Assessment And Modelling Of Reservoir Fluids Pvt Today Wadocx

ASSESSMENT AND MODELLING OF RESERVOIR PVT PROPERTIES

By SAMUEL. P. AKOSA 51448593

A dissertation submitted in partial fulfilment of the requirements of the award of Master of Science in Oil & Gas Engineering at the University of Aberdeen (August, 2015)

ABSTRACT Designing the reservoir simulator usually requires formulating mathematical and numerical models. The mathematical models, describe the behavior of the reservoir fluids and rocks with mathematical equations. While the numerical models solve the mathematical equations in discrete cells and then extend their formulations to the whole extent of the reservoir. Nevertheless, in coupling each cell formulation, changes occur in fluid component phases with movement between neighboring cells. These changes tend to impact on computational time during simulation. Currently, the practice of modelling thermodynamic state variables i.e. the relating pressure, volume and temperature (PVT) has become increasingly important in catering for the cell to cell fluid phase changes and complex volumetric behavior of reservoir fluids. These thermodynamic mathematical models usually follow either a black oil or a compositional approach. The black oil handles the hydrocarbon components as two components phases: a pseudo oil phase and a pseudo gas phase. Whereas compositional models honor the pure components of the fluid and make use of an equation of state to describe the phase equilibrium between these pure components. Compositional models tend to account for this phase changes more than the black oil models. However in considering computational time, most engineers prefer the black oil approach This report aims to develop a fluid phase PVT model which will enable the engineer to perform analysis on a reservoir sample in a short time. The proposed tool also aimed at describing the reservoir fluid with a pseudo split approach as opposed to a black oil modelling approach. The outputs from the formulation of the phase model showed that computational time could be reduced by using a reverse black oil modelling approach. The results also showed that the pseudo split approach could considerably function similar to the black oil models and in turn act like a compositional model in accommodating phase changes.

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DEDICATION This thesis is dedicated to the Almighty God who has made it possible for me to successfully complete this phase of life.

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ACKNOWLEDGEMENTS I will like to express my gratitude to my supervisor Dr. Jefferson Gomes, for his patience, guidance, advice, and scrutiny during the thesis period. Moreover, special thanks goes to Petroleum Technology Development Fund (PTDF) for the help they rendered to me in achieving my life goals.

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CONTENTS ABSTRACT ............................................................................................................................ i DEDICATION ....................................................................................................................... ii ACKNOWLEDGEMENTS .................................................................................................. iii LIST OF TABLES ............................................................... Error! Bookmark not defined. NOMENCLETURE ............................................................................................................. vii CHAPTER 1 ........................................................................................................................... 1 1.1 INTRODUCTION ............................................................................................................ 1 1.2 FLUID PVT MODELLING ............................................................................................. 2 1.3 OBJECTIVE: .................................................................................................................... 4 1.4 CHAPTER BY CHAPTER SUMMARY ..................................................................... 5 OBTAINING THE FLUID PVT PROPERTIES AND ANALYSIS METHODS................. 6 2.1

RESERVOIR FLUID PROPERTIES ...................................................................... 6

2.2

PHASE BEHAVIOR OF RESERVOIR FLUIDS .................................................. 6

2.3

PHASE EQUILIBRIA AND FLASH CALCULATIONS ................................... 10

2.4

PVT LABORATORY EXPERIMENTS .............................................................. 12

2.5

FLUID DATA PROPERTIES ............................................................................... 14

2.6

CHAPTER SUMMARY........................................................................................ 16

CHAPTER THREE .............................................................................................................. 17 OVERVIEW OF RESERVOIR FLUID PVT MODELING APPROACHES ..................... 17 3.1 RESERVOIR FLUID PVT MODELS ....................................................................... 17 3.2 BLACK OIL PVT MODELS ..................................................................................... 17 3.2.1 CLASSICAL BLACK OIL PVT MODEL (CBO): ............................................. 18 3.2.2

MODIFIED BLACK OIL MODELS ............................................................. 20

3.3 METHODS IN BLACK OIL PVT MODELING ....................................................... 22 3.4 COMPOSITIONAL PVT MODELS .......................................................................... 23 3.5 BRIEF OVERVIEW OF SOME THERMODYNAMIC MODELS THAT ARE SUITED FOR SPECIFIC PURPOSES IN COMPOSITIONAL RESERVOIR SIMULATION.................................................................................................................. 24 3.5.1 ACTIVITY COEFFICIENT MODELS ............................................................... 25 3.5.2

EQUATION OF STATE ............................................................................... 27

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3.6 OTHER MODELS: THERMAL MODELS ............................................................... 36 3.7 REGRESSION-TUNING AND CHARACTERIZATION ........................................ 36 3.8 CHOOSING A FLUID PVT MODEL ....................................................................... 37 3.9 CHAPTER SUMMARY............................................................................................. 38 4.1 FLUID PVT MODEL ................................................................................................. 39 4.2 MODEL DESCRIPTION ........................................................................................... 39 4.3 MODEL ARRANGEMENT....................................................................................... 39 4.3.1 RECOMBINATION MODULE (RC) ................................................................. 39 4.3.2 FOR FLASH MODULE ...................................................................................... 40 4.3.3

FOR BLACK OIL PVT MODULE: ............................................................. 40

4.4

FLUID MODEL VALIDATION.......................................................................... 41

4.5

MODEL RESULTS ............................................................................................. 41

4.5.2 FLASH MODULE (F-M): ................................................................................... 43 4.5.3 BLACK OIL PVT CALCULATOR MODULE (BC): ........................................ 44 4.6: ANALYSIS AND DISCUSSION ............................................................................. 46 4.7 SOURCES OF POSSIBLE ERRORS ............................................................................ 49 4.8 CHAPTER SUMMARY............................................................................................. 49 5.1 CONCLUSION ........................................................................................................... 50 APPENDIX A ...................................................................................................................... 62 A ALGORITHM FOR THE THREE MODULES ............................................................. 62 (A.1) RECOMBINATION MODULE ............................................................................. 62 (A.2) PSEUDOIZATION MODULE ............................................................................... 64 (A.2) FLASH MODULE .................................................................................................. 65 (A.3) BUBBLE AND DEW POINT PRESSURE CALCULATIONS ............................ 68 APPENDIX B ................................................................................................................... 72 B: PLOTS AND DIAGRAMS ......................................................................................... 72

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LIST OF FIGURES Figure 1. The Sequence of fluid modelling in reservoir simulation………………….…….…2 Fig 1.2 Assessment of reservoir fluid PVT models………………………………………………5 Fig 2.1 Phase diagram of a binary mixture to illustrate the reservoir fluid phase behavior..9 Figure 2..2 : Principle of pressure temperature flash process for a hydrocarbon reservoir fluid mixture. Diagram extracted from [8] ........................................................................... 10 Fig 3..1 Schematic diagram of a Classical Black oil PVT model. Taken from [22] ............ 18 Figure 4.1 phase envelop of fluid mixture frrom F-M module……………………………..….44 Fig 4.2: Black oil model for reservoir wellstream:…………………..………………..….……45 Fig 4.3: Hypothetical black oil model with Gaussian quadrature……………………………46 Fig: B1: Snapshot of Recombination Module from Matlab…………………………………...70 Fig B2 Phase diagram of reservoir mixture from laboratory (Diagram from Multiflash)……………………………………………………………………………………………71

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LIST OF TABLES

Table 2.1 Equations of PVT properties. Adapted from [13]…………………………………...15 Table 3.1: Cubic equation of state with their formula. Table adapted from [55]…………..35 Table 4.2 Range between values From RC Module and Multiflash ..................................... 42 Table 4.3 Comparison from RC module recombination with Multiflash results.................. 42 Table 4.4: Absolute average error from measured lab recombination ................................ 43 Table 4.5: Black oil PVT parameters from PVTP and BC module………………………..45 Table B.3 : Comparative showing the flash of the combined reservoir stream by Multiflash and the F-M module……………………………………………………………………………………………………………..73

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LIST OF SYMBOLS

NOMENCLETURE Meaning and Unit

Bo,

Oil formation volume factor, rbbl/stb

Bg,

Gas formation volume factor rbbl/scf

Bw,

Water formation volume factor rbbl/stb

Rs,

Solution Gas oil ratio, scf/stb

Rp,

Producing gas oil ratio, scf/stb

rs

Condensate or Vaporized oil gas ratio stb/scf

γ

Gas specific gravity

ρ

Density, lbm/ft3

v

molar volume, ft3/mole

V

Volume, ft3

P

Pressure, psia

T

Temperature, oF

G

Gibbs free energy,

f

Fugacity ,psia

R

Universal gas constant, =10.73 lbmft/oR

μ

Viscosity, cp

μ

Chemical potential

U

Internal energy

M

Molecular weight, lb/lbmole

x

Liquid mole fraction, (%)

y

Vapor mole fraction (%)

z

Feed mole fraction (%)

Z

Compressibility factor

𝛿𝛿𝑖𝑖𝑖𝑖

Binary interaction parameter

co

Isothermal compressibility factor, 1/psi

a

Hermolthz energy

OB

Objective function

CC

Calculated observation

𝜔𝜔

Acentric factor

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OBj

Objective function

Rj

Residual observation

nPR

Number of regression parameters.

nOBS

Number of experimental observations

Ω𝑎𝑎 , Ω𝑏𝑏

EOS critical parameters

a, and b

Attractive and repulsive terms

SUBSCRIPTS o

Oil phase

g

Gas phase

w

Water

v

Vapor phase

l

Liquid phase

API

American petroleum Institute

c

critical

air

air

ABBREVIATION SCN

Single Carbon Number

wj

weight Factor

EOS

Equation of state

PVT

Pressure, volume and temperature

SCN

Single Carbon Number

CVD

Constant volume depletion

CCE

Constant composition Expansion

DLE

Differential liberation experiments

BC

Black oil calculator module

RC

Recombination Module

FM

Flash Module

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CHAPTER 1 1.1 INTRODUCTION Nowadays, the reservoir simulator has been used to tackle lots of problems in reservoir engineering. Due to its sophisticated nature, it has been is relied upon to describe the behavior of the reservoir to any given length of time and also to optimize hydrocarbon production using different operating conditions [1]. Moreover, in designing the simulator, formulating mathematical and numerical models are usually required. The mathematical models describe the behavior of the reservoir fluids and rocks with mathematical equations. While the numerical models divide the whole reservoir into small cells (also called grids) in order to accommodate any minute reservoir description such as wells positioning. Also, these numerical models solve the mathematical equations in these cells and then extend their formulations to the whole extent of the reservoir. Nevertheless, in coupling each cell formulation, changes in fluid component phases and movement do occur between neighboring cells. [2]. Since these cell to cell relationship do experience changes resulting from the fluid composition and state, it is important to adequately account for the fluid properties that lead to such changes. [3]. Thus a sound understanding these fluid properties is fundamental to the reservoir simulator design [4]. Presently, the practice of modelling thermodynamic state variables i.e. the relating pressure, volume and temperature (henceforth called PVT) has become successful in catering for the cell to cell fluid phase changes and complex volumetric behavior of reservoir fluids. This modelling approach has gained wide acceptance because it is capable of yielding lots of fluid properties of interest (e.g. Pressure, volume and temperature can be used to derive the isobaric heat capacity: a property that determines how hot a substance gets) the

that are

useful in determining the nature of the fluid at the reservoir as well as the surface when it is produced [5]. Conversely, to accommodate computing time and to reduce the enormous mathematical relations (used to account for the complex behavior of reservoir fluid) in the cells, huge challenges of improper predictions have been encountered. Thus, retarding the accuracy of

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the simulators overall results and its robustness over time. Therefore, there is the need for a PVT model that will accommodate computing time and also predict fluid behavior to a large extent and for different cases.

1.2 FLUID PVT MODELLING In reservoir simulation, fluid modelling involves describing the physical properties of the reservoir fluids, which depend on the reservoir rock, pressure, temperature as well as the fluid composition. Basically, the sequence followed in fluid modeling begins with collecting the samples from the reservoir, analyzing the samples and then developing the mathematical models that describe the thermodynamic behavior of the fluid. After describing the fluids behavior, the next stage is to describe its flow with regards to the reservoir rock as shown in Fig 1 below

FLUID PVT RESERVOIR SIMULATOR FLUID FLOW Figure 1.1 The Sequence of fluid modelling in reservoir simulation

Diagram adapted from [2]. The volumetric properties of the fluid will first be defined and modelled before the fluid flow properties are handled since fluid flow properties depend on the inherent thermodynamic behavior of the fluid. However, this thesis concentrates on the thermodynamic aspect of the fluid model and thus further discussions on the flow stage will be omitted. Moreover, the mathematical models used in describing the thermodynamic behavior of the fluids are classified into two which include: the Black oil PVT models and the Compositional models.

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The black oil PVT model presumes that properties of phases depend only on pressure at a constant temperature. The black oil model divides the hydrocarbon components into two components: a pseudo oil phase and a pseudo gas phase (the pseudo gas phase is analogous to the separator gas and whereas the pseudo oil is to the stock tank oil). Thus the Separator gas is considered a pseudo-component consisting of hydrocarbons in the reservoir that remain in the gas phase at standard conditions. On the other hand, the stock tank oil is the other pseudo-component consisting of hydrocarbons in the reservoir that remains in the oil phase at standard conditions [6]. However, the PVT properties suited for this modelling approach involves those properties that describe the volumetric relationship between the amount of oil and gas phases at reservoir and at surface. These properties include the gas oil ratio 𝑅𝑅𝑠𝑠, the vaporized oil gas ratio 𝑟𝑟𝑠𝑠 , the oil formation volume factor,𝐵𝐵𝑜𝑜 as well as the gas formation volume factor, 𝐵𝐵𝑔𝑔 [7]. Equations 1.1 to 1.4 shows the relationship of these properties to volume. 𝐵𝐵𝑜𝑜 =

𝑉𝑉𝑜𝑜 𝑉𝑉𝑜𝑜�𝑜𝑜

(1.1)

𝐵𝐵𝑔𝑔 =

𝑉𝑉𝑔𝑔 𝑉𝑉𝑔𝑔�𝑔𝑔

(1.2)

𝑅𝑅𝑠𝑠 =

𝑉𝑉𝑔𝑔�𝑜𝑜 𝑉𝑉𝑜𝑜�𝑜𝑜

(1.3)

𝑟𝑟𝑠𝑠 =

𝑉𝑉𝑜𝑜�𝑔𝑔 𝑉𝑉𝑔𝑔�𝑔𝑔

(1.4)

Where, 𝑉𝑉𝑜𝑜 is the volume of reservoir oil, 𝑉𝑉𝑜𝑜�𝑜𝑜 is the volume of stock tank oil from the reservoir

oil, 𝑉𝑉𝑔𝑔 is the volume of reservoir gas, 𝑉𝑉𝑔𝑔�𝑔𝑔 is the volume of surface gas from reservoir oil, 𝑉𝑉𝑜𝑜�𝑜𝑜 3

is the volume of surface oil from reservoir oil, 𝑉𝑉𝑔𝑔�𝑜𝑜 is the volume of surface gas in reservoir

oil (it is usually the same as 𝑉𝑉𝑔𝑔�𝑔𝑔 ) .

However, if the oil and gas composition starts varying strongly such as in the case of volatile oil reservoir, retrograde gas condensate reservoir, gas injection, solution-gas, and gas-cap reservoirs drive studies. The large number of components will need an equation of state which can provide consistent densities, compositions, and molar volumes, hence the need for a full compositional model. The Compositional model therefore refers to models that make use of an equation of state to describe the phase equilibrium behavior based on its pure components. Equilibrium phase splits and phase properties are determined by blending the properties of the stream constituents. The equation of state (henceforth called EOS) is used to predict the vapor liquid equilibrium (VLE), and associated thermodynamic properties such as gas and liquid enthalpies, gas and liquid densities, gas and liquid viscosities, surface tension and thermal properties. Furthermore, it is impractical to model every component of a reservoir fluid due to the large numbers of components that are present [8]. Generally, for acceptable phase behavior prediction, it is sufficient to specify the mole fractions of the main light end hydrocarbons (typically from methane to decane for black oils). Heavier components are lumped together and handled as pseudo or hypothetical components [9]. Nevertheless, computational considerations and cost can favor companies to choose the black oil model for their simulation studies. On the other hand, it is highly advisable to derive the properties of black oil using the equation of state for interpretation of the results of the analysis to ensure proper understanding and identification of the quality of PVT data obtained. [10] Most modern reservoir flow simulators are usually written with a general compositional formulation whereas the black oil PVT properties are converted internally to a two component compositional model.

1.3 OBJECTIVE: The aim of this report is to describe the reservoir fluid PVT properties and how they are formulated in the simulator design. This description will include the methods used in analyzing fluid properties and their modeling approaches. Furthermore, at the end of the

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project, a fluid phase model will be designed to predict the phase behavior of a gas condensate fluid by combining the black oil and compositional modeling approach.

1.4 CHAPTER BY CHAPTER SUMMARY: This report provides an assessment of the whole process involved in developing a fluid PVT model to be used in the reservoir simulator. The flow of ideas is broadly classified into two sections which include Assessment of PVT properties and their Modeling approaches (Chapters two and three), and then coding of a fluid PVT property model (Chapter four). The introductory chapter gives a brief description of the fluid PVT modeling process as related to reservoir simulation while also stating the aims of the report. Chapter two discusses the methods of analyzing these fluid properties which serves as inputs to the fluid models. Moreover, chapter three discusses the description of the fluid PVT models. The next chapter shows the coding of a black oil PVT model to predict phase behavior and fluid properties. Then Chapter five summarizes the report. A schematic of the whole report process is shown in Fig 1.2. The chapters are arranged in such a way to show the sequence in modeling the fluid properties are developed ASSESSMENT AND MODELING OF RESERVOIR FLUIDS

Chapter one Introducti on Review of fluid models

Chapter two

Chapter three

Nature of reservoir fluid

Fluid Pvt Model’s types

Overview of Phase equilibria and Pvt Analysis

Black oil Pvt Model Compositional

Fluid Properties

Project objectives

Chapter four Overview of Phase behavior pvt model Model validation Critical analysis of

Fig 1.2 Assessment of reservoir fluid PVT models

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Chapter five Conclusion and Recommend ation

CHAPTER TWO OBTAINING THE FLUID PVT PROPERTIES AND ANALYSIS METHODS 2.1 RESERVOIR FLUID PROPERTIES Gaining proper insight of the reservoirs fluid nature and its behavior usually requires obtaining and analyzing representative samples of the reservoir. In order to ensure reliability of results from the fluid models, it is important to understand the results from the analyses that was carried out on the reservoir samples. This chapter will describe the methods used to analyze the fluid samples and how they affect the fluid modelling process. Moreover as a background overview, it will also discuss the phase behavior of the reservoir fluid which affects the analyses performed on the fluid. 2.2 PHASE BEHAVIOR OF RESERVOIR FLUIDS In thermodynamic studies, a phase refers to a continuous homogenous portion of a system which is physically distinct from other parts by definite boundaries. Thus, a reservoir consists of liquid phases i.e. both oil and water phases as well as a vapor or gaseous phase. In some reservoirs the gaseous phase tends to be dissolve in the oil but later presents itself during depletion [11]. Moreover, the state of equilibrium that is where no state changes occur with time is important to the concept of phase behavior at a particular temperature and pressure. This equilibrium is usually attained at minimum Gibbs [12] energy and when the chemical potentials of each component in the phases are equal [13]. Equations 2.1 to 2.8 describe how this energy is related to phase behavior

𝜇𝜇𝑖𝑖 =

𝜕𝜕𝜕𝜕 𝜕𝜕𝑛𝑛𝑖𝑖

(2.1)

Where μi is the chemical potential of component-i, G is the Gibbs free energy and ni is the number of moles of component-i. Thus for a system with two phases the condition for equilibrium is 𝜇𝜇𝑖𝑖(𝑥𝑥) = 𝜇𝜇𝑖𝑖(𝑦𝑦) 6

(2.2)

,

𝑃𝑃𝑖𝑖(𝑥𝑥) = 𝑃𝑃𝑖𝑖(𝑦𝑦)

(2.3)

𝑇𝑇𝑖𝑖(𝑥𝑥) = 𝑇𝑇𝑖𝑖(𝑦𝑦)

(2.4)

Where, P is the pressure, T is the temperature, x and y represents the phases. Chemical potentials are usually expressed in terms of fugacity, fi, 𝑑𝑑𝑑𝑑𝑖𝑖 = 𝑅𝑅𝑅𝑅 ln 𝑓𝑓𝑖𝑖

(2.5)

It is readily shown that equation (2.2) is readily satisfied by the equal-fugacity constraint, 𝑓𝑓𝑖𝑖𝑣𝑣 = 𝑓𝑓𝑖𝑖𝐿𝐿

(2.6)

Where, 𝑓𝑓𝑖𝑖𝐿𝐿 is the fugacity of component-i in the liquid phase and 𝑓𝑓𝑖𝑖𝑣𝑣 is the fugacity of component-i in the vapor phase.

Fugacity is a property of a real substance that is a measure of the tendency of the substance to prefer one phase at a particular temperature and pressure. It has same dimensions as pressure but is related to the partial pressure of a component in a mixture. The ratio of the fugacity to the pressure is referred to as the fugacity coefficient, φ and is given by 𝑓𝑓𝑖𝑖 = 𝜑𝜑𝑖𝑖 𝑥𝑥𝑖𝑖 𝑝𝑝

(2.7)

Where, xip =partial pressure of the component in the mixture, fi, p, x and φi are the fugacity, pressure, phase component molar composition and partial fugacity coefficient of component-i respectively. The fugacity coefficient of a component in a mixture is also related to temperature and volume by the expression

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∞

fi 1 ∂P RT ln φi = ln = �� - � dV- ln Z xi p RT ∂ni V

(2.8)

V

Where R,T,V ,ni, and Z are the universal gas constant, temperature, volume , number of moles of each component and the compressibility factor of component- i Nevertheless, the phase with the lowest fugacity coefficient will thermodynamically be the most favorable i.e. the phase with minimum Gibbs energy. Gibbs free energy can be related to fugacity as shown in equations (2.9) and (2.10) 𝑛𝑛

𝐺𝐺𝑦𝑦∗ = � 𝑦𝑦𝑖𝑖 ln 𝑓𝑓𝑖𝑖𝑣𝑣 𝑖𝑖=1 𝑛𝑛

𝐺𝐺𝑥𝑥∗ = � 𝑥𝑥𝑖𝑖 ln 𝑓𝑓𝑖𝑖𝑙𝑙 𝑖𝑖 𝑖𝑖=1

(2.9)

(2.10)

Where, 𝐺𝐺𝑦𝑦∗ 𝑎𝑎𝑎𝑎𝑎𝑎 𝐺𝐺𝑥𝑥∗ are the normalized Gibbs free energies in the vapor and liquid phases

respectively. Most compositional PVT models rely of the concept of equal fugacities of the phase components and minimum Gibbs energy to determine phase equilibrium state of

multicomponent mixtures. In order to describe the percentage of a component in a phase at a particular pressure and temperature a Pressure-Temperature phase diagram could be used [14] . Fig 2.1 shows a typical pressure temperature phase diagram with characteristic features used in describing fluids. The phase diagram is bounded by the bubble point and dew point curves with the two curves meeting at the critical point (C). The critical point is where all differences between the two phases seize and the phases become indistinguishable. Furthermore, since critical properties are just properties of individual components in a mixture, for fluid modeling the pseudo critical values (e.g. pseudo critical pressure and temperature) for the mixture are used. The pseudo critical value of a mixture is derived from mixing the critical values of individual components by using a mixing rule. A widely used

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mixing rule is the Kay’s [15] mixing rule which is based on molar averaging of the components. Mathematically the Kay’s rule is 𝑛𝑛

.𝑝𝑝 𝜃𝜃𝑐𝑐 = � 𝑧𝑧𝑖𝑖 𝜃𝜃𝑐𝑐𝑐𝑐

(2.11)

𝑖𝑖

Where zi is the mole fraction, .pθc is any pseudo critical property such as temperature, pressure or volume and θci is the critical component of component i and n is the number of components.

Pressure

B C Critica l point D TWO PHASE

Temperature Fig 2.1 Phase diagram of a binary mixture to illustrate the reservoir fluid phase behavior Diagram taken from [14]. The maximum pressure (B) on the phase diagram is called the Cricondenbar and refers to the pressure beyond which no gas can be formed regardless of the temperature. While the maximum temperature (D) beyond which no liquid can be formed regardless of the pressure is the Cricondentherm [14].

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2.3 PHASE EQUILIBRIA AND FLASH CALCULATIONS To determine the phase equilibrium, calculations that estimate the percentage or molar amount of compositions of the mixtures that are vaporized or condensed is paramount. These percentages can also help in determining the phase diagram by calculating the bubble point and the dew point curves. Thus also yielding the temperatures and pressures at which the mixture begins to vaporize of condense [16]. Mixtures molar amounts can be obtained by flashing the mixture at a particular temperature and pressure in a flash separator. Figure 2.2 gives an illustration of a pressure-temperature flash process for a hydrocarbon mixture. If the pressure , temperature and mole fractions in the feed (zi = z1,z2,z3…zn) are known, a flash calculation can provide the number of phases; molar amounts of each phase e.g. FV ,FL and also the individual component molar compositions for each phase i.e. for the vapor phases (y1,y2,y3…..yn) and the liquid phase(x1,x2,x3….xn). Moreover , referring to equation (2.6) and (2.7) at equilibrium

(y1,y2,….yn) FV Feed, 1mole (z1, z2….zn)

Gas

T, P (1-Fv)

Oil (x1, x2,…xn) Figure 2.2 : Principle of pressure temperature flash process for a hydrocarbon reservoir fluid mixture. Diagram extracted from [8] A feed of 1 mole (with mole fractions z1, z2…. zn) entering a separator is flashed at the temperature of the separator, T and pressure P. The flashing yields two phases at equilibrium with the phases being a gaseous (vapor) phase with Fv moles ( containing y1, y2 ….yn mole fractions) and liquid phase with (1-Fv) moles (containing x1, x2,…xn mole fractions),

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𝜑𝜑𝑖𝑖𝐿𝐿 𝑦𝑦𝑖𝑖 = 𝜑𝜑𝑖𝑖𝑉𝑉 𝑥𝑥𝑖𝑖

(2.12)

For i = 1,2,….n. where n is the total number of components The above relation is equal to the equilibrium constant K for each component, hence

𝐾𝐾𝑖𝑖 =

𝜑𝜑𝑖𝑖𝐿𝐿 𝑦𝑦𝑖𝑖 = 𝜑𝜑𝑖𝑖𝑉𝑉 𝑥𝑥𝑖𝑖

(2.13)

Also performing a mass balance on the diagram, Feed F=1 𝐹𝐹𝑣𝑣 + 𝐹𝐹𝐿𝐿 = 1

𝑧𝑧𝑖𝑖 = 𝐹𝐹𝑣𝑣 𝑦𝑦𝑖𝑖 + (1 − 𝐹𝐹𝑣𝑣 )𝑥𝑥𝑖𝑖

(2.14)

(2.15)

Thus, relating zi and Fv to K yields and rearranging

𝑦𝑦𝑖𝑖 = 𝑥𝑥𝑖𝑖 =

𝑧𝑧𝑖𝑖 𝐾𝐾𝑖𝑖 1 + 𝐹𝐹𝑣𝑣 (𝐾𝐾𝑖𝑖 − 1) 𝑧𝑧𝑖𝑖 1 + 𝐹𝐹𝑣𝑣 (𝐾𝐾𝑖𝑖 − 1)

(2.16)

(2.17)

For i =1, 2…n. The sum of the mole fractions of the phases and the entire mixture must all be equal to one or a hundred in percentage form. Thus

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𝑛𝑛

𝑛𝑛

𝑛𝑛

𝑖𝑖=1

𝑖𝑖=1

𝑖𝑖=1

� 𝑧𝑧𝑖𝑖 = � 𝑦𝑦𝑖𝑖 = � 𝑥𝑥𝑖𝑖 = 1.0

(2.18)

This implies that 𝑛𝑛

𝑛𝑛

𝑖𝑖=1

𝑖𝑖=1

� 𝑦𝑦𝑖𝑖 − � 𝑥𝑥𝑖𝑖 = 0

(2.19)

Thus substituting equations 2.16 and 2.17 for yi and xi into the above equation yields the Rachford-Rice equation [13] 𝑛𝑛

ℎ(𝐹𝐹𝑣𝑣 ) = � 𝑖𝑖=1

𝑧𝑧𝑖𝑖 (𝐾𝐾𝑖𝑖 − 1) =0 1 + 𝐹𝐹𝑉𝑉 (𝐾𝐾𝑖𝑖 − 1)

(2.20)

This could be solved to obtain the vapor mole fraction Fv using an iteration scheme. However a stability analysis is usually performed before commencing a flash calculation because it determines the number of phases present. Michelsen (1982) [17] proposed a consistent phase stability check that is based on the minimization of the Gibbs free energy by splitting off a vapor phase [8]. Currently most compositional PVT models usually depend on stability checks and Rachford equation to perform vapor liquid equilibrium calculations. Nevertheless, thorough details of the stability analysis are beyond the scope of this report but an overview of stability criteria is shown in Appendix A.2

2.4 PVT LABORATORY EXPERIMENTS After collecting representative fluid samples (samples that maintain same quality with the reservoir fluids) of the reservoir, the next step involves measuring the fluid properties by analyzing the PVT properties of the sample [18]. Generally, the samples are usually transferred to a laboratory for its analysis but it can also be analyzed On-site. However, to ensure accuracy and reliability, an integration of the both methods is more acceptable as a suitable means [19]. Accurate sampling and analysis 12

techniques often provide critical input to reservoir simulation models and help to optimize processing facility designs while boosting the profitability of an oil or gas field [20]. Laboratory experiments performed determine the phase behavior and physical properties of the fluid. The lab experiments are broadly classified into the following groups [13] namely (i) Constant Composition Expansion (CCE) (ii) Constant Volume Depletion test (CVD) (iii) Fluid composition (iv) Separator test (multistage) (v) Differential liberation test (DLE) (vi) Viscosity measurements (vii) Swelling test The constant Volume depletion (CVD) experiment is generally used for retrograde condensate fluid samples to measure the composition and specific gravity of the fluid. It could provide useful data for the mathematical models depicting the changes in a condensate reservoir. Also Fluid composition test involves measuring the vapor and liquid phase compositions (mole percent and specific gravity [13]) of the components of the sample using a gas chromatograph [21]. The compositions can also be used for phase calculations in the mathematical thermodynamic models.[11] Constant composition expansion or flash expansion test involves placing a known volume of an equilibrated single phase fluid in a windowed PVT cell and monitoring the fluid change with reduction in pressure isothermally which eventually leads to the production of two phases under agitation. It usually consists of stages of equilibration, whereby the cell volume is increased until the next predetermined pressure level is obtained with no gas or liquid removed at any point. This test is used to determine the bubble point pressure [13, 21] Another important laboratory test is the differential liberation or vaporization test. It usually involves equilibration of the reservoir fluid sample in a PVT cell at the bubble point pressure and at its reservoir temperature. The pressure inside the cell is usually reduced by increasing the volume, thus enabling a gas phase to form. Agitation is also employed to equilibrate this gas with the liquid. The gas is then displaced isobarically by reducing the volume of the cell slowly. Parameters that could be measured from this test include the specific gravity, volume of the expelled gas and volume of the liquid remaining in the cell. 13

Other properties like Gas and oil compressibility factors, density of the remaining oil, solution gas-oil ratio, gas and oil formation volume factors could be obtained from this test as calculated parameters. [13] In the separator or flash vaporization test, a known volume of the fluid sample at bubble point and reservoir temperature is charged into the PVT cell. It is displaced at bubble point pressure via two or more stages of separation. Similar equilibration process to the differential liberation test is used however the pressures and temperatures of the laboratory separators are selected to be closed to expected field conditions. The gas oil ratio and formation volume factor at separator could be measured by this test.[10, 13]. Other laboratory tests include swelling or extraction test and viscosity tests are also important in proper characterizing of the fluids. The swell test is carried out to investigate the reaction of the reservoir fluid to gas injection.[8] It is paramount in measuring phase behavior and it is used to determine the reservoir fluid volume -compositions changes with regards to the injected fluid (e.g. CO2) dissolution at reservoir temperature. While the Oil viscosity test are performed by using a capillary viscometer or an electromagnetic viscometer. Measurements are usually done at same pressure levels just as in differential liberation experiments with samples deficient of gas at each pressure level. [13] Alternatively, PVT analysis could be performed on Well-site rather than taking samples to the laboratory as this could save time and ensure early warnings of problematic chemicals like hydrogen sulphide and mercury. This test could also be used to: check levels of oil-based mud contamination in liquids and to analyze of amounts of rare gases present in hydrocarbon gas. A current example of this technique is the Schlumberger’s PVT Express which uses less than 50cm3 volume of a sample to conduct full PVT study.[18]

2.5 FLUID DATA PROPERTIES To handle fluid problems with reservoir simulation, the required PVT properties needed include the following: Specific gravity, molecular weight, density, viscosity, formation volume factors (FVF) for oil and gas, solution gas oil ratios, and compressibility. Table 2.2 shows some important PVT properties that are critical in defining a reservoir fluid. The formulae presented in the tables are just the basis of how such properties are obtained.

14

Table 2.1 Equations of PVT properties. Adapted from [13] PVT

TYPE

SYMBOL

PROPERTY Specific gravity

BASE EQUATION FOR FORMULATION

Oil

γo

Gas

γg

𝛾𝛾𝑜𝑜 =

𝛾𝛾𝑔𝑔 =

Solution gas oil Condensate Rs

Formation

Water

Bw

Oil

Bo

Gas

Bg

Oil

ρo

Gas

ρg

Gas

Cg

Oil

μo

Gas

μg

𝐵𝐵𝑤𝑤 =

volume factor (FVF)

Density

Isothermal

𝐵𝐵𝑜𝑜 =

𝐵𝐵𝑔𝑔 = 𝜌𝜌𝑜𝑜 = 𝜌𝜌𝑔𝑔 = 𝐶𝐶𝑔𝑔 =

Compressibility Viscosity

𝜌𝜌𝑔𝑔� 𝜌𝜌𝑎𝑎𝑎𝑎𝑎𝑎 �����

𝑅𝑅𝑠𝑠 =

ratio

𝜌𝜌𝑜𝑜� 𝜌𝜌𝑤𝑤� 𝑉𝑉𝑔𝑔� 𝑉𝑉𝑜𝑜�

𝑉𝑉𝑤𝑤 𝑉𝑉𝑤𝑤� 𝑉𝑉𝑜𝑜 𝑉𝑉𝑜𝑜�

𝑉𝑉𝑔𝑔 𝑉𝑉𝑔𝑔�

𝑀𝑀𝑜𝑜 𝑉𝑉𝑜𝑜

𝑀𝑀𝑔𝑔 𝑉𝑉𝑔𝑔

1 𝜕𝜕𝜌𝜌𝑔𝑔 � � 𝜌𝜌𝑔𝑔 𝜕𝜕𝜕𝜕 𝑇𝑇

𝜇𝜇𝑜𝑜 = 𝑓𝑓(𝑇𝑇, 𝛾𝛾𝐴𝐴𝐴𝐴𝐴𝐴 ) 𝜇𝜇𝑔𝑔 = 𝑓𝑓(𝑇𝑇, 𝑀𝑀𝑔𝑔)

Where subscripts o, g and w represent reservoir conditions while 𝑜𝑜̅ , 𝑔𝑔̅ , 𝑎𝑎𝑎𝑎𝑎𝑎 ���� and 𝑤𝑤 � represent

surface conditions. Also subscript T is the isotherm temperature and 𝛾𝛾𝐴𝐴𝐴𝐴𝐴𝐴 is the API gravity and is closely related to the specific gravity of oil.

15

2.6 CHAPTER SUMMARY The following are the points what noting from this chapter (i)

Minimizing the Gibbs free energy determines the equilibrium of the phases. At this equilibrium the fugacity's of the individual phases are equal

(ii)

The phase of a mixture is determined by its composition, pressure and temperature

(iii)

PVT laboratory analysis are important deducing the input parameters (fluid properties) needed for the fluid mathematical modeling stage. Integration of on-site analysis and laboratory analysis could yields good result.

16

CHAPTER THREE OVERVIEW OF RESERVOIR FLUID PVT MODELING APPROACHES 3.1 RESERVOIR FLUID PVT MODELS After measuring the composition of the fluid in the laboratory, the next step is to model the volumetric behavior of the hydrocarbons fluids in the reservoir. This step involves representing the physical properties of the fluid by consistent mathematical formulations that describes the samples behavior close to the reservoir condition. The approaches taken in fluid PVT modeling can be broadly classified into two which include (a) Black oil PVT models (b) Compositional PVT models Moreover, it is imperative to note that the chosen approach strongly affects the kind of model that will be used in the fluid flow modeling stage of the simulator

.

3.2 BLACK OIL PVT MODELS Generally, black oil PVT models are models that describe the volumetric behavior of reservoir fluids by using phase’s properties which depend only on pressure at a constant temperature. These properties that depend on pressure include: solution gas oil ratio Rs, reservoir oil and gas formation volume factors Bo and Bg and surface oil and gas gravities 𝛾𝛾𝑜𝑜 and 𝛾𝛾𝑔𝑔 respectively as shown in equation (3.1) They are used to calculate the densities and

relative amounts of oil and gas phases at reservoir conditions . [22] �𝐵𝐵𝑜𝑜 , 𝐵𝐵𝑔𝑔 , 𝑅𝑅𝑠𝑠 , 𝛾𝛾𝑜𝑜 , 𝛾𝛾𝑔𝑔 � = 𝑓𝑓(𝑃𝑃)

(3.1)

Where, subscript o and g refers to oil and gas phase respectively. Moreover, black oil models divides the reservoir hydrocarbons into binary components: a pseudo oil phase and a pseudo gas phase (the pseudo gas is analogous to the separator gas and whereas the pseudo oil is to the stock tank oil). Thus the Separator gas is considered a pseudo-component consisting of hydrocarbons in the reservoir that remain in the gas phase

17

at standard conditions. While the stock tank oil is the other pseudo-component consisting of hydrocarbons in the reservoir that remains in the oil phase at standard conditions [6]. Presently, two approaches have been used in black oil PVT modeling and they include the following [7] (a) Classical or standard black oil model(CBO) (b) Modified black oil models (MBO)

3.2.1 CLASSICAL BLACK OIL PVT MODEL (CBO): This approach assumes that there is no oil component in the gas phase, thus a reservoir gas at surface will remain the same without any liquid (oil) condensing out. Fig 3.1 shows diagrammatically how the reservoir gas remains similar to the gas at the surface. Since the gas never contains any liquid, the model can be referred to as a dry gas black oil model [23]. The dry gas approach means that the solution oil-gas ratio rs is approximately zero, thus no oil comes out of the gas phase. Reservoir condition

Surface condition

G a s

2

2

1

3- Hydrocarbon surface gas

Vapor

2

Liquid 2

2- Hydrocarbon surface gas

Gas

1 Oil

1-Nonvolatile surface oil

Fig 3.1 Schematic diagram of a Classical Black oil PVT model. Taken from [22] Initially at reservoir conditions there is equilibrium between the liquid and the vapor phases. As they are produced to the surface (e.g. at the separator) there is a split into separate phases whereby the gas coming out at surface components contains no oil, thus rs=0. However, the 18

lowermost box in the diagram is analogous to the stock tank where some hydrocarbon gas still comes of the oil at surface contains some gas leaving at the bottom a non-volatile oil. Thus only a non-volatile oil and a hydrocarbon gas is present at the surface in a classical black oil model. Moreover, to compare the changes in volumetric amounts of the phases at surface and at reservoir conditions, three properties of the fluids are used to capture the changes. This properties are the Solution gas oil ratio, Rs; the gas formation volume factor Bg and the oil formation volume factor Bo and are described by equation (3.2) to (3.4) 𝐵𝐵𝑜𝑜 =

𝑉𝑉𝑜𝑜 𝑉𝑉𝑜𝑜�𝑜𝑜

(3.2)

𝐵𝐵𝑔𝑔 =

𝑉𝑉𝑔𝑔 𝑉𝑉𝑔𝑔�𝑔𝑔

(3.3)

𝑅𝑅𝑠𝑠 =

𝑉𝑉𝑔𝑔�𝑜𝑜 𝑉𝑉𝑜𝑜�𝑜𝑜

(3.4)

Where, 𝑉𝑉𝑜𝑜 is the volume of reservoir oil, 𝑉𝑉𝑜𝑜�𝑜𝑜 is the volume of stock tank oil from the reservoir oil, 𝑉𝑉𝑔𝑔 is the volume of reservoir gas, 𝑉𝑉𝑔𝑔�𝑔𝑔 is the volume of surface gas from reservoir oil, 𝑉𝑉𝑜𝑜�𝑜𝑜

is the volume of surface oil from reservoir oil, 𝑉𝑉𝑔𝑔�𝑜𝑜 is the volume of surface gas in reservoir oil (it is usually the same as𝑉𝑉𝑔𝑔�𝑔𝑔 , cause it contains similar elements as shown in the diagram).

Equation (3.2) and (3, 3) refers to the shrinkage of the oil and gas volume (from the reservoir) at the surface respectively, while Equation (3.3) gives the volumetric ratio of the oil and gas at the surface. According to Whitson [7], the assumptions used in formulating the standard black oil model include the following (i) Reservoir oil consists of two surface “components,” stock-tank oil and surface (total separator) gas. (ii) Reservoir gas does not yield liquids when brought to the surface. (iii) Surface gas released from the reservoir oil has the same properties as the reservoir gas. (iv) Properties of stock-tank oil and surface gas do not change during depletion of a reservoir. 19

Furthermore, as stated earlier, these three parameters (Bo, Bg and Rs) are functions of pressure and can be obtained from correlations mentioned in chapter two, whenever PVT laboratory data is not sufficient. Classical black oil formulations are suited for modelling: (a) Reservoir oil with Rsi < 750 SCF/STB. , where Rsi is initial solution gas oil ratio (b) Reservoir oil with productions higher than its bubble point pressure (e.g., strong waterdrive, gas-cap-drive, or water-flooded reservoirs). [7]

3.2.2

MODIFIED BLACK OIL MODELS:

These models are referred to as wet gas models, because they honor the vaporized oil that condenses out of the gas phase. In this approach, new PVT properties are added to the classical formulations in order to account for the vaporized oil or dropped out liquid [7] and they include the solution oil gas ratio, rs and the dry gas formation volume factor Bgd. A useful explanation of the modified black oil formulation is captured by Fig 3.2 [23]

Separator Gas in gas phase Stock tank oil vaporized into gas phase Separator gas dissolved in oil phase Stock tank oil in oil phase

𝑉𝑉𝑔𝑔�𝑔𝑔, 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

Gas phase, Vg (STB)

𝑉𝑉𝑜𝑜�𝑔𝑔 , 𝑆𝑆𝑆𝑆𝑆𝑆 𝑉𝑉𝑔𝑔�𝑜𝑜 , 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀

Oil phase, Vo (STB)

𝑉𝑉𝑜𝑜�𝑜𝑜 , STB

Fig 3.2 Diagram of Modified black oil PVT models. Taken from [23].

The “-“sign on the subscripts indicates surface conditions, while the subscripts g and o represent gas and oil phases. For the subscripts with two alphabets, the first letter in the

20

subscript is the current state of the substance while the second letter shows the origin of the substance. From the diagram above, the volume of the reservoir gas (Vg) is split into a separator gas in gaseous phase (Vgg) with a condensate or a vaporized oil dropping out of the gas phase (Vog). Whereas the reservoir oil volume (Vo) is also separated into a separator gas (Vgo) and an oil phase (Voo). Moreover comparing Fig 3.2 to Fig 3.1, the new term added is the Vog term, while the terms Vgo and Vgg are considered the same i.e. the hydrocarbon gas as in Fig 3.1. Thus, 𝑉𝑉𝑔𝑔�𝑔𝑔 = 𝑉𝑉𝑔𝑔�𝑜𝑜

(21)

Moreover, introducing the volumetric properties of interest that indicate the ratio of the reservoir to stock tank volumes, PVT parameters used here usually the Rs, Bo, rs, Bgd , where[24] 𝐵𝐵𝑔𝑔𝑔𝑔 = 𝑟𝑟𝑠𝑠 =

𝑉𝑉𝑔𝑔 𝑉𝑉𝑔𝑔�𝑔𝑔

(226)

𝑉𝑉𝑜𝑜�𝑔𝑔 𝑉𝑉𝑔𝑔�𝑔𝑔

(3.7)

𝑉𝑉𝑜𝑜�𝑔𝑔, is the stock tank oil produced from reservoir gas and 𝑉𝑉𝑔𝑔�𝑔𝑔 is the volume of surface

gas from reservoir gas. Moreover 𝐵𝐵𝑔𝑔𝑔𝑔 is related to 𝐵𝐵𝑔𝑔 (wet gas formation volume factor) by the relation

𝐵𝐵𝑔𝑔𝑔𝑔 = 𝐵𝐵𝑔𝑔 × [1 + 𝐶𝐶𝑜𝑜�𝑜𝑜 𝑟𝑟𝑠𝑠 ]

(3.8)

Where, in equation (3.8) 𝐶𝐶𝑜𝑜�𝑜𝑜 = 133,000

𝛾𝛾𝑜𝑜�𝑔𝑔 𝑀𝑀𝑜𝑜�𝑔𝑔

(3.9)

Where, 𝑉𝑉𝑜𝑜�𝑜𝑜 is the stock tank oil from reservoir oil, 𝑀𝑀𝑜𝑜�𝑔𝑔 molecular weight and 𝛾𝛾𝑜𝑜�𝑔𝑔 is the

stock tank oil specific gravity. rs is the term that accounts for the condensate that drops out of

21

the separator gas. The Rs and the Bo terms still maintain same forms as in Equations (3.2) and (3.4) If the value of rs =0, then the value of Bgd = Bg which then becomes the classical black oil formulation. The modified approach is suitable for modeling all types of reservoirs especially volatile oils and retrograde condensate reservoirs cause the solution oil gas ratio term helps to account for liquid drop out [25]and retrograde liquid in the gas stream.[7]

3.3 METHODS IN BLACK OIL PVT MODELING Presently, many methods have been proposed for modeling black oil properties but the Walsh-Towler [26] (1995) method, Coats [27] (1985) method and the Whitson-Torps (1983) [28]methods have gained wide importance over the years. Walsh-Towler suggested a procedure using material balance to calculate the PVT properties. Their method did not use equations of state to predict the fluid properties instead it made use of measured quantities from CVD [26]. Also, Fekete [23] black oil model (2003) made modifications to the WalshTower method by honoring consistency checks from other experiments like the CCE test in calculating the black oil properties.[23]. Conversely, Coats method and Whitson method made use of modified (pseudoized) forms of equation of states to model the fluids. Further details of all the methods mentioned above could be viewed from reference 26 to 28. Considering Whitsons [22] method, a reservoir fluid e.g. a condensate with components ranging from N2, CO2, C1 to C7+, is transformed into three components whereby C5, C6 and C7+ fractions are lumped together as a pseudo component analogous to the non-volatile oils due to their high molecular weight. Then a second group consisting of C2, C3 and C4 are considered as a second pseudo component which would account for the vaporized oil in the gas phase. While the last group will contain hydrocarbon gas consisting of + N2, CO2 and C1. (Note the divisions used here are not exactly how Whitson [22] did in his work, but are illustrated to capture the overall concept. C1, C2 …C7+, refers to hydrocarbons with their carbon number starting from methane). Referring to section 2.3, assigning mole percent’s to the above pseudo components will give values such as y1, y2, y3 and x1, x2 and x3 for the vapor and liquid phase of the three 22

components respectively. Thus for a black oil model, the component 1-2 are referred as the oil pseudo phase while component 2-3 is the pseudo gas phase. Moreover, comparing these mole fractions with respect to black oil PVT properties, the following relationships are relevant:

𝑅𝑅𝑠𝑠 =

𝑉𝑉𝑔𝑔�𝑜𝑜 𝑥𝑥3 = 2130 𝑉𝑉𝑜𝑜�𝑜𝑜 𝑥𝑥2 𝑣𝑣2 + 𝑥𝑥1 𝑣𝑣𝑖𝑖

𝑟𝑟𝑠𝑠 =

𝑉𝑉𝑜𝑜�𝑔𝑔 𝑦𝑦2 𝑣𝑣2 = 𝑉𝑉𝑔𝑔�𝑔𝑔 2130(𝑦𝑦3 + 𝑦𝑦4 )

𝑍𝑍𝑣𝑣 𝑇𝑇�𝑃𝑃 𝑉𝑉𝑔𝑔 𝐵𝐵𝑔𝑔𝑔𝑔 = = 0.02827 𝑉𝑉𝑔𝑔�𝑔𝑔 (𝑦𝑦3 + 𝑦𝑦4 ) 𝑉𝑉𝑜𝑜 𝑣𝑣1 𝐵𝐵𝑜𝑜 = = 𝑉𝑉𝑜𝑜�𝑜𝑜 𝑥𝑥1 𝑣𝑣1 + 𝑥𝑥2 𝑣𝑣2

(23.10) (3.1124) (3.1225) (3.1326)

Where v is the molar volume, P T and 𝑍𝑍𝑣𝑣 are the pressure, the temperature and the gas

compressibility factor respectively. Therefore if the molar volume of a phase is known as well as the gas compressibility, at a given temperature and pressure, the black oil PVT properties could be estimated. Also, flash calculations might be needed to obtain individual components (i.e. methane, ethane etc.) of vapor and liquid phase compositions. Then mixing rules are applied to combine them into black oil pseudo forms. (All elements in the above equation are in field units. Thus the constants 2130 and 0.0287 are conversion factors from standard unis to field units). Appendix A contains details of a black oil model approach.

3.4 COMPOSITIONAL PVT MODELS When oil consists of two or more hydrocarbons and those hydrocarbons exhibit distinctly different phase and composition changes relative to temperature and pressure, a more complex definition of the fluid behavior is required for reservoir simulation and predictions of recovery. For example, retrograde gas condensate reservoirs as well as volatile oils and other gas reservoir systems (e.g. gas cap, solution gas and gas injection) usually exhibit such variations between their oil and gas compositions. Therefore, the large number of components in these systems will need a strong formulation which can provide consistent densities, compositions, and molar volumes. Typically, an equation of state is dully suited for such a purpose because it is able to account for the fluids phases, the phase equilibria of 23

the phases and also the thermodynamic properties of the compositions of the phases. The approach to formulating the fluids properties by totally relying on an equation of state to model the components and thus mix them up to yield a single result for the total mixture is what is referred to as compositional modeling [29]. Here properties of phases are a function of not only pressure as in black oil models but also of their composition This approach usually considers the critical properties of individual pure components present in a fluid mixture and their overall effect on the system. These properties are entered for each component, allowing the liquid and vapor phases to be broken down into their constituent components. �

𝐾𝐾𝑖𝑖 = 𝜌𝜌 𝜇𝜇

𝑦𝑦𝑖𝑖 𝑥𝑥𝑖𝑖

� = 𝑓𝑓(𝑃𝑃, 𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 )

(3.14)

For instance, the percentage of butane present in the gas phase and liquid phase could be determined. Equilibrium phase splits and phase properties are determined by blending the properties of the stream constituents. Moreover, it is impractical to model every component of a reservoir fluid due to the large numbers of components that are present. Generally, for acceptable phase behavior prediction, it is sufficient to specify the mole fractions of the main light end paraffin’s, (typically methane to decane for black oils) [8]. Heavier components are handled as pseudo or hypothetical components (a process referred to as pseudoization) [9]. Nevertheless, such a fluid model is more computationally expensive to solve but allows the hydrocarbons of different compositions to mix during the simulation [2]

3.5 BRIEF OVERVIEW OF SOME THERMODYNAMIC MODELS THAT ARE SUITED FOR SPECIFIC PURPOSES IN COMPOSITIONAL RESERVOIR SIMULATION Since the predictions from a compositional PVT model rely on the type of thermodynamic model used will thus dictate the nature of the result from such formulations. Presently, there are many types of thermodynamic models that could be used in reservoir simulators to predict

24

physical properties of the reservoir fluids. These models have been formulated created by many authors and they can be classified broadly into the following (a) Activity coefficient models (b) Equations of state 3.5.1 ACTIVITY COEFFICIENT MODELS An activity coefficient is a dimensionless factor used in thermodynamics to account for deviations of ideal behavior in a mixture of chemical substances [30]. The activity coefficient can be related to the chemical potential for a substance B in a liquid or solid mixture containing mole fractions xB and xc of substances B, C by the relation below RTln(𝑥𝑥𝐵𝐵 𝛾𝛾𝐵𝐵 ) = 𝜇𝜇𝐵𝐵 (𝑇𝑇, 𝑃𝑃, 𝑥𝑥) − 𝜇𝜇𝐵𝐵∗ (𝑇𝑇, 𝑝𝑝)

(3.15)

Where μB is the chemical potential of substance B, x denotes the set of mole fractions xB, xC , … and

T and P represents the temperature and pressure of the system at that point.[30]

As models, they are the simplest possible models for the composition dependence of the phase i-component fugacity fi, representing a standard to which actual behavior may be compared. Liquid activity methods are based on the following equation for the fugacity of the ith- component in the mixture. Mathematically, 𝛾𝛾𝑖𝑖 =

𝑓𝑓𝑖𝑖𝑙𝑙 𝑥𝑥𝑖𝑖 𝑓𝑓𝑖𝑖𝑜𝑜

(3.16)

Where, 𝛾𝛾𝑖𝑖 is the activity coefficient of component i. xi is the liquid phase mole fraction and

𝑓𝑓𝑖𝑖𝑜𝑜 is the fugacity of the pure component which is usually approximated to the vapor pressure

of the pure component, 𝑃𝑃𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠

Thus an activity coefficient of a component can be related to the equilibrium constant by the relation 𝛾𝛾𝑖𝑖 = 𝐾𝐾𝑖𝑖

𝑃𝑃 𝑦𝑦𝑖𝑖 𝑃𝑃 𝑦𝑦𝑖𝑖 𝑃𝑃 𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑥𝑥𝑖𝑖 𝑓𝑓𝑖𝑖𝑜𝑜 𝑃𝑃𝑖𝑖 𝑥𝑥𝑖𝑖 𝑃𝑃𝑖𝑖 25

(3.17)

For i=1, 2….n, where n is the total number of components. Ideal solutions i.e. solutions where liquid fugacity of each component is directly proportional to the mole fraction of each component with activity coefficient equals to unity, assumes that all molecules interact with the same intermolecular potential but this assumption is only reasonable for molecules of similar size and type. Most real mixtures tend to deviate from ideality, thus their activity coefficient is different from unity. Such models are can handle vapor –liquid systems and also perform well in simulating asphaltenes, waxes and other solid systems cause they can be used to handle the solubility. These models also allow the use of binary interaction parameters to alter the predictions until values are obtained similar to experimental data. A few examples of these models include Andersen and Speight [31], Chung[32], Yarranton and Masliyah[33], Zhou et al [34]models, predictive UNIQUAC [35][36], predictive Wilson [37], predictive UNIFAC [38] and regular solution theory [39]. In general, the approaches taken in determining the parameters of the activity coefficient model include: (i) determining the parameters by fitting to experimental vapor liquid equilibrium data on binary mixtures at a single temperature. E.g. the Predictive Wilsons model (ii) Determining the parameters by treating the excess Gibbs free energy e.g. the predictive UNIFAC model. The former is unable to handle liquid-liquid equilibria or vapor liquid- liquid equilibria while the latter is capable of performing such equilibria scenarios. Nevertheless, Al Ghafri [40], showed that the limitations associated with these models are evident around the critical regions. He proposed that the limitation comes from the fact that the activity coefficients are defined based on the mixing of liquid components at conditions of temperature equal to that of the mixture, and so the pure components must be liquid at those conditions. Also, since these coefficients are not written as explicit functions of temperature, pressure and composition they do not have the flexibility of equations of state to calculate densities and other derivative properties.

26

3.5.2 EQUATION OF STATE An equation of state refers to an analytical thermodynamic expression that describes the state of matter by relating state variables i.e. pressure, volume and temperature. An example of an equation of state is the ideal gas equation, which shows how volume and pressure are related to temperature. 𝑃𝑃𝑃𝑃 = 𝑛𝑛𝑛𝑛𝑛𝑛

(3.18)

Where R is the universal gas constant and n is the number of moles. However in reality most gases never obey this arrangement (for pressure and volume against temperature), instead they experience lots of deviations from the ideal. Due to this, different equations of state where developed to capture the real permutation of pressure and volume against temperature. Originally, the equations formed where basically created for describing phase behavior for single pure components, but they have been extended to handle mixtures [41]. The extension of EOS to multicomponent systems required the use of mixing rules (as stated in chapter two) to determine the mixture parameters [42] Currently there are three basic formulations of equations of state used to determine volumetric properties and vapor liquid equilibrium with respect to hydrocarbon system and they include the following: (a) Cubic Equations of state (b) Higher order equations of state (not discussed in this report) (c) Perturbed chain and statistical associating fluid theory equations of state In this report, only the cubic equations of state and their statistical neighbors will be explained here. Although higher order equations of state might be used in parts of the petroleum system like pipelines, they will not be discussed because most current commercial reservoir simulators hardly use them in PVT modeling.

3.5.2.1 CUBIC EQUATIONS OF STATE (CEOS): These refer to equations of state that where originally developed to account for the nonideality of real gases. This non ideal behavior was denoted by a deviation factor, Z (also referred to as the compressibility) and on solution of this factor, three (cubic) real roots are usually obtained hence the name cubic EOS.

27

The first person to propose such a relation was Van der Waals (Vdw) [43]. He proposed the fact that the total pressure of a substance is equal to the difference between the repulsive pressure element and the attractive pressure element as shown below 𝑃𝑃𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑃𝑃𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑃𝑃𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑃𝑃 =

𝑅𝑅𝑅𝑅 𝑎𝑎 − 2 (𝑉𝑉 − 𝑏𝑏) 𝑉𝑉

(3.19)

(3.20)

Where b is the effective molar volume to correct for volume occupied by the molecules, ‘a’ is the attraction parameter to correct for pressures for attraction of molecules. Although his relation proved superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. Other modern equations have been developed and have thus replaced the VdW EOS successfully. Table 3.1 shows the basic formulas for some of these cubic equations of state. In general, for a compositional simulation, two important equations are paramount which include the compressibility and the fugacity equation. 𝑍𝑍 = 𝑍𝑍(𝑃𝑃, 𝑇𝑇, 𝑥𝑥) 𝑓𝑓 = 𝑓𝑓(𝑃𝑃, 𝑇𝑇, 𝑥𝑥)

(3.21) (3.22)

Martin [44] proposed that all cubic equation could be written in the following generalized way 𝑍𝑍 3 + [(𝑚𝑚1 + 𝑚𝑚2 − 1)𝐵𝐵 − 1]𝑍𝑍 2 + [𝐴𝐴 + 𝑚𝑚1 𝑚𝑚2 𝐵𝐵 2 − (𝑚𝑚1 + 𝑚𝑚2 )𝐵𝐵(𝐵𝐵 + 1)]𝑍𝑍 −[𝐴𝐴𝐴𝐴 + 𝑚𝑚1 𝑚𝑚2 𝐵𝐵 2 (𝐵𝐵 + 1)] = 0

28

(3.23)

𝑓𝑓𝑖𝑖 ln 𝜑𝜑𝑖𝑖 = ln � � 𝑃𝑃𝑃𝑃𝑖𝑖

= − ln(𝑍𝑍 − 𝐵𝐵) + +

2 ∑𝑛𝑛𝑗𝑗=1 𝐴𝐴𝑖𝑖𝑖𝑖 𝑥𝑥𝑗𝑗 𝐵𝐵𝑖𝑖 𝐴𝐴 𝑍𝑍 + 𝑚𝑚2 𝐵𝐵 � − � ln (𝑚𝑚1 − 𝑚𝑚2 )𝐵𝐵 𝑧𝑧 + 𝑚𝑚1 𝐵𝐵 𝐴𝐴 𝐵𝐵

(3.24)

𝐵𝐵𝑖𝑖 𝐵𝐵(𝑍𝑍 − 1)

Where, 𝑛𝑛

𝑛𝑛

𝐴𝐴 = � � 𝑥𝑥𝑗𝑗 𝑥𝑥𝑘𝑘 𝐴𝐴𝑗𝑗𝑗𝑗

(3.25)

𝑗𝑗=1 𝑘𝑘=1 𝑛𝑛

𝐵𝐵 = � 𝑥𝑥𝑗𝑗 𝐵𝐵𝑗𝑗

(3.26)

𝑗𝑗=1

𝐴𝐴𝑗𝑗𝑗𝑗 = �1 − 𝛿𝛿𝑗𝑗𝑗𝑗 ��𝐴𝐴𝑗𝑗 𝐴𝐴𝑘𝑘 �

0.5

(3.27)

𝐵𝐵𝑗𝑗 = Ω𝑏𝑏𝑏𝑏

𝑃𝑃𝑟𝑟𝑟𝑟 𝑇𝑇𝑟𝑟𝑟𝑟

(3.28)

𝐴𝐴𝑗𝑗 = Ω𝑎𝑎𝑎𝑎

𝑃𝑃𝑟𝑟𝑟𝑟 2 𝑇𝑇𝑟𝑟𝑟𝑟

(3.279)

In Equation (3.21), 𝛿𝛿𝑗𝑗𝑗𝑗 are the binary interaction coefficients (BICS), symmetric in j and k

with 𝛿𝛿𝑗𝑗𝑗𝑗 = 0. For two identical components e.g. N2-N2, the 𝛿𝛿𝑗𝑗𝑗𝑗 is zero. For two different nonpolar compounds (e.g. CO2-N2) 𝛿𝛿𝑗𝑗𝑗𝑗 is equal to or close to zero. But for a binary pair of at

least one polar component e.g. (N2-CH4), non- zero 𝛿𝛿𝑗𝑗𝑗𝑗 is appropriate [8]. The subscripts j, k

indicate components in the mixture while r represent reduced values. For Redlich-Kwong (RK) [45], and. Soave-Redlich- Kwong (SRK) [46]. For the PR EOS, 𝑚𝑚1 = 1 + √2 and 29

𝑚𝑚2 = 1 − √2. In describing all the parameters used in the above equation, the Peng

Robinson (PREOS) will be considered. The parameters Ω𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 Ω𝑏𝑏𝑏𝑏

represent the

dimensionless constants (particular to an equation of state) and are universal constants independent of component identity and temperature. However, in practice they are treated as functions of temperature and component identity. For PR EOS they are defined by the following relations,

Ω𝑏𝑏𝑏𝑏 = Ω0𝑏𝑏 Ω𝑎𝑎𝑎𝑎 = Ω0𝑎𝑎 �1 + (0.37464 + 1.5422𝜔𝜔𝑖𝑖 − 0.26992𝜔𝜔𝑖𝑖2 ) �1 − 𝑇𝑇𝑟𝑟𝑟𝑟0.5 ��

(28)

2

(3.3129)

Where in equation 3.31 above Ω0𝑎𝑎 = 0.457235529 and Ω0𝑏𝑏 = 0.077796074

30

(30.32)

Table 3.1: Cubic equation of state with their formula. Table adapted from [55]

Equation of state

Model

Van der Waals(VDW EOS)

𝑃𝑃 =

Redlich Kwong[45] (RK EOS)

𝑃𝑃 =

Soave [46] RK(SRK EOS)

Peng Robinson [47] (PR EOS)

Usdin and McAullife (UM EOS) [48]

Heyen EOS[50]

Patel and Teja (PT EOS) [53]

EOS)[54]

and

Wenzel

𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 − (𝑉𝑉 − 𝑏𝑏) [𝑉𝑉(𝑉𝑉 + 𝑏𝑏)]

𝑃𝑃 =

𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 − (𝑉𝑉 − 𝑏𝑏) [𝑉𝑉(𝑉𝑉 + 𝑑𝑑)]

𝑅𝑅𝑅𝑅 𝑎𝑎 − (𝑉𝑉 − 𝑏𝑏) [𝑉𝑉(𝑉𝑉 + 𝑏𝑏)𝑇𝑇 0.5 ]

𝑅𝑅𝑅𝑅 𝑎𝑎 − 2 (𝑉𝑉 − 𝑏𝑏) (𝑉𝑉 + (𝑏𝑏 + 𝑐𝑐)𝑉𝑉 − 𝑏𝑏𝑏𝑏) 𝑃𝑃 =

Adachi and Lu (AL EOS) [52]

Schmidt

𝑃𝑃 =

𝑃𝑃 = 𝑃𝑃 =

Kubic EOS [51]

𝑅𝑅𝑅𝑅 𝑎𝑎 − 𝑉𝑉 − 𝑏𝑏 √𝑇𝑇𝑇𝑇(𝑉𝑉 + 𝑏𝑏)

𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 − (𝑉𝑉 − 𝑏𝑏) [𝑉𝑉(𝑉𝑉 + 𝑏𝑏) + 𝑏𝑏(𝑉𝑉 − 𝑏𝑏)]

𝑃𝑃 =

Stein-Redlich-Kwong [49] EOS

(SW

𝑃𝑃 = 𝑃𝑃 =

𝑅𝑅𝑅𝑅 𝑎𝑎 − 2 (𝑉𝑉 − 𝑏𝑏) 𝑉𝑉

𝑅𝑅𝑅𝑅 𝑎𝑎 − (𝑉𝑉 − 𝑏𝑏) (𝑉𝑉 + 𝑐𝑐)2

𝑃𝑃 =

𝑅𝑅𝑅𝑅 𝑎𝑎 − 2 (𝑉𝑉 − 𝑏𝑏) 𝑉𝑉

𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 − (𝑉𝑉 − 𝑏𝑏) [𝑉𝑉(𝑉𝑉 + 𝑏𝑏) + 𝑐𝑐(𝑉𝑉 − 𝑏𝑏)]

𝑅𝑅𝑅𝑅 𝑎𝑎𝑎𝑎 − 2 (𝑉𝑉 − 𝑏𝑏) [𝑉𝑉 + (1 + 3𝜔𝜔)𝑏𝑏𝑏𝑏 − 3𝜔𝜔𝑏𝑏 2

. All the equations of state are arranged like the Van der Waals equation or similar to equation (3.20). The term 𝜔𝜔 is the acentric factor. The acentric factor is a measure of the sphericity of a molecule and has it values usually between zero and one (spherical molecules have acentric

factor of one) [56]. It is important to note that these equations of state where created for single 31

components but can be extended to multicomponent mixtures by using mixing rules as stated in chapter two. The SRK and PR equations have gained wide acceptance over the years in fluid modeling but PR is more accurate than the former in regions around the critical point. Nevertheless they both do not accurately account for the liquid density. To handle this problem (liquid density prediction), Martins introduced the concept of volume translation to improve the volumetric prediction. Peneloux et al [57] used the volume translation or shift to improve the SRK EOS volumetric predictions while Jhaveri and Youngren [58] extended Peneloux work to the PR EOS. Presently in the reservoir simulation, PR and the SRK and their modifications have been widely accepted for used in fluid PVT modeling because they yield good results over a wide range of conditions. However, Ahmed [55] performed suitable comparisons on eight cubic equations and he outlined their performance capabilities in his work. He concluded that SWEOS exhibits suitable predictive capability for gas condensate systems, PT and SW equations are found to predict reliable gas compressibility factors while the PR, PT and SW equations yield good Vapor liquid equilibrium predictions.

3.5.2.2 PERTURBED CHAIN AND STATISTICAL ASSOCIATING FLUID THEORY EQUATIONS OF STATE: Simple molecules with prevalent intermolecular forces being those of attraction and repulsion (e.g. the Van der Waals force and weak electrostatic forces) can be modelled successfully by means of a cubic equation of state. Most hydrocarbons fall within this category, thus making the wide use of cubic equations of state for their modelling. However, mixtures containing strong intermolecular forces like hydrogen bonds, complex electrolytes, polar solvents and polymers cannot be accurately described by both cubic equations of state and even activity coefficient models. The non-ideality in such systems do require more advanced associating fluid theories to handle them [40]. These theories usually account for the chemical reactions that forms distinct associating chemical species.

32

Associating fluid theories are be broadly classified into two types which include: integral equations models and perturbation models. The integral models considers molecular associations as strong symmetrically spherical attractive forces. On the other hand, the perturbation theories consider a reference fluid with known thermodynamic properties and relates it to an unknown fluid by mean of a correction or perturbation term.(Moreover, this report will be restricted to just the perturbation models and as such integral models will not be discussed further) . Statistical perturbation models are based on the principle that the potential energy of a relative complex molecular fluid can be described as the sum of the potential energy of a simple known reference fluid (e.g. a hard sphere term) and a correction term [59]. Once this correction term exists as a function of pressure temperature or density and composition then other thermodynamic relations could be derived for the physical and derivative properties [59]. Previously, the hard sphere system usually was adopted as the reference term but it was later considered not reliable because it was did not regard chain length (a key feature that affects the fluid structure). Thus, it is inappropriate for modelling mixtures with highly non-spherical associating terms. A more appropriate reference term that considers both chain length and molecular attraction was then sorted after. Wertheim [60] and Chapman [61] proposed a method whereby molecules are treated as different species with respect to the number of associating sites that are bonded together. The latter also extended the principle to modelling molecules as chains of spherical segments (divided monomers) that interact through evenly dispersive and repulsive forces. This proved to be the basis of the statistic associating fluid theories (SAFT). SAFT models are commonly written as summations of the residual Helmholtz free energy terms that occur due to differences in molecular interactions of pure components in the system [59]. They usually employ the Helmholtz energy term because a differentiation of it could yield properties of interest such as pressure, isothermal compressibility coefficient, isobaric heat capacity and so on [62]. For a system consisting of associated chains, the SAFT equations can be expressed as

33

Where,

𝐴𝐴𝑟𝑟𝑟𝑟𝑟𝑟 𝑎𝑎𝑟𝑟𝑟𝑟𝑟𝑟 (𝑇𝑇, 𝜌𝜌) 𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐴𝐴𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎 𝐴𝐴𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = = + + + 𝑁𝑁𝑘𝑘𝐵𝐵 𝑇𝑇 𝑘𝑘𝐵𝐵 𝑇𝑇 𝑁𝑁𝑘𝑘𝐵𝐵 𝑇𝑇 𝑁𝑁𝑘𝑘𝐵𝐵 𝑇𝑇 𝑁𝑁𝑁𝑁𝐵𝐵 𝑇𝑇 𝑁𝑁𝑘𝑘𝐵𝐵 𝑇𝑇

(3.33)

N is the number of molecules and 𝑘𝑘𝐵𝐵 is the Boltzmann’s constant. “a” is the Helmholtz free energy per mole and the superscripts res, ideal, mono, chain and assoc refer to residual, ideal, monomer reference fluid (e.g. hard sphere), chain dispersion and association respectively, 𝐴𝐴𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 is the ideal free energy of the fluid to which three residual contribution are added: the monomeric contribution (𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) due to the repulsion-dispersion segment-segment

interactions; the contribution due to the chain formation (𝐴𝐴𝑐𝑐ℎ𝑎𝑎𝑎𝑎𝑎𝑎 ) and the contribution that takes into account the short range intermolecular association (𝐴𝐴𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 ).

Different possible choices for the reference fluid has given rise to lot of SAFT versions. Examples of these versions include: the Statistical Associating Fluid theory (SAFT) model developed by Wertheim [60, 63, 64] , Huang-Radosz (HR- SAFT also called CK-SAFT) [65], SAFT-VR (Variable Range) [66] , simplified SAFT [67], SAFT-LJ (Lennard-Jones) [61] and the Perturbed Chain statistical Associating fluid theory (PC-SAFT) model developed by Gross and Sadowski [68, 69]. In the next lines details of PC-SAFT model would be discussed since it is a modification of the SAFT model. PC-SAFT usually employs a hard sphere reference system as opposed to a monomer reference fluid. The PC-SAFT model can be expressed in terms of compressibility factor, Z as 𝑍𝑍 = 1 + 𝑍𝑍 ℎ𝑐𝑐 + 𝑍𝑍 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑍𝑍 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

(3.34)

Where superscripts hc is the contribution hard chain part, assoc and disp give the respective contribution from the association and dispersion terms. The contribution from the ideal term is 1, thus 𝑍𝑍 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 1. Moreover for two pure components parameters with association sites

𝐴𝐴𝑖𝑖 and𝐵𝐵𝑗𝑗 , the characteristic energy ∈ , determining their association interactions is given by ∈𝐴𝐴𝑖𝑖 𝐵𝐵𝑗𝑗 =

1 𝐴𝐴 𝐵𝐵 (∈ 𝑖𝑖 𝑖𝑖 +∈𝐴𝐴𝑗𝑗𝐵𝐵𝑗𝑗 ) 2 34

(3.35)

While the association volume k is

𝑘𝑘 𝐴𝐴𝑖𝑖 𝐵𝐵𝑗𝑗 = �𝑘𝑘 𝐴𝐴𝑖𝑖 𝐵𝐵𝑖𝑖 𝑘𝑘 𝐴𝐴𝑗𝑗𝐵𝐵𝑗𝑗 � Where 𝜖𝜖

𝐴𝐴𝑖𝑖 𝐵𝐵𝑖𝑖� 𝑘𝑘

�𝜎𝜎𝑖𝑖𝑖𝑖 𝜎𝜎𝑗𝑗𝑗𝑗

1� �𝜎𝜎 + 𝜎𝜎 � 𝑖𝑖𝑖𝑖 2 𝑗𝑗𝑗𝑗

�

9

(3.36)

and 𝑘𝑘 𝐴𝐴𝑖𝑖 𝐵𝐵𝑖𝑖 are the effective association energy and effective association

volume. 𝜎𝜎 is the monomeric segment. Equation (3.35) and (3.36) shows the relationship between the characteristic energy, the monomeric segment, and the association volume. According to Senol [62] , perturbation in PC-SAFT applies to connected chains for the hard sphere segment rather between disconnected segments. This approach is similar to considering attractive (dispersion) interactions between connected segments which could really represent the behavior of hydrocarbons and polymers in solution. The PC-SAFT equation of state model is capable of predicting thermo- physical properties and also vaporliquid equilibriums for virtually different types of mixtures ranging from associated to nonassociating components as well as others mentioned above. However, for a system to be modeled with it, the segment number, diameter and energy of the pure component must be estimated and validated for a certain range of pressure and temperature. [62]

Al Ghafri [40] outlined generalized requirements for SAFT models and they are (1) SAFT models requires a minimum of two parameters, the characteristic energy, ∈ and the monomeric segment 𝜎𝜎

(2) A third parameter for the number of segments per molecule, m is required to describe the non-sphericity of the fluids. (3) For associating fluids, two additional parameters must be assigned to characterize both the association energy and the volume available for bonding. (4) For each additional specie the associating sites and their bonding correspondence (site which to bond with) must be defined. (5) Parameters for new systems can be estimated from those of previously modelled systems. (6) Mixing rules are needed in the dispersion term while combining rules are needed for the

35

segment energy and volume parameters where a correcting interacting parameter 𝑘𝑘𝑖𝑖𝑖𝑖 is often

used similar to binary interaction parameter in CEOS.

3.6 OTHER MODELS: THERMAL MODELS A Thermal PVT model refers to a special kind of compositional model developed to accommodate temperature variation [70]. It is suited for system involving non isothermal temperature such as heavy oil recovery, water flooding, steam, air and other chemical injection. Not only do they allow temperature variation, they also account for pressure and phase changes effects during simulation [29]. The EOS used in this models are usually tuned with distillation or other separation data. This is imperative because additional pressure volume temperature requirements like density, viscosity and even K-values have dependence on the temperature.[71]. Further details of thermal models can be viewed in reference (73). 3.7 REGRESSION-TUNING AND CHARACTERIZATION It is important to note that all equations of state, activity coefficient models, black oil models and even thermal models are not overly accurate formulas and thus work needs to be done in aligning them to suit laboratory observed data.[70] To achieve this, regression is applied to match the models parameters to closely represent the experimental data in such a way that deviations between the measured and simulated data are eliminated [8] . Typically in a reservoir simulation, a non- linear regression model with a Chi-squared distribution test might be employed to ensure a good fit. Parameters of compositional models that render themselves suitable for regression include the critical parameters (i.e. Critical pressure, critical Temperature, Critical Volume), Acentric Factors, omega A and B (in cubic equations of state) Binary interaction coefficient and Volume shift factors. While for black oil models, their correlations are the ones suited for regression [70]. . Conversely from experimental data parameters such as saturation points, gas phase compressibility factors, liquid drop outcurves or liquid phase densities at reservoir temperature as well as gas oil ratios are usually the data aimed at matching. Christensen [8] proposed a procedure for regression capable of obtaining match to PVT data properties. It is also necessary to note that the number of regression parameters must not exceed the 36

number of data points or observations. Thus, for a given number of n experimental observations, 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛

2

𝑟𝑟𝑗𝑗 𝑂𝑂𝑂𝑂 = � � � 𝑤𝑤𝑗𝑗 𝑟𝑟𝑗𝑗 =

(3.37)

𝑗𝑗=1

𝑂𝑂𝑂𝑂𝑂𝑂𝑗𝑗 − 𝐶𝐶𝐶𝐶𝑗𝑗 𝑂𝑂𝑂𝑂𝑂𝑂𝑗𝑗

𝑛𝑛𝑛𝑛𝑛𝑛 = 1 + ln(𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛)

(31) (3.39)

Where OB is the objective function to be minimized during regression calculation, OBS is the experimental observations used in the regression, CC is the calculated observation, wj is the weight factor for the j- observation, rj is the residual observation of the jth term and nPR is the number of regression parameters [8]. Moreover, in laboratory analysis of reservoir fluids, heavy hydrocarbons are lumped together + ). Nevertheless, using this plus fraction and classified as a plus fraction, 𝐶𝐶𝑛𝑛+ e.g. (𝐶𝐶7+ 𝑜𝑜𝑜𝑜 𝐶𝐶10

in analysis could lead to inaccuracies in predicting PVT properties because only few data such as molecular weight and specific gravity of these pseudo fractions are usually reported. A true boiling point analysis for single carbon number (SCN) groups by distillation usually annihilates such problems but since it is rarely available in most PVT reports other techniques had to be developed. Amongst others, two modern techniques have been widely used to perform this split. The first, proposed by Whitson [72, 73] and the later by Behrens and Sadler [74]. The former uses a probability density function to split up the plus fractions and regroup them into multiple carbon number groups (MCN) with various averaging techniques. While the latter employed semi-continous thermodynamic approach and has gained wide acceptance in the petroleum industry. Continous thermodynamics theory is based on representing a fluid mixture with identifiable, discrete components while also using a continous distribution function to represent the remaining components in the mixture.[9] 3.8 CHOOSING A FLUID PVT MODEL The decision of using a black oil PVT model over a compositional model or vice versa might prove to be worthwhile or even turn out to be detrimental. However, Schlumberger’s PVTi [70] had outlined some consideration that will aid the decision making process and full details 37

will be given in reference. Moreover, Fevang et al [75] proposed some guidelines in for choosing black oils models or compositional models in volatile oils and retrograde condensate systems. (Further details of these guidelines could be viewed from reference 66)

3.9 CHAPTER SUMMARY There fluid modeling process in reservoir simulation might be approached by using either a black oil formulation or a fully compositional model. Black oil models tend to approximate the physical properties of the fluid as functions of only pressure, while the compositional models honor the fluid compositions as well as the pressure in defining the fluids physical properties. Regression-tuning and characterizing of the hydrocarbon heavy end is also necessary in ensure quality results from the simulator.

38

CHAPTER FOUR 4.1 FLUID PVT MODEL The major task here is to develop a phase behavior PVT model. The model was developed by assembling all the ideas in the previous chapters to model PVT properties for reservoir simulation. The design of the model is suited for gas condensate and volatile reservoirs with sufficient data from a flash separation test.

4.2 MODEL DESCRIPTION The proposed fluid PVT model has been developed using Matlab software. The workflow begins with obtaining classical black-oil PVT properties (GOR, specific gravity and molecular weight), which is based on standard PVT laboratory results such as chromatograph separation analysis. Then combining the data from the separation and describing the fluid behavior with an equation of state. After that, black oil PVT properties are then calculated from the model and other test scenarios could be considered. Moreover the designed routines for the model are classified into three modules which include (1)

A Recombination module (RC)

(2)

A Flash Module (FM) and

(3)

A Black oil PVT properties calculator (BC) module

In a nutshell, the recombination module accepts inputs such as composition from a separation test (which are usually split into gas and liquid compositional streams) and gas oil ratio of separator and stock tank, and then combines them to form an original reservoir stream. The flash module then performs vapor- liquid equilibrium calculations on the recombined stream. Then the black oil calculator calculates black oil properties 4.3 MODEL ARRANGEMENT 4.3.1 RECOMBINATION MODULE (RC) (a) Input parameters: The input parameters for the recombination module include the following: Separation gas and liquid compositions, Specific gravities of heavy fractions, surface Gas gravity (separator and stock tank) , API gravity of tock tank oil, GOR of separator and stock tank oil, Molecular weight of heavy fractions, reservoir and surface(both separator and stock tank) pressures and temperature.

39

(b) Output parameters: Specific gravity of oil, average molecular weight gas molecular weight, total molecular weight of reservoir stream, total reservoir specific gas gravity, vapor and liquid phases of reservoir well stream. Elmabrouk’s [76] bubble pressure, Standings [77] bubble pressure, and vapor mole fraction of reservoir fluid and well stream or recombined composition of the reservoir fluid (i.e. zi as used in chapter two). Also the recombination module is responsible for splitting the heavy fractions of the fluid mixture to three to five pseudo fractions. It uses Whitson method stated in chapter three to perform this operation. Outputs from here include the critical properties (acentric factor, critical temperature, pressure and volume) of the pseudo fractions, the molecular weights and the mole fractions of each fraction.

4.3.2 FOR FLASH MODULE: (a) Input parameters: Input data here consists of vapor mole fraction (same as Fv in chapter 2) from recombination modules, Elmabrouk’s bubble pressure, Nemeth’s dew point pressure, reference pressure, temperature, recombined composition, critical properties, binary interaction coefficients (estimated from Chueh-Prausnitz [78] correlation for hydrocarbon interactions and from Whitson’s [7] Monograph for non-hydrocarbon systems) and. PengRobinson[47] Equation-of-State (PR EOS, 1976) is used to calculate Pressure-VolumeTemperature (PVT) relationship of the reservoir fluid (b) Output parameters: The output from flash subroutine consists of number of phases, bubble point pressure, dew point pressure, molar fraction, composition, molecular weight, compressibility factor, density, fugacity, density and viscosity of each fluid phase.

4.3.3 FOR BLACK OIL PVT MODULE: (a) Input parameters: Data to be inputted here involve pressure values to compute the black oil values, output and input values from the recombination module, temperature range. (b) Output parameters include: Elmabrouk bubble pressure, Standings bubble pressure, Glaso bubble pressure, Petrosky and Farshad [79] formation volume factors for oil, Beggs[80] viscosity for oil and gas, Glaso [81] GOR. (For proper details of each models algorithm see Appendix A.4).

40

4.4 FLUID MODEL VALIDATION To validate the model, a gas condensate sample from Gulf of Mexico, USA M-4/M-4A reservoir taken from reference 9 [9] was used. Results gotten from the recombination module (RM) and flash modules (FM) where compared with Multiflash Infochem results. Multiflash infochem (2013) is a recent commercial PVT modelling software created by Infochem Computer services ltd. Multiflash is a fully integrated toolkit with multiple equations of state (ranging from cubic, higher order to statistical equations of state). This feature enabled it to be adopted by lots of oil companies adopted it for PVT modeling. However, it does not pay much attention to black oil modelling therefore another thermodynamic package was chosen for comparing the output from the black oil model. PVTP (2013) fluid thermodynamics package (created by Petroleum Experts) was chosen for this purpose, since it honors black oil as well as compositional modelling approaches. It is imperative to note that PVTP has only two equations of state (SRK and PR equation of state without their variations) and does not show vapor liquid equilibrium splits, thus is not considered suitable for comparing the flash module. 4.5 MODEL RESULTS This section shows the results from each module tabulated alongside the results from commercial software and also the laboratory results 4.5.1 RECOMBINATION MODULE (RC) (a) Results from a Heptane fraction split

Table 4.1 Heptane plus Characterization (pseudo Fraction split) PSEUDO FRACTION Pc (pa) Tc (K) MW ω Tb (K) RECOMBINATION MODULE (RC) PSEUDO FRACTION PS-1 2982327.26 573.44 106.47 0.30 386.58 PS-2 2005615.95 688.33 171.85 0.48 500.25 PS-3 1227059.95 863.00 310.92 0.83 688.39 -------------------------MULTIFLASH FLASH PSEUDO FRACTION --------------PS-1 3010000.00 593.73 114.44 0.38 408.18 PS-2 2540000.00 656.04 149.00 0.46 465.25 PS-3 2380000.00 677.02 163.00 0.50 486.32

41

The table above shows a list of the critical values for the pseudo fractions pseudo fractions PS-1, PS-2, and PS-3 pseudo fractions for a heptane characterization with Gaussian quadrature (Whitsons [7] method). Where Pc- critical pressure in psi, Tc is the critical temperature in (K), MW is the molecular weight, ω is the acentric factor while Tb is the true boiling temperature (K). Table 4.1 Range between values From RC Module and Multiflash Range (Pc) 27672.739 534384.055 1152940.047 27672.739 571665.6137

Range (Tc) 20.2855097 32.2933884 185.979137 20.2855097 79.51934503

Range(MW) 7.97 22.85 147.92 7.97 59.58

Range(Ac) 0.08284 0.01296 0.32477 0.08284 0.14019

Range(Tb) 21.6026088 35.00004 202.068944 21.6026088 86.22386427

PS-1 PS-2 PS-3 PS-1 Mean range Table shows the deviation of values of RC model to Multiflash critical values in table 4.1 Range refers to the absolute range between the values of the critical parameters in Multiflash to the RC model. Mean range is reported as a single value for easy analysis

(b) Table 4.2 Comparison from RC module recombination with Multiflash results LAB MULTIFLASH RC MODULE RECOMBINATION RECOMBINATION RECOMBINATION component Mole Component Mole percent component Mole percent percent N2 0.19 N2 0.19274309 N2 0.19553 CO2 0.93 CO2 1.1159177 CO2 0.93023 CH4 88.01 CH4 87.158423 CH4 88.701 C2 5.3 C2 6.2681336 C2 5.297 C3 1.82 C3 2.2325292 C3 1.7849 i-C4 0.42 i-C4 0.5028291 i-C4 0.39909 n-C4 0.5 n-C4 0.57266648 n-C4 0.47533 i-C5 0.23 0.15866919 i-C5 0.21249 C5 n-C5 0.19 n-C5 0.16802 C6 0.24 C6 0.14580306 C6 0.2136 C7+ 2.17 C7+ 1.6522855 C7+ 1.6226 The table indicates comparative result between the laboratory , Multiflash and the RC Module recombination

42

Table 3.4: Absolute average error from measured lab recombination

Component

Multiflash (AARE)i

RC MODULE (AARE)i

NITROGEN CO2 METHANE ETHANE PROPANE ISOBUTANE N-BUTANE C5 C6 C7+

1.443731579 19.99115054 0.967591183 18.2666717 22.66643956 19.72121429 14.533296 62.22162143 39.248725 23.85781106

2.910526316 0.024731183 0.785138052 0.056603774 1.928571429 4.978571429 4.934 19.18146453 11 25.22580645

TOTAL AARE (%)

222.9182523 20.26529567

71.02541316 6.456855742

Calculating the Absolute Average Error (AARE) for both Multiflash and RC module. The numbers in each row represent the individual average error for each row while the absolute average error is calculated based on equation 𝑛𝑛

1 |𝐶𝐶𝐶𝐶 − 𝑀𝑀𝑀𝑀| 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = � × 100 𝑛𝑛 𝑀𝑀𝑀𝑀 𝑖𝑖=1

(4.1)

Where , n is the number of items, CC is the calculated property and MP is the measured property. The AARE helps to check the magnitude of errors in a set of results

4.5.2 FLASH MODULE (F-M): The recombined sample from the laboratory was flashed at standard conditions and the Vapor–liquid equilibrium phase envelop (Pressure Temperature diagram) plotted on Multiflash (see diagram in Appendix B.3). Whereas the RC modules recombined sample was flashed with the F-M module and also the phase envelop plotted on Matlab. The result of the phase envelop is shown below. (for results of the flashing splits see Appendix B.2)

43

PHASE DIGRAM OF RC MODULE SAMPLE FROM F-M MODULE FOR

CP

GAS

2500

Pressure, psia

2000

1500

1000

X: 246.5 Y: 588.8

500

0 -100

-50

0

50

100

150

250

200 CT

Temperature, F

Figure 4.1 phase envelop of fluid mixture frrom F-M module Diagram for gas mole equals to 1. The Cricondentherm and Cricondenbar values predicted here where 246.5oF and 2830psia respectively.

4.5.3 BLACK OIL PVT CALCULATOR MODULE (BC): Results from this module are tables with black oil properties. However further work was done to devise their critical values and combine these surface terms to relate to reservoir terms to be used in the flash module. Table showing black oil PVT properties developed with regard to correlations (see appendices for list of correlations)

44

Table 4.5: Black oil PVT parameters from PVTP and BC module Tempe rature (F)

Press Gas FVF (psia) (ft3/scf)

Gas Gas OIL Visco Density, FVF, (cp) (lb/ft3) (rb/stb) PVTP BLACK OIL OUTPUT 0.0166737 0.015771 0.001041 1.2527 0.0094481 0.017425 0.000589 1.4190 0.0037528 0.028207 0.000234 2.6160 0.0032118 0.034301 0.000201 3.4525 0.0029365 0.040875 0.000183 1.2089 0.0026804 0.047135 0.000167 1.2089

Oil Visco, (cp)

Oil Density, (lb/ft3)

267 267 267 267 267 267

1155 2000 6000 8000 9912 12688

0.3264 0.2678 0.1506 0.1218 1.0214 1.0215

4.3E+10 2.83E+03 14.4E+05 6.9E+09 5.1E+12 3.7E+15

BC MODULE RESULTS 267 1155 0.02396 0.0049 0.001983 1.2639 0.6039 171.50 267 2000 0.009871 0.0052 0.004814 1.3401 0.5602 198.08 267 6000 0.003886 0.0058 0.012229 1.8936 0.4505 217.21 267 8000 0.003326 0.0059 0.014288 2.3433 0.4193 198.19 267 9912 0.002996 0.0061 0.015862 2.945 0.3959 172.98 267 12688 0.002726 0.0062 0.017433 4.3136 0.3693 131.71 The table shows a comparison between the black oil PVT results from PVTP with those from BC module. black oil model for the reservoir wellstream fluid 700

600

pressure, psia

500

400

300

200

100

0 0

100

200

300

400

500

600

700

temp, F

Fig 4.2: Black oil model for reservoir wellstream: Cricondentherm: 602.31F, Cricondentherm: 683.633psia: here recombined wellstream was modelled with a modified

45

black oil approach (see appendix A.4 for full algorithm) fluid was modelled with a conventional black oil

Hypothetical pseudo black oil model envelop with gaussian distribution for two pseudo components 1000

900

800

700

pressure, psia

600

500

400

300

200

100

0 0

100

200

300

400

500

600

700

temperature, F

Fig 4.3: Hypothetical black oil model with Gaussian quadrature

Cricondentherm: 607.71F, Cricondenbar 996.848 psia: A hypothetical black oil model was develop by means of a Gaussian quadrature. Here the reservoir recombined stream is split into two with the same technique Whitson proposed for in splitting heavy fractions (Note no regression was performed).

4.6: ANALYSIS AND DISCUSSION The three modules formed were all created to ensure that the mathematical model (equations) that describes the fluid is consistent and reliable. The first module (Recombination module) aimed at recombining the different phase of a fluid back to its original reservoir composition with a consistent formulation. The underlying principle for recombination (see full algorithm in appendix a) was a reverse black oil modelling approach. This reverse black oil modelling approach makes use of the solution gas oil ratio at separator and stock tank, the specific gravities of the gas and oil phases at the surface and the molecular weight to recreate the reservoir stream. The major drive in using this means is to develop the initial vapour mole fraction (FV), and the saturation pressure (both bubble and dew point pressures) from 46

correlations that would be plugged into the next module the F-M module. The presumption here is that the initial Fv value is supposed to be closed to the Fv value that would be returned from the Rachford- Rice solution in the F-M module (The Rachford rice solution usually requires an initial guess for the start of the iterative process as mentioned in chapter two with initial guesses sometimes far from the actual value). Thus this method aims at reducing the amount of iterations to be performed at this stage, (considering a cell in the reservoir simulator). From the R-C module, recombination yielded an initial Fv value of 0.977 which was plugged into the Flash module and the final Fv value returned was 0.98 after performing just two iteration (using Newton Raphson’s process). Therefore this method is able to achieve convergence for Fv in short time. Moreover, considering other aspects of the recombination module which include solving for saturation pressures by updating saturation pressures gotten from correlations(for bubble point pressure: Elmabrouk or Standing pressure is used, whereas for dew point pressure Nemeth and Kennedy dew point pressure is used). The recombination module yields the saturation pressure (by using correlations) which is then passed on to the Flash module (FM) for updating. The Flash module uses this inserted pressure value as the guess pressure for the bubble point or dew point pressure calculation. Such calculation of saturation pressure where employed in phase envelop plotting. The phase envelop plots took a calculation time of was less than a second to complete plot. Thus method is consider to be employed in thermodynamics modelling of reservoir fluids in reservoir simulators. An important feature to also consider is the nature of the result from the first two modules. In comparing the results to Multiflash prediction, for pseudo splitting of heavy fractions the linear deviations (as shown in table 4.2) of the RC model values was not considerably far since the only the critical pressures where the only parameter with wide deviation. This deviation is possibly due to the method used in split. (The RC-module uses a Guassian quadrature or Whitsons method for splitting as shown in appendix A.2). Moreover, the values of the calculated wellstream (from RC module) where compared to Multiflash result and the original recombination from the PVT laboratory. The RC-modules had an absolute average error value of about 6% from the original laboratory data while the Multiflash module had an AARE value of 20.2% (see Table 4.4 above). Thus, the RC-module predicts the reservoir wellstream composition with more accuracy than Multiflash. 47

Multiflash has this deviation from the laboratory sample because it uses few data points (i.e. CO2, H2S N2, and C1 to C4) in developing its recombined fluid, (a feature properly accounted for in the R-M module). For the flash module, the value of the Cricondentherm and Cricondenbar are 246.5oF and 2830 psia respectively on the phase envelop (Pressure –Temperature diagram) for the mixture. The value is comparative to the original reservoir mixture from the laboratory which had 195.931oF and 2388.1 psia respectively. The little deviations comes from the recombination average absolute error. Thus signifying that the models where suitable for representing the fluid behavior. Furthermore, output results from the black oil module are based on internal correlations and the values it predicted for the PVT properties were close in some cases to the values from PVTP. Wherever the values where similar to those from PVTP, it is assumed that the correlation used is similar or the same. Also, it found that some correlations underestimate values and thus they require regression. For instance the Nemeth correlation for dew point yielded a value of approximately 6000psia which was far less than the dew point of the system (9912 psia). Moreover, the BC module was used to test a hypothetical formulated fluid to a modified black oil model fluid. The hypothetical fluid was formed based on Whitson [7] heavy fraction split which utilizes a Gaussian distribution (pseudo split) to break a component into subsequent fractions. Here the hydrocarbon from the wellstream is considered as a single mixture and is then broken down into two pseudo fractions (a hypothetical oil and gas stream). Then critical properties are calculated for each stream so that they can be modelled with an equation of state.(see Appendix A.2 for full algorithm) To test the resemblance of this hypothesis with a modified black oil model. The phase envelop was considered so as to see the volumetric behavior of the two fluids. The results of the test showed for the hypothetical fluid a Cricondentherm and Cricondenbar of about 607.71oF, Cricondenbar 996.848 psia (see fig 4.3) whereas the modified black oil fluid had a value of 602.31oF, 683.633psia (see Fig 4.2) respectively. Thus, indicating that the deviation between the two models (hypothetical and modified approach) are not quite far. Further adjustments to the pressure might be needed in the hypothetical model

48

4.7 SOURCES OF POSSIBLE ERRORS The possible errors considered were errors that regression analysis would solve such as differences in the binary interaction coefficients (BICs) and other critical parameters. Moreover, to avoid large deviations, the BICs values where not set at zero instead their values were taken from a text while for hydrocarbon to hydrocarbon interactions they were calculated using the Prausnitz [78] equation. Moreover since these models did not honor regression on the overall, the values from the flash module cannot be used directly in simulator cell.

4.8 CHAPTER SUMMARY The following points could be derived from this chapter (1) The Recombination modules were used to ensure a close to accurate representation of the reservoir fluid as well as develop possible values for the Rachford Rice equation (2) The Flash module helps to update saturation pressure values gotten from correlations (3) Black oil calculator modules capability usually rely on the type of correlation used (4) Guassian quadrature (using Whitsons Characterization method) could be hypothetically used to replace the modified black oil approach (5) Regression analysis is key to the success of any model in order to ensure that errors in predictions are reduced.

49

CHAPTER FIVE 5.1 CONCLUSION A phase behavior model able to perform analyses and also to reduce computational time in a simulation was developed. The model proved capable of predicting phase equilibrium on a gas condensate sample. The proposed model formulated with three routines modules on Matlab software had each module created for a define purpose. The first routine yielded a recombination sample from separation analysis that was close to the actual reservoir samples composition. Moreover it furnished the initial value of the vapour mole fraction Fv that would be used for flashing the reservoir mixture at any time. This proves to be vital in reducing the number of iterations would performed by the simulator in obtaining the vapour mole fraction. Thus helping in reducing computation time for compositional modelling. The second module was created for flashing reservoir fluids mixtures to obtain their vapour and liquid fractions. However this module coupled with results from the first module provided a means of updating saturation pressures (both bubble and dew points) predicted from correlations. Moreover, the third module predicted the PVT properties of the gas condensate sample. This module also showed that using Gaussian quadrature means as predicted by Whitson could be a possible replacement for the black oil modelling approach. Since its output was similar to the output from of the black oil model. 5.2 RECOMMENDATION The following are the recommendations proposed for future works with this model (i) A recombination module formulated by the means above will be helpful in reducing iteration time in cells during simulation. Also, obtaining proper predictions of saturation pressures by updating a correlated bubble point or dew point (using such techniques as in the flash module) when using a black oil modelling approach... (ii) The proposed method of splitting hydrocarbon phases by means of a pseudo split (i.e same method Whitson proposed for characterizing heptane’s C7+) should be further tested for its accuracy. If found consistent (i.e. to resemble the modified back oil method closely) could replace the Modified black oil model approach and could provide other dimensions for applications of black oil models. 50

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[57] Peneloux, A., Rauzy, E. and Fréze, R., (1982). A consistent correction for RedlichKwong-Soave volumes, Fluid Phase Equilibria, 8 (1), pp.7-23.

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[64] Wertheim, M., (1986). Fluids with highly directional attractive forces. IV. Equilibrium polymerization, Journal of Statistical Physics, 42 (3-4), pp.477-492.

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[79] Petrosky Jr, G. and Farshad, F., (1998). Pressure-volume-temperature correlations for Gulf of Mexico crude oils, SPE Reservoir Evaluation & Engineering, 1 (05), pp.416420.

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61

APPENDIX A A ALGORITHM FOR THE THREE MODULES (A.1) RECOMBINATION MODULE (A) STEP 1: START STEP 2: Input the following (i)Compositions of separator gas (ysp) and oil (xsp) taken from a chromatograph sampling in the order N2,CO2,C1,C2,C3,i-C4,n-C4,i-C5,n-C5,C6,C7+ (ii) separator and reservoir conditions i.e. Pressure and temperatures (pressures in psia and temperatures in Fahrenheit) (iii ) Separator and Stock tank values i.e. Separator and stock tank :GOR(Rsp and Rst), Gas gravity(SGsp and SGst) and API or condensate gravity (API_grav) STEP 3 : Solve the following;𝐺𝐺𝑝𝑝𝑝𝑝 = 𝑅𝑅𝑠𝑠𝑠𝑠 × 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠

141.5

(i)Specific gravity of oil or stock tank liquid : 𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 = (𝐴𝐴𝐴𝐴𝐼𝐼

𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 +131.5)

(ii) Average molecular weight of oil Mol_Oil =0.95*((42.42.*SGoil)./(1.008-SGoil)) 𝑀𝑀_𝑜𝑜𝑜𝑜𝑜𝑜 = (

42.42 × 𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 ) 1.008 − 𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

(iii) Condensate molecular weight of oil from gas, Mog from Cragoes Correlation

(iv);

𝑀𝑀𝑜𝑜𝑜𝑜 =

6084 (𝐴𝐴𝐴𝐴𝐼𝐼𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 − 5.9) 𝑆𝑆𝑆𝑆

𝐶𝐶𝑜𝑜 = 133316 × 𝑀𝑀_𝑜𝑜𝑜𝑜𝑜𝑜

, 𝐶𝐶𝑜𝑜𝑜𝑜 = 133316 ×

𝑜𝑜𝑜𝑜𝑜𝑜

𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜

,

𝑀𝑀𝑜𝑜𝑜𝑜

𝐺𝐺𝑝𝑝𝑝𝑝 = 𝑅𝑅𝑠𝑠𝑠𝑠 × 𝑆𝑆𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔 , 𝑉𝑉𝑒𝑒𝑒𝑒 = 𝑅𝑅𝑠𝑠𝑠𝑠 + 𝐶𝐶𝑜𝑜

, 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 = 𝑉𝑉𝑒𝑒𝑒𝑒 + 𝑅𝑅𝑠𝑠𝑠𝑠 (v) Total reservoir specific gas gravity 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 =

�𝑅𝑅𝑠𝑠𝑠𝑠 ×𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠 �+(4620×𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 )+𝐺𝐺𝑝𝑝𝑝𝑝 𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒

(vi)Average specific gas gravity of surface stream, 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠 = 62

(𝐺𝐺𝑝𝑝𝑝𝑝 +𝐺𝐺𝑝𝑝𝑝𝑝 ) (𝑅𝑅𝑠𝑠𝑠𝑠 +𝑅𝑅𝑠𝑠𝑠𝑠 )

(vii)Elmabrouk’s bubble pressure, (Epb) 𝐸𝐸𝑝𝑝𝑝𝑝 = �𝑅𝑅𝑠𝑠𝑠𝑠 0.683 � × �𝑃𝑃𝑠𝑠𝑠𝑠 0.18 � × �𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 4.98 � × �𝑇𝑇𝑟𝑟 0.658 �

(viii) Calculate vaporized gas ratio from 𝑟𝑟𝑠𝑠 =

�𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 −𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠 �

4600×𝑆𝑆𝑆𝑆𝑜𝑜𝑜𝑜𝑜𝑜 −𝐶𝐶𝑜𝑜 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠

STEP 4: Assume undersaturated conditions, let Rp= Rsp, where Rp is the producing gas oil ratio. Solve for mole fraction of wellstream that comes from reservoir gas and oil (i.e Fg and Fo respectively) using the relations below Rp = Rsp; Km = 1 − Rsp.× rs; jm =

Fgg =

Rsp ; Rp

qm = rs × Rp;

1 − 𝑗𝑗𝑚𝑚 , 𝐾𝐾𝑚𝑚

Foo =

gm = 1 − Foo;

1 − qm ; Km

gk = Co + Rsp;

gt = Cog + rs.−1 ; jk = gm × gt;

jg = Foo.× gk;

1 Fk = �1 + �jg. �� jk Fg = Fk −1 ;

Fo = 1 − Fg

STEP 5:Calculate the approximate wellstream vapor (yr) and liquid (xr) compositions by the following relations : 𝑙𝑙𝑙𝑙 == Cog × rs ;

𝑙𝑙𝑙𝑙 = 𝑙𝑙𝑙𝑙𝑥𝑥𝑥𝑥𝑥𝑥 ,

𝑔𝑔𝑔𝑔 = 𝑦𝑦𝑦𝑦𝑦𝑦 + 𝑙𝑙𝑙𝑙 ,

𝑗𝑗𝑗𝑗 = 1 + 𝑙𝑙𝑙𝑙 , Co

lb = Rsp ,

lc = lb × xsp , gb = ysp + lc , 63

𝑗𝑗𝑗𝑗 = 1 + 𝑙𝑙𝑙𝑙

xr = gb/jb

STEP 6: Calculate the approximate reservoir well stream composition (zi) by the relations zg = Fg × yr

zo = Fo.×xr

Calculate zi with the relations:

zi = zg + zo

Co FyG = 1/(1 + � �) Rsp

Where, FyG is the mole fraction of wellstream that becomes separator gas. Then zi = FyG × ysp + (1 − FyG)xsp

STEP 7: Calculate Nemeth-Kennedy’s dew point pressure using the relations in reference [13] STEP 7: disp all calculated values STEP 8: END

(A.2) PSEUDOIZATION MODULE (using Whitson’s method) STEP 1:START STEP 2: Input the following: Laboratory C7+ molecular weight in stock tank oil (M_cn), laboratory C7+ mole fraction in mixture (z_cn), laboratory C7+ specific gravity in mixture (SG_cn) STEP 3 Determine the number of Cn + split fractions, n. obtain the Guassian quadrature values: function variables (Xi) and the weight factors (Wi). STEP 4: Input the following Lowest molecular weight in plus fraction𝜂𝜂 = 92. Also input shape of distribution as 𝛼𝛼 = 1

STEP 5: Estimate the heaviest molecular weight of fraction n as Mn by the formula 𝑀𝑀𝑀𝑀 = 2.5 × 𝑀𝑀_𝑐𝑐𝑐𝑐

Step6: calculate the parameters 𝛿𝛿 𝑎𝑎𝑎𝑎𝑎𝑎 𝛽𝛽 by the equations

64

𝛿𝛿 = 𝑒𝑒

𝛼𝛼𝛼𝛼 −1� 𝑀𝑀_𝑐𝑐𝑐𝑐 −𝜂𝜂

�

𝛽𝛽 =

and

(𝑀𝑀𝑀𝑀 − 𝜂𝜂) 𝑋𝑋𝑋𝑋

Step 7: Calculate the Cn+ mole fraction (zik) and molecular weight (Mi) for each fraction by the formula 𝑧𝑧𝑧𝑧𝑧𝑧 = 𝑧𝑧_𝑐𝑐𝑐𝑐 × [𝑊𝑊𝑊𝑊𝑊𝑊(𝑋𝑋𝑋𝑋)] , 𝑀𝑀𝑖𝑖 = 𝜂𝜂 + 𝛽𝛽𝛽𝛽𝛽𝛽 and 𝑓𝑓(𝑋𝑋) =

(𝑋𝑋)𝛼𝛼−1 (1+ln 𝛿𝛿)𝛼𝛼 Γ(𝛼𝛼) ×𝛿𝛿 𝑥𝑥

Step8: Check the condition ∑𝑛𝑛𝑖𝑖=1 𝑧𝑧𝑧𝑧𝑘𝑘 𝑀𝑀𝑖𝑖 = 𝑀𝑀𝑐𝑐𝑐𝑐 is satisfied. Else modify the value of 𝛿𝛿 and repeat steps % and 6 until a reasonable match is achieved.

Step 9: calculate Watson characterization factor 𝐾𝐾𝑤𝑤 and the specific gravities of each split

SGi by the relations

0.15178 −0.84573 𝐾𝐾𝑤𝑤 = 4.5579 × 𝑀𝑀_𝑐𝑐𝑐𝑐 × 𝑆𝑆𝑆𝑆_𝑐𝑐𝑐𝑐

𝑆𝑆𝑆𝑆𝑆𝑆 = 6.0108 × 𝑀𝑀𝑖𝑖0.17947 × 𝐾𝐾𝑤𝑤−1.18241

Step 10: calculate the boiling points (Tb) and the critical pressure (Pc), temperature (Tc) and volume (Vc) from Riazi Daubert correlations. Also estimate the acentric factors (𝜔𝜔) from Edminster’s correlation 𝑇𝑇𝑏𝑏𝑏𝑏 = (𝐾𝐾𝑤𝑤 𝑆𝑆𝑆𝑆𝑖𝑖 )3

−2.3125 𝑃𝑃𝑐𝑐𝑐𝑐 = (3.12281 × 109 ) × 𝑇𝑇𝑏𝑏𝑏𝑏 × 𝑆𝑆𝑆𝑆𝑖𝑖2.3201 0.58848 × 𝑆𝑆𝑆𝑆𝑖𝑖0.3596 𝑇𝑇𝑐𝑐𝑐𝑐 = 24.27871 × 𝑇𝑇𝑏𝑏𝑏𝑏

𝑃𝑃𝑐𝑐 3 log � �14.7� 𝜔𝜔 = � × − 1� 𝑇𝑇𝑐𝑐 7 �� �𝑇𝑇 � − 1�

𝑉𝑉𝑐𝑐 = (7.0434 ×

𝑏𝑏

2.3829 10−7 )𝑇𝑇𝑏𝑏𝑏𝑏

× 𝑆𝑆𝑆𝑆𝑖𝑖−1.683

Volume is in ft3/lb mole, pressure in psia and temperatures (boiling and critical) are in degrees Rankine. Step 11: END (A.2) FLASH MODULE Step 1: Start Step 2: input value of vapor mole fraction gotten from recombination module Step 3: Enter pressure and temperature to flash

65

Step 4: Input values of critical pressure, temperature, acentric factors, critical volumes and molecular weight for the mixture in the order N2, CO2, C1, C2….Ps-1, Ps-2, Ps-3… Step 5: Input value for the binary interaction coefficient 𝑘𝑘𝑖𝑖𝑖𝑖 from Whitsons Monograph and for hydrocarbon element interactions (C1 to C7+ pairs) use Chueh-Prausnitz correlation given below 𝑘𝑘𝑖𝑖𝑖𝑖 = 𝐴𝐴 �1 − With A=0.18 and B=6

𝐵𝐵 1� 1� 6 6 2𝑉𝑉𝑐𝑐𝑐𝑐 𝑉𝑉𝑐𝑐𝑐𝑐 � 1 1� � � �3 𝑉𝑉𝑐𝑐𝑐𝑐 + 𝑉𝑉𝑐𝑐𝑐𝑐 3

Step 6: Estimate K values by using the Wilson equation or use K values gotten from recombination. Wilsons equation is given as −1

𝑒𝑒 [5.37(1+𝜔𝜔𝑖𝑖 �1−𝑇𝑇𝑟𝑟𝑟𝑟 𝐾𝐾𝑖𝑖 = 𝑃𝑃𝑟𝑟𝑟𝑟

)�]

Where Tr and Pr are the reduced temperatures and pressures. (see chapter two)

Step 7: calculate the minimum and maximum values ok K i.e Kmin and Kmax respectively and set Fv to be limited by 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 < 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣 > 1. Thus 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 < 𝐹𝐹𝑣𝑣 < 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 and the values are obtained by

𝐹𝐹𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 =

1 1 − 𝐾𝐾𝑚𝑚𝑚𝑚𝑚𝑚 1

and 𝐹𝐹𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 1−𝐾𝐾

𝑚𝑚𝑚𝑚𝑚𝑚

Step 8: Solve the Rachford-Rice phase split equation (2.20) at start value of Fv being the Fv from recombination using Newton-Raphson’s method. Step 9: Calculate the phase compositions x and y as given by equations (2.19 and 2.18) Step 10: Calculate phase Z factors ZL , Zv and components fugacities fLi and fvi using PengRobinson’s equation of state Step 11: Calculate the phase Gibbs energy functions 𝐺𝐺𝑦𝑦∗ 𝑎𝑎𝑎𝑎𝑎𝑎 𝐺𝐺𝑥𝑥∗ as in equation (2.9). Determine the correct Z-factor roots of each phase (if multiple roots exist) and calculate the mixture Gibbs energy given by the relation 66

∗ 𝐺𝐺𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐺𝐺𝑥𝑥∗ (1 − 𝐹𝐹𝑣𝑣 ) + 𝐺𝐺𝑦𝑦∗ 𝐹𝐹𝑣𝑣

Step 12: Check the equal fugacities constraints from equation () Step 13: (a) if a convergence is reached, stop (b) if convergence is not reached update the K values with the fugacity ratio using Newton Raphson methods for K update Step 14: Check for convergence at a trivial solution (𝐾𝐾𝑖𝑖 → 1) with the condition 𝑛𝑛

�(ln 𝐾𝐾𝑖𝑖 )2 < 10−4 𝑖𝑖=1

Step 15: if a trivial solution is not detected return to step 6. Otherwise perform a stability check. Step 16: For stability check : calculate mixture fugacities fzi (use normalized zi), with the multiple Z-factor roots, choose root with the lowest g Step 17: use Wilson equation to estimate the K values Step18: calculate the second phase mole fractions (yi), with the assumption that zi is equal to the other phase. Then it follows that (𝑦𝑦𝑖𝑖 )𝑣𝑣 = 𝑧𝑧𝑖𝑖 (𝐾𝐾)𝑣𝑣

Step 18: add up the mole numbers to get 𝑆𝑆𝑣𝑣

𝑛𝑛

𝑆𝑆𝑣𝑣 = �(𝑦𝑦𝑖𝑖 )𝑣𝑣 𝑗𝑗=1

Step 19: calculate the fugacities of the second phase (fyi)v from Peng-Robinson equation of state with multiple Z-factor roots (for the given phase) , choose root with the minimum Gibbs energy 𝐺𝐺 ∗

Step 20: Calculate the fugacity-ratio corrections (𝑅𝑅𝑅𝑅)𝑣𝑣 for subsequent substitution to update

the K values ,

(𝑅𝑅𝑖𝑖 )𝑣𝑣 =

𝑓𝑓𝑧𝑧 𝑖𝑖

1 �𝑓𝑓𝑦𝑦𝑦𝑦 �𝑣𝑣 𝑆𝑆𝑣𝑣

Step 21: Test for convergence with the criteria 𝑒𝑒 < 1 × 10−12 , thus 67

𝑛𝑛

�(𝑅𝑅𝑖𝑖 − 1)2 < 𝑒𝑒 𝑖𝑖=1

If convergence is not obtained , update K values with the relation below (𝑛𝑛+1)

𝐾𝐾𝑖𝑖

(𝑛𝑛) (𝑛𝑛)

= 𝐾𝐾𝑖𝑖 𝑅𝑅𝑖𝑖

Step 22: Test for a trivial solution by using the criteria below 𝑛𝑛

�(ln 𝐾𝐾𝑖𝑖 )2 < 1 × 10−4 𝑖𝑖=1

If trivial solution is not indicated return to step 17 Step 23: when S< 1 stability has been achieved Step 24: END

(A.3) BUBBLE AND DEW POINT PRESSURE CALCULATIONS STEP 1 : Start: Step 2: For bubble point pressure estimate use the Elmabrouk’s bubble point pressure reported by recombination module. (For dew point use the dew point pressure estimated Nemeth and Kennedy correlation given by the recombination module). The next lines describe the bubble point pressure scenario. Step 3: Estimate K-values using Wilson’s equation 𝑗𝑗

Step 4: Calculate the vapor phase compositions from 𝑦𝑦𝑖𝑖𝑖𝑖+1 = 𝑧𝑧𝑖𝑖 𝐾𝐾𝑖𝑖 , i=1, 2….N, where j is an iteration counter.

Step 5: Calculate the vapor and liquid phase fugacity coefficients ((𝜑𝜑𝑖𝑖𝑣𝑣 , 𝑖𝑖 = 1,2 … . 𝑁𝑁) and

(𝜑𝜑𝑖𝑖𝑙𝑙 , 𝑖𝑖 = 1,2, … . 𝑁𝑁) using estimates for bubble point pressure and vapor composition. The

liquid composition equals the feed composition

Step 6: Calculate the new K factors from equation (2.13) Step 7: Calculate h(FV) in the equation ℎ(𝐹𝐹𝑣𝑣 ) = ∑𝑛𝑛𝑖𝑖=1 𝑧𝑧𝑖𝑖 𝐾𝐾𝑖𝑖 − 1

Step 8: Calculate the expression

𝑑𝑑(ℎ(𝐹𝐹𝐹𝐹)) 𝑑𝑑𝑑𝑑

𝜕𝜕 ln 𝜑𝜑𝑖𝑖𝑙𝑙

= ∑𝑛𝑛𝑖𝑖=1 𝑧𝑧𝑖𝑖 𝐾𝐾𝑖𝑖 �

𝜕𝜕𝜕𝜕

−

𝜕𝜕 ln 𝜑𝜑𝑖𝑖𝑣𝑣 𝜕𝜕𝜕𝜕

�

Step 9: Calculate the (j+1)th estimate of bubble point pressure from 𝑃𝑃𝑗𝑗+1 = 𝑃𝑃𝑗𝑗 − Step 10: if solution did not converge return to step 4

68

(ℎ(𝐹𝐹𝑣𝑣 ))𝑖𝑖

𝑑𝑑(ℎ(𝐹𝐹𝐹𝐹))𝑖𝑖 𝑑𝑑𝑑𝑑

Else Step 11: END Note: Phase envelope algorithm followed for this report was same as in Pedersen book (reference )

(A.4) BLACK OIL PROPERTIES MODULE: (i) CORRELATION CALCULATOR STEP 1: START STEP 2:calculate the following black oil properties by the correlations listed below: full equations of these correlations can be found in their reference (i) Bubble point pressure : Glaso (ii) Dew point pressure by Nemeth and Kennedy correlation (iii) Bubble point pressure using Standings correlation (iv) solution gas oil ratio by Glaso correlation (v) Viscosity by Ng and Egbogah correlation (for dead oil viscosity), Beggs and Robinson correlation for saturated oil (vi) Oil formation volume factor by Petrosky and Farshad correlation. (vii) coefficient of isothermal Petrosky and Farshad correlation STEP 3: END

(A.5) MODIFIED BLACK OIL PARAMETERS CORRELATION STEP 1: START STEP 2: Calculate the composition of the oil and gas phases at reservoir by the relation For oil at surface 𝑥𝑥𝑜𝑜 =

𝐶𝐶𝑜𝑜 𝑅𝑅𝑠𝑠 + 𝐶𝐶𝑜𝑜

𝑥𝑥𝑔𝑔� = 1 − 𝑥𝑥𝑜𝑜

For reservoir oil

𝑀𝑀𝑔𝑔 = 𝑆𝑆𝑆𝑆𝑔𝑔𝑔𝑔𝑔𝑔 × 𝑀𝑀𝑎𝑎𝑎𝑎𝑎𝑎 69

𝑀𝑀𝑜𝑜 = 𝑀𝑀𝑜𝑜 𝑥𝑥𝑜𝑜� + 𝑀𝑀𝑔𝑔 𝑥𝑥𝑔𝑔�

𝜌𝜌𝑜𝑜 =

𝑉𝑉𝑜𝑜 =

𝑀𝑀𝑜𝑜 𝜌𝜌𝑜𝑜

62.48𝛾𝛾𝑜𝑜 + 0.0136𝛾𝛾𝑔𝑔 𝑅𝑅𝑠𝑠 𝐵𝐵𝑜𝑜

STEP 3: calculate reservoir gas phase Reservoir gas densities 𝜌𝜌𝑔𝑔 =

𝜌𝜌𝑔𝑔𝑠𝑠𝑠𝑠 + 𝜌𝜌𝑜𝑜𝑜𝑜𝑜𝑜 𝑟𝑟𝑠𝑠 𝐵𝐵𝑔𝑔𝑔𝑔

Mole fraction of surface oil in reservoir gas 𝑦𝑦𝑜𝑜� =

𝐶𝐶𝑜𝑜

1� + 𝐶𝐶 𝑜𝑜 𝑟𝑟𝑠𝑠

Mole fraction of surface gas in reservoir gas 𝑦𝑦𝑔𝑔� = 1 − 𝑦𝑦𝑜𝑜�

Molecular weight of reservoir gas 𝑀𝑀𝑔𝑔 = 𝑀𝑀𝑜𝑜� 𝑦𝑦𝑜𝑜� + 𝑀𝑀𝑔𝑔� 𝑦𝑦𝑔𝑔� Molar volume of reservoir gas 𝑣𝑣𝑔𝑔 =

𝑀𝑀𝑔𝑔 𝜌𝜌𝑔𝑔

STEP 4: calculate EOS parameters using the Flash module For oil critical properties use Matthews relation 𝑇𝑇𝑐𝑐 = 608 + 364 log(𝑀𝑀𝑜𝑜�− 71.2) + (2450 log 𝑀𝑀𝑜𝑜� − 3800) log 𝛾𝛾𝑜𝑜

𝑃𝑃𝑐𝑐 = 1188 − 431 log(𝑀𝑀𝑜𝑜� − 61.1) + [2319 − 852 log(𝑀𝑀𝑜𝑜� − 53.7)](𝛾𝛾𝑜𝑜 − 0.8) 𝛼𝛼 = �1 + 𝑚𝑚�1 − �𝑇𝑇𝑟𝑟 ��

2

𝜔𝜔 = 0.000003𝑀𝑀𝑜𝑜2� + 0.004𝑀𝑀𝑜𝑜� − 0.039

For gas critical properties use the Standings correlation If 𝑦𝑦𝑔𝑔 < 0.75 If 𝑦𝑦𝑔𝑔 ≥ 0.75

𝑇𝑇𝑐𝑐 = 168 + 325𝛾𝛾𝑔𝑔 − 12.5𝛾𝛾𝑔𝑔2

𝑃𝑃𝑐𝑐 = 667 + 15.0𝛾𝛾𝑔𝑔 − 37.5𝛾𝛾𝑔𝑔2 𝑇𝑇𝑐𝑐 = 187 + 330𝛾𝛾𝑔𝑔 − 71. 5𝛾𝛾𝑔𝑔2 70

𝑃𝑃𝑐𝑐 = 706 + 51.7𝛾𝛾𝑔𝑔 − 11.1𝛾𝛾𝑔𝑔2 𝑤𝑤 = 01637𝛾𝛾𝑔𝑔 − 0.0792 STEP 5:Calculate the EOS constants 𝑎𝑎 = 𝑎𝑎𝑜𝑜� 𝑥𝑥𝑜𝑜2� + 𝑎𝑎𝑔𝑔� 𝑥𝑥𝑔𝑔2� + 2�𝑎𝑎𝑜𝑜� 𝑎𝑎𝑔𝑔� � 𝑏𝑏 = 𝑏𝑏𝑜𝑜� 𝑥𝑥𝑜𝑜� + 𝑏𝑏𝑔𝑔�́ 𝑥𝑥𝑔𝑔�

0.5

𝑥𝑥𝑜𝑜� 𝑥𝑥𝑔𝑔�

EOS a and b still maintain the same forms as in Peng _Robinson (ref.) STEP 8: Run the flash module STEP: 7 END

Note: The algorithm for the hypothetical model using Gaussian distribution is the same as the Pseudoization model in RM module. But the molecular weight used is the weight of the total well stream, while the normalized mole fraction and specific gravity are 1 and the total specific gravity of the well stream.

71

APPENDIX B B: PLOTS AND DIAGRAMS

Fig: B1: Snapshot of Recombination Module from Matlab

B2

72

Fig B2: Phase diagram of reservoir mixture from laboratory (Diagram from Multiflash)

B3 Table B.3 : Comparative showing the flash of the combined reservoir stream by Multiflash and the F-M module Component

feed

NITROGEN CO2 METHANE ETHANE PROPANE ISOBUTANE N-BUTANE ISOPENTANE N-PENTANE C6 C7+

0.0019 0.0095 0.8996 0.0542 0.0186 0.0043 0.0051 0.0024 0.0019 0.0004 0.0021

Multiflash results Vapour liquid 0.0019 0.0000 0.0095 0.0002 0.9014 0.0042 0.0543 0.0015 0.0186 0.0018 0.0043 0.0010 0.0051 0.0018 0.0024 0.0025 0.0019 0.0028 0.0004 0.0112 0.0001 0.9730

73

F_M module vapor liquid 0.0019 0.0000 0.0092 0.0003 0.8720 0.0054 0.0525 0.0021 0.0181 0.0028 0.0042 0.0017 0.0050 0.0030 0.0023 0.0038 0.0019 0.0042 0.0026 0.0197 0.0303 0.9569