Fundamentals of Numerical Reservoir Simulation (eBook)

Front Cover 1
Fundamentals of Numerical Reservoir Simulation 4
Copyright Page 5
CONTENTS 10
PREFACE 6
CHAPTER 1. DIFFERENTIAL EQUATIONS FOR FLOW IN RESERVOIRS 16
Introduction 16
Single-phase flow 17
Types of second-order differential equations 27
Two-phase flow 29
Three-phase flow 38
Flow with change of phase 39
The black-oil model 42
Limited compositional model 45
Summary 47
Nomenclature 48
CHAPTER 2. ELEMENTARY FINITE DIFFERENCES 50
Introduction 50
First-difference quotients 50
Second-difference quotients 52
Grid systems 53
Truncation error 56
Nomenclature 58
CHAPTER 3. NUMERICAL SOLUTION OF PARABOLIC PROBLEMS IN ONE DEPENDENT VARIABLE 60
The forward-difference equation 60
Stability by harmonic analysis (the von Neumann criterion) 61
Implicit difference equations 64
Other explicit difference equations 68
Multidimensional problems 70
Alternating-direction methods 72
Nomenclature 79
CHAPTER 4. NUMERICAL SOLUTION OF FIRST-ORDER HYPERBOLIC PROBLEMS IN ONE DEPENDENT VARIABLE 80
Introduction 80
Difference equations 80
Stability 83
Truncation error analysis–numerical dispersion 86
Example calculations 90
Summary 96
Nomenclature 97
CHAPTER 5. NUMERICAL SOLUTION OF ELLIPTIC PROBLEMS IN ONE DEPENDENT VARIABLE 98
Elliptic difference equations 98
Direct solution of band matrix equations by factorization 103
Application of band algorithm t o two-dimensional problems 105
Iterative methods for solving elliptic problems 106
Point relaxation methods 107
Line relaxation methods 127
Alternating-direction iteration (A.D.I.) 135
Strongly implicit procedure (S.I.P.) 143
Summary 149
Nomenclature 150
CHAPTER 6. NUMERICAL SOLUTION OF TWO-PHASE FLOW PROBLEMS 154
Introduction 154
Differential equations 154
Difference notation 156
Simultaneous numerical solution 159
Sequential numerical solution 168
Implicit and semi-implicit mobilities 175
Numerical dispersion 179
Well rates 179
Nomenclature 181
REFERENCES 184
SUBJECT INDEX 188