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

Theory of Gas Injection Processes
Franklin M. Orr, Jr.
Stanford University
Stanford, California
2005
Library of Congress Cataloging-in-Publication Data
Orr, Franklin M., Jr.
Theory of Gas Injection Processes / Franklin M. Orr, Jr.
Bibliography: p.
Includes index.
ISBN xxxxxxxxxxx
1. Enhanced recovery of oil. I. Title. XXXXX XXXXX
c
2005 Franklin M. Orr, Jr.
All rights reserved. No part of this book may be reproduced, in any form or by an means, without
permission in writing from the author.
To Susan
.
i
Preface
This book is intended for graduate students, researchers, and reservoir engineers who want to
understand the mathematical description of the chromatographic mechanisms that are the basis
for gas injection processes for enhanced oil recovery. Readers familiar with the calculus of partial
derivatives and properties of matrices (including eigenvalues and eigenvectors) should have no
trouble 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 book

will 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 is
expended to illuminate the mathematical structure of the flow problems considered. In addition, I
hope some of the material will be of interest to mathematicians who will find that many interesting
questions of mathematical rigor remain to be investigated for multicomponent, multiphase flow in
porous media.
Readers already familiar with the subject of this book will recognize the work of many students
and colleagues with whom I have been privileged to work in the last twenty-five years. I am
much 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’s
Bellaire 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 research
engineer at that laboratory. Gary Pope and Larry Lake were also part of that Shell group of future
academics who have made extensive use of the theoretical approach used here in their work with
students at the University of Texas. I have benefited greatly from many conversations with them
over the years about the material discussed here. Thormod Johansen patiently explained to me
his mathematician’s point of view concerning the Riemann problems considered in detail in this
book. All of them have contributed substantially to the development of a rigorous description of
multiphase, 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 ex-
planations given here in a course taught for graduate students since 1985. Their questions over the
years 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 exceptionally
talented group of graduate students: Wes Monroe, Kiran Pande, Jeff Wingard, Russ Johns, Birol
Dindoruk, Yun Wang, Kristian Jessen, Jichun Zhu, and Pavel Ermakov. Wes Monroe obtained the
first four-component solutions for dispersion-free flow in one dimension. Kiran Pande solved for
the interactions of phase behavior, two-phase flow, and viscous crossflow. Jeff Wingard considered
problems with temperature variation and three-phase flow. Russ Johns and Birol Dindoruk greatly
extended our understanding of flow of four or more components with and without volume change
on mixing. Yun Wang extended the theory to systems with an arbitrary number of components,

and Kristian Jessen, who visited for six months with our research group during the course of his
PhD work at the Danish Technical University, contributed substantially to the development of
efficient 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 first
solutions for arbitrary numbers of components with volume change on mixing. Jichun Zhu and
Pavel Ermakov contributed substantially to the derivation of compact versions of key proofs. Birol
Dindoruk, Russ Johns, Yun Wang, and Kristian Jessen kindly allowed me to use example solutions
ii
and figures from their dissertations. This book would have little to say were it not for the work of
all those students. Marco Thiele and Rob Batycky developed the streamline simulation approach
for gas injection processes. Their work allows the application of the one-dimensional descriptions of
the 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 for
teaching 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 those
chapters, 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 of
Science, Technology and Medicine for providing a quiet place to write during the fall of 2000 and
for reading an early draft of the manuscript. I thank my colleagues Margot Gerritsen and Khalid
Aziz, Stanford University, for their careful readings of the draft manuscript. They and the other
faculty of the Petroleum Engineering Department at Stanford have provided a wonderful place to
try to understand how gas injection processes work. The students and faculty associated with
the SUPRI-C gas injection research group, particularly Martin Blunt, Margot Gerritsen, Kristian
Jessen, Hamdi Tchelepi, and Ruben Juanes, and our dedicated staff, Yolanda Williams and Thuy
Nguyen, have done all the useful work in that quest, of course. It is my pleasure to report on a
part of that research effort here.
And finally, I thank Mark Walsh for asking questions about the early work that caused us to

think about these problems in a whole new way. I also thank an anonymous proposal reviewer who
said that the problem of finding analytical solutions to multicomponent, two-phase flow problems
could not be solved and even if it could, the solutions would be of no use. That challenge was too
good to pass up.
The financial support for the graduate students who contributed so much to the material pre-
sented here was provided by grants from the U.S. Department of Energy, and by the member
companies of the Stanford University Petroleum Research Institute Gas Injection Industrial Affili-
ates program. That support is gratefully acknowledged.
Lynn Orr
Stanford, California
March, 2005
Contents
Preface i
1 Introduction 1
2 Conservation Equations 5
2.1 GeneralConservationEquations 5
2.2 One-DimensionalFlow 10
2.3 PureConvection 12
2.4 NoVolumeChangeonMixing 13
2.5 ClassificationofEquations 14
2.6 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Convection-DispersionEquation 15
2.8 AdditionalReading 17
2.9 Exercises 17
3 Calculation of Phase Equilibrium 21
3.1 ThermodynamicBackground 21
3.1.1 CalculationofThermodynamicFunctions 22
3.1.2 ChemicalPotentialandFugacity 24
3.2 CalculationofPartialFugacity 26
3.3 Phase Equilibrium from an Equation of State . . . . . . . . . . . . . . . . . . . . . . 27

3.4 FlashCalculation 31
3.5 PhaseDiagrams 34
3.5.1 BinarySystems 34
3.5.2 TernarySystems 35
3.5.3 QuaternarySystems 37
3.5.4 ConstantK-Values 38
3.6 AdditionalReading 40
3.7 Exercises 40
4 Two-Component Gas/Oil Displacement 43
4.1 SolutionbytheMethodofCharacteristics 44
4.2 Shocks 48
4.3 VariationsinInitialorInjectionComposition 56
4.4 VolumeChange 61
iii
iv CONTENTS
4.4.1 FlowVelocity 62
4.4.2 CharacteristicEquations 62
4.4.3 Shocks 63
4.4.4 ExampleSolution 64
4.5 ComponentRecovery 67
4.6 Summary 69
4.7 AdditionalReading 70
4.8 Exercises 71
5 Ternary Gas/Oil Displacements 73
5.1 CompositionPaths 75
5.1.1 EigenvaluesandEigenvectors 78
5.1.2 Tie-LinePaths 81
5.1.3 Nontie-LinePaths 81
5.1.4 SwitchingPaths 87
5.2 Shocks 90

5.2.1 Phase-ChangeShocks 90
5.2.2 ShocksandRarefactionsbetweenTieLines 92
5.2.3 Tie-LineIntersectionsandTwo-PhaseShocks 97
5.2.4 EntropyConditions 98
5.3 ExampleSolutions:VaporizingGasDrives 99
5.4 ExampleSolutions:CondensingGasDrives 106
5.5 StructureofTernaryGas/OilDisplacements 110
5.5.1 EffectsofVariationsinInitialComposition 117
5.6 Multicontact Miscibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.6.1 VaporizingGasDrives 118
5.6.2 CondensingGasDrives 119
5.6.3 Multicontact Miscibility in Ternary Systems . . . . . . . . . . . . . . . . . . . 119
5.7 VolumeChange 120
5.8 ComponentRecovery 127
5.9 Summary 129
5.10AdditionalReading 130
5.11Exercises 131
6 Four-Component Displacements 135
6.1 Eigenvalues,Eigenvectors,andCompositionPaths 135
6.1.1 TheEigenvalueProblem 135
6.1.2 CompositionPaths 137
6.2 SolutionConstructionforConstantK-values 144
6.3 SystemswithVariableK-values 149
6.4 Condensing/VaporizingGasDrives 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 VolumeChange 172

6.7 Summary 176
6.8 AdditionalReading 176
6.9 Exercises 177
7 Multicomponent Gas/Oil Displacements 179
byF.M.Orr,Jr.andK.Jessen 179
7.1 KeyTieLines 180
7.1.1 InjectionofaPureComponent 180
7.1.2 MulticomponentInjectionGas 183
7.2 SolutionConstruction 185
7.2.1 FullySelf-SharpeningDisplacements 193
7.2.2 Solution Routes with Nontie-line Rarefactions . . . . . . . . . . . . . . . . . . 198
7.3 SolutionConstruction:VolumeChange 201
7.4 DisplacementsinGasCondensateSystems 204
7.5 CalculationofMMPandMME 206
7.6 Summary 210
7.7 AdditionalReading 212
8 Compositional Simulation 213
byF.M.Orr,Jr.andK.Jessen 213
8.1 NumericalDispersion 213
8.2 ComparisonofNumericalandAnalyticalSolutions 215
8.3 SensitivitytoNumericalDispersion 221
8.4 CalculationofMMPandMME 230
8.5 Summary 237
8.6 AdditionalReading 238
Nomenclature 241
Bibliography 244
Appendix A: Entropy Conditions in Ternary Systems 255
Appendix B: Details of Gas Displacement Solutions 266
Index 280
vi CONTENTS

Chapter 1
Introduction
When a gas mixture is injected into a porous medium containing an oil (another mixture of hy-
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. The
liquid 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 and
viscosity). As those phases encounter the oil present in the reservoir or more injected gas, new
mixtures form and come to equilibrium. The result is a set of component separations that occur
during flow, with light components propagating more rapidly than heavy ones. These separations
are 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 the
mathematical representation of those chromatographic separations and the resulting compositional
changes that occur in such processes.
Gas injection processes are among the most widely used of enhanced oil recovery processes
[62, 117]. CO
2
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 a
very large project is underway in the Prudhoe Bay field in Alaska [74]. At Prudhoe Bay, dry gas is
injected into the upper portion of the reservoir to vaporize light hydrocarbon liquids and remaining
oil, 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 of
Canadian 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 behavior
and 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 mathematical
description of the physical mechanisms that produce high local displacement efficiency. While the
approach 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 displacement
pressures required for miscible displacement and a very efficient semianalytical calculation method
for solving one-dimensional compositional displacement problems.
While the focus here is on gas/oil displacements in porous media, the ideas, and the math-
ematical approaches apply to physical processes that range from flow of traffic on a highway to
chemical reactions in a tubular reactor to compressible fluid flow. Chapter 1 of First Order Partial
1
2 CHAPTER 1. INTRODUCTION
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 mixture
separate 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 the
carrier 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 to
leaching of minerals [9]. Many of these ideas also apply to the area of geologic storage of carbon
dioxide [85], or CO
2
sequestration, as it is sometimes called. These processes are intended to reduce
the rate of increase of the concentration of CO
2
in the atmosphere by injecting CO
2
that would
otherwise be released to the atmosphere into subsurface formations such as deep saline aquifers or
coalbeds [139].
In the area of enhanced oil recovery, theoretical descriptions of the displacement of oil by water
containing polymer and displacement of oil by surfactant solutions are closely linked to the theory

described 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 injection
fluid that lower interfacial tension between oil and water. Effects of volume change as components
transfer between phases were not considered in that work, a completely reasonable assumption for
the 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 four
component systems. A detailed review by Johansen [50] summarizes the relevant papers published
through 1990. Lake’s [62] comprehensive description of enhanced oil recovery also cites the large
body of work related to polymer and surfactant flooding processes.
This book applies the one-dimensional theory of multicomponent, multiphase flow to gas/oil
displacements. In Chapter 2, the appropriate material balance equations are derived, and the
assumptions that lead to the limiting cases explored in detail are stated. An introduction to the
representation of phase equilibria with an equation of state is given in Chapter 3. Chapter 4
considers two-phase flow of two components that are mutually soluble. When effects of volume
change 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 hence
its 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 three-component gas/oil displacements is developed in Chapter 5. The three-
component theory leads directly and rigorously to the ideas of “multicontact miscible” displacement
via condensing or vaporizing gas drives. Extensions of the analysis to systems with more than three
components are considered in Chapters 6 and 7. That treatment shows that there are important
features of gas injection processes that cannot be represented by three-component descriptions of
the 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
3

condensing/vaporizing gas drives, which turn out to be relevant to many gas injection projects now
underway in field applications. Chapter 7 extends the theory to systems with an arbitrary number
of components in the oil or the injection gas. Chapter 7 also describes how the one-dimensional
theory can be applied to create a rigorous method for calculating the so-called minimum miscibility
pressure, the displacement pressure required to achieve high displacement efficiency, for multi-
component systems. Thus, all the mathematical effort does pay off with a calculation method of
considerable practical value.
Effects of dispersive mixing are ignored in the development of the theory presented in Chapters
4-7, 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 and
adversely 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 it
describes how displacement behavior changes when dispersion also acts. Chapter 8 also explains
when and why numerical schemes that have been proposed for calculating the minimum miscibility
pressure fail to give accurate estimates.
Flow is never one-dimensional in actual field-scale gas injection projects, and hence, many
additional 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 the
flow in heterogeneous reservoirs [119, 121, 8, 120, 16, 43]. The resulting compositional streamline
approach 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, three-
phase flow of CO
2
/hydrocarbon mixtures is also observed at temperatures about 50 C and pressures
near the critical pressure of CO
2

[25, 94, 86, 61]. The approach used here to study the mechanisms
of gas/oil displacements has also been applied to the flow of three immiscible phases [137, 24, 27].
LaForce and Johns obtained solutions for three-phase flow for ternary systems with composition
variation 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 chromato-
graphic separations that occur in both one-dimensional and multidimensional flows. Hence the
analysis 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 high
displacement 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.
4 CHAPTER 1. INTRODUCTION
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 with
varying 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 the
transport of each component. For gas/oil systems, the physical mechanisms that are most important
are:
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). Dispersion
during 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 state
the assumptions required to reduce the general material balance equations to the special cases
considered in detail below. Effects of chemical reactions are not included in the flow problems
considered here, nor are effects of adsorption or temperature variation. Derivations that include

such effects are given by Lake [62].
2.1 General Conservation Equations
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
Rate of change of
amount of compo-
nent i in V (t)
=
Net rate of in-
flow of component
i into V (t) due to
flow of phases
+
Net rate of in-
flow of component
i into V (t) due
to hydrodynamic
dispersion
5
6 CHAPTER 2. CONSERVATION EQUATIONS
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)andthe
net inflow that arises from the diffusion-like process of hydrodynamic dispersion.
Accumulation Terms
The amount of phase j present in a differential element of volume, dV is
⎧
⎨
⎩
Moles of
phase j

in dV
⎫
⎬
⎭
= φρ
j
S
j
dV, (2.1.1)
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
⎧
⎨
⎩
Moles of com-
ponent i in
phase j in dV
⎫
⎬
⎭
= φx
ij
ρ
j

S
j
dV, (2.1.2)
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
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Total moles
of compo-
nent i in
dV
⎫
⎪
⎪
⎬
⎪
⎪
⎭
= φ
n
p


j=1
x
ij
ρ
j
S
j
dV. (2.1.3)
Integration of Eq. 2.1.3 gives the total amount of component i in the control volume, V (t),
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Total moles
of compo-
nent i in
V (t)
⎫
⎪
⎪
⎬
⎪
⎪
⎭
=


V (t)
φ
n
p

j=1
x
ij
ρ
j
S
j
dV, (2.1.4)
and hence the rate of accumulation of component i in V is
Rate of change of
moles of compo-
nent i in V
=
d
dt

V
φ
n
p

j=1
x
ij
ρ

j
S
j
dV. (2.1.5)
Convection Terms
Part of the accumulation of component i in V is due to the transport of component i in the
phases that flow in and out of the control volume. At any differential element of area dS,the
convective molar flux (moles of component i per unit area per unit time) of component i in the j
th
phase is
⎧
⎨
⎩
Molar flux of
component i in
phase j
⎫
⎬
⎭
= x
ij
ρ
j
v
j
, (2.1.6)
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
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Rate of inflow
of component
i in phase j
across dS
⎫
⎪
⎪
⎬
⎪
⎪
⎭
= −n ·x
ij
ρ
j
v
j
dS, (2.1.7)
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,toobtain
Net rate of inflow
of component i by
convection
= −

S(t)
n ·
n
p

j=1
x
ij
ρ
j
v
j
dS, (2.1.8)
Dispersion Terms
Diffusion and hydrodynamic dispersion are independent physical mechanisms. Diffusion can
occur due to concentration gradients in the absence of any flow, and the local fluctuations in
flow velocity that cause hydrodynamic dispersion [26] would occur even if diffusion were absent.
Nevertheless, the mathematical representations of the two mechanisms are essentially the same,
and both contributions are generally lumped together [100]. Hence the flux due to diffusion and
dispersion is taken to be given by [5]
⎧
⎨
⎩

Dispersive flux
of component i
in phase j
⎫
⎬
⎭
= −φ


K
ij
·∇ρ
j
x
ij
, (2.1.9)
where the dispersion tensor,


K
ij
, for component i in phase j includes contributions due to diffusion
and dispersion. The diffusion contribution is represented by a molecular diffusion coefficient, D
ij
,
and the dispersion contribution is usually taken to be linear in the local displacement velocity.
Typical forms for the longitudinal and transverse dispersion coefficients are given by Bear [5]:
(K

)

ij
= D
ij
+
(α
j
−α
tj
)
φS
j
|v
j
|
(v

)
2
j
+
α
tj
|v
j
|
φS
j
, (2.1.10)
(K
t

)
ij
=
(α
j
−α
tj
)
φS
j
|v
j
|
|(v

)(v
t
)|, (2.1.11)
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
8 CHAPTER 2. CONSERVATION EQUATIONS
Net rate of in-
flow of component
i due to hydrody-
namic dispersion
=

S
n · φ

n
p

j=1


K
ij
·∇ρ
j
x
ij
dS. (2.1.12)
Continuity Equation
The accumulation, convection, and dispersion terms can be combined to yield an integral ma-
terial balance for component i,
d
dt

V (t)
φ
n
p

j=1
x
ij
ρ
j
S

j
dV = −

S
n ·
n
p

j=1
x
ij
ρ
j
v
j
dS +

S
n · φ
n
p

j=1
ρ
j


K
ij
·∇x

ij
dS. (2.1.13)
Eq. 2.1.13 is a balance on component i in the full control volume, V (t). If information about
the spatial distribution of component i is needed, then a differential material balance known as
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

V (t)
GdV =

V (t)
∂G
∂t
dV +

S(t)
n ·v
s
GdS. (2.1.14)
The left side of Eq. 2.1.14 is the total rate of change of G in V (t). The first term on the right
side of Eq. 2.1.14 represents the change due to the local rate of change of G, and the second term
represents the rate of change due to the surface, S(t), overtaking G as it moves. In the examples
considered here, we will choose the velocity, v
s
, to be zero. Hence for a stationary control volume,

Eq. 2.1.14 reduces to
d
dt

V
GdV =

V
∂G
∂t
dV. (2.1.15)
The divergence theorem relates volume integrals to surface integrals. For a vector quantity,

H,
it states that

V
∇·

HdV =

S
n ·

HdS. (2.1.16)
Application of the Reynolds transport and divergence theorems gives a set of volume integral
material balances, one for each component,

V
⎧

⎨
⎩
∂
∂t
φ
n
p

j=1
x
ij
ρ
j
S
j
+ ∇·
n
p

j=1
x
ij
ρ
j
v
j
−∇·φ
n
p


j=1


K
ij
·∇ρ
j
x
ij
⎫
⎬
⎭
dV =0,i=1,n
c
. (2.1.17)
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 negative
portion of the original control volume. If so, Eq. 2.1.17 would not be satisfied for the new control
volume. Hence, the final form of the continuity equations for multicomponent, multiphase flow is
∂
∂t
φ
n
p

j=1
x

ij
ρ
j
S
j
+ ∇·
n
p

j=1
x
ij
ρ
j
v
j
−∇ · φ
n
p

j=1


K
ij
·∇ρ
j
x
ij
=0,i=1,n

c
. (2.1.18)
To complete specification of the flow problem, a number of additional functions and conditions
must be available. The flow velocity is the most important part of Eq. 2.1.18 yet to be determined,
because it controls the convective part of the flow. In principle, we could derive and solve a set
of balance equations for momentum, which must also be conserved. In practice, however, the
solution of the resulting Navier-Stokes equations for the detailed velocity distributions within the
pores 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 momentum
equations 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
v
j
= −
kk
rj
µ
j
(∇P
j
+ ρ
mj
g) , (2.1.19)
where k is the permeability, and k
rj
, µ

j
, ρ
mj
,andP
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 nonzero
interfacial 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 are
taken to be properties of the fluids and the porous medium that can be measured in independent
experiments.
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 that
flow can be driven by departures from that equilibrium. A similar assumption can be made about
chemical equilibrium . The time required for diffusion of components over the length scale of a

pore is often small enough compared to the time required for flow to change the compositions
significantly within the pore that it is reasonable to assume that fluids present are in chemical
10 CHAPTER 2. CONSERVATION EQUATIONS
equilibrium. If so, then the statement of chemical equilibrium gives an additional set of relations
between 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 chemical
potential 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 must
sum to one,
n
p

j=1
S
j

=1, (2.1.22)
as must the mole fractions in each of the phases,
n
c

i=1
x
ij
=1,j=1,n
p
. (2.1.23)
Finally, the functions that describe the properties of each phase and its relative permeability
must be given:
ρ
j
= ρ
j
(x
1j
,x
2j
, ,x
n
c
−1,j
,P
j
,T), (2.1.24)
µ
j

= µ
j
(x
1j
,x
2j
, ,x
n
c
−1,j
,P
j
,T), (2.1.25)
k
rj
= k
rj
(S
1
,S
2
, ,S
n
p
−1
), (2.1.26)
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 that
models the effects of convection, dispersion, and phase equilibrium, as the inventory of equations
and 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 relations
that specify capillary and phase equilibrium, Darcy’s law, and the mole fraction and saturation
summations. As Table 2.1 demonstrates, the number of equations exactly equals the number of
unknowns. That equality is required if a solution is to exist, though it does not guarantee that a
solution exists or is unique.
2.2 One-Dimensional Flow
Eq. 2.1.18 is complex enough that it must be solved numerically unless additional simplifying
assumptions 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
∂
∂t
φ
n
p

j=1
x
ij
ρ
j
S
j
+
∂
∂x
n
p

j=1


x
ij
ρ
j
v
j
− φK
ij
∂ρ
j
x
ij
∂x

=0,i=1,n
c
. (2.2.1)
2.2. ONE-DIMENSIONAL FLOW 11
Table 2.1: Inventory of Equations and Unknowns
Unknowns Equations
Variable Number Equation Eq. Number
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)
Pressures, P
j
n
p
Capillary pressure 2.1.20 n
p
−1
Velocities, v
j
n
p
Darcy’s law 2.1.19 n
p
Saturation sum 2.1.22 1
Mole fraction sum 2.1.23 n
p
Total n
p
(n
c
+3) Total n

p
(n
c
+3)
If capillary pressure differences are neglected, Eq. 2.2.1 can be simplified by eliminating the pressure
gradient from the expressions for the flow velocities. Phase flow velocities can then be written easily
in terms of fractional flow functions , f
j
defined by
v
j
= f
j
v = f
j
n
p

k=1
v
k
,j=1,n
p
. (2.2.2)
In Eq. 2.2.2 v is the total flow velocity , defined as the sum of the phase flow velocities, v
j
.In
one-dimensional flow in the absence of capillary pressure Darcy’s law (Eq. 2.1.19) becomes
v
j

= −
kk
rj
µ
j

∂P
∂x
+ ρ
mj
g sin θ

,j=1,n
p
. (2.2.3)
In Eq. 2.2.3, θ is the dip angle measured as the angle between the flow direction and a horizontal
line. An expression for the fractional flow function, f
j
, is obtained by eliminating the pressure
gradient from Eqs. 2.2.2 and Eq. 2.2.3. The expression for the pressure gradient can be obtained
from any one of the n
p
expressions for the phase flow velocities (Eq. 2.2.3). For the j
th
phase, for
example,
∂P
∂x
= −
µ

j
v
j
kk
rj
− ρ
mj
g sin θ. (2.2.4)
Substitution of Eq. 2.2.4 into Eq. 2.2.3 written for the n
th
phase gives
v
n
=
k
rn
k
rj
µ
j
µ
n
v
j
+
kk
rn
µ
n
g sin θ(ρ

mj
−ρ
mn
). (2.2.5)
Substitution of Eq. 2.2.5 into Eq. 2.2.2 gives the expression for the fractional flow of phase j [62] ,
f
j
=
k
rj
/µ
j

n
p
n=1
(k
rn
/µ
n
)

1 −
kg sin θ
v
n
p

n=1
k

rn
µ
n
(ρ
mj
−ρ
mn
)

. (2.2.6)
Substitution of Eq. 2.2.2 into Eq. 2.2.1 gives the one-dimensional version of the convection-
dispersion equation for multicomponent, multiphase flow,
12 CHAPTER 2. CONSERVATION EQUATIONS
∂
∂t
φ
n
p

j=1
x
ij
ρ
j
S
j
+
∂
∂x
n

p

j=1

x
ij
ρ
j
f
j
v − φK
ij
∂ρ
j
x
ij
∂x

=0,i=1,n
c
. (2.2.7)
2.3 Pure Convection
If the effects of dispersion can be neglected, then Eq. 2.2.7 reduces to a set of equations that
describes the interaction of pure convection with equilibrium phase behavior,
∂
∂t
φ
n
p


j=1
x
ij
ρ
j
S
j
+
∂
∂x
v
n
p

j=1
x
ij
ρ
j
f
j
=0,i=1,n
c
. (2.3.1)
It is also convenient to write Eq. 2.3.1 in dimensionless form based on the following scaled variables:
τ =
v
inj
t
φL

, (2.3.2)
ξ =
x
L
, (2.3.3)
v
D
=
v
v
inj
, (2.3.4)
ρ
jD
=
ρ
j
ρ
inj
. (2.3.5)
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 displace
one 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

thetimerequiredtoflowlengthL 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
∂
∂τ
n
p

j=1
x
ij
ρ
jD
S
j
+
∂
∂ξ
v
D
n
p

j=1
x
ij
ρ
jD

f
j
=0,i=1,n
c
. (2.3.6)
The notation of Eq. 2.3.6 can be simplified by defining two additional functions, G
i
and H
i
,as
G
i
=
n
p

j=1
x
ij
ρ
jD
S
j
, (2.3.7)
and
H
i
= v
D
n

p

j=1
x
ij
ρ
jD
f
j
. (2.3.8)
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
∂G
i
∂τ
+
∂H
i
∂ξ
=0,i=1,n
c
. (2.3.9)
The local flow velocity , v
D

, in the definition of H
i
(Eq. 2.3.8) can vary with spatial location
because volume is not conserved if components change volume as they transfer between phases.
If, for example, CO
2
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 local
flow velocity can vary substantially over the displacement length [22, 82, 19]. Thus, for some gas
displacements, it will be important to include the effects of volume change on mixing .
2.4 No Volume Change on Mixing
If the displacement pressure is high enough, then the volume occupied by a component in the gas
phase may not change greatly when that component transfers to the liquid phase. Components in
the liquid/liquid systems that describe surfactant flooding processes also exhibit minimal volume
change on mixing. In such systems, it is reasonable to assume that the partial molar volume of
each component is a constant (independent of composition or phase) and hence that ideal mixing
applies. In other words, the volume occupied by a given amount of a component is constant no
matter 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
i in phase j is

c
ij
=
x
ij
/ρ
ci

n
c
k=1
x
kj
/ρ
ck
. (2.4.1)
The molar density of a phase is
ρ
j
=

n
c

i=1
x
ij
/ρ
ci


−1
. (2.4.2)
Comparison of Eqs. 2.4.1 and 2.4.2 indicates that
ρ
ci
c
ij
= ρ
j
x
ij
. (2.4.3)
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 volume
change on mixing,
∂C
i
∂τ
+
∂F
i
∂ξ

=0,i=1,n
c
− 1. (2.4.4)
where C
i
is an overall volume fraction of component i given by
C
i
=
n
p

j=1
c
ij
S
j
, (2.4.5)
14 CHAPTER 2. CONSERVATION EQUATIONS
and F
i
is an overall fractional volumetric flow of component i given by
F
i
=
n
p

j=1
c

ij
f
j
. (2.4.6)
2.5 Classification of Equations
The convection-dominated conservation equations, Eqs. 2.3.6 and 2.4.4 are systems of first order
partial differential equations. Those equations have the general form,
P (x, t, z)
∂z
∂t
+ Q(x, t, z)
∂z
∂x
= R(x, t, z). (2.5.1)
Equations like 2.5.1 are called quasilinear.IfP, Q,andR 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, quasilinear

systems of first order equations.
2.6 Initial and Boundary Conditions
Before Eqs. 2.3.9 and 2.4.4 can be solved, initial and boundary conditions must be imposed. In the
chapters 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
init
i
, 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
inj
i
,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.
2.7 Convection-Dispersion Equation
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
∂C
∂x
− K

∂
2
C
∂x
2
=0, (2.7.1)
where C is the volume fraction of one component, and K

is the longitudinal dispersion coefficient,
assumed here to be independent of composition. If Eq. 2.7.1 is made dimensionless with the scaled
length and time given in Eqs. 2.3.2 and 2.3.3, the result is
∂C
∂τ
+
∂C
∂ξ
−
1
Pe
∂
2
C
∂ξ
2
=0, (2.7.2)

where Pe = vL/φK

is the Peclet number . The Peclet number is a ratio of a characteristic time
for dispersion, L
2
/K

, 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)=1is
injected for τ>0, the solution is [12]
C(ξ,τ)=
1
2
erfc

√
Pe(ξ − τ )
2
√
τ

+
1
2

exp(Peξ)erfc

√
Pe(ξ + τ )
2
√
τ

. (2.7.3)
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 Pe > 1000), however, the second term can be neglected. Hence, many
investigators have used the approximate solution,
C(ξ,τ)=
1
2
erfc

√
Pe(ξ −τ)
2
√
τ

. (2.7.4)
Fig. 2.1 is a plot of Eq. 2.7.3 for three Peclet numbers (Pe = 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 Pe

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