Theory of Gas Injection Processes Franklin M. Orr, Jr. Stanford University Stanford ppt

For flow in porous media, the approach applies to many physical systems in which the convection of one or more phases dominates the flow, and the effects of dispersive mixing can be neglect

Theory of Gas Injection Processes

Franklin M Orr, Jr.

Stanford UniversityStanford, California

2005

To Susan

Preface

This book is intended for graduate students, researchers, and reservoir engineers who want tounderstand the mathematical description of the chromatographic mechanisms that are the basisfor gas injection processes for enhanced oil recovery Readers familiar with the calculus of partialderivatives and properties of matrices (including eigenvalues and eigenvectors) should have notrouble following the mathematical development of the material presented The emphasis here

is on the understanding of physical mechanisms, and hence the primary audience for this bookwill be engineers Nevertheless, the mathematical approach used, the method of characteristics, is

an essential part of the understanding of those physical mechanisms, and therefore some effort isexpended to illuminate the mathematical structure of the flow problems considered In addition, Ihope some of the material will be of interest to mathematicians who will find that many interestingquestions of mathematical rigor remain to be investigated for multicomponent, multiphase flow inporous media

Readers already familiar with the subject of this book will recognize the work of many studentsand colleagues with whom I have been privileged to work in the last twenty-five years I ammuch indebted to Fred Helfferich (now at the Pennsylvania State University) and George Hirasaki(now at Rice University), working then (in the middle 1970’s) at Shell Development Company’sBellaire Research Center They originated much of the theory developed here and introduced me

to the ideas of multicomponent, multiphase chromatography when I was a brand new researchengineer at that laboratory Gary Pope and Larry Lake were also part of that Shell group of futureacademics who have made extensive use of the theoretical approach used here in their work withstudents at the University of Texas I have benefited greatly from many conversations with themover the years about the material discussed here Thormod Johansen patiently explained to mehis mathematician’s point of view concerning the Riemann problems considered in detail in thisbook All of them have contributed substantially to the development of a rigorous description ofmultiphase, multicomponent flow and to my education about it in particular

Thanks are also due to many Stanford students, who listened to and helped me refine the planations given here in a course taught for graduate students since 1985 Their questions over theyears have led to many improvements in the presentation of the important ideas Much of the ma-terial in this book that describes flow of gas/oil mixtures follows from the work of an exceptionallytalented group of graduate students: Wes Monroe, Kiran Pande, Jeff Wingard, Russ Johns, BirolDindoruk, Yun Wang, Kristian Jessen, Jichun Zhu, and Pavel Ermakov Wes Monroe obtained thefirst four-component solutions for dispersion-free flow in one dimension Kiran Pande solved forthe interactions of phase behavior, two-phase flow, and viscous crossflow Jeff Wingard consideredproblems with temperature variation and three-phase flow Russ Johns and Birol Dindoruk greatlyextended our understanding of flow of four or more components with and without volume change

ex-on mixing Yun Wang extended the theory to systems with an arbitrary number of compex-onents,and Kristian Jessen, who visited for six months with our research group during the course of hisPhD work at the Danish Technical University, contributed substantially to the development ofefficient algorithms for automatic solution of problems with an arbitrary number of components

in the oil or injection gas Kristian Jessen and Pavel Ermakov independently worked out the firstsolutions for arbitrary numbers of components with volume change on mixing Jichun Zhu andPavel Ermakov contributed substantially to the derivation of compact versions of key proofs BirolDindoruk, Russ Johns, Yun Wang, and Kristian Jessen kindly allowed me to use example solutions

and figures from their dissertations This book would have little to say were it not for the work ofall those students Marco Thiele and Rob Batycky developed the streamline simulation approachfor gas injection processes Their work allows the application of the one-dimensional descriptions ofthe interactions of flow and phase to model the behavior of multicomponent gas injection processes

in three-dimensional, high resolution simulations All those students deserve my special thanks forteaching me much more than I taught them

Kristian Jessen deserves special recognition for his contributions to teaching this material with

me and to the completion of Chapters 7 and 8 He contributed heavily to the material in thosechapters, and he constructed many of the examples

I am indebted to Chick Wattenbarger for providing a copy of his “gps” graphics software All

of the figures in the book were produced with that software

I am also indebted to Martin Blunt at the Centre for Petroleum Studies at Imperial College ofScience, Technology and Medicine for providing a quiet place to write during the fall of 2000 andfor reading an early draft of the manuscript I thank my colleagues Margot Gerritsen and KhalidAziz, Stanford University, for their careful readings of the draft manuscript They and the otherfaculty of the Petroleum Engineering Department at Stanford have provided a wonderful place totry to understand how gas injection processes work The students and faculty associated withthe SUPRI-C gas injection research group, particularly Martin Blunt, Margot Gerritsen, KristianJessen, Hamdi Tchelepi, and Ruben Juanes, and our dedicated staff, Yolanda Williams and ThuyNguyen, have done all the useful work in that quest, of course It is my pleasure to report on apart of that research effort here

And finally, I thank Mark Walsh for asking questions about the early work that caused us tothink about these problems in a whole new way I also thank an anonymous proposal reviewer whosaid that the problem of finding analytical solutions to multicomponent, two-phase flow problemscould not be solved and even if it could, the solutions would be of no use That challenge was toogood to pass up

The financial support for the graduate students who contributed so much to the material sented here was provided by grants from the U.S Department of Energy, and by the membercompanies of the Stanford University Petroleum Research Institute Gas Injection Industrial Affili-ates program That support is gratefully acknowledged

pre-Lynn Orr

Stanford, California

March, 2005

2.1 General Conservation Equations 5

2.2 One-Dimensional Flow 10

2.3 Pure Convection 12

2.4 No Volume Change on Mixing 13

2.5 Classification of Equations 14

2.6 Initial and Boundary Conditions 14

2.7 Convection-Dispersion Equation 15

2.8 Additional Reading 17

2.9 Exercises 17

3 Calculation of Phase Equilibrium 21 3.1 Thermodynamic Background 21

3.1.1 Calculation of Thermodynamic Functions 22

3.1.2 Chemical Potential and Fugacity 24

3.2 Calculation of Partial Fugacity 26

3.3 Phase Equilibrium from an Equation of State 27

3.4 Flash Calculation 31

3.5 Phase Diagrams 34

3.5.1 Binary Systems 34

3.5.2 Ternary Systems 35

3.5.3 Quaternary Systems 37

3.5.4 Constant K-Values 38

3.6 Additional Reading 40

3.7 Exercises 40

4 Two-Component Gas/Oil Displacement 43 4.1 Solution by the Method of Characteristics 44

4.2 Shocks 48

4.3 Variations in Initial or Injection Composition 56

4.4 Volume Change 61

iii

4.4.1 Flow Velocity 62

4.4.2 Characteristic Equations 62

4.4.3 Shocks 63

4.4.4 Example Solution 64

4.5 Component Recovery 67

4.6 Summary 69

4.7 Additional Reading 70

4.8 Exercises 71

5 Ternary Gas/Oil Displacements 73 5.1 Composition Paths 75

5.1.1 Eigenvalues and Eigenvectors 78

5.1.2 Tie-Line Paths 81

5.1.3 Nontie-Line Paths 81

5.1.4 Switching Paths 87

5.2 Shocks 90

5.2.1 Phase-Change Shocks 90

5.2.2 Shocks and Rarefactions between Tie Lines 92

5.2.3 Tie-Line Intersections and Two-Phase Shocks 97

5.2.4 Entropy Conditions 98

5.3 Example Solutions: Vaporizing Gas Drives 99

5.4 Example Solutions: Condensing Gas Drives 106

5.5 Structure of Ternary Gas/Oil Displacements 110

5.5.1 Effects of Variations in Initial Composition 117

5.6 Multicontact Miscibility 117

5.6.1 Vaporizing Gas Drives 118

5.6.2 Condensing Gas Drives 119

5.6.3 Multicontact Miscibility in Ternary Systems 119

5.7 Volume Change 120

5.8 Component Recovery 127

5.9 Summary 129

5.10 Additional Reading 130

5.11 Exercises 131

6 Four-Component Displacements 135 6.1 Eigenvalues, Eigenvectors, and Composition Paths 135

6.1.1 The Eigenvalue Problem 135

6.1.2 Composition Paths 137

6.2 Solution Construction for Constant K-values 144

6.3 Systems with Variable K-values 149

6.4 Condensing/Vaporizing Gas Drives 155

6.5 Development of Miscibility 158

6.5.1 Calculation of Minimum Miscibility Pressure 161

6.5.2 Effect of Variations in Initial Oil Composition on MMP 162

6.5.3 Effect of Variations in Injection Gas Composition on MMP 169

CONTENTS v

6.6 Volume Change 172

6.7 Summary 176

6.8 Additional Reading 176

6.9 Exercises 177

7 Multicomponent Gas/Oil Displacements 179 by F M Orr, Jr and K Jessen 179

7.1 Key Tie Lines 180

7.1.1 Injection of a Pure Component 180

7.1.2 Multicomponent Injection Gas 183

7.2 Solution Construction 185

7.2.1 Fully Self-Sharpening Displacements 193

7.2.2 Solution Routes with Nontie-line Rarefactions 198

7.3 Solution Construction: Volume Change 201

7.4 Displacements in Gas Condensate Systems 204

7.5 Calculation of MMP and MME 206

7.6 Summary 210

7.7 Additional Reading 212

8 Compositional Simulation 213 by F M Orr, Jr and K Jessen 213

8.1 Numerical Dispersion 213

8.2 Comparison of Numerical and Analytical Solutions 215

8.3 Sensitivity to Numerical Dispersion 221

8.4 Calculation of MMP and MME 230

8.5 Summary 237

8.6 Additional Reading 238

Chapter 1

Introduction

When a gas mixture is injected into a porous medium containing an oil (another mixture of drocarbons), a fascinating set of interactions begins Components in the gas dissolve in the oil,and components in the oil transfer to the vapor as local chemical equilibrium is established Theliquid and vapor phases move under the imposed pressure gradient at flow velocities that depend(nonlinearly) on the saturations (volume fractions) of the phases and their properties (density andviscosity) As those phases encounter the oil present in the reservoir or more injected gas, newmixtures form and come to equilibrium The result is a set of component separations that occurduring flow, with light components propagating more rapidly than heavy ones These separationsare similar to those that occur during the chemical analysis technique known as chromatography,and they are the basis for a variety of enhanced oil recovery processes This book describes themathematical representation of those chromatographic separations and the resulting compositionalchanges that occur in such processes

hy-Gas injection processes are among the most widely used of enhanced oil recovery processes[62, 117] CO2 floods are being conducted on a commercial scale in the Permian Basin oil fields

of west Texas (see references [90, 81, 118, 116] for examples of the many active projects), and avery large project is underway in the Prudhoe Bay field in Alaska [74] At Prudhoe Bay, dry gas isinjected into the upper portion of the reservoir to vaporize light hydrocarbon liquids and remainingoil, and in other portions of the field a gas mixture that is enriched in intermediate components

is being injected to displace the oil Large-scale gas injection is also underway in a variety ofCanadian projects [110, 72] and in the North Sea [124] In all these processes, there are transfers

of components between flowing phases that strongly affect displacement performance The goal

of this book is to develop a detailed description of the interactions of equilibrium phase behaviorand two-phase flow, because it is those interactions that make possible the efficient displacement

of oil by gas known as “miscible flooding [112].” We will examine in some detail the mathematicaldescription of the physical mechanisms that produce high local displacement efficiency While theapproach involves considerable mathematical effort, the effort expended on that analysis will pay

off in the development of rigorous ways to calculate the injection gas compositions and displacementpressures required for miscible displacement and a very efficient semianalytical calculation methodfor solving one-dimensional compositional displacement problems

While the focus here is on gas/oil displacements in porous media, the ideas, and the ematical approaches apply to physical processes that range from flow of traffic on a highway to

math-chemical reactions in a tubular reactor to compressible fluid flow Chapter 1 of First Order Partial

1

Differential Equations: Vol I by Rhee, Amundson and Aris [106] describes these and other physical

systems for which the equations solved have many similarities to those considered here

For flow in porous media, the approach applies to many physical systems in which the convection

of one or more phases dominates the flow, and the effects of dispersive mixing can be neglected.The basis for the theory is the description of chromatography, in which components in a mixtureseparate as they flow through a column because the components adsorb (and subsequently desorb)with different affinities onto a stationary phase [108, 30] In chromatography, however, only thecarrier fluid moves, and hence there is no nonlinearity that results when two or more phases flow.Similar theory applies to ion exchange [102], diagenetic alteration of porous rocks [34, 63] and toleaching of minerals [9] Many of these ideas also apply to the area of geologic storage of carbondioxide [85], or CO2 sequestration, as it is sometimes called These processes are intended to reducethe rate of increase of the concentration of CO2 in the atmosphere by injecting CO2 that would

otherwise be released to the atmosphere into subsurface formations such as deep saline aquifers orcoalbeds [139]

In the area of enhanced oil recovery, theoretical descriptions of the displacement of oil by watercontaining polymer and displacement of oil by surfactant solutions are closely linked to the theorydescribed here In fact, the theory for three-component systems was developed first for applications

to surfactant flooding [31, 35, 65], processes that make use of chemical constituents in the injectionfluid that lower interfacial tension between oil and water Effects of volume change as componentstransfer between phases were not considered in that work, a completely reasonable assumption forthe liquid/liquid phase equilibria of surfactant/oil/water mixtures In gas/oil systems, however,some components can change volume quite substantially as they move between liquid and vapor

phases Dumore et al [22] worked out the extension of the three-component theory to include the effects of volume change Monroe et al [82] reported the first solutions for four-component gas/oil

displacements

Many other investigators contributed to the development of the full theory for three and fourcomponent systems A detailed review by Johansen [50] summarizes the relevant papers publishedthrough 1990 Lake’s [62] comprehensive description of enhanced oil recovery also cites the largebody of work related to polymer and surfactant flooding processes

This book applies the one-dimensional theory of multicomponent, multiphase flow to gas/oildisplacements In Chapter 2, the appropriate material balance equations are derived, and theassumptions that lead to the limiting cases explored in detail are stated An introduction to therepresentation of phase equilibria with an equation of state is given in Chapter 3 Chapter 4considers two-phase flow of two components that are mutually soluble When effects of volumechange are ignored, a modest generalization of the familiar Buckley-Leverett solution [10] results.That simple two-phase flow reappears in more complex flows involving more components, and henceits description is the basis for understanding multicomponent systems The most important effects

of volume change as components transfer between phases are also illustrated in Chapter 4

The theory of component gas/oil displacements is developed in Chapter 5 The component theory leads directly and rigorously to the ideas of “multicontact miscible” displacementvia condensing or vaporizing gas drives Extensions of the analysis to systems with more than threecomponents are considered in Chapters 6 and 7 That treatment shows that there are importantfeatures of gas injection processes that cannot be represented by three-component descriptions ofthe phase behavior Chapter 6 describes the construction of solutions for four-component displace-ments and explores the resulting implications for multicontact miscible displacements known as

Effects of dispersive mixing are ignored in the development of the theory presented in Chapters4-7, though, of course, some dispersion will be present in all real displacements Furthermore,finite difference compositional simulations of gas/oil displacements normally include some effects

of numerical dispersion In fact, many finite difference compositional simulations are strongly andadversely affected by numerical dispersion Chapter 8 shows that numerical solutions for the one-dimensional flow equations converge to the analytical solutions, with sufficiently fine grids, and itdescribes how displacement behavior changes when dispersion also acts Chapter 8 also explainswhen and why numerical schemes that have been proposed for calculating the minimum miscibilitypressure fail to give accurate estimates

Flow is never one-dimensional in actual field-scale gas injection projects, and hence, manyadditional factors influence the performance of those multidimensional flows: viscous instability,gravity segregation, reservoir heterogeneity, and crossflow due to viscous and capillary forces [62].Even so, the one-dimensional theory can be used effectively to describe the behavior of three-dimensional flows by coupling one-dimensional solutions with streamline representations of theflow in heterogeneous reservoirs [119, 121, 8, 120, 16, 43] The resulting compositional streamlineapproach can be orders of magnitude faster than conventional finite difference reservoir simulation,and it is more accurate because it is affected much less by numerical dispersion [109]

Water is also always present and is often flowing in addition to oil and gas In addition, phase flow of CO2/hydrocarbon mixtures is also observed at temperatures about 50 C and pressuresnear the critical pressure of CO2 [25, 94, 86, 61] The approach used here to study the mechanisms

three-of gas/oil displacements has also been applied to the flow three-of three immiscible phases [137, 24, 27].LaForce and Johns obtained solutions for three-phase flow for ternary systems with compositionvariation in the two-phase regions that bound the three-phase region on a ternary phase diagram

In any real displacement, of course, all these physical mechanisms interact with the graphic separations that occur in both one-dimensional and multidimensional flows Hence theanalysis given here of one-dimensional flow is only a first step toward full understanding of field-scale displacements It is an important first step, however, because it reveals how and why highdisplacement efficiency can be achieved in gas injection processes, and thus it provides the under-standing needed to design an essential part of any gas injection process for enhanced oil recovery

chromato-Chapter 2

Conservation Equations

The fundamental principle that underlies any description of flow in a porous medium is conservation

of mass The amount of a component present at any location is changed by the motion of fluid withvarying composition through the porous medium Thus, the first issue to be faced in constructing

a model of a flow process is to define and describe the flow mechanisms that contribute to thetransport of each component For gas/oil systems, the physical mechanisms that are most importantare:

1 Convection – the flow of a phase carries components present in the phase along with the flow,

2 Diffusion – the random motions of molecules act to reduce any sharp concentration gradients

that may exist, and

3 Dispersion – small-scale random variations in flow velocity also cause sharp fronts to be

smeared (when transversely averaged concentrations are calculated or measured) Dispersionduring flow in a porous medium is always modeled as if it were qualitatively like diffusion.For a detailed discussion of the relationship between diffusion and dispersion, see [100] or [5]

In this chapter, we derive the differential equations solved in subsequent chapters, and we statethe assumptions required to reduce the general material balance equations to the special casesconsidered in detail below Effects of chemical reactions are not included in the flow problemsconsidered here, nor are effects of adsorption or temperature variation Derivations that includesuch effects are given by Lake [62]

Consider an arbitrary volume, V (t), of the porous medium bounded by a surface, S(t) A material balance on component i in the control volume can be stated as

5

Thus, the rate at which the amount of component i in V changes is exactly balanced by the net inflow of component i carried with the flow of each phase (often referred to as convection) and the

net inflow that arises from the diffusion-like process of hydrodynamic dispersion

in dV

⎫

⎬

where φ is the porosity, and ρ j and S j are the molar density and saturation (volume fraction) of

phase j The amount of the i th component present in phase j is

where x ij is the mole fraction of component i in phase j The total amount of component i present

in dV is obtained by summation over the n p phases present, which gives

⎫

⎬

2.1 GENERAL CONSERVATION EQUATIONS 7

where  v j is the Darcy flow velocity of phase j, the volume of phase j flowing per unit area of porous medium per unit time The flux vector may or may not be normal to the surface, S(t), and hence

the magnitude of the vector component of the flow crossing the element of surface is

where  n is the outward-pointing normal to the surface at the location of the differential element

of area, dS, and the negative sign gives positive accumulation for flow in the opposite direction

of the normal vector, which is flow into the control volume The net rate of convective inflow of

component i is obtained by summing the contributions for flow of each phase and integrating over the full surface, S, to obtain

Net rate of inflow

where α is a material constant known as the dispersivity, and the subcripts  and t refer to the

longitudinal and transverse flow directions By arguments similar to those given for convection,

the net rate of transfer of component i due to dispersion is

Net rate of flow of component

The accumulation, convection, and dispersion terms can be combined to yield an integral

ma-terial balance for component i,

the continuity equation must be derived To do so, we make use of the Reynolds transport and

divergence theorems For some scalar quantity G that is conserved, and a control volume, V (t), that moves with velocity,  v s, the Reynolds transport theorem states that

d dt

Eq 2.1.14 reduces to

d dt

2.1 GENERAL CONSERVATION EQUATIONS 9

Recall that the control volume, V , was chosen arbitrarily If the integral in Eq 2.1.17 is to

be zero for any choice of the control volume, the integrand of Eq 2.1.17 must be identically zero everywhere If it were negative for some portion of V and positive for the remainder, for example,

it would be possible to choose a new control volume that included either the positive or negativeportion of the original control volume If so, Eq 2.1.17 would not be satisfied for the new controlvolume Hence, the final form of the continuity equations for multicomponent, multiphase flow is

of balance equations for momentum, which must also be conserved In practice, however, thesolution of the resulting Navier-Stokes equations for the detailed velocity distributions within thepores of the medium would be intractably and unnecessarily complex Instead, an averaged version

of the momentum equation is used For single-phase flow, volume averaging of the momentumequations yields a form equivalent to Darcy’s law [37, 111], which states that the local flow velocity

is proportional to the pressure gradient Flow of more than one phase is always assumed to be

similarly related to the pressure gradient, and hence, the flow velocity of a phase,  v j, is assumed to

be given by

 j =− kk rj

µ j

where k is the permeability, and k rj , µ j , ρ mj , and P j are the relative permeability, viscosity, mass

density, and pressure of phase j.

The phase subscript, j, on the pressure in Eq 2.1.19 implies that pressures are different in

different phases, as they must be if the phases are separated by curved interfaces with nonzerointerfacial tension The relationships between those pressures are always assumed to be represented

by capillary pressure functions of the form,

P j − P k = P ckj , j = 1, n p , k = 1, n p , k = j. (2.1.20)These equilibrium capillary pressure functions are usually assumed to be functions of the saturations

of the phases (and sometimes, they are scaled with respect to interfacial tension), and they aretaken to be properties of the fluids and the porous medium that can be measured in independentexperiments

The use of equilibrium capillary pressure functions is really an implicit assumption that flow

in a porous medium can be represented in terms of phases in local capillary equilibrium and thatflow can be driven by departures from that equilibrium A similar assumption can be made aboutchemical equilibrium The time required for diffusion of components over the length scale of apore is often small enough compared to the time required for flow to change the compositionssignificantly within the pore that it is reasonable to assume that fluids present are in chemical

equilibrium If so, then the statement of chemical equilibrium gives an additional set of relationsbetween the compositions of the phases,

µ ij = µ ik , j = 1, n p , k = 1, n p , k = j. (2.1.21)

Eq 2.1.21 states that the chemical potential of component i in phase j equals its chemical potential

in all the other phases present (note that according to standard usage, the chemical potential of

component i in phase j is µ ij , while the viscosity of phase j is µ j) The calculation of chemicalpotential of a component in a phase given the composition of that phase is reviewed in Section 3.3

In addition, the following auxiliary relations hold The volume fractions of the phases mustsum to one,

and appropriate initial and boundary conditions must be stated for solution of Eq 2.1.18

Eqs 2.1.18–2.1.26 provide enough information to determine the solution to a flow problem thatmodels the effects of convection, dispersion, and phase equilibrium, as the inventory of equationsand unknowns in Table 2.1 indicates The unknowns are the phase compositions, saturations, pres-sures and velocities The equations are the material balance equations and the auxiliary relationsthat specify capillary and phase equilibrium, Darcy’s law, and the mole fraction and saturationsummations As Table 2.1 demonstrates, the number of equations exactly equals the number ofunknowns That equality is required if a solution is to exist, though it does not guarantee that asolution exists or is unique

Eq 2.1.18 is complex enough that it must be solved numerically unless additional simplifyingassumptions are made In the remainder of this book, we will consider flow in one space dimension,and we will assume that the effects of pressure differences between phases can be neglected.For one-dimensional flow in a Cartesian coordinate system, Eq 2.1.18 reduces to

2.2 ONE-DIMENSIONAL FLOW 11

Table 2.1: Inventory of Equations and Unknowns

Compositions, x ij n p n c Material balances 2.1.18 n c

Saturations, S j n p Phase equilibrium 2.1.21 n c (n p − 1)

from any one of the n p expressions for the phase flow velocities (Eq 2.2.3) For the j th phase, forexample,

where v inj and ρ inj are the flow velocity and density of the injected fluid, and L is the length of the

one-dimensional flow system In Eq 2.3.2, the time scale is the length of time required to displaceone pore volume of fluid at the flow velocity and density of the injected fluid (the volume per unit

of area available for flow over length L is φL, and the volumetric flow rate per unit area is v inj, so

the time required to flow length L is φL/v inj ) Thus, τ is a dimensionless time measured in pore volumes For simplicity, we also assume that the porosity, φ, is constant, though that assumption

can be relaxed easily The result is

G i is an overall concentration (in moles per unit volume) of component i H i is an overall molar

flow of component i The final version of the equations for multicomponent, multiphase convection

is, therefore,

2.4 NO VOLUME CHANGE ON MIXING 13

If, for example, CO2 displaces oil at modest pressure, it often occupies much less volume when

dissolved in a liquid hydrocarbon phase than it does in a vapor phase In those systems, the localflow velocity can vary substantially over the displacement length [22, 82, 19] Thus, for some gasdisplacements, it will be important to include the effects of volume change on mixing

If the displacement pressure is high enough, then the volume occupied by a component in the gasphase may not change greatly when that component transfers to the liquid phase Components inthe liquid/liquid systems that describe surfactant flooding processes also exhibit minimal volumechange on mixing In such systems, it is reasonable to assume that the partial molar volume ofeach component is a constant (independent of composition or phase) and hence that ideal mixingapplies In other words, the volume occupied by a given amount of a component is constant nomatter what phase the component appears in Under the assumption that each component has a

constant molar density, ρ ci, in any phase Eq 2.3.9 can be simplified further The local flow velocity

is constant everywhere and equal to the injection velocity, so v D = 1 Furthermore, the volume

occupied by component i in one mole of phase j is x ij /ρ ci, and the volume fraction of component

Division of Eq 2.4.3 by ρ inj followed by substitution of Eq 2.4.3 into Eq 2.3.6, with v D = 1, and

division by ρ ci /ρ inj yields the set of conservation equations for pure convection with no volumechange on mixing,

and F i is an overall fractional volumetric flow of component i given by

Equations like 2.5.1 are called quasilinear If P , Q, and R are independent of z, the equation is

strictly linear It is called linear if R depends linearly on z and semilinear if R is a nonlinear function

of z In the problems considered here, R(x, t, z) = 0 Such equations are called homogeneous.

In Eqs 2.3.9 and 2.4.4 the dependent variables that correspond to z in Eq 2.5.1 are the overall concentrations G i or C i Because the phase saturations, S j , densities, ρ j , and fractional flows, f j,all depend nonlinearly on those concentrations, Eqs 2.3.9 and 2.4.4 are homogeneous, quasilinearsystems of first order equations

Before Eqs 2.3.9 and 2.4.4 can be solved, initial and boundary conditions must be imposed In thechapters that follow, solutions will be derived for initial compositions that are constant throughout

a semi-infinite domain,

G i (ξ, 0) = G init i , 0 < ξ < ∞, i = 1, n c , (2.6.1)or

C i (ξ, 0) = C i init , 0 < ξ < ∞, i = 1, n c (2.6.2)The only boundary condition required is the composition of the injected fluid,

G i (0, τ ) = G inj i , τ > 0, i = 1, n c , (2.6.3)or

C i (0, τ ) = C i inj , τ > 0, i = 1, n c (2.6.4)

Thus, at time τ = 0, the composition of the fluid at the inlet changes discontinuously from the

initial value to the injected value

Problems in which the initial state (sometimes referred to as the right state) is constant and the upstream boundary condition (sometimes called the left state) is also constant are known

as Riemann problems Such problems can be viewed as a description of the propagation of a discontinuity, initially placed at ξ = 0, between constant initial states for −∞ < ξ < 0, the

2.7 CONVECTION-DISPERSION EQUATION 15

injection composition, and for 0 < ξ < ∞, the initial composition Given the fact that the flow

problem begins with the propagation of a discontinuity, it is no surprise that the solutions may also

display discontinuities known as shocks At a shock, the differential material balances derived in

this chapter must be replaced by integral balances across the shock The properties and behavior

of shocks are considered in some detail in Chapter 4 for two-component flow problems, and again

in Chapter 5 for multicomponent problems

If only two components and one phase are present, and the assumptions of constant porosity and

no volume change on mixing apply, then Eq 2.2.1 simplifies considerably to

∂C

∂t +

v φ

, to a characteristic time for convection, φL/v When the Peclet number is

large, the effects of dispersion are small, and convection dominates Thus, Eqs 2.3.9 and 2.4.4 can

be viewed as applicable in the limit of large Peclet number

If K l is a constant (independent of composition) then the Peclet number is a constant as well,and Eq 2.7.2 can be solved easily by Laplace transforms If the domain is chosen to be 0≤ ξ ≤ ∞,

the initial concentration is C(ξ, 0) = 0 for 0 ≤ ξ ≤ ∞, and fluid with concentration C(0, τ) = 1 is

injected for τ > 0, the solution is [12]



The first term on the right side of Eq 2.7.3 is usually significantly larger than the second term.The second term is significant only at early times near the inlet when the Peclet number is small

For large Peclet number (say P e > 1000), however, the second term can be neglected Hence, many

investigators have used the approximate solution,



Fig 2.1 is a plot of Eq 2.7.3 for three Peclet numbers (P e = 10, 100, and 1000) at times, τ

= 0.25 and 0.75 Fig 2.1 shows that at each Peclet number, a transition zone from the injected

composition (c = 1) to the initial composition (c = 0) moves downstream and increases in length

as the flow proceeds The width of the transition zone increases as the Peclet number is reduced

At τ = 0.75, for example, detectable amounts of the injected fluid have reached the outlet for P e

= 10 and 100 but have not yet done so for P e = 1000.

0.0 0.2 0.4 0.6 0.8 1.0

Figure 2.1: Solutions to the convection-dispersion equation at times τ = 0.25 and 0.75 for P e =

10, 100, and 1000

The solution for P e = 1000 is nearly symmetric about the location ξ = τ Eq 2.7.4 describes

a concentration distribution that is exactly symmetric about ξ = τ , with c = 0.5 at ξ = τ The full solution for P e = 1000 is very close to Eq 2.7.4, but the solutions at lower P e display some

asymmetry that reflects the contribution of the second term on the right side of Eq 2.7.3 In the

limit, P e → ∞, the solution approaches piston-like displacement of single-phase, dispersion-free

2.8 ADDITIONAL READING 17

τ = 1 for Pe = 2500 indicates that the length of the transition zone (defined as the distance

between the 90 and 10 percent concentrations) is only about 7 percent of the displacement length.Hence, at those flow conditions dispersion has a relatively small effect on the displacement Stilllarger Peclet numbers are likely in many field-scale flows If dispersion coefficients are taken to

be a property of the porous medium that is independent of scale, then the Peclet number growslinearly with the displacement length Measurements of field-scale dispersion coefficients suggeststhat they grow linearly with the length scale of the system in which the displacement takes place[2] That result is due to the fact that flows that are multidimensional in heterogeneous porousmedia are being represented with a one-dimensional model [2][62, pp 163-168] Flows at the largerscales sample volumes with greater variability in permeability, and the flow within those volumes

is nonuniform At the scale at which fluids are mixed and at chemical equilibrium, however, themagnitudes of dispersion coefficients should be in the range of 10−3-10−4 cm2/s (depending on the

pressure through the diffusions coefficient) If so then Peclet numbers for a displacement with a

spacing between wells as short as 100 m and a flow velocity of 3 ×10 −4 cm/s would be in the range,

3000 to 30,000 Thus, the convection-dominated theory presented here is relevant in many physicalsituations of practical interest

Derivations of the conservation equations used here are available from a variety of sources See Lake[62] for another version of many of the limiting cases considered here For application of similarideas to the problem of reacting geochemical flows in which the local equilibrium assumption can

be made, see Lake et al [63].

1 Consider the flow velocity of a compressible gas in a cylindrical region around a well-bore

with radius r w The mass density of the gas is a known function of pressure, ρ g (P ) Derive a

differential equation for the flow velocity and express it in cylindrical polar coordinates Notethat the gradient and divergence operators in cylindrical polar coordinates are:

2 A core sample has been obtained for measurement of the permeability The porosity of the

sample is φ Consider the following experiment:

(a) Evacuate the core so that P = 0.

(b) Open the valve at the upstream end of the core at time t = 0 to flow helium into the core At that time, the pressure at the upstream end of the core jumps instantaneously

to P1.

(c) Measure the pressure, P , somewhere in the middle of the core, for t > 0.

To analyze the experiment, calculate how the pressure changes with time after the upstreamvalve is opened Assume that the core is indefinitely long Also assume that the permeability

in the core has constant value k, that the viscosity of the helium is a constant, µ, and that

the helium is an ideal gas

P V = nRT

Derive a differential equation for the pressure in the core as a function of time and space

3 Hot water (temperature T h ) is being injected into a cold core (temperature T c ) of length L The flow velocity in the core is v, and it is constant at all times during the displacement The heat capacities of the water and rock are c w and c r respectively, and are constant [units =

watt/(gram · ◦ K)] Also the mass density of the rock, ρ

r, and the mass density of the water,

ρ w, are constant

(a) Consider a volume element of rock, dV If the reference temperature is T = 0, what is the heat energy stored in the volume dV of rock filled with water with constant porosity,

φ? Note that the water and rock are at the same temperature.

(b) What is the net inflow of heat due to convection only? Note: only water flows

(c) What is the net inflow of heat due to conduction only? The thermal conductivity of the

rock is α [units = joules/(cm · sec · K)], and the thermal conductivity of the fluid is

zero

(d) Write an integral energy balance for the flow in the core

(e) Write the differential energy balance

(f ) Use the following scaled variables:

T = (T − T c )/(T h − T c ),

x D = x/L,

t D = t/t ∗ .

Make the differential equation dimensionless, and choose t ∗ appropriately What

dimension-less groups are obtained?

4 Certain heavy metal pollutants are carried in the sediment load of the Sacramento River.When the river reaches San Francisco Bay, the sediment is deposited on the floor of the bay

at a rate R (mass/area/time) The organometallic pollutants partition between the water

and the solid particles according to the relation

ω w = K(T )ω sed

where ω w and ω sed are weight fractions, and K(t) is a known coefficient that describes the

partitioning

2.9 EXERCISES 19

The bottom of the bay is at temperature T b while the river water, warmed by effluent power

plant cooling water, is at temperature T s As sediment depth increases, the porosity decreases

from the value at the sediment surface, φ s, according to the relation

φ = φ s[1− β (P − P s )] , where φ s , β and P s are constants The heat capacity of the solid and water, C sed and C w,can be assumed to be independent of temperature and pressure over the range of conditions

present, as can the respective mass densities, ρ sed and ρ w The thermal conductivity, α, of

sediment containing water in the pore space is constant throughout the sediment bed Useappropriate integral balance equations to derive a set of differential equations that could besolved to find the concentration distribution of organometallic pollutants in the sediment bed

at any time Also state the appropriate boundary conditions

(a) Assume that the concentration of pollutants in the river is constant, so the concentration

at the sediment bed surface is also constant

(b) Assume that water flows only in the vertical direction in the sediment bed and neglectdispersion in the flow of pollutants in the water phase

(c) Assume that Darcy’s law describes the difference in the flow velocities between the waterand sediment,

Cm∆T Consider the effect of compaction Is the sediment stationary if it is compacting?

Consider how the total depth of sediment present at any time varies Is a differential or anintegral balance appropriate for that calculation?

5 A displacement experiment, in which the injected fluid can be mixed in any proportion withthe fluid in place in the core without forming a second phase, is performed in a long cylinder

core (L = 1 m) The experiment is performed by injecting fluid at a velocity of 10 cm/day.

From the results given in the Table 2.2, estimate the pore volume of the core, and thedispersion coefficient

Table 2.2: Data for Problem 5

Injected Volume (cc) Effluent Concentration

Chapter 3

Calculation of Phase Equilibrium

The representation of equilibrium phase behavior is a key part of the models of dominated flows described by Eqs 2.3.9 and 2.4.5, because those conservation equations are based

convection-on the assumpticonvection-on of local chemical equilibrium Under that assumpticonvection-on, the compositiconvection-ons of thephases that form at a particular location in the porous medium are determined by the pressureand temperature at that location and, of course, the overall composition of the fluid present Ingeneral, the pressure at each spatial position will differ, because a pressure gradient is required ifflow is to take place In many situations, however, the magnitude of the pressure drop due to flow

is small compared to the pressure level For example, in a slim tube displacement, the pressure

drop over the full length of the slim tube is often less than 350 kP a (50 psi), while the pressure maintained at the outlet of the slim tube is typically in the range of 7–35 M P a (1000–5000 psi).

In field-scale displacements, it is commonly observed that pressure gradients near injection andproduction wells are significant, but there are large portions of the reservoir for which gradientsare small Under those circumstances it is reasonable to evaluate equilibrium phase behavior at asingle pressure (and temperature, already assumed to be constant), a step that also makes possiblethe construction of analytical solutions to Eqs 2.4.1 and 2.4.5 In this section, we review calcu-lation of phase equilibrium for gas/oil systems with an equation of state (EOS), and we describebriefly the phase diagrams used to report equilibrium phase compositions We make use here of theremarkable fact, shown originally by J Willard Gibbs, that if the relationship between pressure,temperature, volume, and composition can be specified, as it can by an equation of state, then thecomposition of equilibrium phases can be calculated To see why that statement is true, we need

to review first how thermodynamic functions are calculated from volumetric data

The calculation of phase equilibrium solely from information about the volumetric behavior of amixture is a remarkable demonstration of the power of thermodynamics The fundamental idea ofall thermodynamics is that energy is conserved A second key assumption, completely consistentwith experience, is that once we set the temperature, pressure and composition of a mixture, allother properties (internal energy, molar density, viscosity, heat capacity, thermal conductivity, etc.)

are also determined In other words, the state of a mixture of given composition is fixed by setting

its temperature and pressure

21

Internal energy, U , is one thermodynamic property that is set when the temperature, pressure

and composition of a mixture is specified It represents the energy associated with molecular

motions and vibrations If we neglect kinetic and potential energy contributions, any change in U

for a mixture of fixed composition must be the result of addition of heat or removal of energy aswork is done by the fluid, a statement that can be written

Heat that is transferred to or from a system, Q, is not a state function, but Q/T is, so it is convenient

to define that state function as the entropy, dS = Q/T Work is obtained by displacement through

some distance against a force For the expansion of a fluid against a pressure, the work done is

W = P dV , where V is the molar volume, and P is the pressure Hence the statement of conservation

of energy, also known as the fundamental property relation, is

In the discussion that follows, it will be convenient to make use of other related thermodynamic

functions, the enthalpy, H , Helmholtz function, A, and Gibbs function, G, defined by

3.1.1 Calculation of Thermodynamic Functions

It is important to recognize that the change in any of the thermodynamic state functions can becalculated if enough volumetric and heat capacity data are available (For more detailed derivations

of the expressions given in this section, see Chapter 2 of Classical Thermodynamics of Nonelectrolyte

Solutions by van Ness and Abbott [123]) For example, the enthalpy can be evaluated as follows

for a mixture of fixed composition According to the assumption that the state of the mixture is

determined if T and P are fixed, H = H (T , P ), and we can write

The derivative of enthalpy with respect to temperature (at constant pressure) is a property known

as the constant pressure heat capacity, C P, which can be measured,

The entropy, S, associated with a mixture is not an easily measured quantity, so it is useful to

express Eq 3.1.8 in terms of variables such as volume, temperature, and pressure It can be shownfrom the properties of state functions [123, pp 21-22] that

Eq 3.1.9 is one of the Maxwell relations (See [123, Appendix A-1] for a derivation.) Substitution

of Eqs 3.1.7–3.1.9 into Eq 3.1.6 gives an expression for dH in terms of measurable quantities,

Because H is a state function that depends only on the endpoint values of T and P , any convenient

path of integration can be used An approach that is frequently used is to integrate at constant

temperature to a pressure where heat capacity data are available (usually P at or near zero,

where the ideal gas law applies), to integrate then in temperature at constant pressure to the newtemperature, and finally to integrate at constant temperature to the new pressure That sequence

3.1.2 Chemical Potential and Fugacity

The discussion of thermodynamic functions so far has dealt only with closed systems with fixedcomposition For phase equilibrium calculations, however, each phase is an open system, because

components can transfer between phases If G is the Gibbs energy per mole of a mixture, then the total Gibbs energy is nG, where n is the number of moles of mixture Any change in the total Gibbs energy for the j th phase must be given by

The chemical potential measures how much Gibbs energy is added to a mixture by an infinitesimal

amount of component i The chemical potential can also be defined in terms of any of the other

commonly used thermodynamic functions:

Eqs 3.1.19 and 3.1.20 indicate that the chemical potential can be calculated by differentiating any

of the thermodynamic functions For example, the first expression of Eq 3.1.20 can be used to

determine µ ij by differentiation of Eq 3.1.11

3.1 THERMODYNAMIC BACKGROUND 25

The chemical potential is of fundamental importance because the requirements for phase librium can be stated concisely in terms of the chemical potential Standard thermodynamicarguments [123, Section 2-2] lead to the statement of chemical equilibrium,

equi-µ ij = µ ik , i = 1, n c , j = 1, n p , k = 1, n p , k = j. (3.1.21)

where µ ij is the chemical potential of component i in phase j Eq 3.1.21 states that at a given

temperature and pressure, the compositions of any equilibrium phases that form must be such thatthe chemical potential of each component is the same in all the phases present

Chemical potential has units of energy per mole, units that carry much less physical informationfor most people than does pressure As a result, phase equilibrium calculations frequently make

use of fugacity, which is scaled to have units of pressure The fugacity function is defined by analogy with the behavior of an ideal gas The Gibbs function, G ideal, for an ideal gas at constanttemperature is given by

Thus, the fugacity function takes whatever values are required to reproduce the nonideal Gibbs

function At constant temperature and composition, Eq 3.1.18 indicates that dG = V dP tution for dG in Eq 3.1.23, subtraction of RT dP/P from both sides, and integration gives

Substi-ln

f P

= ln

f P

where the subscript ref indicates a reference state To complete the definition of fugacity, therefore,

a reference state must be chosen Again, the limit of zero pressure is useful because gas behaviorbecomes ideal, and therefore,

lim

P →0

f P

where φ is the fugacity coefficient, a dimensionless version of the fugacity scaled by the pressure.

The integrand of Eq 3.1.26 is just the departure from ideal gas volumetric behavior at a givenpressure Hence the integral is a summation of the departures from ideality If the gas is ideal, the

fugacity equals the pressure, and ln(f /P ) = 0.

Eq 3.1.26 gives the fugacity for a single component or a mixture as a whole For individual

components in a phase, however, the quantity of interest is the partial fugacity, ˆ f ij, which is definedby

ˆij = ˆf ik , i = 1, n c , j = 1, n p , k = 1, n p , k = j (3.1.30)Thus, the task that remains is to calculate the partial fugacity of each component in each phasefrom a description of the volumetric behavior of the mixture.

Equations of state in common use for gas/oil systems give the pressure as a function of temperature,

volume and composition, a representation that is known as a pressure–explicit equation of state,

P = P (T , V, n1, n2, · · ·). (3.2.1)For equations of state of the form of Eq 3.2.1, it is convenient to differentiate the Helmholtzfunction to obtain the chemical potential or partial fugacity According to Eq 3.1.18, the chemical

potential of phase j is given by

Eq 3.2.4 shows why use of the Helmholtz function is convenient If P is given as a function of

volume, temperature, and composition, as it is in a pressure-explicit equation of state, that functioncan be inserted directly in Eq 3.2.4 Integration of Eq 3.2.4 gives

3.3 PHASE EQUILIBRIUM FROM AN EQUATION OF STATE 27

V d(nV ) on the right side of Eq 3.2.5 yields n(A − A ref ) = n

V

P − RT V

If V ref is chosen to be large enough that the mixture behaves as an ideal gas, then the pressure in

the last integral on the right side of Eq 3.2.6 is P = RT /V Substitution for P in Eq 3.2.6 gives

the final expression for the Helmholtz function,

d(V ) + nRT ln RT

If the variable of integration is changed to the total volume, V t = nV , and the differentiation in

Eq 3.2.2 is carried out, the result, after some rearrangement, is

to evaluate the partial fugacity of a component in the mixture Michelsen and Mollerup [80, p 60]have pointed out that a more convenient version of Eq 3.2.8 can be obtained by interchanging theorder of integration and differentiation, which yields

Here again the relationship between P , V , T , and composition given by the equation of state

provides the information needed to evaluate component partial fugacity coefficients

The use of an equation of state for phase equilibrium calculations has its roots in the work of vander Waals [122] , who proposed in his 1873 PhD dissertation the following equation of state for a

pure component i:

P = RT

V − b i − a i

In Eq 3.3.1, the parameters a i and b irepresent in a simple way the forces of attraction and repulsion

between molecules Repulsion is represented by b i As the molar volume, V , approaches b i, the

pressure increases rapidly, so b ican be viewed as an estimate of molecular volume Attractive forces

between molecules are represented by a i The negative sign of the second term on the right side of

Eq 3.2.7 indicates that the pressure is reduced by attractions between molecules Attraction forces

must be present if a fluid is to form a liquid phase In the limit of large molar volume, molecules aretoo far apart for repulsions or attractions to have any effect, and van der Waals equation reduces tothe appropriate limit, the ideal gas law Eq 3.3.1 is the simplest equation of state that representsthe limiting behavior of both liquid and vapor phases [1].

The attraction and repulsion parameters can be evaluated from the behavior of the equation

of state at the critical point, the pressure and temperature at which vapor and liquid phases have identical compositions and properties In addition, a fluid at its critical pressure, P c, and critical

b i = RT ci

8P ci

Eq 3.3.1 is written for a pure component, but it can be applied to a mixture if suitable average

values, a mj and b mj , can be calculated for the j th phase, for example For van der Waals equation,the mixing rules always used are [126]:

where x ij is the mole fraction of component i in phase j, and a i , and b i are the attraction and

repulsion parameters for component i.

Phase equilibrium calculations can be performed with van der Waals equation of state, andwhile it gives qualitatively reasonable predictions, it does not give good quantitative predictions

of phase compositions and densities As a result, many investigators have proposed modifications

to Eq 3.3.1 designed to improve agreement with experimental observations [126] Most of theequations of state in widespread use for gas/oil systems are based on van der Waals equation, butwith more complex representations of the attraction terms An example of a widely used equation

of state is the Peng-Robinson equation [99] ,

where the attraction and repulsion parameters, a mj and b mj, for a mixture with the composition

of phase j are given by the mixing rules,

3.3 PHASE EQUILIBRIUM FROM AN EQUATION OF STATE 29

where δ ik is a binary interaction parameter chosen to improve the agreement between calculated and

measured phase compositions In Eqs 3.3.9 and 3.3.11, the attraction and repulsion parameters

for the individual components, a i and b i, are given by

where the numerical coefficients in Eqs 3.3.12 and 3.3.13 are obtained by evaluation of Eq 3.3.7

at the critical point with the conditions of Eq 3.3.2 In the Peng-Robinson equation, the tion term has been modified to include temperature dependence and information about the vapor

attrac-pressure of a component through the function α i,

α i=



1 + α ∗ i

Table 3.1 reports values of T ci , P ci , and ω i for a variety of components commonly present in

gas/oil systems For the hydrocarbons ethane and larger, P ci decreases and T ciincreases as the size

of the molecule increases ω i also increases with the size of the molecule All the values of ω i are

positive because 10P i sat < P ci The positive values of ω i indicate that attractions between pairs ofhydrocarbon molecules increase with the size of the molecule

Values of a i and b i for the normal alkanes are also reported in Table 3.1 The repulsion

pa-rameter, b i, can be viewed as an excluded volume, which increases with the size of the molecule

Table 3.1: Critical Properties and Acentric Factors [105]

The attraction parameter, a i, also increases with the size of the hydrocarbon molecule The effect

of acentric factor on the magnitude of the attraction term can be seen by examining Eq 3.3.15,

which is the equation of a parabola with a maximum at ω i ≈ 2.9 The increase in ω i with thesize of the molecule, which arises from the estimates of attractions between pairs of molecules inthe pure saturated liquid, also increases the magnitude of the attraction term The temperaturedependence of the attraction term can be seen easily in Eq 3.3.14, which indicates that the value

of α i decreases with increasing temperature and is largest for temperatures far below the criticaltemperature As temperature increases, molecules move at higher velocities, and the attractiveforces have smaller effect

In the expression for the attraction parameter of the mixture, Eq 3.3.8, (aα) m includes one

additional parameter, known as a binary interaction parameter, δ ik, for each pair of components(although values for pairs of hydrocarbon components that have similar size are often taken to bezero) The use of binary interaction parameters is an acknowledgement that the simple mixingrules do not represent fully the attractions between dissimilar molecules The introduction ofbinary interaction parameters provides additional empirical constants that can be used to improvethe accuracy of phase equilibrium calculations (if enough phase behavior data are available todetermine all the additional parameters) Calculated phase compositions are often quite sensitive

to the values of δ ij (particularly for pairs of large and small components), and hence, those valuescan be selected to make the equation of state predictions match measured phase equilibrium datafor two-component systems The resulting representation of attractions based on binary systemscan then be used to predict behavior of systems containing more than two components Table 3.2

reports values of δ ij used in the phase equilibrium calculations performed for a variety of systems

Equations of state in common use for gas/ oil systems give the pressure as a function of temperature,... [126] Most of theequations of state in widespread use for gas/ oil systems are based on van der Waals equation, butwith more complex representations of the attraction terms An example of a widely... pressure–explicit equation of state,

P = P (T , V, n1, n2, · · ·). (3.2.1)For equations of state of the form of Eq 3.2.1,