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 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

iv CONTENTS

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

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]. 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 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 CO2 sequestration, as it is sometimes called. These processes are intended to reduce

the 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 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 CO2/hydrocarbon mixtures is also observed at temperatures about 50 C and pressures

near the critical pressure of CO2 [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