). The result is that the total electrical conductivity T is

(9.46)

[(1 ⫹ eT)/(eT ⫺ ec)]2 1 ⫹ (Tc /T)2 [(1 ⫹ ec)2 /ec(eT ⫺ ec)] Z⫽a

冉

ec

冊冉 冊

eT ⫺ ec

(9.45)

T Tc

(9.47)

2

(9.48)

in which T is the intercluster tortuosity, Tc is the intracluster tortuosity, and a is the effective cluster ‘‘contact area.’’ The cluster contact area is very small except for heavily consolidated systems. This model successfully describes the flow of current in soils saturated with high conductivity water. In such systems, the contribution of the surface conduc-

Copyright © 2005 John Wiley & Sons

(9.49)

in which s is a surface conductivity term, and X is a constant analogous to the reciprocal of the formation factor that represents the internal geometry. This approach yields better fits of T versus W data for clay-bearing soils. However, it assumes a constant value for the contribution of the surface ions that is independent of the electrolyte concentration in the pore water, and it fails to include a contribution for the surface conductance and pore water conductance in a series path. Three-Element Network Model A third path is included in this formulation that considers flow along particle surfaces and through pore water in series in addition to the paths included in the two-parallelresistor model. The flow paths and equivalent electrical circuit are shown in Fig. 9.15. Analysis of the electrical network for determination of T gives T ⫽

aWs ⫹ bs ⫹ cW (1 ⫺ e)W ⫹ es

(9.50)

If the surface conductivity s is negligible, the simple formulation proposed by Archie (1942) for sands is obtained; that is, T ⫽ constant ⫻ W. Some of the geometric parameters a, b, c, d, and e can be written as functions of porosity and degree of saturation; others are obtained through curve regression analysis of T versus W data. Soil conductivity as a function of pore fluid conductivity is shown in Fig. 9.16 for a silty clay. The three-element model fits the data well over the full range, the two-element model gives good predictions for the higher values of conductivity, and the simple formation factor relationship is a reasonable average

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9

CONDUCTION PHENOMENA

Co py rig hte dM ate ria l

270

Figure 9.15 Three-element network model for electrical conductivity: (a) current flow paths

and (b) equivalent electrical circuit.

for conductivity values in the range of about 0.3 to 0.6 S/m. Alternating Current Conductivity and Dielectric Constant

The electrical response of a soil in an AC field is frequency dependent owing to the polarizability properties of the system constituents. Several scale-dependent polarization mechanisms are possible in soils, as shown in Fig. 9.17. The smaller the element size the higher the polarization frequency. At the atomic and molecular scales, there are polarizations of electrons [electronic resonance at ultraviolet (UV) frequencies], ions [ionic resonance at infrared (IR) frequencies], and dipolar molecules (orientational relaxation at microwave frequencies). A mixture of components (like water and soil particles) having different polarizabilities and conductivities produces spatial polarization by charge accumulation at interfaces (called Maxwell– Wagner interfacial polarization). The ions in the Stern layer and double layer are restrained (Chapter 6), and hence they also exhibit polarization. This polarization results in relaxation responses at radio frequencies. Further details of the polarization mechanisms are given by Santamarina et al. (2001). The effective AC conductivity eff is expressed as eff ⫽  ⫹ !ⴖ"0

(9.51)

where  is the conductivity, !ⴖ is the polarization loss (called the imaginary relative permittivity), " is the

Copyright © 2005 John Wiley & Sons

frequency, and 0 is the permittivity of vacuum [8.85 ⫻ 10⫺12 C2 /(Nm2)]. The frequency-dependent effective conductivities of deionized water and kaolinite–water mixtures at two different water contents (0.2 and 33 percent) are shown in Fig. 9.18a. The complicated interactions of different polarization mechanisms are responsible for the variations shown. A material is dielectric if charges are not free to move due to their inertia. Higher frequencies are needed to stop polarization at smaller scales. The dielectric constant (or the real relative permittivity !5) decreases with increasing frequency; more polarization mechanisms occur at lower frequencies. The frequency-dependent dielectric constants of deionized water and kaolinite–water mixtures are shown in Fig. 9.18b. The value for deionized water is about 79 above 10 kHz. Below this frequency, the values increase with decrease in frequency. This is attributed to experimental error caused by an electrode effect in which charges

5 To describe the out-of-phase response under oscillating excitation, the electrical properties of a material are often defined in the complex plane:

 ⫽  ⫺ jⴖ

where  is the complex permittivity, j is the imaginary number (兹⫺1), and  and ⴖ are real and imaginary numbers describing the electrical properties. The permittivity  is often normalized by the permittivity of vacuum 0 as !⫽

 ⫽ ! ⫺ j!ⴖ 0

where ! is called the relative permittivity.

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ELECTRICAL CONDUCTIVITY

271

100

σeff (S/m)

Deionized Water 10–2 33% No Data Available

10–4

102

104 106 Frequency (Hz) (a)

Co py rig hte dM ate ria l

0.2% 10–6 100

106 33%

Figure 9.16 Soil electrical conductivity as a function of pore

fluid conductivity and comparisons with three models.

κ

108

1010

Electrode Effect

104 0.2%

Deionized Water

No Data Available

102

100

accumulate at the electrode–specimen interface (Klein and Santamarina, 1997). Similarly to the observations made for the effective conductivities, the real permittivity values of the mixtures show complex trends of frequency dependency. For analysis of AC conductivity and dielectric constant as a function of frequency in an AC field, Smith and Arulanandan (1981) modified the three-element model shown in Fig. 9.15 by adding a capacitor in parallel with each resistor. The resulting equations can be fit to experimental frequency dispersions of the con-

100

102

104 106 Frequency (Hz) (b)

1010

Figure 9.18 (a) Conductivity and (b) relative permittivity as a function of frequency for deionized water and kaolinite at water contents of 0.2 and 33 percent (from Santamarina et al., 2001).

Figure 9.17 Frequency ranges associated with different polarization mechanisms (from Santamarina et al., 2001).

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108

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CONDUCTION PHENOMENA

ductivity and apparent dielectric constant by computer optimization of geometrical and compositional parameters. The resulting parameter values are useful for characterizing mineralogy, porosity, and fabric. More detailed discussions on electrical models, data interpretation, and correlations with soil properties are given by Santamarina et al. (2001).

DIFFUSION

Anion (1)

D0 ⫻ 1010(m2 /s) (2)

Cation (3)

D0 ⫻ 1010(m2 /s) (4)

OH⫺ F⫺ Cl⫺ Br⫺ I⫺ HCO3⫺ NO3⫺ SO42⫺ CO32⫺ — — — — — — — — — — —

52.8 14.7 20.3 20.8 20.4 11.8 19.0 10.6 9.22 — — — — — — — — — — —

H⫹ Li⫹ Na⫹ K⫹ Rb⫹ Cs⫹ Be2⫹ Mg2⫹ Ca2⫹ Sr2⫹ Ba2⫹ Pb2⫹ Cu2⫹ Fe2⫹a Cd2⫹a Zn2⫹ Ni2⫹a Fe3⫹a Cr3⫹a Al3⫹a

93.1 10.3 13.3 19.6 20.7 20.5 5.98 7.05 7.92 7.90 8.46 9.25 7.13 7.19 7.17 7.02 6.79 6.07 5.94 5.95

Co py rig hte dM ate ria l

9.7

Table 9.3 Self-Diffusion Coefficients for Ions at Infinite Dilution in Water

Chemical transport through sands is dominated by advection, wherein dissolved and suspended species are carried with flowing water. However, in fine-grained soils, wherein the hydraulic flow rates are very small, for example, kh less than about 1 ⫻ 10⫺9 m/s, chemical diffusion plays a role and may become dominant when kh becomes less than about 1 ⫻ 10⫺10 m/s. Fick’s law, Eq. (9.5), is the controlling relationship, and D(L2T⫺1), the diffusion coefficient, is the controlling parameter. Diffusive chemical transport is important in clay barriers for waste containment, in some geologic processes, and in some forms of chemical soil stabilization. Comprehensive treatments of the diffusion process, values of diffusion coefficients and methods for their determination, and applications, especially in relation to chemical transport and waste containment barrier systems, are given by Quigley et al. (1987), Shackelford and Daniel (1991a, 1991b), Shincariol and Rowe (2001) and Rowe (2001). Diffusive flow is driven by chemical potential gradients, but for most applications chemical concentration gradients can be used for analysis. The diffusion coefficient is measured and expressed in terms of chemical gradients. Maximum values of the diffusion coefficient D0 are found in free aqueous solution at infinite dilution. Self-diffusion coefficients for a number of ion types in water are given in Table 9.3. Usually cation–anion pairs are diffusing together, thereby slowing down the faster and speeding up the slower. This may be seen in Table 9.4, which contains values of some limiting free solution diffusion coefficients for some simple electrolytes. Diffusion through soil is slower and more complex than diffusion through a free solution, especially when adsorptive clay particles are present. There are several reasons for this (Quigley, 1989): 1. Reduced cross-sectional area for flow because of the presence of solids 2. Tortuous flow paths around particles 3. The influences of electrical force fields caused by the double-layer distributions of charges

Copyright © 2005 John Wiley & Sons

a

Values from Li and Gregory (1974). Reprinted with permission from Geochimica et Cosmochimica Acta, Vol. 38, No. 5, pp. 703–714. Copyright  1974, Pergamon Press.

4. Retardation of some species as a result of ion exchange and adsorption by clay minerals and organics or precipitation 5. Biodegradation of diffusing organics 6. Osmotic counterflow 7. Electrical imbalance, possibly by anion exclusion

The diffusion coefficient could increase with time of flow through a soil as a result of such processes as (Quigley, 1989): 1. K⫹ fixation by vermiculite, which would decrease the cation exchange capacity and increase the free water pore space 2. Electrical imbalances that act to pull cations or anions 3. The attainment of adsorption equilibrium, thus eliminating retardation of some species

In an attempt to take some of these factors, especially geometric tortuosity of interconnected pores, into account, an effective diffusion coefficient D* is

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DIFFUSION

transient diffusion, that is, the time rate of change of concentration with distance:

Table 9.4 Limiting Free Solution Diffusion Coefficients for Some Simple Electrolytes D0 ⫻ 1010(m2 /s) (2)

Electrolyte (1)

33.36 34.00 13.66 13.77 16.10 16.25 16.14 19.93 20.16 19.99 20.44 13.35 13.85

c 2c ⫽ D* 2 t x

Reported by Shackelford and Daniel, 1991a after Robinson and Stokes, 1959. Reprinted from the Journal of Geotechnical Engineering, Vol. 117, No. 3, pp. 467–484. Copyright  1991. With permission of ASCE.

used. Several definitions have been proposed (Shackelford and Daniel, 1991a) in which the different factors are taken into account in different ways. Although these relationships may be useful for analysis of the importance of the factors themselves, it is sufficient for practical purposes to use D* ⫽ a D0

(9.52)

in which a is an ‘‘apparent tortuosity factor’’ that takes several of the other factors into account, and use values of D* measured under representative conditions. The effective coefficient for diffusion of different chemicals through saturated soil is usually in the range of about 2 ⫻ 10⫺10 to 2 ⫻ 10⫺9 m2 /s, although the values can be one or more orders of magnitude lower in highly compacted clays and clays, such as bentonite, that can behave as semipermeable membranes (Malusis and Shackelford, 2002b). Values for compacted clays are rather insensitive to molding water content or method of compaction (Shackelford and Daniel, 1991b), in stark contrast to the hydraulic conductivity, which may vary over a few orders of magnitude as a result of changes in these factors. This suggests that soil fabric differences have relatively minor influence on the effective diffusion coefficient. Whereas Fick’s first law, Eq. (9.5), applies for steady-state diffusion, Fick’s second law describes

Copyright © 2005 John Wiley & Sons

(9.53)

For transient diffusion with constant effective diffusion coefficient D*, the solution for this equation is of exactly the same form as that for the Terzaghi equation for clay consolidation and that for one-dimensional transient heat flow. An error function solution for Eq. (9.53) (Ogata, 1970; Freeze and Cherry, 1979), for the case of onedimensional diffusion from a layer at a constant source concentration C0 into a layer having a sufficiently low initial concentration that it can be taken as zero at t ⫽ 0, is

Co py rig hte dM ate ria l

HCl HBr LiCl LiBr NaCl NaBr NaI KCl KBr KI CsCl CaCl2 BaCl2

273

C x x ⫽ erfc ⫽ 1 ⫺ erf C0 2兹D*t 2兹D*t

(9.54)

where C is the concentration at any time at distance ⫻ from the source. Curves of relative concentration as a function of depth for different times after the start of chloride diffusion are shown in Fig. 9.19a (Quigley, 1989). An effective diffusion coefficient for chloride of 6.47 ⫻ 10⫺10 m2 /s was assumed. Also shown (Fig. 9.19b) is the migration velocity of the C/C0 front within the soil as a function of time. As chloride is one of the more rapidly diffusing ionic species, Fig. 9.19 provides a basis for estimating maximum probable migration distances and concentrations as a function of time that result solely from diffusion. When there are adsorption–desorption reactions, chemical reactions such as precipitation–solution, radioactive decay, and/or biological processes occurring during diffusion, the analysis becomes more complex than given by the foregoing equations. For adsorption– desorption reactions and the assumption that there is linearity between the amount adsorbed and the equilibrium concentration, Eq. (9.53) is often written as c D* 2c ⫽ t Rd x2

(9.55)

where Rd is termed the retardation factor, and it is defined by Rd ⫽ 1 ⫹

d K d

(9.56)

in which d is the bulk dry density of the soil, is the

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CONDUCTION PHENOMENA

tailed discussions of distribution coefficients and their determination are given by Freeze and Cherry (1979), Quigley et al., (1987), Quigley (1989), and Shackelford and Daniel (1991a, b).

9.8 TYPICAL RANGES OF FLOW PARAMETERS

Co py rig hte dM ate ria l

Usual ranges for the values of the direct flow conductivities for hydraulic, thermal, electrical, and diffusive chemical flows are given in Table 9.5. These ranges are for fine-grained soils, that is, silts, silty clays, clayey silts, and clays. They are for full saturation; values for partly saturated soils can be much lower. Also listed in Table 9.5 are values for electroosmotic conductivity, osmotic efficiency, and ionic mobility. These properties are needed for analysis of coupling of hydraulic, electrical, and chemical flows, and they are discussed further later.

9.9 SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS

Figure 9.19 Time rate of chloride diffusion (from Quigley,

1989). (a) Relative concentration as a function of depth after different times and (b) velocity of migration of the front having a concentration C / C0 of 0.5.

volumetric water content, that is, the volume of water divided by the total volume (porosity in the case of a saturated soil), and Kd is the distribution coefficient. The distribution coefficient defines the amount of a given constituent that is adsorbed or desorbed by a soil for a unit increase or decrease in the equilibrium concentration in solution. Other reactions influencing the amount in free solution relative to that fixed in the soil (e.g., by precipitation) may be included in Kd, depending on the method for measurement and the conditions being modeled. Distribution coefficients are usually determined from adsorption isotherms, and they may be constants for a given soil–chemical system or vary with concentration, pH, and temperature. More de-

Copyright © 2005 John Wiley & Sons

Usually there are simultaneous flows of different types through soils and rocks, even when only one type of driving force is acting. For example, when pore water containing chemicals flows under the action of a hydraulic gradient, there is a concurrent flow of chemical through the soil. This type of chemical transport is termed advection. In addition, owing to the existence of surface charges on soil particles, especially clays, there are nonuniform distributions of cations and anions within soil pores resulting from the attraction of cations to and repulsion of anions from the negatively charged particle surfaces. The net negativity of clay particles is caused primarily by isomorphous substitutions within the crystal structure, as discussed in Chapter 3, and the ionic distributions in the pore fluid are described in Chapter 6. Because of the small pore sizes in fine-grained soils and the strong local electrical fields, clay layers exhibit membrane properties. This means that the passage of certain ions and molecules through the clay may be restricted in part or in full at both microscopic and macroscopic levels. Owing to these internal nonhomogeneities in ion distributions, restrictions on ion movements caused by electrostatic attractions and repulsions, and the dependence of these interactions on temperature, a variety of microscopic and macroscopic effects may be observed when a wet soil mass is subjected to flow

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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND SALTS THROUGH SOIL-COUPLED FLOWS

Table 9.5

275

Typical Range of Flow Parameters for Fine-Grained Soilsa

Parameter

Symbol

Units

Minimum

Maximum

Porosity Hydraulic conductivity Thermal conductivity Electrical conductivity Electro osmotic conductivity Diffusion coefficient Osmotic efficiencyb Ionic mobility

n kh

— m s⫺1

0.1 1 ⫻ 10⫺11

0.7 1 ⫻ 10⫺6

kt

W m⫺1 K⫺1

0.25

2.5

e

siemens m⫺1

0.01

1.0

m2 s⫺1 V⫺1

1 ⫻ 10⫺9

1 ⫻ 10⫺8

D

m2 s⫺1

2 ⫻ 10⫺10

2 ⫻ 10⫺9

—

0

1.0

m2 s⫺1 V⫺1

3 ⫻ 10⫺9

1 ⫻ 10⫺8

Co py rig hte dM ate ria l ke

"

u

a

The above values of flow coefficients are for saturated soil. They may be much less in partly saturated soil. b 0 to 1.0 is the theoretical range for the osmotic efficiency coefficient. Values greater than about 0.7 are unlikely in most fine-grained materials of geotechnical interest.

gradients of different types. A gradient of one type Xj can cause a flow of another type Ji, according to Ji ⫽ Lij Xj

(9.57)

The Lij are termed coupling coefficients. They are properties that may or may not be of significant magnitude in any given soil, as discussed later. Types of coupled flow that can occur are listed in Table 9.6, along with terms commonly used to describe them.6 Of the 12 coupled flows shown in Table 9.6, several are known to be significant in soil–water systems, at least under some conditions. Thermoosmosis, which is water movement under a temperature gradient, is important in partly saturated soils, but of lesser importance in fully saturated soils. Significant effects from thermally driven moisture flow are found in semiarid and arid areas, in frost susceptible soils, and in expansive soils. An analysis of thermally driven moisture

6

Mechanical coupling also occurs in addition to the hydraulic, thermal, electrical, and chemical processes listed in Table 9.6. A common manifestation of this in geotechnical applications is the development of excess pore pressure and the accompanying fluid flow that result from a change in applied stress. This type of coupling is usually most easily handled by usual soil mechanics methods. A few other types of mechanical coupling may also exist in soils and rocks (U.S. National Committee for Rock Mechanics, 1987).

Copyright © 2005 John Wiley & Sons

flow is developed later. Electroosmosis has been used for many years as a means for control of water flow and for consolidation of soils. Chemicalosmosis, the flow of water caused by a chemical gradient acting across a clay layer, has been studied in some detail recently, owing to its importance in waste containment systems. Isothermal heat transfer, caused by heat flow along with water flow, has caused great difficulties in the creation of frozen soil barriers in the presence of flowing groundwater. Electrically driven heat flow, the Peltier effect, and chemically driven heat flow, the Dufour effect, are not known to be of significance in soils; however, they appear not to have been studied in any detail in relation to geotechnical problems. Streaming current, the term applied to both hydraulically driven electrical current and ion flows, has importance to both chemical flow through the ground (advection) and the development of electrical potentials, which may, in turn, influence both fluid and ion flows as a result of additional coupling effects. The complete roles of thermoelectricity and diffusion and membrane potentials are not yet known; however, electrical potentials generated by temperature and chemical gradients are important in corrosion and in some groundwater flow and stability problems. Whether thermal diffusion of electrolytes, the Soret effect, is important in soils has not been evaluated;

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9

Table 9.6

CONDUCTION PHENOMENA

Direct and Coupled Flow Phenomena Gradient X Hydraulic Head

Fluid

Heat

Current

Ion

Temperature

Hydraulic conduction Darcy’s law Isothermal heat transfer or thermal filtration Streaming current

Thermoosmosis

Electroosmosis

Chemical osmosis

Thermal conduction Fourier’s law Thermoelectricity Seebeck or Thompson effect

Peltier effect

Dufour effect

Streaming current ultrafiltration (also known as hyperfiltration)

Thermal diffusion of electrolyte or Soret effect

Electrophoresis

however, since chemical activity is highly temperature dependent, it may be a significant process in some systems. Finally, electrophoresis, the movement of charged particles in an electrical field, has been used for concentration of mine waste and high water content clays. The relative importance of chemically and electrically driven components of total hydraulic flow is illustrated in Fig. 9.20, based on data from tests on kaolinite given by Olsen (1969, 1972). The theory for description of coupled flows is given later. A practical form of Eq. (9.57) for fluid flow under combined hydraulic, chemical, and electrical gradients is qh ⫽ ⫺kh

H

L

Chemical Concentration

Electrical

Co py rig hte dM ate ria l

Flow J

A ⫹ kc

log(CB /CA)

E A ⫺ ke A (9.58) L L

in which kh, kc, and ke are the hydraulic, osmotic, and electroosmotic conductivities, H is the hydraulic head difference, E is the voltage difference, and CA and CB are the salt concentrations on opposite sides of a clay layer of thickness L. In the absence of an electrical gradient, the ratio of osmotic to hydraulic flows is

冉冊

qhc k log(CB /CA) ⫽⫺ c qh kh

H

( E ⫽ 0)

(9.59)

and, in the absence of a chemical gradient, the ratio of electroosmotic flows to hydraulic flows is

Copyright © 2005 John Wiley & Sons

Electric conduction Ohm’s law

冉冊

qhe ke E ⫽ qh kh H

Diffusion and membrane potentials or sedimentation current Diffusion Fick’s law

( C ⫽ 0)

(9.59a)

The ratio (kc /kh) in Fig. 9.20 indicates the hydraulic head difference in centimeters of water required to give a flow rate equal to the osmotic flow caused by a 10fold difference in salt concentration on opposite sides of the layer. The ratio ke /kh gives the hydraulic head difference required to balance that caused by a 1 V difference in electrical potentials on opposite sides of the layer. During consolidation, the hydraulic conductivity decreases dramatically. However, the ratios kc /kh and ke /kh increase significantly, indicating that the relative importance of osmotic and electroosmotic flows to the total flow increases. Although the data shown in Fig. 9.20 are shown as a function of the consolidation pressure, the changes in the values of kc /kh and ke /kh are really a result of the decrease in void ratio that accompanies the increase in pressure, as may be seen in Fig. 9.20c. These results for kaolinite provide a conservative estimate of the importance of osmotic and electroosmotic flows because coupling effects in kaolinite are usually smaller than in more active clays, such as montmorillonite-based bentonites. In systems containing confined clay layers acted on by chemical and/or electrical gradients, Darcy’s law by itself may be an insufficient basis for prediction of hydraulic flow rates, particularly if the clay is highly plastic and at a very low void ratio. Such conditions can be found in deeply buried clay

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277

Co py rig hte dM ate ria l

QUANTIFICATION OF COUPLED FLOWS

Figure 9.20 Hydraulic, osmotic, and electroosmotic conductivities of kaolinite (data from

Olsen 1969, 1972): (a) consolidation curve, (b) conductivity values, and (c) conductivities as a function of void ratio.

and clay shale and in densely compacted clays. For more compressible clays, the ratios kc /kh and ke /kh may be sufficiently high to be useful for consolidation by electrical and chemical means, as discussed later in this chapter.

9.10

QUANTIFICATION OF COUPLED FLOWS

Quantification of coupled flow processes may be done by direct, empirical determination of the relevant parameters for a particular case or by relationships derived from a theoretical thermodynamic analysis of the complete set of direct and coupled flow equations.

Copyright © 2005 John Wiley & Sons

Each approach has advantages and limitations. It is assumed in the following that the soil properties remain unchanged during the flow processes, an assumption that may not be justified in some cases. The effects of flows of different types on the state and properties of a soil are discussed later in this chapter. However, when properties are known to vary in a predictable manner, their variations may be taken into account in numerical analysis methods. Direct Observational Approach

In the general case, there may be fluid, chemical, electrical, and heat flows. The chemical flows can be sub-

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CONDUCTION PHENOMENA

qw ⫽ LHH(⫺H) ⫹ LHE(⫺E) ⫹ LHC(⫺C)

(9.60)

I ⫽ LEH(⫺H) ⫹ LEE(⫺E) ⫹ LEC(⫺C)

(9.61)

JC ⫽ LCH(⫺H) ⫹ LCE(⫺E) ⫹ LCC(⫺C)

(9.62)

where qw I Jc H E C Lij

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

ductivity coefficient kh is readily determined.7 The coefficient of electroosmotic hydraulic conductivity is usually determined by measuring the hydraulic flow rate developed in a known DC potential field under conditions of ih ⫽ 0. The electrical conductivity e is obtained from the same experiment through measurement of the electrical current. The main advantage of this empirical, but direct, approach is simplicity. It is particularly useful when only a few of the possible couplings are likely to be important and when some uncertainty in the measured coefficients is acceptable.

Co py rig hte dM ate ria l

divided according to the particular chemical species present. Each flow type may have contributions caused by gradients of another type, and their importance depends on the values of Lij and Xj in Eq. (9.57). A complete and accurate description of all flows may be a formidable task. However, in many cases, flows of only one or two types may be of interest, some of the gradients may not exist, and/or some of the coupling coefficients may be either known or assumed to be unimportant. The matrix of flows and forces then reduces significantly, and the determination of coefficients is greatly simplified. For example, if simple electroosmosis under isothermal conditions is considered, then Eq. (9.57) yields

water flow rate electrical current chemical flow rate hydraulic head electrical potential chemical concentration coupling coefficients; the first subscript indicates the flow type and the second denotes the type of driving force

If there are no chemical concentration differences across the system, then the last terms on the right-hand side of Eqs. (9.60), (9.61), and (9.62) do not exist. In this case, Eqs. (9.60) and (9.61) become, when written in more familiar terms, qw ⫽ khih ⫹ keie

I ⫽ hih ⫹ eie

(9.63)

(9.64)

where kh ⫽ hydraulic conductivity ke ⫽ electroosmotic hydraulic conductivity h ⫽ electrical conductivity due to hydraulic flow e ⫽ electrical conductivity ih ⫽ hydraulic gradient ie ⫽ electrical potential gradient If permeability tests are done in the absence of an electrical potential difference, then the hydraulic con-

Copyright © 2005 John Wiley & Sons

General Theory for Coupled Flows

When several flows are of interest, each resulting from several gradients, a more formal methodology is necessary so that all relevant factors are accounted for properly. If there are n different driving forces, then there will be n direct flow coefficients Lii and n(n ⫺ 1) coupling coefficients Lij(i ⫽ j). The determination of these coefficients is best done within a framework that provides a consistent and correct description of each of the flows. Irreversible thermodynamics, also termed nonequilibrium thermodynamics, offers a basis for such a description. Furthermore, if the terms are properly formulated, then Onsager’s reciprocal relations apply, that is, Lij ⫽ Lji

(9.65)

and the number of coefficients to be determined is significantly reduced. In addition, the derived forms for the coupling coefficients, when cast in terms of measurable and understood properties, provide a basis for rapid assessment of their importance. The theory of irreversible thermodynamics as applied to transport processes in soils is only outlined here. More comprehensive treatments are given by DeGroot and Mazur (1962), Fitts (1962), Katchalsky and Curran (1967), Greenberg, et al. (1973), Yeung and Mitchell (1992), and Malusis and Shackelford (2002a). Irreversible thermodynamics is a phenomenological, macroscopic theory that provides a basis for descrip-

7 Note that unless the ends of the sample are short circuited to prevent the development of a streaming potential, there will be a small electroosmotic counterflow contributed by the keie term in Eq. (9.63). Streaming potentials may be up to a few tens of millivolts in soils. Streaming potential is one of four types of electrokinetic phenomena that may exist in soils, as discussed in more detail in Section 9.16.

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SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS

tion of systems that are out of equilibrium. It is based on three postulates, namely,

in the formulation of the flow equations. And # is also the sum of products of fluxes and driving forces:

1. Local equilibrium, a criterion that is satisfied if local perturbations are not large. 2. Linear phenomenological equations, that is,

冘L X n

Ji ⫽

ij

j

( j ⫽ 1,2, . . . , n)

(9.66)

冘JX n

#⫽

i

i

3. Validity of the Onsager reciprocal relations, a condition that is satisfied if the Ji and Xj are formulated properly (Onsager, 1931a, 1931b). Experimental verification of the Onsager reciprocity for many systems and processes has been obtained and is summarized by Miller (1960). Both the driving forces and flows vanish in systems that are in equilibrium, so the deviations of thermodynamic variables from their equilibrium values provide a suitable basis for their formulation. The deviations of the state parameters Ai from equilibrium are given by i ⫽ Ai ⫺ A 0i

(9.67)

where A 0i is the value of the state parameter at equilibrium and Ai is its value in the disturbed state. Criteria for deriving the forces and flows are then developed on the basis of the second law of thermodynamics, which states that at equilibrium, the entropy S is a maximum, and i ⫽ 0. The change in entropy

S that results from a change in state parameter gives the tendency for a variable to change. Thus S/ i is a measure of the force causing i to change, and is called Xi. The flows Ji, termed fluxes in irreversible thermodynamics, are given by i / t, the time derivative of i. On this basis, the resulting entropy production  per unit time becomes ⫽

dS ⫽ dt

冘JX

(9.69)

i⫽1

The units of # are energy per unit time, and it is a measure of the rate of local free energy dissipation by irreversible processes. Application of the thermodynamic theory of irreversible processes requires the following steps:

Co py rig hte dM ate ria l

j⫽1

279

1. Finding the dissipation function # for the flows 2. Defining the conjugated flows Ji and driving forces Xi from Eq. (9.69) 3. Formulating the phenomenological equations in the form of Eq. (9.66) 4. Applying the Onsager reciprocal relations 5. Relating the phenomenological coefficients to measurable quantities

When the Onsager reciprocity is used, the number of independent coefficients Lij reduces from n2 to [(n ⫹ 1)n]/2. Application

The quantitative analysis and prediction of flows through soils, for a given set of boundary conditions, depends on the values of the various phenomenological coefficients in the above flow equations. Unfortunately, these are not always known with certainty, and they may vary over wide ranges, even within an apparently homogeneous soil mass. The direct flow coefficients, that is, the hydraulic, electrical, and thermal conductivities, and the diffusion coefficient, exhibit the greatest ranges of values. Thus, it is important to examine these properties first before detailed analysis of coupled flow contributions. For many problems, it may be sufficient to consider only the direct flows, provided the factors influencing their values are fully appreciated.

n

i

i

(9.68)

i⫽1

The entropy production can be related explicitly to various irreversible processes in terms of proper forces and fluxes (Gray, 1966; Yeung and Mitchell, 1992). If the choices satisfy Eq. (9.68), then the Onsager reciprocity relations apply. It has been found more useful to use # ⫽ T, the dissipation function, in which T is temperature, than 

Copyright © 2005 John Wiley & Sons

9.11 SIMULTANEOUS FLOWS OF WATER, CURRENT, AND CHEMICALS

Use of irreversible thermodynamics for the description of coupled flows as developed above is straightforward in principle; however, it becomes progressively more difficult in application as the numbers of driving forces and different flow types increase. This is because of (1) the need for proper specification of the different coupling coefficients and (2) the need for independent

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methods for their measurement. Thus, the analysis of coupled hydraulic and electrical flows or of coupled hydraulic and chemical flows is much simpler than the analysis of a system subjected to electrical, chemical, and hydraulic gradients simultaneously. Relationships for the volume flow rate of water for several cases and for thermoelectric and thermoosmotic coupling in saturated soils are given by Gray (1966, 1969). The simultaneous flows of liquid and charge in kaolinite and the fluid volume flow rates under hydraulic, electric, and chemical gradients were studied by Olsen (1969, 1972). The theory for coupled salt and water flows was developed by Greenberg (1971) and applied to flows in a groundwater basin (Greenberg et al., 1973) and to chemicoosmotic consolidation of clay (Mitchell et al., 1973). Equations for the simultaneous flows of water, electricity, cations, and anions under hydraulic, electrical, and chemical gradients were formulated by Yeung (1990) using the formalism of irreversible thermodynamics as outlined previously. The detailed development is given by Yeung and Mitchell (1993). The results are given here. The chemical flow is separated into its anionic and cationic components in order to permit determination of their separate movements as a function of time. This separation may be important in some problems, such as chemical transport through the ground, where the fate of a particular ionic species, a heavy metal, for example, is of interest. The analysis applies to an initially homogeneous soil mass that separates solutions of different concentrations of anions and cations, at different electrical potentials and under different hydraulic heads, as shown schematically in Fig. 9.21. Only one anion and one cation species are assumed to be present, and no adsorption or desorption reactions are occurring. The driving forces are the hydraulic gradient (⫺P), the electrical gradient (⫺E), and the concentrationdependent parts of the chemical potential gradients of the cation (cc) and of the anion (ca). The fluxes are the volume flow rate of the solution per unit area Jv, the electric current I, and the diffusion flow rates of the cation Jdc and the anion Jda per unit area relative to the flow of water. These diffusion flows are related to the absolute flows according to

Figure 9.21 Schematic diagram of system for analysis of

simultaneous flows of water, electricity, and ions through a soil.

Jv ⫽ L11(⫺P) ⫹ L12(⫺E) ⫹ L13(⫺cc) ⫹ L14(⫺ca)

(9.71)

I ⫽ L21(⫺P) ⫹ L22(⫺E) ⫹ L23(⫺cc) ⫹ L24(⫺ca)

(9.72)

Jcd ⫽ L31(⫺P) ⫹ L32(⫺E) ⫹ L33(⫺cc) ⫹ L34(⫺ca)

(9.73)

Jad ⫽ L41(⫺P) ⫹ L42(⫺E) ⫹ L43(⫺cc) ⫹ L44(⫺ca)

(9.74)

These equations contain 4 conductivity coefficients Lii and 12 coupling coefficients Lij. As a result of Onsager reciprocity, however, the number of independent coupling coefficients reduces because L12 ⫽ L21 L13 ⫽ L31 L14 ⫽ L41 L23 ⫽ L32

Ji ⫽ Jid ⫹ ci Jv

(9.70)

L24 ⫽ L42 L34 ⫽ L43

in which ci is the concentration of ion i. The set of phenomenological equations that relates the four flows and driving forces is

Copyright © 2005 John Wiley & Sons

Thus there are 10 independent coefficients needed for a full description of hydraulic, electrical, anionic,

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e " w cc ca u* c u* a D* c D* a n R T

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

bulk electrical conductivity of the soil coefficient of osmotic efficiency unit weight of water concentration of cation concentration of anion effective ionic mobility of the cation effective ionic mobility of the anion effective diffusion coefficient of the cation effective diffusion coefficient of the anion soil porosity universal gas constant (8.314 J K⫺1 mol⫺1) absolute temperature (K)

Co py rig hte dM ate ria l

and cationic flows through a system subjected to hydraulic, electrical, and chemical gradients. If three of the four forces can be set equal to zero during a measurement of the flow under the fourth force, then the ratio of the flow rate to that force will give the value of its corresponding Lij. However, such measurements are not always possible or convenient. Accordingly, two forces and one flow are usually set to zero and the appropriate Lij are evaluated by solution of simultaneous equations. For measurements of hydraulic conductivity, electroosmotic hydraulic conductivity, electrical conductivity, osmotic efficiency, and effective diffusion coefficients done in the usual manner in geotechnical and chemical laboratories, the detailed application of irreversible thermodynamic theory led Yeung (1990) and Yeung and Mitchell (1993) to the following definitions for the Lij. It was assumed in the derivations that the solution is dilute and there are no interactions between cations and anions.8 k L L L11 ⫽ h ⫹ 12 21 wn L22

(9.75)

L33 ⫽ cc

L44 ⫽ ca

L12 ⫽ L21 ⫽

ke n

(9.76)

L13 ⫽ L31 ⫽

⫺"cckh L L ⫹ 12 23 wn L22

(9.77)

L14 ⫽ L41 ⫽

⫺"cakh L L ⫹ 12 24 wn L22

(9.78)

L22 ⫽

e n

(9.79)

L23 ⫽ L32 ⫽ ccu* c

(9.80)

L24 ⫽ L42 ⫽ ⫺cau*a

(9.81)

D* c cc RT

(9.82)

L33 ⫽

L34 ⫽ L43 ⫽ 0 L44 ⫽

D* a ca RT

Subsequently, Manassero and Dominijanni (2003) pointed out that the practical equations for diffusion L33 and L44 do not take the osmotic efficiency " (Section 9.13) into account, so Eqs. (9.82) and (9.84) more properly should be

冋 冋

where kh ⫽ hydraulic conductivity as usually measured (no electrical short circuiting) ke ⫽ coefficient of electroosmotic hydraulic conductivity 8

The Lij coefficients in Eqs. (9.75) to (9.84) were derived in terms of the cross-sectional area of the soil voids. They may be redefined in terms of the total cross-sectional area by multiplying each term on the right-hand side by the porosity, n.

Copyright © 2005 John Wiley & Sons

册 册

(1 ⫺ ")D* c k"2 a ⫹ a RT wn

(9.85) (9.86)

This modification becomes important in clays wherein osmotic efficiency, that is, the ability of the clay to restrict the flow of ions, is high. As the flows of ions relative to the soil are of more interest than relative to the water, Eq. (9.70) and Eqs. (9.73) and (9.74) can be combined to give Jc ⫽ (L31 ⫹ ccL11) w(⫺h) ⫹ (L32 ⫹ ccL12)(⫺E) ⫹ (L33 ⫹ ccL13)

RT (⫺cc) cc

⫹ (L34 ⫹ ccL14)

RT (⫺ca) ca

(9.83)

(9.84)

(1 ⫺ ")D* c k"2 c ⫹ c RT wn

(9.87)

Ja ⫽ (L41 ⫹ caL11) w(⫺h) ⫹ (L42 ⫹ caL12)(⫺E) ⫹ (L43 ⫹ caL13)

RT (⫺cc) cc

⫹ (L44 ⫹ caL14)

RT (⫺ca) ca

(9.88)

where (⫺h) is the hydraulic gradient. In Eqs. (9.87) and (9.88) the gradient of the chemical potential has been replaced by the gradient of the concentration according to

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(⫺ci ) ⫽

RT (⫺ci) ci

(9.89)

Co py rig hte dM ate ria l

These equations reduce to the known solutions for special cases such as chemical diffusion, advection– dispersion, osmotic pressure according to the van’t Hoff equation [see Eq. (9.98)], osmosis, and ultrafiltration. They predict reasonably well the distribution of single cations and anions as a function of time and position in compacted clay during the simultaneous application of hydraulic, electrical, and chemical gradients (Mitchell and Yeung, 1990). The analysis of multicomponent systems is more complex. The use of averaged chemical properties and the assumption of composite single species of anions and cations may yield reasonable approximate solutions in some cases. Malusis and Shackelford (2002a) present a more general theory for coupled chemical and hydraulic flow, based on an extension of the Yeung and Mitchell (1993) formulation, which accounts for multicomponent pore fluids and ion exchange processes occurring during transport.9 The flow equations can be incorporated into numerical models for the solution of transient flow problems. Conservation of mass of species i requires that

At the pore scale level, the fluid particles carrying dissolved chemicals move at different speeds because of tortuous flow paths around the soil grains and variable velocity distribution in the pores, ranging from zero at the soil particle surfaces to a maximum along the centerline of the pore. This results in hydrodynamic dispersion and a zone of mixing rather than a sharp boundary between two flowing solutions of different concentrations. Mathematically, this is accounted for by adding a dispersion term to the diffusion coefficient in the L33 and L44 terms to account for the deviation of actual motion of fluid particles from the overall or average movement described by Darcy’s law. More details can be found in groundwater and contamination textbooks such as Freeze and Cherry (1979) and Dominico and Schwartz (1997). Numerical models are available for groundwater flow and contaminant transport into which the above flow equations can be introduced (e.g., Anderson and Woessner, 1992; Zheng and Bennett, 2002). The most widely used groundwater flow numerical code is MODFLOW developed by the United States Geological Survey (USGS); various updated versions are available (e.g., Harbaugh et al., 2000). To solve single-species contaminant transport problems in groundwater, MT3DMS (Zheng and Wang, 1999) can be used. The code utilizes the flow solutions from MODFLOW. More complex multispecies reactions can be simulated by RT3D (Clement, 1997). POLLUTE (Rowe and Booker, 1997) provides ‘‘one- and onehalf-dimensional’’ solution to the advection–dispersion equation and is widely used in landfill design. A variety of public domain groundwater flow and contaminant transport codes is available from the web sites of the USGS, the U.S. Environmental Protection Agency (U.S. EPA), and the U.S. Salinity Laboratory.

ci ⫽ ⫺Ji ⫺ Gi t

(9.90)

in which Gi is a source–sink term describing the addition or removal rate of species i from the solution. As commonly used in groundwater flow analyses of contaminant transport, Gi is given by

冋

Gi ⫽ 1 ⫹

册

Kd Kd ci ici ⫹ n n t

(9.90a)

where i is the decay constant of species i,  is the bulk dry density of the soil, Kd is the distribution coefficient, and n is the soil porosity. As defined previously, the distribution constant is the ratio of the amount of chemical adsorbed on the soil to that in solution. The quantity in the brackets on the right-hand side of Eq. (9.90) is the retardation factor Rd defined by Eq. (9.56). Advection rather than diffusion is the dominant chemical transport mechanism in coarse-grained soils.

9.12

ELECTROKINETIC PHENOMENA

Coupling between electrical and hydraulic flows and gradients can generate four related electrokinetic phenomena in materials such as fine-grained soils, where there are charged particles balanced by mobile countercharges. Each involves relative movements of electricity, charged surfaces, and liquid phases, as shown schematically in Fig. 9.22. Electroosmosis

9

Malusis and Shackelford (2002a) defined parameters in terms of the total cross-sectional area for flow rather than the cross-sectional area of voids as used in the development of Eqs. (9.75) through (9.84).

Copyright © 2005 John Wiley & Sons

When an electrical potential is applied across a wet soil mass, cations are attracted to the cathode and anions to the anode (Fig. 9.22a). As ions migrate, they

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ELECTROKINETIC PHENOMENA

283

Figure 9.22 Electrokinetic phenomena: (a) electroosmosis, (b) streaming potential, (c) elec-

trophoresis, and (d) migration or sedimentation potential.

carry their water of hydration and exert a viscous drag on the water around them. Since there are more mobile cations than anions in a soil containing negatively charged clay particles, there is a net water flow toward the cathode. This flow is termed electroosmosis, and its magnitude depends on ke, the coefficient of electroosmotic hydraulic conductivity and the voltage gradient, as considered in more detail later. Streaming Potential

When water flows through a soil under a hydraulic gradient (Fig. 9.22b), double-layer charges are displaced in the direction of flow. This generates an electrical potential difference that is proportional to the hydraulic flow rate, called the streaming potential, between the opposite ends of the soil mass. Streaming potentials up to several tens of millivolts have been measured in clays. Electrophoresis

If a DC field is placed across a colloidal suspension, charged particles are attracted electrostatically to one

Copyright © 2005 John Wiley & Sons

of the electrodes and repelled from the other. Negatively charged clay particles move toward the anode as shown in Fig. 9.22c. This is called electrophoresis. Electrophoresis involves discrete particle transport through water; electroosmosis involves water transport through a continuous soil particle network. Migration or Sedimentation Potential

The movement of charged particles such as clay relative to a solution, as during gravitational settling, for example, generates a potential difference, as shown in Fig. 9.22d. This is caused by the viscous drag of the water that retards the movement of the diffuse layer cations relative to the particles. Of the four electrokinetic phenomena, electroosmosis has been given the most attention in geotechnical engineering because of its practical value for transporting water in fine-grained soils. It has been used for dewatering, soft ground consolidation, grout injection, and the containment and extraction of chemicals in the ground. These applications are considered in a later section.

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9.13 TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS

ui ⫽

Di兩zi兩F RT

(9.91)

in which zi is the ionic valence and F is Faraday’s constant. Similarly to the diffusion coefficients, the ionic mobilities are considerably less in a soil than in a free solution, especially in a fine-grained soil. The importance of coupled flows to fluid, electrical current, and chemical transport through soil under different conditions can be examined by study of the contributions of the different terms in Eqs. (9.71), (9.72), (9.87), and (9.88). For this purpose, the equations have been rewritten in one-dimensional form and in terms of the hydraulic, electrical, and chemical concentration gradients: ih ⫽ ⫺dh/dx, ie ⫽ ⫺dV/dx, and ic ⫽ ⫺dc/ dx, respectively. In addition, the chemical flows have been represented by a single equation. This assumes that all dissolved species are moving together. Terms involving the ionic mobility u do not exist in such a formulation because the cations and anions move together, with the effects of electrical fields assumed to accelerate the slower moving ions and to retard the faster moving ions. Thus there is no net transfer of electric charge due to ionic movement. The Lij coefficients have been replaced by the physical and chemical quantities that determine them, as given by Eqs. (9.74) through (9.85). The resulting equations are the following. For fluid flow: Jv ⫽

冋

冋 册 冋册

e ke w ih ⫹ i n n e

(9.93)

For chemical flow relative to the soil: Jc ⫽

冋

册 冋 册 冋 册

(1 ⫺ ")ckh ck2e w cke ⫹ ih ⫹ i n ne n e ⫹

D* ⫺

"ckh RT ic n w

(9.94)

Co py rig hte dM ate ria l

To assess conditions where coupled chemical, electrical, and hydraulic flows will be significant relative to direct flows, it is necessary to know the values of the Lij relative to the Lii. Estimates can be made by considering the probable values of the soil state parameters and the several flow and transport coefficients given in Eqs. (9.75) to (9.84). Typical ranges are given in Table 9.5. In Table 9.5 the diffusion coefficients and ionic mobilities for cations and anions are considered together since they lie within similar ranges for most species. Values of ionic mobility for specific ions in dilute solution are given in standard chemical references, for example, Dean (1973), and values of diffusion coefficients are given in Tables 9.3 and 9.4. Ionic mobility is related to the diffusion coefficient according to

I⫽

册 冋册

冋

册

kh k2 k "kh ⫹ e w ih ⫹ e ie ⫹ RT ⫺ i n en n wn c

Coupling Influences on Hydraulic Flow

In the absence of applied electrical and chemical gradients, flow under a hydraulic gradient is given by the first bracketed term on the right-hand side of Eq. (9.92). It contains the quantity k2e w /ne, which compensates for the electroosmotic counterflow generated by the streaming potential, which causes the measured value of kh to be slightly less than the true value of L11. As it is not usual practice to short-circuit between the ends of samples during hydraulic conductivity testing, the second bracketed term on the right-hand side of Eq. (9.92) is not zero. This term represents an electroosmotic counterflow that results from the streaming potential and acts in the direction opposite to the hydraulically driven flow. Analysis based on the values of properties in Table 9.5, as well as the results of measurements, for example, Michaels and Lin (1954) and Olsen (1962) show that this counterflow is negligible in most cases, but it may become significant relative to the true hydraulic conductivity for soils of very low hydraulic conductivity, for example, kh ⬍ 1 ⫻ 10⫺10 m/s. For example, for a value of ke of 5 ⫻ 10⫺9 m2 /s-V, an electrical conductivity of 0.01 mho/m, and a porosity of 35 percent, the counterflow term is 0.7 ⫻ 10⫺10 m/s. In the presence of an applied DC field the second bracketed term on the right-hand side of Eq. (9.92) can be very large relative to hydraulic flow in soils finer than silts, as ke, which typically ranges within only narrow limits, is large relative to kh; that is, kh is less than 1 ⫻ 10⫺8 m/s in these soils. The relative effectiveness of hydraulic and electrical driving forces for water movement can be assessed by comparing gradients needed to give equal flow rates. They will be equal if keie ⫽ khih

(9.95)

(9.92) The hydraulic gradient required to balance the electroosmotic flow then becomes

For electrical current flow:

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TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS

ih ⫽

ke i kh e

(9.96)

Co py rig hte dM ate ria l

As the hydraulic conductivity of soils in which electroosmosis is likely to be used is usually of the order of 1 ⫻ 10⫺9 m/s or less, whereas ke is in the range of 1 ⫻ 10⫺9 to 1 ⫻ 10⫺8 m2 /s  V, it follows that even small electrical gradients can balance flows caused by large hydraulic gradients. Because of this, and because ke is insensitive to particle size while kh decreases rapidly with decreasing particle size, electroosmosis is effective in fine-grained soils, as discussed further in Section 9.15. Chemically driven hydraulic flow is given by the last term on the right-hand side of Eq. (9.92). It depends primarily on the osmotic efficiency ". Osmotic efficiency has an important influence on the movement of chemicals through a soil, the development of osmotic pressure, and the effectiveness of clay barriers for chemical waste containment. Osmotic Efficiency The osmotic efficiency of clay, a slurry wall, a geosynthetic clay liner (GCL), or other seepage and containment barrier is a measure of the material’s effectiveness in causing hydraulic flow under an osmotic pressure gradient and of its ability to act as a semipermeable membrane in preventing the passage of ions, while allowing the passage of water. The osmotic pressure concept can be better appreciated by rewriting the last term in Eq. (9.92): "

kh k RT c 1 RTic ⫽ " h wn n w x

(9.97)

This form is analogous to Darcy’s law, with the quantity RT c/ w being the head difference. The osmotic efficiency is a measure of the extent to which this theoretical pressure difference actually develops. Theoretical values of osmotic pressure, calculated using the van’t Hoff equation, as a function of concentration difference for different values of osmotic efficiency are shown in Fig. 9.23. The van’t Hoff equation for osmotic pressure is  ⫽ kT

冘 (n

iA

⫺ niB) ⫽ RT(ciA ⫺ ciB)

285

(9.98)

where k is the Boltzmann constant (gas constant per molecule), R is the gas constant per molecule, T is the absolute temperature, ni is concentration in particles per unit volume, and ci is the molar concentration. The van’t Hoff equation applies for ideal and relatively dilute solution concentrations (Malusis and Shackelford, 2002c). According to Fritz (1986) the error is low (⬍5%) for 1⬊1 electrolytes (e.g., NaCl, KCl) and concentrations 1.0 M.

Copyright © 2005 John Wiley & Sons

Figure 9.23 Theoretical values of osmotic pressure as a function of concentration difference across a clay layer for different values of osmotic efficiency coefficient, ". (T ⫽ 20C).

Values of osmotic efficiency coefficient, ", or membrane efficiency (" expressed as a percentage), have been measured for clays and geosynthetic clay liners; for example, Kemper and Rollins (1966), Letey et al. (1969), Olsen (1969), Kemper and Quirk (1972), Bresler (1973), Elrick et al. (1976), Barbour and Fredlund (1989), and Malusis and Shackelford (2002b, 2002c). Values of membrane efficiency from 0 to 100 percent have been determined, depending on the clay type, porosity, and type and concentration of salts in solution. The results of many determinations were summarized by Bresler (1973) as shown in Fig. 9.24. The efficiency is shown as a function of a normalizing parameter, the half distance between particles b times the square root of the solution concentration 兹c. To put these relations into more familiar terms for use in geotechnical studies, the half spacings were converted to water contents on the assumption of uniform water layer thicknesses on all particles, using specific surface areas corresponding to different clay types and noting that volumetric water content equals surface area times layer thickness. The relationship between specific surface area and liquid limit (LL) obtained by Farrar and Coleman (1967) for 19 British clays LL ⫽ 19 ⫹ 0.56As (20%)

(9.99)

in which the specific surface area As is in square meters per gram, was then used to obtain the relationships shown in Fig. 9.25. The computed efficiencies shown in Fig. 9.25 should be considered upper bounds because the assumption of uniform water distribution over the full surface area underestimates the effective particle spacing in most cases. In most clays, espe-

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concentrations on the inside of a lined repository should be greater than on the outside, osmotically driven water flow should be directed from the outside toward the inside. The greater the osmotic efficiency the greater the driving force for this flow. Furthermore, if the efficiency is high, then outward diffusion of contained chemicals is restricted (Malusis and Shackelford, 2002b). In diffusion-dominated containment barriers, the effect of solute restriction on reducing solute diffusion is likely substantially more significant than the effect of osmotic flow (Shackelford et al., 2001). Coupling Influences on Electrical Flow

Figure 9.24 Osmotic efficiency coefficient as a function of b兹c where c is concentration of monovalent anion in nor-

Substitution of values for the parameters in Eq. (9.93) indicates, as would be expected, that electrical current flow is dominated completely by the electrical gradient ie. In the presence of an applied voltage difference, the other terms are of little importance, even if the movements of anions and cations are considered separately and the contributions due to ionic mobility are taken into account. On the other hand, when a soil layer behaves as an open electrical circuit, small electrical potentials, measured in millivolts, may exist if there are hydraulic and/or chemical flows. This may be seen by setting I ⫽ 0 in Eq. (9.93) and solving for ie, which must have value if ih has value. These small potentials and flows are important in such processes as corrosion and electroosmotic counterflow.

mality and 2b is the effective spacing between particle surfaces (from Bresler, 1973).

Coupling Influences on Chemical Flow

cially those with divalent adsorbed cations, individual clay plates associate in clusters giving an effective specific surface that is less than that determined by most methods of measurement. This means that the curves in Fig. 9.25 should in reality be displaced to the left. High osmotic efficiencies are developed at low water contents, that is, in very dense, low-porosity clays, and in dilute electrolyte systems. Malusis and Shackelford (2002a, 2002b, 2002c) found that the osmotic efficiency decreases with increasing solute concentration and attribute this to compression of the diffuse double layers adjacent to the clay particles. Water flow by osmosis can be significant relative to hydraulically driven water flow in heavily overconsolidated clay and clay shale, where the void ratio is low and the hydraulic conductivity is also very low. Such flow may be important in geological processes (Olsen 1969, 1972). Densely compacted clay barriers for waste containment, usually composed of bentonite, possess osmotic membrane properties. As the chemical

Equation (9.94) provides a description of chemical transport relative to the soil. It contains two terms that influence chemical flow under a hydraulic gradient; one for chemical transport under an electrical gradient, and one for transport of chemical under a chemical gradient. The first term in the first bracket of the righthand side of Eq. (9.94) describes advective transport. As would be expected, the smaller the osmotic efficiency, the more chemical flow through the soil is possible. The second term in the same bracket simply reflects the advective flow reduction that would result from electroosmotic counterflow caused by development of a streaming potential. As noted earlier, this flow will be small, and its contribution to the total flow will be small, except in clays of very low hydraulic and electrical conductivities. Advective transport is the dominant means for chemical flow for soils having a hydraulic conductivity greater than about 1 ⫻ 10⫺9 m/s. The importance of an electrical driving force for chemical flow depends on the electrical potential gradient. For a unit gradient, that is, 1 V/m, chemical flow

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TRANSPORT COEFFICIENTS AND THE IMPORTANCE OF COUPLED FLOWS

287

Figure 9.25 Osmotic efficiency of clays as a function of water content.

quantities are comparable to those by advective flow under a unit hydraulic gradient in a clay having a hydraulic conductivity of about 1 ⫻ 10⫺9 m/s. Electrically driven chemical flow is relatively less important in higher permeability soils and more important in soils with lower kh. In cases where the electrically driven chemical transport is of interest, as in electrokinetic waste containment barrier applications, anion, cation, and nonionic chemical flows must be considered separately using expanded relationships such as given by Eqs. (9.87) and (9.88). The last bracketed quantity of Eq. (9.94) represents diffusive flow under chemical gradients. The quantity D*ic gives the normal diffusive flow rate. The second term represents a restriction on this flow that depends on the clay’s osmotic efficiency, "; that is, if the clay acts as an effective semipermeable membrane, diffusive flow of chemicals is restricted. However, even un-

Copyright © 2005 John Wiley & Sons

der conditions where the value of " is low such that the second term in the bracket is negligible, chemical transport by diffusion is significant relative to advective chemical transport in soils with hydraulic conductivity values less than about 1 ⫻ 10⫺9 to 1 ⫻ 10⫺10 m/s for chemicals with diffusion coefficients in the range given by Table 9.7, that is, 2 ⫻ 10⫺10 to 2 ⫻ 10⫺9 m2 /s. This is illustrated by Fig. 9.26 from Shackelford (1988), which shows the relative importance of advective and diffusive chemical flows on the transit time through a 0.91-m-thick compacted clay liner having a porosity of 0.5 acted on by a hydraulic gradient of 1.33. A diffusion coefficient of 6 ⫻ 10⫺10 m2 /s was assumed. The transit time is defined as the time required for the solute concentration on the discharge side to reach 50 percent of that on the upstream side. For hydraulic conductivity values less than about 2 ⫻

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CONDUCTION PHENOMENA

term stability of clay liners are discussed by Mitchell and Jaber (1990). Rigid wall, flexible wall, and consolidometer permeameters are used for compatibility testing in the laboratory. These three types of test apparatus are shown schematically in Fig. 9.27. Tests done in a rigid wall system overestimate hydraulic conductivity whenever chemical–clay interactions cause shrinkage and cracking; however, a rigid wall system is well suited for qualitative determination of whether or not there may be adverse interactions. In the flexible wall system the lateral confining pressure prevents cracks from opening; thus there is risk of underestimating the hydraulic conductivity of some soils. The consolidometer permeameter system allows for testing clays under a range of overburden stress states that are representative of those in the field and for quantitative assessment of the effects of chemical interactions on volume stability and hydraulic conductivity. More details of these permeameters are given by Daniel (1994). The effects of chemicals on the hydraulic conductivity of high water content clays such as used in slurry walls are likely to be much greater than on lower water content, high-density clays as used in compacted clay liners. This is because of the greater particle mobility and easier opportunity for fabric changes in a higher water content system. A high compactive effort or an effective confining stress greater than about 70 kPa can make properly compacted clay invulnerable to attack by concentrated organic chemicals (Broderick and Daniel, 1990). However, it is not always possible to ensure high-density compaction or to maintain high confining pressures, or eliminate all construction defects, so it is useful to know the general effects of different types of chemicals on hydraulic conductivity. The influences of inorganic chemicals on hydraulic conductivity are consistent with (1) their effects on the double-layer and interparticle forces in relation to flocculation, dispersion, shrinkage, and swelling, (2) their effects on surface and edge charges on particles and the influences of these charges on flocculation and deflocculation, and (3) their effects on pH. Acids can dissolve carbonates, iron oxides, and the alumina octahedral layers of clay minerals. Bases can dissolve silica tetrahedral layers, and to a lesser extent, alumina octahedral layers of clay minerals. Removal of dissolved material can cause increases in hydraulic conductivity, whereas precipitation can clog pores and reduce hydraulic conductivity. The most important factors controlling the effects of organic chemicals on hydraulic conductivity are (1) water solubility, (2) dielectric constant, (3) polarity, and (4) whether or not the soil is exposed to the pure organic or a dilute solution. Exposure of clay barriers

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288

Figure 9.26 Transit times for chemical flow through a 0.91-

m-thick compacted clay liner having a porosity of 50 percent and acted on by a hydraulic gradient of 1.33 (from Shackelford, 1988).

10⫺9 m/s the transit time in the absence of diffusion would be very long. For diffusion alone the transit time would be about 47 years. Most compacted clay barriers and geosynthetic clay liners are likely to have hydraulic conductivity values in the range of 1 ⫻ 10⫺11 to 1 ⫻ 10⫺9 m/s, with the latter value being the upper limit allowed by the U.S. EPA for most waste containment applications. In this range, diffusion reduces the transit time significantly in comparison to what it would be due to advection alone. This is shown by the curve labeled advection– dispersion in Fig. 9.26. The calculations were done using the well-known advection–dispersion equation (Ogata and Banks, 1961) in which the dispersion term includes both mechanical mixing and diffusion. Mechanical mixing is negligible in low-permeability materials such as compacted clay. 9.14 COMPATIBILITY—EFFECTS OF CHEMICAL FLOWS ON PROPERTIES

Chemical Compatibility and Hydraulic Conductivity

The compatibility between waste chemicals, especially liquid organics, and compacted clay liners and slurry wall barriers constructed to contain them must be considered in the design of waste containment barriers. Numerous studies have been done to evaluate chemical effects on clay hydraulic conductivity because of fears that prolonged exposure may compromise the integrity of the liners and barriers and because tests have shown that under some conditions clay can shrink and crack when permeated by certain classes of chemicals. Summaries of the results of chemical compatibility studies are given by Mitchell and Madsen (1987) and Quigley and Fernandez (1989), and factors controlling the long-

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COMPATIBILITY—EFFECTS OF CHEMICAL FLOWS ON PROPERTIES

289

Figure 9.27 Three types of permeameter for compatibility testing: (a) rigid wall, (b) flexible wall, and (c) consolidometer permeameter (from Day, 1984).

to water-insoluble pure or concentrated organics is likely only in the case of spills, leaking tanks, and with dense non-aqueous-phase liquids (DNAPLs) or ‘‘sinkers’’ that accumulate above low spots in liners. Some general conclusions about the influences of organics on the hydraulic conductivity are: 1. Solutions of organic compounds having a low solubility in water, such as hydrocarbons, have no large effect on the hydraulic conductivity. This is in contrast to dilute solutions of inorganic compounds that may have significant effects as a result of their influence on flocculation and dispersion of the clay particles. 2. Water-soluble organics, such as simple alcohols and ketones, have no effect on hydraulic conductivity at concentrations less than about 75 to 80 percent.

Copyright © 2005 John Wiley & Sons

3. Many water-insoluble organic liquids (i.e., nonaquoues-phase liquids, NAPLs) can cause shrinkage and cracking of clays, with concurrent increases in hydraulic conductivity. 4. Hydraulic conductivity increases caused by permeation by organics are partly reversible when water is reintroduced as the permeant. 5. Concentrated hydrophobic compounds (like many NAPLs) permeate soils through cracks and macropores. Water remains within mini- and micropores. 6. Hydrophilic compounds permeate the soil more uniformly than NAPLs, as the polar molecules can replace the water in hydration layers of the cations and are more readily adsorbed on particle surfaces. 7. Organic acids can dissolve carbonates and iron oxides. Buffering of the acid can lead to precip-

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CONDUCTION PHENOMENA

itation and pore clogging downstream. However, after long time periods these precipitates may be redissolved and removed, thus leading to an increase in hydraulic conductivity. 8. Pure bases can cause a large increase in the hydraulic conductivity, whereas concentrations at or below the solubility limit in water have no effect. 9. Organic acids do not cause large-scale dissolution of clay particles.

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The combined effects of confining pressure and concentration, as well as permeant density and viscosity, are illustrated by Fig. 9.28 (Fernandez and Quigley, 1988). The data are for water-compacted, brown Sarnia clay permeated by solutions of dioxane in domestic landfill leachate. Increased hydrocarbon concentration caused a decrease in hydraulic conductivity up to concentrations of about 70 percent, after which the hydraulic conductivity increased by about three orders of magnitude for pure dioxane (Fig. 9.28a), for samples that were unconfined by a vertical stress (v ⫽ 0). On the other hand, the data points for samples maintained under a vertical confining stress of 160 kPa indicated no effect of the dioxane on hydraulic conductivity rel-

ative to that measured with water. The decreases in hydraulic conductivity for dioxane concentrations up to 70 percent can be accounted for in terms of fluid density and viscosity, as may be seen in Fig. 9.28b where the intrinsic values of permeability are shown. As noted earlier in this chapter, the intrinsic permeability is defined by K ⫽ k / . Although many chemicals do not have significant effect on the hydraulic conductivity of clay barriers, this does not mean that they will not be transported through clay. Unless adsorbed by the clay or by organic matter, the chemicals will be transported by advection and diffusion. Furthermore, the actual transit time through a barrier by advection, that is, the time for chemicals moving with the seepage water, may be far less than estimated using the conventional seepage velocity. The seepage velocity is usually defined as the Darcy velocity khih, divided by the total porosity n. In systems with unequal pore sizes the flow is almost totally through mini- and macropores, which comprise the effective porosity ne, which may be much less than the total porosity. Thus effective compaction of clay barriers must break down clods and aggregates to decrease the effective pore size and increase the propor-

Figure 9.28 (a) Hydraulic conductivity and (b) intrinsic permeability of compacted Sarnia clay permeated with leachate–dioxane mixtures. Initial tests run using water (●) followed by leachate–chemical solution (䉱). (from Fernandez and Quigley, 1988). Reproduced with per-

mission from the National Research Council of Canada.

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ELECTROOSMOSIS

tion of the porosity that is effective porosity, thereby increasing the transit time. 9.15

ELECTROOSMOSIS

Helmholtz and Smoluchowski Theory

Table 9.7

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.



v

E ⫽ 

L

10

A derivation using a Poisson–Boltzmann distribution of counterions adjacent to the wall gives the same result.

Coefficients of Electroosmotic Permeability

Material

London clay Boston blue clay Kaolin Clayey silt Rock flour Na-Montmorillonite Na-Montmorillonite Mica powder Fine sand Quartz powder ˚ s quick clay A Bootlegger Cove clay Silty clay, West Branch Dam Clayey silt, Little Pic River, Ontario

(9.100)

or

This theory, based on a model introduced by Helmholtz (1879) and refined by Smoluchowski (1914), is one of the earliest and most widely used. A liquidfilled capillary is treated as an electrical condenser with

No.

charges of one sign on or near the surface of the wall and countercharges concentrated in a layer in the liquid a small distance from the wall, as shown in Fig. 9.29.10 The mobile shell of counterions is assumed to drag water through the capillary by plug flow. There is a high-velocity gradient between the two plates of the condenser as shown. The rate of water flow is controlled by the balance between the electrical force causing water movement and friction between the liquid and the wall. If v is the flow velocity and  is the distance between the wall and the center of the plane of mobile charge, then the velocity gradient between the wall and the center of positive charge is v / ; thus, the drag force per unit area is  dv /dx ⫽ v / , where  is the viscosity. The force per unit area from the electrical field is  E/

L, where  is the surface charge density and E/ L is the electrical potential gradient. At equilibrium

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The coefficient of electroosmotic hydraulic conductivity ke defines the hydraulic flow velocity under a unit electrical gradient. Measurement of ke is made by determination of the flow rate of water through a soil sample of known length and cross section under a known electrical gradient. Alternatively, a null indicating system may be used or it may be deduced from a streaming potential measurement. From experience it is known that ke is generally in the range of 1 ⫻ 10⫺9 to 1 ⫻ 10⫺8 m2 /s V (m/s per V/m) and that it is of the same order of magnitude for most soil types, as may be seen by the values for different soils and a freshwater permeant given in Table 9.7. Several theories have been proposed to explain electroosmosis and to provide a basis for quantitative prediction of flow rates.

291

Water Content (%)

ke in 10⫺5 (cm2 /s-V)

Approximate kh (cm/s)

52.3 50.8 67.7 31.7 27.2 170 2000 49.7 26.0 23.5 31.0 30.0 32.0 26.0

5.8 5.1 5.7 5.0 4.5 2.0 12.0 6.9 4.1 4.3 20.0–2.5 2.4–5.0 3.0–6.0 1.5

10⫺8 10⫺8 10⫺7 10⫺6 10⫺7 10⫺9 10⫺8 10⫺5 10⫺4 10⫺4 2.0 ⫻ 10⫺8 2.0 ⫻ 10⫺8 1.2 ⫻ 10⫺8 –6.5 ⫻ 10⫺8 2 ⫻ 10⫺5

ke and water content data for Nos. 1 to 10 from Casagrande (1952). kh estimated by authors; no. 11 from Bjerrum et al. (1967); no. 12 from Long and George (1967); no. 13 from Fetzer (1967); no. 14 from Casagrande et al. (1961).

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CONDUCTION PHENOMENA

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292

Figure 9.29 Helmholtz–Smoluchowski model for electrokinetic phenomena.

 ⫽ v

L

E

(9.101)

From electrostatics, the potential across a condenser  is given by ⫽

 D

(9.102)

where D is the relative permittivity, or dielectric constant of the pore fluid. Substitution for  in Eq. (9.102) gives v⫽

冉 冊

D E  L

(9.103)

The potential  is termed the zeta potential. It is not the same as the surface potential of the double-layer 0 discussed in Chapter 6, although conditions that give high values of 0 also give high values of zeta potential. A common interpretation is that the actual slip plane in electrokinetic processes is located some small, but unknown, distance from the surface of particles; thus  should be less than 0. Values of  in the range of 0 to ⫺50 mV are typical for clays, with the lowest values associated with high pore water salt concentrations. For a single capillary of area a the flow rate is qa ⫽ va ⫽

D E a  L

(9.104a)

and for a bundle of N capillaries within total crosssectional area A normal to the flow direction

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qA ⫽ Nqa ⫽

D E Na  L

(9.104b)

If the porosity is n, then the cross-sectional area of voids is nA, which must equal Na. Thus, qA ⫽

D E n A 

L

(9.105)

By analogy with Darcy’s law we can write Eq. (9.105) as qA ⫽ keie A

(9.106)

in which ie is the electrical potential gradient E/ L and ke the coefficient of electroosmotic hydraulic conductivity is ke ⫽

D n 

(9.107)

According to the Helmholtz–Smoluchowski theory and Eq. (9.107), ke should be relatively independent of pore size, and this is borne out by the values listed in Table 9.7. This is in contrast to the hydraulic conductivity kh, which varies as the square of some effective pore size. Because of this independence of pore size, electroosmosis can be more effective in moving water through fine-grained soils than flow driven by a hydraulic gradient. This is illustrated by the following simple example. Consider a fine sand and a clay of hydraulic conductivity kh of 1 ⫻ 10⫺5 m/s and 1 ⫻ 10⫺10 m/s, respectively. Both have ke values of 5 ⫻ 10⫺9 m2 /s V. For equal hydraulic flow rates khih ⫽ keie, so

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ELECTROOSMOSIS

ih ⫽

ke i kh e

E

L

Schmid Theory

The Helmholtz–Smoluchowski theory is essentially a large-pore theory because it assumes a negligible extension of the counterion layer into the pore. Also, it does not account for an excess of ions over those needed to balance the surface charge. A model that overcomes the first of these problems was proposed by Schmid (1950, 1951). It can be considered a smallpore theory. The counterions are assumed to be distributed uniformly throughout the fluid phase in the soil. The electrical force acts uniformly over the entire pore cross section and gives the same velocity profile as shown by Fig. 9.29. The hydraulic flow rate through a single capillary of radius r is given by Poiseuille’s law: q⫽

r i 8 w h 4

(9.109)

The hydraulic seepage force per unit length causing flow is FH ⫽ r 2 wih

(9.112)

where A0 is the concentration of wall charges in ionic equivalents per unit volume of pore fluid, and F0 is the Faraday constant. Replacement of FH by FE in Eq. (9.111) gives qa ⫽

r 4

E F A A F ⫽ 0 0 r 2iea 8 0 0 L 8

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If an electrical potential gradient of 20 V/m is used, substitution in Eq. (9.108) shows that ih is 0.01 for the fine sand and 1000 for the clay. This means that a hydraulic gradient of only 0.01 can move water as effectively as an electrical gradient of 20 V/m in fine sand. However, for the clay, a hydraulic gradient of 1000 would be needed to offset the electroosmotic flow. However, it does not follow that electroosmosis will always be an efficient means to move water in clays because the above analysis does not take into account the power requirement to develop the potential gradient of 20 V/m or energy losses in the system. These points are considered further later.

so

FE ⫽ A0 F0r 2

(9.108)

293

(9.113)

so for a total cross section of N capillaries and area A qA ⫽

A0 F0r 2 nie A 8

(9.114)

This equation shows that ke should vary as r 2, whereas the Helmholtz–Smoluchowski theory leads to ke independent of pore size, as previously noted. Of the two theories, the Helmholtz gives the better results for soils, perhaps because most clays have a cluster or aggregate structure with electroosmotic flow controlled more by the larger pores than by the intracluster pores. Spiegler Friction Model

A completely different concept for electrokinetic processes takes into account the interactions of the mobile components (water and ions) on each other and of the frictional interactions of these components with pore walls (Spiegler, 1958). This theory provides insight into conditions leading to high electroosmotic efficiency. The assumptions include: 1. Exclusion of coions,11 that is, the medium behaves as a perfect perm-selective membrane, admitting ions of only one sign 2. Complete dissociation of pore fluid ions

The following equation for electroosmotic transport of water across a fine-grained porous material containing adsorbed and free ions can be derived:

(9.110)

⫽ (W ⫺ H) ⫽

C3 C1 ⫹ C3(X34 /X13)

(9.115)

(9.111)

in which is the true electroosmotic water flow (moles/faraday), W is the measured water transport

The electrical force per unit length FE is equal to the charge times the potential, that is,

11 Ions of the opposite sign to the charged surface are termed counterions. Ions of the same sign are termed coions.

q⫽

r2 F 8 H

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CONDUCTION PHENOMENA

opposite sign. The greater the difference between the concentrations of cations and anions, the greater the net drag on the water in the direction toward the cathode. The efficiency and economics of the process depend on the volume of water transported per unit electrical charge passed. If the volume is high, then more water is transported for a given expenditure of electrical energy than if it is low. This volume may vary over several orders of magnitude depending on such factors as soil type, water content, and electrolyte concentration. In a low exchange capacity soil at high water content in a low electrolyte concentration solution, there is much more water per cation than in a high exchange capacity, low water content soil having the same pore water electrolyte concentration. This, combined with cation-to-anion ratio considerations, leads to the predicted water transport–water content–soil type–electrolyte concentration relationships shown schematically in Fig. 9.30, where increasing electrolyte concentration in the pore water results in a much

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(moles/faraday), H is the water transport by ion hydration (moles/faraday), C3 is the concentration of free water in the material (mol/m3), C1 is the concentration of mobile counterions m2, X34 is the friction coefficient between water and the solid wall, and X13 is the friction coefficient between cation and water. Concentrations C1 and C3 are hypothetical and probably less than values measured by chemical analysis because some ions may be immobile. Evaluation of X13 and X34 requires independent measurements of diffusion coefficients, conductance, transference numbers, and water transport. Thus Eq. (9.115) is limited as a predictive equation. Its real value is in providing a relatively simple physical representation of a complex process. From Eq. (9.115), ⫽ (W ⫺ H) ⫽

1 (C1 /C3 ⫹ X34 /X13)

(9.116)

At high water contents and for large pores, X34 /X13 → 0 because X34 becomes negligible. Then X34→0

⫽ C3 /C1

(9.117)

This relationship indicates that a high water-to-cation ratio implies a high rate of electroosmotic flow. At low water contents and for small pores, X34 will not be zero, thus reducing the flow. An increase in C1 reduces the flow of water per faraday of current passed because there is less water per ion. An increase in X13 increases the flow because there is greater frictional drag on the water by the ions. Ion Hydration

Water of hydration is carried along with ions in a direct current electric field. The ion hydration transport H is given by H ⫽ t⫹N⫹ ⫺ t N

(9.118)

where t⫹ and t are the transport numbers, that is, numbers that represent the fraction of current carried by a particular ionic species. The numbers N⫹ and N are the number of moles of hydration water per mole of cation and anion, respectively.

9.16

ELECTROOSMOSIS EFFICIENCY

Electroosmotic water flow occurs if the frictional drag between the ions of one sign and their surrounding water molecules exceeds that caused by ions of the

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Figure 9.30 Schematic prediction of water transport by elec-

troosmosis in various clays according to the Donnan concept (from Gray, 1966).

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ELECTROOSMOSIS EFFICIENCY

R⫽ where

y⫽

2C0  A0 

C⫹ 1 ⫹ (1 ⫹ y2)1 / 2 ⫽ C⫺ ⫺1 ⫹ (1 ⫹ y2)1 / 2

(9.119)

A0 ⫽

(CEC)w w

(9.121)

where w is the density of water and w is the water content. The higher R, the greater is the electroosmotic water transport, all other things equal. From Eqs. (9.119) to (9.121) it may be deduced that exclusion of anions is favored by a high exchange capacity (active clay), a low water content, and low salinity in the external solution. However, the concentration of anions in the double layer builds up more

Figure 9.31 Electroosmotic water transport versus concentration of external electrolyte solution for homoionic kaolinite and illite at various water content (from Gray, 1966).

Copyright © 2005 John Wiley & Sons

(9.120)

The concentration C0 is in the external solution, is the mean molar activity coefficient in the external solution, is the mean activity coefficient in the double layer, and A0 is the surface charge density per unit pore volume. The parameter A0 is related to the cation exchange capacity (CEC) by

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greater decrease in efficiency for inactive clay than more plastic, active clay. Tests on sodium kaolinite (inactive clay) and sodium illite (more active clay) gave the results shown in Fig. 9.31, which agree well with the predictions in Fig. 9.30. The slopes and locations of the curves can be explained more quantitatively in the following way. Alternatively to the double-layer theory given in Chapter 6, the Donnan (1924) theory can be used to describe equilibrium ionic distributions in fine-grained materials. The basis for the Donnan theory is that at equilibrium the potentials of the internal and external solutions are equal and that electroneutrality is required in both phases. It may be shown (Gray, 1966; Gray and Mitchell, 1967) that the ratio R of cations to anions in the internal phase for the case of a symmetrical electrolyte (z⫹ ⫽ z⫺) is given by

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CONDUCTION PHENOMENA

E L ⫽ ⫺ EH

P LEE

(9.124)

In electroosmosis P ⫽ 0, so Eq. (9.122) is qh ⫽ LHE E

(9.125)

and Eq. (9.122) becomes I ⫽ LEE E

(9.126)

qh LHE ⫽ I LEE

(9.127)

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rapidly as the salinity of the external solution increases in inactive clays than in active clays. As a result the efficiency, as measured by volume of water per unit charge passed, decreases much more rapidly with increasing electrolyte concentration than in the more active clay. The results of electroosmosis measurements on a number of different materials are summarized in Fig. 9.32, which shows water flow rate as a function of water content. This figure may be used as a guide for prediction of electroosmotic flow rates. The flow rates shown are for open systems, that is, solution was admitted at the anode at the same time it was extracted from the cathode. Electrochemical effects (Section 9.18) and water content changes were minimized in these tests. Thus, the values can be interpreted as upper bounds on the flow rates to be expected in practice. Values of water content, electrolyte concentration in the pore water, and type of clay are required for electroosmosis efficiency estimation. Water content is readily measured, the electrolyte concentration is easily determined using a conductivity cell, and the clay type can be determined from plasticity and grain size information if mineralogical data are not available. Electroosmotic flow rates of 0.03 to 0.06 gal/h/amp are predicted using Fig. 9.32 for soils 11, 13, and 14 in Table 9.7. Electrical treatment for consolidation and ground strengthening was effective in these soils. For soil 12, however, a flow rate of 0.008 to 0.012 gal/h/ amp was predicted, and electroosmosis was not effective. Saxen’s Law Prediction of Electroosmosis from Streaming Potential

Streaming potential can be measured directly during a measurement of hydraulic conductivity by using a high-impedance voltmeter and reversible electrodes. Equivalence between streaming potential and electroosmosis may be derived. Expansion of Eq. (9.57) for coupled hydraulic and current flows gives qh ⫽ LHH P ⫹ LHE E

(9.122)

I ⫽ LEH P ⫹ LEE E

(9.123)

in which qh is the hydraulic flow rate, I is the electric current, LHH and LEE are the direct flow coefficients, LHE and LEH are the coupling coefficients for hydraulic flow due to an electrical gradient and electrical flow due to a hydraulic gradient, P is the pressure drop, and E is the electrical potential drop. In a usual hydraulic conductivity measurement, there is no electrical current flow, so I ⫽ 0, and E is the streaming potential. Equation (9.123) then becomes

Copyright © 2005 John Wiley & Sons

so

By Onsager’s reciprocity theorem LEH ⫽ LHE so

冉冊 qh I

冉 冊

⫽⫺

P⫽0

E

P

(9.128)

I⫽0

This equivalence between streaming potential and electroosmosis was first shown experimentally by Saxen (1892) and is known as Saxen’s law. It has been verified for clay–water–electrolyte systems. Care must be taken to ensure consistency in units. For example, the electroosmotic flow rate in gallons per hour per ampere is equal to 0.0094 times the streaming potential in millivolts per atmosphere. Energy Requirements

The preceding analysis leads to a prediction of the amount of water moved per unit charge passed, for example, gallons or cubic meters of water per hour per ampere or moles per faraday. If this quantity is denoted by ki, then qh ⫽ ki I

(9.129)

Unlike ke, ki varies over a wide range, as may be seen in Fig. 9.32. The power consumption P is P ⫽ E  I ⫽

Eqh

ki

(in W)

(9.130)

for E in volts and I in amperes. The power consumption per unit volume of flow is P

E ⫽ ⫻ 10⫺3 qh ki

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(in kWh)

(9.131)

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Figure 9.32 Electroosmotic water transport as a function of water content, soil type, and electrolyte concentration: (a) homoionic kaolinite and illite, (b) illitic clay and collodion membrane, and (c) silty clay, illitic clay, and kaolinite.

297

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CONDUCTION PHENOMENA

Relationship Between ke and ki

From Eqs. (9.108) and (9.129), the electroosmotic flow rate is given by

equations in place of Darcy’s law in consolidation theory. Assumptions

qh ⫽ ki I ⫽ ke

E A

L

(9.132)

The following idealizing assumptions are made: 1. There is homogeneous and saturated soil. 2. The physical and physicochemical properties of the soil are uniform and constant with time.12 3. No soil particles are moved by electrophoresis. 4. The velocity of water flow by electroosmosis is directly proportional to the voltage gradient. 5. All the applied voltage is effective in moving water.13 6. The electrical field is constant with time. 7. The coupling of hydraulic and electrical flows can be formulated by Eqs. (9.63) and (9.64). 8. There are no electrochemical reactions.

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Because E/I is resistance and L/(resistance ⫻ A) is specific conductivity , Eq. (9.132) becomes ki ⫽

ke 

(9.133)

As ke varies within relatively narrow limits, Eq. (9.133) shows that the electroosmotic efficiency, measured by ki, is a sensitive function of the electrical conductivity of the soil. For soils 11, 13, and 14 in Table 9.7,  is in the range of 0.02 to 0.03 S. For soil 12, in which electroosmosis was not effective,  is 0.25 S. In essence, a high value of electrical conductivity means that the current required to develop the voltage is too high for economical movement of water. In addition, if high current is used, the generation of gas, heat, and electrochemical effects become excessive.

Governing Equations

9.17

for the flow rate per unit area. For radial flow for the conditions shown in Fig. 9.33b and a layer of unit thickness

CONSOLIDATION BY ELECTROOSMOSIS

If, in a compressible soil, electroosmosis draws water to a cathode where it is drained away and no water is allowed to enter at the anode, then consolidation of the soil between the electrodes occurs in an amount equal to the volume of water removed. Water movement away from the anode causes consolidation in the vicinity of the anode. The effective stress must increase concurrently. Because the total stress in the vicinity of the anode remains essentially unchanged, the pore water pressure must decrease. Water drains at the cathode where there is no consolidation. Therefore, the total, effective, and pore water pressures at the cathode remain unchanged. As a result, hydraulic gradient develops that tends to cause water flow from cathode to anode. Consolidation continues until the hydraulic force that drives water back toward the anode exactly balances the electroosmotic force driving water toward the cathode. The usefulness of consolidation by electroosmosis as a means for soil stabilization was established by a number of successful field applications, for example, Casangrande (1959) and Bjerrum et al. (1967). Two questions are important: (1) How much consolidation will there be? and (2) How long will it take? Answers to these questions are obtained using the coupled flow

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For one-dimensional flow between plate electrodes (Fig. 9.33a), Eq. (9.63) becomes k u V qh ⫽ ⫺ h ⫺ ke w x x

k u V qh ⫽ ⫺ h  2r ⫺ ke  2r w r r

(9.134)

(9.135)

Introduction of Eq. (9.134) in place of Darcy’s law in the derivation of the diffusion equation governing consolidation in one dimension leads to kh 2u 2V u ⫹ k ⫽ mv e 2 2 w x x t

(9.136)

and

12

Flow of water away from anodes toward cathodes causes a nonuniform decrease in water content along the line between electrodes. This leads to changes in hydraulic conductivity, electroosmotic hydraulic conductivity, compressibility, and electrical conductivity with time and position. To account for these effects, which are discussed by Mitchell and Wan (1977) and Acar et al. (1990), would greatly complicate the analysis because it would be highly nonlinear. Similar problems arise in classical consolidation theory, but the simple linear theory developed by Terzaghi is adequate for most cases. 13 In most cases some of the electrical energy will be consumed by generation of heat and gases at the electrodes. To account for those losses, an effective voltage can be used (Esrig and Henkel, 1968).

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CONSOLIDATION BY ELECTROOSMOSIS

299

kh u V ⫽ ⫺ke w x x

(9.139)

k du ⫽ ⫺ e w dV kh

(9.140)

or

Co py rig hte dM ate ria l

The solution of this equation is k u ⫽ ⫺ e w V ⫹ C kh

(9.141)

At the cathode, V ⫽ 0 and u ⫽ 0; therefore, C ⫽ 0, and the pore pressure at equilibrium at any point is given by k u ⫽ ⫺ e w V kh

Figure 9.33 Electrode geometries for analysis of consoli-

dation by electroosmosis: (a) one-dimensional flow and (b) radial flow.

2u ke 2V 1 u ⫹ ⫽ w 2 2 x kh x cv t

(9.137)

where mv is the compressibility and cv is the coefficient of consolidation. For radial flow, the use of Eq. (9.135) gives 2u ke 2V 1 ⫹ ⫹ w 2 2 r kh r r

冉

冊

u k V ⫹ e w r kh r

⫽

1 u cv t

(9.138)

Both V and u are functions of position, as shown in Fig. 9.34; V is assumed constant with time, whereas u varies.

where the values of u and V are those at any point of interest. A similar result is obtained from Eq. (9.135) for radial flow. Equation (9.142) indicates that electroosmotic consolidation continues at a point until a negative pore pressure, relative to the initial value, develops that is proportional to the ratio ke /kh and to the voltage at the point. For conditions of constant total stress, there must be an equal and opposite increase in the effective stress. This increase in effective stress causes the consolidation. For the one-dimensional case, consolidation by electroosmosis is analogous to the loading shown in Fig. 9.35. For a given voltage, the magnitude of effective stress increase that develops depends on ke /kh. As ke only varies within narrow limits for different soils, the total consolidation that can be achieved depends largely on kh. Thus, the potential for consolidation by electroosmosis increases as soil grain size decreases because the finer grained the soil, the lower is kh. However, the amount of consolidation in any case depends on the soil compressibility as well as on the change in effective stress. For linear soil compression with increase in effective stress, the coefficient of compressibility av is

Amount of Consolidation

When the hydraulic gradient that develops in response to the differing amounts of consolidation between the anode and cathode generates a counterflow (kh / w)/ (u/ x) that exactly balances the electroosmotic flow ke(V/ x) in the opposite direction, consolidation is complete. As there then is no flow, qh in Eqs. (9.14) and (9.135) is zero. Thus Eq. (9.134) is

Copyright © 2005 John Wiley & Sons

(9.142)

de de av ⫽ ⫺ ⫽ d du

(9.143)

de ⫽ av du ⫽ ⫺av d

(9.144)

or

in which d is the increase in effective stress.

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9

CONDUCTION PHENOMENA

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300

Figure 9.34 Assumed variation of voltage with distance during electroosmosis: (a) onedimensional flow and (b) radial flow.

Thus, the more compressible the soil, the greater will be the amount of consolidation for a given stress increase, just as in the case of consolidation under applied loads. It follows, also, that electroosmosis will be of little value in an overconsolidated clay unless the effective stress increases are large enough to bring the material back into the virgin compression range. The consolidation loading of any small element of the soil is isotropic, as it is done by increasing the effective stress through reduction in the pore water pressure. The entire soil mass being treated is not consolidated isotropically or uniformly, however, because the amount of consolidation varies with position, de-

Copyright © 2005 John Wiley & Sons

pendent on the voltage at the point. Accordingly, properties at the end of treatment vary along a line between the anode and cathode, as shown, for example, by the posttreatment variations in shear strength and water content shown in Fig. 9.36. Values of these properties before treatment are also shown for comparison. More uniform property distributions between electrodes can be obtained if the polarity of electrodes is reversed after partial completion of consolidation (Wan and Mitchell, 1976). The results shown in Fig. 9.36 were obtained at a site in Norway where electroosmosis was used for the consolidation of quick clay (Bjerrum et al., 1967). The

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CONSOLIDATION BY ELECTROOSMOSIS

301

Figure 9.35 Consolidation by electroosmosis and by direct loading, one-dimensional case: (a) electroosmosis and (b) direct loading.

variations in strength and water content after treatment are consistent with the patterns to be expected based on the predicted variation of pore pressure decrease and vertical strain stress increase with voltage and position shown in Fig. 9.35.

voltage, and TV is the time factor, defined in terms of the distance between electrodes L and real time t as

Rate of Consolidation

where cv is the coefficient of consolidation, given by

Solutions for Eqs. (9.137) and (9.138) have been obtained for several cases (Esrig, 1968, 1971). For the one-dimensional case, and assuming a freely draining (open) cathode and a closed anode (no flow), the pore pressure is u⫽

ke 2k V V(x) ⫹ e w 2 m kh w kh 

n⫽0

cv ⫽

n

冋冉 冊 册 1 2 2  TV 2

(9.145)

where V(x) is the voltage at x, Vm is the maximum

Copyright © 2005 John Wiley & Sons

(9.146)

kh mv w

4 3

冘 ⬁

n⫽0

(9.147)

冋冉 冊 册

(⫺1)n 1 exp ⫺ n ⫹ (n ⫹ 1/2)3 2

2

 exp ⫺ n ⫹

cvt L2

The average degree of consolidation U as a function of time is U⫽1⫺

x 冘 (n (⫹⫺1)1/2) sin冋(n ⫹ 1/2) 册 L ⬁

TV ⫽

2

 2TV

(9.148)

Solutions for Eqs. (9.145) and (9.148) are shown in Figs. 9.37 and 9.38. They are applied in the same way as the theoretical solution for classical consolidation theory.

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CONDUCTION PHENOMENA

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302

˚ s, Norway Figure 9.36 Effect of electroosmosis treatment on properties of quick clay at A (from Bjerrum et al., 1967): (a) Undrained shear strength, (b) remolded shear strength, (c) water content, and (d) Atterberg limits.

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ELECTROCHEMICAL EFFECTS

Figure 9.39 Average degree of consolidation as a function

of dimensionless time for radial consolidation by electroosmosis (from Esrig, 1968). Reprinted with permission of ASCE.

Figure 9.37 Dimensionless pore pressure as a function of

dimensionless time and distance for one-dimensional consolidation by electroosmosis.

Figure 9.38 Average degree of consolidation versus dimen-

sionless time for one-dimensional consolidation by electroosmosis.

A numerical solution to Eq. (9.138) gives the results shown in Fig. 9.39 (Esrig, 1968, 1971). For the case of two pipe electrodes, a more realistic field condition than the radial geometry of Fig. 9.33b, Fig. 9.39 cannot be expected to apply exactly. Along a straight line between two pipe electrodes, however, the flow pattern is approximately the same as for the radial case for a considerable distance from each electrode. A solution for the rate of pore pressure buildup at the cathode for the case of no drainage (closed cathode) is shown in Fig. 9.40. This condition is relevant

Copyright © 2005 John Wiley & Sons

to pile driving, pile pulling, reduction of negative skin friction, and recovery of buried objects. Special solutions for in situ determination of soil consolidation properties by electroosmosis measurements have also been developed (Banerjee and Mitchell, 1980). One of the most important points to be noted from these solutions is that the rate of consolidation depends completely on the coefficient of consolidation, which varies directly with kh, but is completely independent of ke. Low values of kh, as is the case in highly plastic clays, mean long consolidation times. Thus, whereas a low value of kh means a high value of ke /kh and the potential for a high effective consolidation pressure, it also means longer required consolidation times for a given electrode spacing. The optimum situation is when ke /kh is high enough to generate a large pore water tension for reasonable electrode spacings (2 to 3 m) and maximum voltage (50 to 150 V DC), but kh is high enough to enable consolidation in a reasonable time. The soil types that best satisfy these conditions are silts, clayey silts, and silty clays. Most successful field applications of electroosmosis for consolidation have been in these types of materials. As noted earlier, the electrical conductivity of the soil is also important; if it is too high, as in the case of high-salinity pore water, adverse electrochemical effects and unfavorable economics may preclude use of electroosmosis for consolidation. 9.18

ELECTROCHEMICAL EFFECTS

The measured strength increases in the quick clay at ˚ s, Norway (Fig. 9.36), were some 80 percent greater A

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CONDUCTION PHENOMENA

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304

Figure 9.40 Dimensionless pore pressure at the face of a cylindrical electrode as a function

of dimensionless time for the case of a closed cathode (a swelling condition) (from Esrig and Henkel, 1968).

than can be accounted for solely by reduction in water content. Also, the liquid and plastic limits were changed as a result of treatment. Consolidation alone should have no effect on the Atterberg limits because changes in mineralogy, particle characteristics, and/or pore solution characteristics are needed to do this. In addition to movement of water when a DC voltage field is applied between metal electrodes inserted into a wet soil, the following effects may develop: ion diffusion, ion exchange, development of osmotic and pH gradients, desiccation by heat generation at the electrodes, mineral decomposition, precipitation of salts or secondary minerals, electrolysis, hydrolysis, oxidation, reduction, physical and chemical adsorption, and fabric changes. As a result, continuous changes in soil properties that are not readily accounted for by the simplified theory developed previously must be expected. Some of them, such as electrochemical hardening of the soil that results in permanent changes in plasticity and strength, may be beneficial; others, such as heating and gas generation, may impair the efficiency of electroosmosis. For example, heat and gas generation were so great that a field test of consolidation by electroosmosis for foundation stabilization of the leaning Tower of Pisa was unsuccessful. A simplified mechanism for some of the processes during electroosmosis is as follows. Oxygen gas is evolved at the anode by hydrolysis 2H2O ⫺ 4e⫺ → O2 ↑ ⫹ 4H⫹

(9.149)

Anions in solution react with freed H⫹ to form acids.

Copyright © 2005 John Wiley & Sons

Chlorine may also form in a saline environment. Some of the exchangeable cations on the clay may be replaced by H⫹. Because hydrogen clays are generally unstable, and high acidity and oxidation cause rapid deterioration of the anodes, the clay will soon alter to the aluminum or iron form depending on the anode material. As a result, the soil is usually strengthened in the vicinity of the anode. If gas generation at the anode causes cavitation and heat causes desiccation, cracking may occur. This will limit the negative pore pressure that can develop to a value less than 1 atm, and also the electrical resistance will increase, leading to a loss in efficiency. Hydrogen gas is generated at the cathode 4H2O ⫹ 4e⫺ → 2H2 ↑ ⫹ 4OH⫺

(9.150)

Cations in solution are drawn to the cathode where they combine with (OH)⫺ that is left behind to form hydroxides. The pH may rise to values as high as 12 at the cathode. Some alumina and silica may go into solution in the high pH environment. More detailed information about electrochemical reactions during electroosmosis can be found in Titkov et al. (1965), Esrig and Gemeinhardt (1967), Chilingar and Rieke (1967), Gray and Schlocker (1969), Gray (1970), Acar et al. (1990), and Hamed et al. (1991). Soil strength increases resulting from consolidation by electroosmosis and the concurrent electrochemical hardening have application for support of foundations on and in fine-grained soil. Pile capacity for a bridge

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SELF-POTENTIALS

9.19

9.20

SELF-POTENTIALS

Natural DC electrical potential differences of up to several tens of millivolts exist in the earth. These selfpotentials are generated by differing chemical conditions in adjacent soil layers, fluid flow, subsurface chemical reactions, and temperature differences. The self-potential (SP) method is one of the oldest geophysical methods for characterization of the subsurface (National Research Council, 2000). Self-potentials may be the source of phenomena of importance in geotechnical problems as well. The magnitude of self-potential between different soil layers depends on the contents of oxidizing and reducing substances in the layers (F. Hilbert, in Veder, 1981). These potentials can cause a natural electroosmosis in which water flows in the direction from the higher to the lower potential, that is, toward the cathode. The process is shown schematically in Fig. 9.41. An oxidizing soil layer is positive relative to a reducing layer, thus inducing an electroosmotic water flow toward the interface. If water accumulates at the interface, there can be swelling and loss of strength, leading ultimately to formation of a slip surface.

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foundation in varved clay at a site in Canada was well below the design value and inadequate for support of the structure (Soderman and Milligan, 1961; Milligan, 1994). Electrokinetic treatment using the piles as anodes resulted in sufficient strength increase to provide the needed support. Recently reported model tests by Micic et al. (2003) on the use of electrokinetics in soft marine clay to increase the load capacity of skirt foundations for offshore structures resulted in increases in soil strength and supporting capacity of up to a factor of 3.

305

ELECTROKINETIC REMEDIATION

The transport of dissolved and suspended constituents into and out of the ground by electroosmosis and electrophoresis, as well as electrochemical, reactions have become of increasing interest because of their potential applications in waste containment and removal of contaminants from fine-grained soils. The electrolysis reactions at the electrodes described in the preceding section, wherein acid is produced at the anode and base at the cathode, are of particular relevance. After a few days of treatment the pH in the vicinity of the anode may drop to less than 2, and that at the cathode increase to more than 10 (Acar and Alshewabkeh, 1993). Toxic heavy metals are preferentially adsorbed by clay minerals and they precipitate except at low pH. Iron or aluminum cations from decomposing anodes can replace heavy-metal ions from exchange sites, the acid generated at the anode can redissolve precipitated material, and the acid front that moves across the soil can keep the metals in solution until removed at the cathode. Geochemical reactions in the soil pores impact the efficiency of the process. Among them are complexation effects that reverse ion charge and reverse flow directions, precipitation/dissolution, sorption, desorption and dissolution, redox, and immobilization or precipitation of metal hydroxides in the high pH zone near the cathode. Some success has been reported in the removal of organic pollutants from soils, at least in the laboratory, as summarized by Alshewabkeh (2001). However, it is unlikely that large quantities of non-aqueous-phase liquids can be effectively transported by electrokinetic processes, except as the NAPL may be present in the form of small bubbles that move with the suspending water. An in-depth treatment of the fundamentals of electrokinetic remediation and the practical aspects of its implementation are given by Alshewabkeh (2001) and the references cited therein.

Copyright © 2005 John Wiley & Sons

Generation of Self-Potentials in Soil Layers

Soils in an oxidizing environment are usually yellow or tan to reddish brown and are characterized by oxides and hydrates of trivalent iron and a low pH relative to reducing soils, which are usually dark gray to bluegray in color and contain sulfides and oxides and hydroxides of divalent iron. The local electrical potential of the soil  depends on the iron concentrations and can be calculated from Nernst’s equation:

Figure 9.41 Natural electroosmosis due to self-potential dif-

ferences between oxidizing and reducing soil layers. The oxidizing soil layer is positive relative to the reducing layer (redrawn from Hilbert, in Veder, 1981).

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306

9

CONDUCTION PHENOMENA

 ⫽ 0.771 ⫹

冉 冊

RT c3⫹ ln Fe F c2⫹ Fe

(9.151)

u ⫽ 50 ⫻ 9.81 ⫻ 0.05 ⬇ 25 kPa is generated, which is not an insignificant value. If water that is driven toward the interface cannot escape or be absorbed by the soil, then the effective stress will be reduced by this amount. If the water is absorbed into the clay layer, then softening will result. Either way, the resistance to sliding along the interface will be reduced.

Co py rig hte dM ate ria l

in which the concentrations are of Fe in solution in moles/liter pore water. The difference in potentials between two layers gives the driving potential for electroosmosis. Values calculated using the Nernst equation are too high for actual soil systems because it applies for conditions of no current flow, and the flowing current also generates a diffusion potential acting in the opposite direction. Hilbert, in Veder (1981), gives the electrical potential as a function of the in situ pH, that is,

then ke /hh ⫽ 50 m/V. If the self-potential difference is 50 mV, then from Eq. (9.142) a pore pressure value of

 ⫽ 0.186 ⫺ 0.059 pH

(9.152)

Reasonable agreement has been obtained between measured and calculated values of  for different soil layers. The end result is that potential differences of up to 50 mV or so are developed between different layers. Potentials measured in a trench excavated in a slide zone are shown in Fig. 9.42. Excess Pore Pressure Generation by Self-Potentials

The pore pressure that may develop at an interface between two different soil layers is given by Eq. (9.142) in which V is the difference in self-potentials between the layers. For a given value of V, the magnitude of pore pressure depends directly on ke /kh. For example, if ke ⫽ 5 ⫻ 10⫺9 m2 /s V and kh ⫽ 1 ⫻ 10⫺10 m/s,

Landslide Stabilization Using Short-Circuit Conductors

If slope instability is caused by a slip surface between reducing and oxidizing soil layers, then a simple means for stabilization can be used (Veder, 1981). Shortcircuiting conductors, such as steel rods, are driven into the soil so that they extend across the slip surface and about 1 to 2 m into the soil below. The mechanism that is then established is shown in Fig. 9.43. Electric current generated by reduction reactions in the oxidizing soil layer and oxidizing reactions in the reducing layer flows through the conductors. Because of the presence of oxidizing agents such as ferric iron, oxygen, and manganese compounds, in the upper oxidizing layer that take up electrons, electrons pass from the metal conductor to the soil. That is, the introduction of electrons initiates reducing reactions. In the reducing layer, on the other hand, there is already a

Figure 9.42 Electrical potentials measured in a trench cut into a slide (from Veder, 1981). Reprinted with permission of Springer-Verlag.

Copyright © 2005 John Wiley & Sons

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THERMALLY DRIVEN MOISTURE FLOW

307

for use of short-circuiting conductors are (1) intact cohesive soils with a low hydraulic conductivity, (2) shear between oxidizing and reducing clay layers, and (3) a relatively thin, well-defined shear zone.

9.21

THERMALLY DRIVEN MOISTURE FLOW

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Thermally driven flows in saturated soils are rather small. Gray (1969) measured thermoelectric currents on the order of 1 to 10 A/ C cm, with the warm side positive relative to the cold side. Thermoosmotic pressures of only a few tenths of a centimeter water head per degree Celsius were measured in saturated soil. Net flows in different directions have been measured in different investigations, evidently because of different temperature dependencies of chemical activity coefficients. These small thermoelectric and thermoosmotic effects in saturated soils may be of little practical significance in geotechnical problems. On the other hand, thermally driven moisture flows in partly saturated soils can be large, and that these flows can be very important in subgrade stability, swelling soils, and heat transfer and storage problems of various types. Theoretical representations of moisture flow through partly saturated soils based solely on the application of irreversible thermodynamics, such as developed by Taylor and Cary (1964), have not been completely successful. They underestimate the flows substantially, perhaps because of the inability to adequately represent all the processes and interactions. A widely used theory for coupled heat and moisture flow through soils was developed by Philip and De Vries (1957). It accounts for both liquid- and vaporphase flows. Vapor-phase flow depends on the thermal and isothermal vapor diffusivities and is driven by temperature and moisture content gradients. The liquidphase flow depends on the thermal and isothermal liquid diffusivities and is driven by the temperature gradient, the moisture content gradient, and gravity. The two governing equations are:

Figure 9.43 Mechanism for slide stabilization using shortcircuiting conductors (adapted from Veder, 1981).

surplus of electrons. If these pass into the conductor, then the environment becomes favorable for oxidation reactions. Thus, positive charges are generated in the reducing soil layer as the conductor carries electrons away. The oxidizing soil layer then takes up these electrons. Completion of the electrical circuit requires current flow through the soil pore water in the manner shown in Fig. 9.43, where adsorbed cations, shown as Na⫹, plus the associated water, flow away from the soil layer interface. This electroosmotic transport of water reduces the water content in the slip zone. Thus, shortcircuit conductors have three main effects (Veder, 1981):

1. Natural electroosmosis is prevented because the short-circuiting conductors eliminate the potential difference between the two soil layers. 2. Electrochemical reactions produce electroosmotic flow in the opposite direction, thus helping to drain the shear zone. 3. Corrosion of the conductors produces high valence cations that exchange for lower valence adsorbed cations, for example, iron for sodium, which leads to soil strengthening.

Several successful cases of landslide stabilization using short-circuiting conductors have been described by Veder (1981) and the references cited therein. Typically, steel rods about 25 mm in diameter are used, spaced a maximum of 3 to 4 m apart in grid patterns covering the area to be stabilized. Conditions favorable

Copyright © 2005 John Wiley & Sons

For vapor-phase flow:

qvap ⫽ ⫺DTVT ⫺ D V w

(9.153)

and for liquid-phase flow:

qliq ⫽ ⫺DTLT ⫺ D L ⫺ k i w where qvap ⫽ vapor flux density (M/L2 /T) w ⫽ density of water (M/L3)

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(9.154)

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CONDUCTION PHENOMENA

T ⫽ temperature (K) ⫽ volumetric water content (L3 /L3) DTV ⫽ thermal vapor diffusivity (L2 /T/K) D V ⫽ isothermal vapor diffusivity (L2 /T) qliq ⫽ liquid flux density (M/L2 /T) DTL ⫽ thermal liquid diffusivity (L2 /T/K) D L ⫽ isothermal liquid diffusivity (L2 /T) k ⫽ unsaturated hydraulic conductivity (L/T) i ⫽ unit vector in vertical direction

DTV ⫽

1. Hydraulic conductivity as a function of water content 2. Thermal conductivity as a function of water content 3. Volumetric heat capacity (see Table 9.2) 4. Suction head as a function of water content

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The thermal vapor diffusivity is given by

in which  is the surface tension of water (F/L). Use of the above equations requires knowledge of four relationships to describe the properties of the soils in the system:

冉冊

冉 冊

D0 d 0 v[a ⫹ ƒ(a)  ]h w dT

(9.155)

The isothermal vapor diffusivity is given by D V ⫽

冉 冊 冉 冊冉 冊  hg D0 va 0 w RT

d d

(9.156)

where D0 ⫽ molecular diffusivity of water vapor in air (L2 /T) v ⫽ mass flow factor ⫽ P/(P ⫺ p) P ⫽ total gas pressure in pore space p ⫽ partial pressure of water vapor in pore space  ⫽ tortuosity factor a ⫽ volumetric air content (L3 /L3) h ⫽ relative humidity of air in pores  ⫽ ratio of average temperature gradient in the air-filled pores to the overall temperature gradient g ⫽ acceleration of gravity (L/T2) R ⫽ gas constant (FL/M/K) 0 ⫽ density of saturated water vapor (M/L3)  ⫽ suction head of water in the soil (negative head) (L) ƒ(a) ⫽ a/ak for 0 ⬍ a ⬍ ak ⫽ 1 for a  ak ak ⫽ a at which liquid conductivity is lost or at which the hydraulic conductivity falls below some arbitrary fraction of the saturated value The thermal liquid diffusivity is given by DTL ⫽ k

冉 冊冉 冊  

d dT

(9.157)

The isothermal diffusivity is given by

冉冊

D L ⫽ k

d d

(9.158)

Copyright © 2005 John Wiley & Sons

The hydraulic conductivity and suction relationships are hysteretic; that is, they depend on whether the soil is wetting or drying. Examples of the variations of the different properties needed for the analysis are shown in Fig. 9.44 as a function of degree of saturation and volumetric water content. The data are for a crushed limestone that is used for a trench backfill around buried electrical transmission cables. This material is used because of its low thermal resistivity, which makes it suitable for effective dissipation of heat from the buried cable, provided the saturation does not fall below about 40 percent. The vapor flow is made up of a flow away from the high-temperature side that is driven by a vapor density gradient and a return flow caused by variation in the pore vapor humidity as reflected by variations in soil suction. At moderate soil suction values, for example, a few meters for sand and several tens of meters for clay, the thermal vapor diffusivity predominates, and moisture is driven away from the heat source (McMillan, 1985). The isothermal diffusivity term only becomes important at very high suction levels. The liquid flow consists of a capillarity-driven flow toward the heat source and an outward liquid flow due to variations in water surface tension with temperature. McMillan’s analysis showed that for both sand and clay the isothermal liquid diffusivity term was 4 to 5 orders of magnitude greater than the thermal liquid diffusivity term. Thus capillarity-driven flow predominates for any significant gradient in the volumetric moisture content. The very small thermal liquid diffusivity is consistent with the observations noted earlier for saturated soils in which measured water flows under thermal gradients are small. The total water flow q in an unsaturated soil under the action of a temperature gradient and its resulting water content gradient equals the sum of the vaporphase and liquid-phase movements. Thus, from Eqs. (9.153) to (9.158),

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309

Figure 9.44 Examples of properties used for analysis of thermally driven moisture flow in a partially saturated, compacted, crushed limestone: (a) particle size distribution, (b) suction head as a function of volumetric water content, (c) hydraulic conductivity as a function of degree of saturation and volumetric water content, (d) isothermal liquid diffusivity as a function of degree of saturation and volumetric water content, (e) isothermal vapor diffusivity as a function of degree of saturation and volumetric water content, and (f) Thermal water diffusivity as a function of degree of saturation and volumetric water content. Thermal resistivity as a function of water content for this soil is shown in Fig. 9.14.

q ⫽ ⫺(DTV ⫹ DTL)T ⫺ (D V ⫹ D L ) ⫺ k i w ⫽ ⫺DTT ⫺ D  ⫺ k i

in which

(9.159)

D ⫽ DTV ⫹ DTL ⫽ thermal water diffusivity

(9.160)

and

Equation (9.159) is the governing equation for moisture movement under a thermal gradient in unsaturated soils as proposed by Philip and De Vries (1957). Differentiation of this equation and application of the continuity requirement gives the general differential equation for moisture flow:  k ⫽ (DTT) ⫹ (D  ) ⫹ t z

(9.162)

The heat conduction equation for the soil is

D ⫽ D V ⫹ D L ⫽ isothermal water diffusivity (9.161)

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冉 冊

T k ⫽  t T t C

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(9.163)

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CONDUCTION PHENOMENA

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310

Figure 9.44 (Continued )

where kt ⫽ thermal conductivity C ⫽ volumetric heat capacity

The ratio of thermal conductivity to the volumetric heat capacity is the thermal diffusivity A. Both transient and steady-state temperature distributions computed using the Philip and De Vries theory incorporated into numerical models have agreed well with measured values in a number of cases. The actual moisture movements and distributions have not agreed as well, for example, Abdel-Hadi and Mitchell (1981) and Cameron (1986). The numerical simulations have been done using transform methods, finite difference methods, the finite element method, and the integrated finite difference method. Cameron (1986) reformulated the equations in terms of suction head rather than moisture content and incorporated them into the finite element model of Walker et al. (1981) for solution of two-dimensional problems. 9.22

GROUND FREEZING

Heat conduction in soils and rocks is discussed in Section 9.5, and values for thermal properties are given in

Copyright © 2005 John Wiley & Sons

Table 9.2. Three topics are considered in this section: (1) the depth of frost penetration, which illustrates the application of transient heat flow analysis, (2) frost action in soils, a phenomenon of great practical importance that can be understood through consideration of interactions of the physical and physicochemical properties of the soil, and (3) some effects of freezing on the behavior and properties of the soil after thawing. These topics are also covered in some detail by Konrad (2001) and the references therein. Depth of Frost Penetration

Accurate estimation of the depth of ground freezing during the winter, the depth of thawing in permafrost areas during the summer, and the refrigeration and time requirements for artificial ground freezing for temporary ground stabilization are all problems involving transient heat flow analysis. They differ from the conduction analyses in the preceding sections in that the phase change of water to ice must be taken into account. Prediction of the maximum depth of frost penetration illustrates this type of problem. Theoretical solutions of this problem are based on a mathematical

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GROUND FREEZING

analysis developed by Neumann in about 1860 (Berggren, 1943; Aldrich, 1956; Brown, 1964; Konrad, 2001). The relationship between thermal energy u and temperature T for a soil mass at constant water content is shown in Fig. 9.45. In the absence of freezing or thawing

(9.168)

where a ⫽ kt /C is the thermal diffusivity (L2 /T). Equation (9.168) is the one-dimensional, transient heat flow equation. At the interface between frozen and unfrozen soil, z ⫽ Z, and the equation of heat continuity is

(9.164) Ls

dZ ⫽ q ƒ ⫺ qu dt

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u ⫽C T

T 2T ⫽a 2 t z

311

(9.169)

The Fourier equation for heat flow is T qt ⫽ ⫺kt z

(9.165)

In the absence of freezing or thawing, thermal continuity and conservation of thermal energy require that the rate of change of thermal energy of an element plus the rate of heat transfer into the element equal zero, that is, for the one-dimensional case u q ⫹ ⫽0 t z

(9.166)

Using Eqs. (9.164) and (9.165), Eq. (9.166) may be written C

or

T 2T ⫽ kt 2 t z

(9.167)

where Ls is the latent heat of fusion of water and qƒ ⫺ qu is the net rate of heat flow away from the interface. Equation (9.169) can be written Ls

dZ T T ⫽ kƒ ƒ ⫺ ku u dt z z

(9.170)

where the subscripts u and f pertain to unfrozen and frozen soil, respectively. Simultaneous solution of Eqs. (9.168) and (9.170) gives the depth of frost penetration. Stefan Formula The simplest solution is to assume that the latent heat is the only heat to be removed during freezing and neglect the heat that must be removed to cool the soil water to the freezing point, that is, the thermal energy stored as volumetric heat is neglected. This condition is shown by Fig. 9.46. For this case Eq. (9.168) does not exist, and Eq. (9.170) becomes Ls

dZ T ⫽ kƒ s dt Z

(9.171)

where Ts is the surface temperature. The solution of this equation is

Figure 9.45 Thermal energy as a function of temperature Figure 9.46 Assumed conditions for the Stefan equation.

for a wet soil.

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CONDUCTION PHENOMENA

冉

2kƒ

Z⫽

冊

冕 T dt s

Ls

1/2

(9.172)

冉 冊

Z⫽

2kTst Ls

1/2

⫽

T0 Ts

(9.174)

and the fusion parameter  is ⫽

C T Ls s

(9.175)

An averaged value for the volumetric heats of frozen and unfrozen soil can be used for C in Eq. (9.175). In application, the quantity Tst in Eq. (9.173) is replaced by the freezing index, and Ts in (9.175) is given by F/t, where t is the duration of the freezing period. The coefficient  corrects the Stefan formula for neglect of volumetric heat. For soils with high water content C is small relative to Ls; therefore,  is small and

Figure 9.47 Freezing index in relation to the annual temperature cycle.

Copyright © 2005 John Wiley & Sons

(9.173)

where k is taken as an average thermal conductivity for frozen and unfrozen soil. The dimensionless correction coefficient  depends on the two parameters shown in Fig. 9.49. The thermal ratio  is given by

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The integral of Ts dt is a measure of freezing intensity. It can be expressed by the freezing index F, which has units of degrees ⫻ time. Index F is usually given in degree-days. It is shown in relation to the annual temperature cycle in Fig. 9.47. Freezing index values are derived from meteorological data. Methods for determination of freezing index values are given by Linell et al. (1963), Straub and Wegmann (1965), McCormick (1971), and others. Maps showing mean freezing index values are available for some areas. It is important when using such data sources to be sure that there are not local deviations from the average values that are given. Different types of ground cover, local topography and vegetation, and solar radiation all influence the net heat flux at the ground surface. The Stefan equation can also be used to estimate the summer thaw depth in permafrost; that is, the thickness of the active layer. In this case the ground thawing index, also in degree-days and derived from meteorological data, is used in Eq. (9.172) in place of the freezing index (Konrad, 2001). Modified Berggren Formula The Stefan formula overpredicts the depth of freezing because it neglects the removal of the volumetric heats of frozen and unfrozen soil. Simultaneous solution of Eqs. (9.168) and

(9.170) has been made for the conditions shown in Fig. 9.48, assuming that the soil has a uniform initial temperature that is T0 degrees above freezing and that the surface temperature drops suddenly to Ts below freezing (Aldrich, 1956). The solution is

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GROUND FREEZING

313

Figure 9.48 Thermal conditions assumed in the derivation of the modified Berggren for-

mula.

the Stefan formula is reasonable. For arctic climates, where T0 is not much above the freezing point,  is small,  is greater than 0.9, and the Stefan formula is satisfactory. However, in more temperate climates and in relatively dry or well-drained soils, the correction becomes important. A comparison between theoretical freezing depths and a design curve proposed by the Corps of Engineers is shown in Fig. 9.50 for several soil types. The theoretical curves were developed by Brown (1964) using the modified Berggren equation and the thermal properties given in Fig. 9.13. Consideration should be given to the effect of different types of surface cover on the ground surface temperature because air temperature and ground temperature are not likely to be the same, and the effects of thermal radiation may be important. Observed

Copyright © 2005 John Wiley & Sons

depths of frost penetration may be misleading if estimates for a proposed pavement or other structure are needed because of differences in ground surface characteristics and because the pavement or foundation base will be at different water content and density than the surrounding soil. The solutions do not account for flow of water into or out of the soil or the formation of ice lenses during the freezing period. This may be particularly important when dealing with frost heave susceptible soils or when developing frozen soil barriers for the cutoff of groundwater flow. Methods for prediction of frost depth in soils susceptible to ice lens formation and the rate of heave are given by Konrad (2001). The initiation of freezing of flowing groundwater requires that the rate of volumetric and latent heat removal be high enough so that ice can form during the residence time

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CONDUCTION PHENOMENA

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314

Figure 9.49 Correction coefficients for use in the modified Berggren formula (from Aldrich,

1956).

of an element of water moving between the boundaries of the specified zone of solidification. Frost Heaving

Freezing of some soils is accompanied by the formation of ice layers or ‘‘lenses’’ that can range from a millimeter to several centimeters in thickness. These lenses are essentially pure ice and are free from large numbers of contained soil particles. The ground surface may ‘‘heave’’ by as much as several tens of centimeters, and the overall volume increase can be many times the 9 percent expansion that occurs when water freezes. Heave pressures of many atmospheres are common. The freezing of frost-susceptible soils beneath pavements and foundations can cause major distress or failure as a result of uneven uplift during freezing and loss of support on thawing, owing to the presence of large water-filled voids. Ordinarily, ice lenses are oriented normal to the direction of cold-front movement and become thicker and more widely separated with depth. The rate of heaving may be as high as several millimeters per day. It depends on the rate of freezing in

Copyright © 2005 John Wiley & Sons

a complex manner. If the cooling rate is too high, then the soil freezes before water can migrate to an ice lens, so the heave becomes only that due to the expansion of water on freezing. Three conditions are necessary for ice lens formation and frost heave: 1. Frost-susceptible soil 2. Freezing temperature 3. Availability of water

Frost heaving can occur only where there is a water table, perched water table, or pocket of water reasonably close to the freezing front. Frost-Susceptible Soils Almost any soil may be made to heave if the freezing rate and water supply are controlled. In nature, however, the usual rates of freezing are such that only certain soil types are frost susceptible. Clean sands, gravels, and highly plastic intact clays generally do not heave. Although the only completely reliable way to evaluate frost susceptibility is by some type of performance test during freezing, soils that contain more than 3 percent of their particles finer than 0.02 mm are potentially frost susceptible.

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GROUND FREEZING

Figure 9.50 Predicted frost penetration depths compared with the Corps of Engineers’ de-

sign curve (Brown, 1964). Curve a—sandy soil: dry density 140 lb / ft3, saturated, moisture content 7 percent. Curve b—silt, clay: dry density 80 lb / ft3, unsaturated, moisture content 2 percent. Curve c—sandy soil; dry density 140 lb / ft3, unsaturated, moisture content 2 percent. Curve d—silt, clay: dry density 120 lb / ft3, moisture content 10 to 20 percent (saturated). Curve e—silt, clay: dry density 80 lb / ft3 saturated, moisture content 30 percent. Curve f—Pure ice over still water.

Frost-susceptible soils have been classified by the Corps of Engineers in the following order of increasing frost susceptibility:

Group (increasing susceptibility) F1 F2 F3

F4

Soil Types

Gravelly soils with 3 to 20 percent finer than 0.02 mm Sands with 3 to 15 percent finer than 0.02 mm a. Gravelly soils with more than 20 percent finer than 0.02mm sands, except fine silt sands with more than 15 percent finer than 0.02 mm b. Clays with PI greater than 12 percent, except varved clays a. Silts and sandy silts b. Fine silty sands with more than 15 percent finer than 0.02 mm c. Lean clays with PI less than 12 percent d. Varved clays

Copyright © 2005 John Wiley & Sons

A method for the evaluation of frost susceptibility that takes project requirements and acceptable risks and freezing conditions into account as well as the soil type is described by Konrad and Morgenstern (1983). Mechanism of Frost Heave The formation of ice lenses is a complex process that involves interrelationships between the phase change of water to ice, transport of water to the lens, and general unsteady heat flow in the freezing soil. The following explanation of the physics of frost heave is based largely on the mechanism proposed by Martin (1959). Although the Martin (1959) model may not be correct in all details in the light of subsequent research, it provides a logical and instructive basis for understanding many aspects of the frost heave process. The ice lens formation cycle involves four stages: 1. 2. 3. 4.

Nucleation of ice Growth of the ice lens Termination of ice growth Heat and water flow between the end of stage 3 and the start of stage 1 again

In reality, heat and water flows continue through all four stages; however, it is convenient to consider them separately. The temperature for nucleation of an ice crystal, Tn, is less than the freezing temperature, T0. In soils, T0 in

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CONDUCTION PHENOMENA

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pore water is less than the normal freezing point of water because of dissolved ions, particle surface force effects, and negative pore water pressures that exist in the freezing zone. The freezing point decreases with decreasing distance to particle surfaces and may be several degrees lower in the double layer than in the center of a pore. Thus, in a fine-grained soil, there is an unfrozen film on particle surfaces that persists until the temperature drops below 0C. The face of an ice front has a thin film of adsorbed water. Freezing advances by incorporation of water molecules from the film into the ice, while additional water molecules enter the film to maintain its thickness. It is energetically easier to bring water to the ice from adjacent pores than to freeze the adsorbed water on the particle or to propagate the ice through a pore constriction. The driving force for water transport to the ice is an equivalent hydrostatic pressure gradient that is generated by freezing point depression, by removal of the water from the soil at the ice front, which creates a higher effective stress in the vicinity of the ice than away from it, by interfacial tension at the ice–water interface, and by osmotic pressure generated by the high concentration of ions in the water adjacent to the ice front. Ice formation continues until the water tension in the pores supplying water becomes great enough to cause cavitation, or decreased upward water flow from below leads to new ice lens formation beneath the existing lens. The processes of freezing and ice lens formation proceed in the following way with time according to Martin’s theory. If homogeneous soil, at uniform water content and temperature T0 above freezing, is subjected to a surface temperature Ts below freezing, then the variation of temperature with depth at some time is as shown in Fig. 9.51. The rate of heat flow at any point is ⫺kt(dT/dz). If dT/dz at point A is greater than at point B, the temperature of the element will drop. When water goes to ice, it gives up its latent heat, which flows both up and down and may slow or stop changes in the value of dT/dz for some time period, thus halting the rate of advance of the freezing front into the soil. Ground heave results from the formation of a lens at A, with water supplied according to the mechanisms indicated above. The energy needed to lift the overlying material, which may include not only the soil and ice lenses above, but also pavements and structures, is available because ice forms under conditions of supercooling at a temperature T X ⬍ TFP, where TFP is the freezing temperature. The available energy is

F ⫽

L(TFP ⫺ T X) TFP

(9.176)

Copyright © 2005 John Wiley & Sons

Figure 9.51 Temperature versus depth relationships in a

freezing soil.

The quantity L is the latent heat. Supercooling of 1C is sufficient to lift 12.5 kg a distance of 10 mm. Alternatively, the energy for heave may originate from the thin water films at the ice surface (Kaplar, 1970). As long as water can flow to a growing ice lens fast enough, the volumetric heat and latent heat can produce a temporary steady-state condition so that (dT/ dz)A ⫽ (dT/dz)B. For example, silt can supply water at a rate sufficient for heave at 1 mm/h. After some time the ability of the soil to supply water will drop because the water supply in the region ahead of the ice front becomes depleted, and the hydraulic conductivity of the soil drops, owing to increased tension in the pore water. This is illustrated in Fig. 9.52, where hydraulic conductivity data as a function of negative pore water pressure are shown for a silty sand, a silt, and a clay, all compacted using modified AASHTO effort, at a water content about 3 percent wet of optimum. A small negative pore water pressure is sufficient to cause water to drain from the pores of the silty sand, and this causes a sharp reduction in hydraulic conductivity. Because the clay can withstand large negative pore pressures without loss of saturation, the hydraulic conductivity is little affected by increasing reductions in the pore pressure (increasing suction). The small decrease that is observed results from the consolidation needed to carry the increased effective stress required to balance the reduction in the pore pressure. For the silt, water drainage starts when the suction reaches

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GROUND FREEZING

317

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which have now reduced the distance that water can be from a particle surface. The temperature drop must reach a depth where there is sufficient water available after nucleation to supply a growing lens. The thicker the overlying lens, the greater the distance, thus accounting for the increased spacings between lenses with depth. The greater the depth, the smaller the thermal gradient, as may be seen in Fig. 9.51, where (dT/dz)A ⬎ (dT/dz)A where A is on the temperature distribution curve for a later time t2. Because of this, the rate of heat extraction is slowed, and the temporary steady-state condition for lens growth can be maintained for a longer time, thus enabling formation of a thicker lens. More quantitative analyses of the freezing and frost heaving processes in terms of segregation potential, rates, pressures, and heave amounts are available. The Proceedings of the International Symposia on Ground Freezing, for example, Jones and Holden (1988), Nixon (1991), and Konrad (2001) provide excellent sources of information on these issues. Thaw Consolidation and Weakening

Figure 9.52 Hydraulic conductivity as a function of negative pore water pressure (from Martin and Wissa, 1972).

about 40 kPa; however, a significant continuous water phase remains until substantially greater values of suction are reached. In sand, the volume of water in a pore is large, and the latent heat raises the freezing temperature to the normal freezing point. Hence, there is no supercooling and no heave. Negative pore pressure development at the ice front causes the hydraulic conductivity to drop, so water cannot be supplied to form ice lenses. Thus sands freeze homogeneously with depth. In clay, the hydraulic conductivity is so low that water cannot be supplied fast enough to maintain the temporary steadystate condition needed for ice lens growth. Heave in clay only develops if the freezing rate is slowed to well below that in nature. Silts and silty soils have a combination of pore size, hydraulic conductivity, and freezing point depression that allow for large heave at normal freezing rates in the field. The freezing temperature penetrates ahead of a completed ice lens, and a new lens will start to form only after the temperature drops to the nucleation temperature. The nucleation temperature for a new lens may be less than that for the one before because of reduced saturation and consolidation from the previous flows,

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When water in soil freezes, it expands by about 9 percent of its original volume. Thus a fully saturated soil increases in volume by 9 percent of its porosity, even in the absence of ice segregation and frost heave. The expansion associated with freezing disrupts the original soil structure. When thawed, the water returns to its original volume, the melting of segregated ice leaves voids, and the soil can be considerably more deformable and weaker that before it was frozen. Under drained conditions and constant applied overburden stress, the soil may consolidate to a denser state than it had prior to freezing. The lower the density of the soil, the greater is the amount of thaw consolidation. The total settlement of foundations and pavements associated with thawing is the sum of that due to (1) the phase change, (2) melting of segregated ice, and (3) compression of the weakened soil structure. Testing of representative samples under appropriate boundary conditions is the most reliable means for evaluating thaw consolidation. Samples of frozen soil are allowed to thaw under specified levels of applied stress and under defined drainage conditions, and the decrease in void ratio or thickness is determined. An example of the effects of freezing and thawing on the compression and strength of initially undisturbed Boston blue clay is shown in Fig. 9.53 from Swan and Greene (1998). These tests were done as part of a ground freezing project for ground strengthening to enable jacking of tunnel sections beneath operating rail lines during construction of the recently completed Central Artery/Tunnel Project in Boston. Detailed

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0 2 C1-UF e0 = 1.064

6 C4-FT e0 = 1.171

8 10 12 14

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Vertical Strain, εv (%)

4

16 18 20

22 10

100 1000 Effective Stress, σc (kPa)

10000

(a)

120

Deviator Stress, σ1 – σ3 (kPa)

100

UUC1-UF (σ1–σ3)max = 109.6 kPa ε1 = 2.3% su/σ3cell = 0.36 e0 = 1.02; w = 37.5%

80

60

40

UUC4-FT (σ1– σ3)max = 42.4 kPa ε1 = 12.8% su/σ3cell = 0.14 e0 = 1.13; w = 43.2%

20

0

0

5

10

15

20

25

Axial Strain. % (b)

Figure 9.53 (a) Comparison between the compression behavior of unfrozen (C1-UF) and frozen then thawed (C4-FT) samples of Boston blue clay. (b) Deviator stress vs. axial strain in unconsolidated–undrained triaxial compression of unfrozen (UUC1-UF) and frozen and thawed (UUC4-FT) Boston blue clay (from Swan and Greene, 1998).

analysis of the thaw consolidation process and its analytical representation is given by Nixon and Ladanyi (1978) and Andersland and Anderson (1978). Ground Strengthening and Flow Barriers by Artificial Ground Freezing

Artificial ground freezing has applications for formation of seepage cutoff barriers in situ, excavation support, and other ground strengthening purposes. These appli-

Copyright © 2005 John Wiley & Sons

cations are usually temporary, and they have the advantage that the ground is not permanently altered, except for such property changes as may be caused by the freeze–thaw processes. Returning the ground to its pristine state may be important for environmental reasons where alternative methods for stabilization could permanently change the state and composition of the subsoil. Freezing is usually accomplished by installation of freeze pipes and circulation of a refrigerant. For emer-

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CONCLUDING COMMENTS

9.23

160

120 Second Stage

First Stage

Third Stage

CONCLUDING COMMENTS

Conductivity properties are one of the four key dimensions of soil behavior that must be understood and quantified for success in geoengineering. The other three dimensions are volume change, deformation and strength, and the influences of time. They form the subjects of the following three chapters of this book. Water flows through soils and rocks under fully saturated conditions have been the most studied, and hydraulic conductivity properties, their determination and application for seepage studies of various types, construction dewatering, and the like are central to geotechnical engineering. One objective of this chapter has been to elucidate the fundamental factors that control

Copyright © 2005 John Wiley & Sons

Natural Strain, ε -%

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gency and rapid ground freezing, expendable refrigerants such as liquid nitrogen or carbon dioxide in an open pipe can be used. The thermal energy removal and time requirements for freezing the ground can be calculated using the appropriate thermal conductivity, volumetric heat, and latent heat properties for the ground and heat conduction theory in conjunction with the characteristics of the refrigeration system (Sanger, 1968; Shuster, 1972; Sanger and Sayles, 1979). For many applications the energy required to freeze the ground in kcal/m3 will be in the range of 2200 to 2800 times the water content in percent (Shuster, 1972). However, if the rate of groundwater flow exceeds about 1.5 m/day, it may be difficult to freeze the ground without a very high refrigeration capacity to ensure that the necessary temperature decrease and latent heat removal can be accomplished within the time any element of water is within the zone to be frozen. The long-term strength and stress–strain characteristics of frozen ground depend on the ice content, temperature, and duration of loading. The short-term strength under rapid loading, which can be up to 20 MPa at low temperature, may be 5 to 10 times greater than that under sustained stresses. That is, frozen soils are susceptible to creep strength losses (Chapter 12). The deformation behavior of frozen soil is viscoplastic, and the stress and temperature have significant influence on the deformation at any time. The creep curves in Fig. 9.54 illustrate these effects. The onset of the third stage of creep indicates the beginning of failure. The evaluation of stability of frozen soil masses, the prediction of creep deformation, and the possibility of creep rupture are complex problems because of heterogeneous ground conditions, irregular geometries, and temperature and stress variations throughout the frozen soil mass. Design and implementation considerations for use of ground freezing in construction are given by Donohoe et al. (1998).

319

80

Pa

55

T

=

0

,σ °C

=

M

Temperature Effect

0.

40

T=

,σ= –2.2 °C

Pa

0.55 M

Stress effect

T = –2.2 °C, σ = 0.138 MPa

0

0

tf

10

20

30

Time, t (hr)

Figure 9.54 Creep curves for a frozen organic silty clay (from Sanger and Sayles, 1979).

the permeability of soils to water and how this property depends on soil type, especially gradation, and is sensitive to testing conditions, soil fabric, and environmental factors. The understanding of these fundamentals is important, not only because of the insights provided but also because many of the same considerations apply to the several other types of flows that are known to be important—chemical, electrical, and thermal. Knowledge of one is helpful in the understanding and quantification of the other because the mathematical descriptions of the flows follow similar force-flux relationships. At the same time it is necessary to take into account that the flows of fluids of different composition and the application of hydraulic, chemical, electrical, and thermal driving forces to soils can cause changes in compositions and properties, with differing consequences, depending on the situation. Furthermore, as examined in considerable detail in this chapter, flow coupling can be important, especially advective and diffusive chemical transport, electroosmotic water and chemical flow, and thermally driven moisture flow. Considerable impetus for research on these processes has been generated by geoenvironmental needs, including enhanced

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and more economical waste containment and site remediation strategies. Ground freezing, in addition to its importance in engineering and construction in cold regions, is seeing new applications for temporary ground stabilization needed for underground construction in sensitive urban areas. QUESTIONS AND PROBLEMS

7. Two parallel channels, one with flowing water and the other with contaminated water, are 100 ft apart. The surface elevation of the contaminated channel is 99 ft, and the surface elevation of the clean water channel is at 97 ft. The soil between the two channels is sand with a hydraulic conductivity of 1 ⫻ 10⫺4 m/s, a dry unit weight of 100 pcf, and a specific gravity of solids of 2.65. Estimate the time it will take for seepage from the contaminated channel to begin flowing into the initially clean channel. Make the following assumptions and simplifications: a. Seepage is one dimensional. b. The only subsurface reaction is adsorption onto the soil particles. c. The soil–water partitioning coefficient is 0.4 cm3 /g. d. Hydrodynamic dispersion can be ignored.

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1. A uniform sand with rounded particles has a void ratio of 0.63 and a hydraulic conductivity, k, of 2.7 ⫻ 10⫺4 m/s. Estimate the value of k for the same sand at a void ratio of 0.75.

6. How can the effects of incompatibility between chemicals in a waste repository and a compacted clay liner best be minimized?

2. The soil profile at a site that must be dewatered consists of three homogeneous horizontal layers of equal thickness. The value of k for the upper and lower layers is 1 ⫻ 10⫺6 m/s and that of the middle layer is 1 ⫻ 10⫺4 m/s. What is the ratio of the average hydraulic conductivity in the horizontal direction to that in the vertical direction? 3. Consider a zone of undisturbed San Francisco Bay mud free of sand and silt lenses. Comment on the probable effect of disturbance on the hydraulic conductivity, if any. Would this material be expected to be anisotropic with respect to hydraulic conductivity? Why?

4. Assume the specific surface of the San Francisco Bay mud in Question 3 is 50 m2 /g and prepare a plot of the hydraulic conductivity in meters/second as a function of water content over the range of 100 percent decreased to 25 percent by consolidation using the Kozeny–Carman equation. Would you expect the actual variation in hydraulic conductivity as a function of water content to be of this form? Why? Sketch the variation you would expect and explain why it has this form. 5. At a Superfund site a plastic concrete slurry wall was proposed as a vertical containment barrier against escape of liquid wastes and heavily contaminated groundwater. The subsurface conditions consist of horizontally bedded mudstone and siltstone above thick, very low permeability clay shale. The cutoff wall was to extend into the slay shale, which has been shown to be able to serve as a very effective bottom barrier. For the final design and construction, however, a 3-ft-wide gravel trench was used instead of the slurry wall. Sumps and pumps placed in the bottom of the trench are used to collect liquids. Explain how this trench can serve as an effective cutoff and discuss the pros and cons of the two systems.

Copyright © 2005 John Wiley & Sons

8. For the compacted clay waste containment liner shown below and assuming steady-state conditions: a. What is the contaminant transport for pure molecular diffusion? b. What is the contaminant transport rate for pure advection? c. What is the contaminant transport rate for advection plus diffusion? d. Why don’t the answers to parts (a) and (b) add up to (c)?

NOTE: Advection and diffusion are in the same direction; therefore, J ⬎ 0, and the solution will be in the form c ⫽ a1ea2x ⫹ a3

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QUESTIONS AND PROBLEMS

9. One-dimensional flow is occurring by electroosmosis between two electrodes spaced at 3.0 m with a potential drop of 100 V (DC) between them. What should the water flow rate be if the coefficient of electroosmotic permeability, ke, is 5 ⫻ 10⫺9 m2 /s V assuming an open system? If no water is resupplied at the anode, what maximum consolidation pressure should develop at a point midway between electrodes if the hydraulic conductivity of the soil is 1 ⫻ 10⫺8 m/s?

Assume that the water pressure at the top of the leachate collection layer is atmospheric and that the only fluxes across the liner are water and electricity. The characteristics of the compacted clay liner are:

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10. a. A soil has a coefficient of electroosmotic permeability equal to 0.3 ⫻ 10⫺8 m/s per V/m and a hydraulic conductivity of 6 ⫻ 10⫺9 m/s. Starting from the general relationship

321

Hydraulic conductivity

Ji ⫽ Lij Xj

kh ⫽ 1 ⫻ 10⫺7 m/s

Electroosmotic coefficients

derive an expression for the pore water tension that may be developed under ideal conditions for consolidation of the clay by electroosmosis and compute the value that should develop at a point where the voltage is 25 V. Be sure to indicate correct units with your answers. b. In the absence of electrochemical effects or cavitation, would you consider your answer to part (a) to represent an upper or lower bound estimate of the pore water tension? Why? (HINT: Consider the influence of consolidation on the soil properties that are used to predict the pore water tension.) 11. In 1892 Saxen established that there is equivalence between electroosmosis and streaming potential such that the results of a hydraulic conductivity test in which streaming potential is measured can be used to predict the volume flow rate during electroosmosis in terms of the electrical current. Starting with the general equations for coupled electrical and hydraulic flow, derive Saxen’s law. What will be the drainage rate from a soil, in m3 /h amp, if the streaming potential is 25 mV/ atm? What will be the cost of electrical power per cubic meter of water drained if electricity costs $0.10 per kWh and a maximum voltage of 75 V is used? 12. It might be possible to prevent leakage of hazardous and toxic chemicals through waste impoundment and landfill clay or geosynthetic-clay liners by means of an electroosmosis counterflow barrier against hydraulically driven seepage. Consider the impoundment and liner system shown below.

Copyright © 2005 John Wiley & Sons

ke ⫽ 2 ⫻ 10⫺9 m2 /s V

ki ⫽ 0.2 ⫻ 10⫺6 m3 /s amp a. Wire mesh is proposed for use as electrodes. Where would you place the anode and cathode meshes? b. If the waste pond is to be filled to an average depth of 6 m, what voltage drop should be maintained between the electrodes? c. What will the power cost be per hectare of impoundment per year? Power costs $0.09 per kWh. d. Assume that the leachate collection layer is flushed continuously with freshwater and that the liquid waste contains dissolved salts. Write the complete set of equations that would be required to describe all the flows across the liner during electroosmosis. Define all terms. e. Will maintenance of a no hydraulic flow condition ensure that no leachate will escape through the clay liner? Why?

13. a. Estimate the minimum footing depths for structures in a Midwestern city where the freezing index is 750 degree-days and the duration of the freezing index is 100 days. The mean annual air temperature is 50F. The soil is silty clay with a water content of 20 percent and a dry unit weight of 110 lb/ft3. Assume no ice segregation and compare values according to the Stefan and modified Berggren formulas. b. What will be the depth of frost penetration below original ground surface level if a surface heave of 6 inches develops due to ice lens for-

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through the liner as a function of the hydraulic conductivity. Show in the same diagram the proportions of the total that are attributable to diffusion and advection. Assume that the leachate collection layer is fully drained, but for purposes of analysis the fluid level can be considered at the bottom of the clay. Determine the leakage rate through the liner per unit area as a function of the hydraulic conductivity and show it on a diagram. 15. The diagram below shows the cross section of a tunnel and underlying borehole in which waste canisters for spent nuclear fuel are located. Such an arrangement is proposed for deep (e.g., several hundred meters) burial of nuclear waste in crystalline rock. The surrounding rock can be assumed fully saturated, and the groundwater table will be within a few tens of meters of the ground surface. Thermal studies have shown that the temperature of the waste canister will rise to as high as 150C at its surface. A canister life of about 100 years is anticipated using either stainless steel or copper for the material. The surrounding environment must be safe against leakage of radionuclides from the repository for a minimum of 100,000 years.

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mation? Assume a frozen ground temperature of 32F. c. If a pavement is to be placed over the soil, what thickness of granular base course should be used to prevent freezing of the subgrade? The base course will be compacted to a dry density of 125 lb/ft3 at a water content of 15 percent. If the pavement structure is to contain an 8inch-thick Portland cement concrete surface layer, will your result tend to overestimate or underestimate the base thickness required? Why? 14. A compacted fine-grained soil is to be used as a liner for a chemical waste storage area. Free liquid leachate and possibly some heavier than water free phase nonsoluble, nonpolar organic liquids (DNAPLs) may accumulate in some areas as a result of rupturing and corrosion of the drums in which they were stored. Two sources of soil for use in the liner are available. They have the following properties: Property

Soil A

Soil B

Unified class Liquid limit (%) Plastic limit (%) Clay size (%) Silt size (%) Sand size (%) Predominant clay mineral Cation exchange capacity (meg/100 g)

(CH) 90 30 50 30 20 Smectite

(CL) 45 25 30 40 30 Illite

60

20

a. Which of the two soils would be best suited for use in the liner? Why? b. What tests would you use to validate your choice? Why? c. Assume that you have confirmed that it will be possible to compact the soil to states that will have hydraulic conductivities in the range of 1 ⫻ 10⫺8 to 1 ⫻ 10⫺11 m/s. A liner thickness of 0.6 m is proposed. Leachate is likely to accumulate to a depth of 1.0 m above the top of the liner. A leachate collection layer will underlie the liner. d. If the concentration of dissolved salts in the leachate is 1.0 M and the average diffusion coefficient is 5 ⫻ 10⫺10 m2 /s, determine for the steady state the total amount of dissolved chemical per unit area per year that will escape

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QUESTIONS AND PROBLEMS

c. Assess the probable natures and directions of heat and fluid flows that will develop, if any. d. What alterations might occur in the material during the life of the repository if any? Consider the effects of groundwater from the surrounding ground, corrosion of the canister, and the prolonged exposure to high temperature. Would each of these alternations be likely to enhance or impair the effectiveness of the clay pack?

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Clay or a mix of clay with other materials such as sand and crushed rock is proposed for use as the fill both around the canisters and in the tunnel. a. What are the most important properties that the backfill should possess to ensure isolation and buffering of the waste from the outside environment? b. What clay material would you propose for this application and under what conditions would you place it?

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