ELSEVIER SCIENTIFIC PUBLISHING COMPANY, Amsterdam, 1977. - 191
pages

Developments in Petroleum Science, 6

Over the past decade, the use of numerical reservoir simulation with high-speed electronic computers has gained wide acceptance throughout the petroleum industry for making engineering studies of a wide variety of oil and gas reservoirs throughout the world. These reservoir simulators have been designed for use by reservoir engineers who may possess little or no background in the numerical mathematics upon which they are based. Yet in spite of our best efforts to improve numerical methods so as to make reservoir simulators as reliable, efficient, and automatic as possible, the user of a simulator is constantly faced with a myriad of decisions that have nothing to do with the problem he really wants to solve. He must decide on various numerical questions not directly germane to the problem at hand. For example, he may have a choice among several simulators that use different numerical methods. He may have to pick an iteration method. He definitely will have to choose the grid spacing as part of the reservoir description, and probably will also have to select the time step size. And perhaps the biggest bugaboo of all is the choice of iteration parameters.

It is this engineer-user that I have had in mind while writing this book, one who wants to lea how to deal more effectively with the numerical decisions mentioned above. I hope he also has some curiosity about the inner workings of the black box that is a reservoir simulator, and I have tried to satisfy that curiosity, as well as to prepare him to read the literature, should he wish to study recent developments and future research in greater depth than I have been able to provide here.

The first chapter combines a review of some basic reservoir mechanics with the derivation of the differential equations that reservoir simulators are designed to solve. The next four chapters provide basic theory on the numerical solution of simple partial differential equations. The final chapter brings together this basic theory as it applies to the numerical solution of multidimensional, multiphase flow problems.

I have attempted to make this book as self-contained as possible. The reader is assumed to have some knowledge of partial differential equations and simple matrix algebra; additional mathematical tools are provided where needed. In developing the numerical theory, I have tried to serve the engineer’s needs better than do the standard textbooks on numerical analysis, which tend to be either too rigorous or too general. I have not attempted to be completely rigorous in the mathematical proofs, but I have included sufficient derivations so as to make the various mathematical arguments as plausible as possible.

But the engineer-user is not the only reader I had in mind in writing this book. The mathematician skilled in numerical analysis will, of course, find much material already familiar to him. However, the first chapter will introduce him to the basic principles of reservoir mechanics, and the remainder of the book will indicate to him those topics in numerical analysis that I consider significant in the numerical solution of reservoir flow problems.

Furthermore, I have included some material which appears not to be well known. For example, the section on successive overrelaxation methods discusses the effect of Neumann boundary conditions and the effect of anisotropy. whereas standard textbooks confine themselves to Dirichlet boundary conditions and isotropic problems. Finally, the material in the last chapter should be new to most numerical analysts, as it is quite special to the area of multiphase reservoir flow. It is my hope this book will provide food for thought leading to further progress in numerical reservoir simulation.

While much work is now being done on the application of variational methods to the solution of partial differential equations, little of this has reached the stage of practical application in reservoir simulation. Most practical reservoir simulators now in use are based on finite-difference methods. For this reason, only finitedifference methods are covered in this book.

The reader familiar with reservoir engineering will note a departure from a practice I feel is

all too common in the field, namely, the inclusion of numerical constants in equations involving flow. All the equations in this book are free from numerical constants (which are dependent on the units being used) and are valid for any consistent set of units. The use of dimension-free equations should become more common as the industry moves to the adoption of the SI (Systeme Inteational) standard of units, as is now being proposed. Accordingly, in the nomenclature following each chapter, I have specified the units of various quantities in the basic SI units of kilograms, meters, and seconds, together with the derived units of the newton for force (which equals kg * m/s2) and the pascal for pressure (which equals N/mZ). These form a consistent set of units. If the reader prefers, any other consistent set of units can be used, and the equations will still be correct.

The material in this book is based primarily on notes prepared for a series of lectures I was privileged to give at a NATO-sponsored Summer School on Hydrocarbon Reservoir Simulation by Computer, held in Milan, Italy, in May, 1969. I have given the same lectures within Exxon Production Research Company and for a Continuing Education Group of the Los Angeles Basin SPE Section in January, 1974. Many who have seen these lecture notes have urged me to enlarge and publish them. I felt that the original notes were somewhat incomplete and so added several sections. The most significant additions were the section (in Chapter 1) on alteative differential equations for two-phase flow, all of Chapter 4 on solution of hyperbolic problems, the section (in Chapter 5 ) on relaxation methods, and the sections (in Chapter 6) on sequential solution methods, semi-implicit mobility, and well rates.

I am indebted to the management of Exxon Production Research Company for permission to publish this book and for the encouragement they gave. In particular, I want to thank C.C. Mattax, Manager of the Reservoir Divison, for his encouragement and many helpful suggestions about the revision of the original lecture notes. Thanks are due, also, to J.W. Watts and R.P. Kendall, of EPR, for the helpful discussions I had with them while writing the section on relaxation methods. Discussions with J.G. Hillestad and H.L. Stone, of EPR, also were of great help.

I owe much to the skill of Dodi Fenner for the preparation of the figures, to Winn Alms for typing the original lecture notes and part of the book manuscript, and to Altha Frazier and the Word Processing Group of EPR for typing the major portion of the manuscript for the book. I owe a large debt of gratitude to John Colby, Supervising Editor at EPR, for carefully editing the first draft of the book and checking the printer’s proofs.

Finally, I would like to dedicate this book to the three women in my life, who have provided inspiration to me and many others around them: to my mother, Ida, of blessed memory; to my wife, Ruth; and to my daughter, Caren.

Developments in Petroleum Science, 6

Over the past decade, the use of numerical reservoir simulation with high-speed electronic computers has gained wide acceptance throughout the petroleum industry for making engineering studies of a wide variety of oil and gas reservoirs throughout the world. These reservoir simulators have been designed for use by reservoir engineers who may possess little or no background in the numerical mathematics upon which they are based. Yet in spite of our best efforts to improve numerical methods so as to make reservoir simulators as reliable, efficient, and automatic as possible, the user of a simulator is constantly faced with a myriad of decisions that have nothing to do with the problem he really wants to solve. He must decide on various numerical questions not directly germane to the problem at hand. For example, he may have a choice among several simulators that use different numerical methods. He may have to pick an iteration method. He definitely will have to choose the grid spacing as part of the reservoir description, and probably will also have to select the time step size. And perhaps the biggest bugaboo of all is the choice of iteration parameters.

It is this engineer-user that I have had in mind while writing this book, one who wants to lea how to deal more effectively with the numerical decisions mentioned above. I hope he also has some curiosity about the inner workings of the black box that is a reservoir simulator, and I have tried to satisfy that curiosity, as well as to prepare him to read the literature, should he wish to study recent developments and future research in greater depth than I have been able to provide here.

The first chapter combines a review of some basic reservoir mechanics with the derivation of the differential equations that reservoir simulators are designed to solve. The next four chapters provide basic theory on the numerical solution of simple partial differential equations. The final chapter brings together this basic theory as it applies to the numerical solution of multidimensional, multiphase flow problems.

I have attempted to make this book as self-contained as possible. The reader is assumed to have some knowledge of partial differential equations and simple matrix algebra; additional mathematical tools are provided where needed. In developing the numerical theory, I have tried to serve the engineer’s needs better than do the standard textbooks on numerical analysis, which tend to be either too rigorous or too general. I have not attempted to be completely rigorous in the mathematical proofs, but I have included sufficient derivations so as to make the various mathematical arguments as plausible as possible.

But the engineer-user is not the only reader I had in mind in writing this book. The mathematician skilled in numerical analysis will, of course, find much material already familiar to him. However, the first chapter will introduce him to the basic principles of reservoir mechanics, and the remainder of the book will indicate to him those topics in numerical analysis that I consider significant in the numerical solution of reservoir flow problems.

Furthermore, I have included some material which appears not to be well known. For example, the section on successive overrelaxation methods discusses the effect of Neumann boundary conditions and the effect of anisotropy. whereas standard textbooks confine themselves to Dirichlet boundary conditions and isotropic problems. Finally, the material in the last chapter should be new to most numerical analysts, as it is quite special to the area of multiphase reservoir flow. It is my hope this book will provide food for thought leading to further progress in numerical reservoir simulation.

While much work is now being done on the application of variational methods to the solution of partial differential equations, little of this has reached the stage of practical application in reservoir simulation. Most practical reservoir simulators now in use are based on finite-difference methods. For this reason, only finitedifference methods are covered in this book.

The reader familiar with reservoir engineering will note a departure from a practice I feel is

all too common in the field, namely, the inclusion of numerical constants in equations involving flow. All the equations in this book are free from numerical constants (which are dependent on the units being used) and are valid for any consistent set of units. The use of dimension-free equations should become more common as the industry moves to the adoption of the SI (Systeme Inteational) standard of units, as is now being proposed. Accordingly, in the nomenclature following each chapter, I have specified the units of various quantities in the basic SI units of kilograms, meters, and seconds, together with the derived units of the newton for force (which equals kg * m/s2) and the pascal for pressure (which equals N/mZ). These form a consistent set of units. If the reader prefers, any other consistent set of units can be used, and the equations will still be correct.

The material in this book is based primarily on notes prepared for a series of lectures I was privileged to give at a NATO-sponsored Summer School on Hydrocarbon Reservoir Simulation by Computer, held in Milan, Italy, in May, 1969. I have given the same lectures within Exxon Production Research Company and for a Continuing Education Group of the Los Angeles Basin SPE Section in January, 1974. Many who have seen these lecture notes have urged me to enlarge and publish them. I felt that the original notes were somewhat incomplete and so added several sections. The most significant additions were the section (in Chapter 1) on alteative differential equations for two-phase flow, all of Chapter 4 on solution of hyperbolic problems, the section (in Chapter 5 ) on relaxation methods, and the sections (in Chapter 6) on sequential solution methods, semi-implicit mobility, and well rates.

I am indebted to the management of Exxon Production Research Company for permission to publish this book and for the encouragement they gave. In particular, I want to thank C.C. Mattax, Manager of the Reservoir Divison, for his encouragement and many helpful suggestions about the revision of the original lecture notes. Thanks are due, also, to J.W. Watts and R.P. Kendall, of EPR, for the helpful discussions I had with them while writing the section on relaxation methods. Discussions with J.G. Hillestad and H.L. Stone, of EPR, also were of great help.

I owe much to the skill of Dodi Fenner for the preparation of the figures, to Winn Alms for typing the original lecture notes and part of the book manuscript, and to Altha Frazier and the Word Processing Group of EPR for typing the major portion of the manuscript for the book. I owe a large debt of gratitude to John Colby, Supervising Editor at EPR, for carefully editing the first draft of the book and checking the printer’s proofs.

Finally, I would like to dedicate this book to the three women in my life, who have provided inspiration to me and many others around them: to my mother, Ida, of blessed memory; to my wife, Ruth; and to my daughter, Caren.