Petroleum Reservoir Simulation

CONTENTS
Preface
Nomenclature

INTRODUCTION
What is a Computer Model?
Other Models
What Questions Can a Computer Model Answer?
Concluding Remarks

FLUID FLOW EQUATIONS
Introduction
The Law of Mass Conservation
Single-Phase Flow
Multiphase Flow
Darcy’s Law
Single-Phase Flow
Multiphase Flow
The Basic Flow Equations
Single-Phase Flow
Multiphase Flow
Use of Pseudopotential
Boundary Conditions
Alternative Forms of Multiphase Flow Equations
Formulation in ‘Parabolic’ Form
Formulation in ‘Hyperbolic’ Form
Flow Equations Which Include Non-Darcy Effects
High Flow Rates (Inertial and Turbulent Effects)
Threshold and Slip Phenomena
Non-Newtonian Flow
Other Effects
Fluid and Rock Properties
Fluid Properties
Rock Properties
Concluding Remarks
Exercises

FLOW OF A SINGLE FLUID IN ONE DIMENSION
Introduction
Finite-Difference Approximations
Discretisation in Space
Discretisation in Time
Discretisation Errors
Other Selected Methods
Other Explicit Methods
Other Implicit Methods
ODE Methods
Comparison of Methods
Grid Systems and Boundary Conditions
Two Methods of Grid Construction
Boundary Conditions
Discretisation of One-Dimensional Flow Equations in Cartesian Co-ordinates
Difference Equations for Irregular Grid
Difference Equations in Matrix Form
Treatment of Variable Coefficients
Discretisation of One-Dimensional Flow Equations in Radial Cylindrical Co-ordinate
Difference Equations for Irregular Grid
Difference Equations in Matrix Form
Treatment of Variable Coefficients
Some Properties of Finite-Difference Equations
Existence of Solution and Material Balance
Treatment of Nonlinearities
Concluding Remarks
Exercises

SOLUTION OF TRIDIAGONAL MATRIX EQUATIONS
Introduction
Methods of Solution
Thomas’ Algorithm
Tang\'s Algorithm
Solution of Symmetric Tridiagonal Matrix Equations
Special Cases of Non-Unique Solution
Other Special Cases
Exercises

MULTIPHASE FLOW IN ONE DIMENSION
Introduction
The Simultaneous Solution (SS) Method
The SS Method for Two-Phase Flow
Extension of the SS Method to Three-Phase Flow
Other Formulations of the SS Method
The Implicit Pressure-Explicit Saturation (IMPES) Method IMPES Method for Three-Phase Flow
Other Derivations of the IMPES Method
Analysis of SS and IMPES Methods
Stability
Existence and Uniqueness of Solution
Convergence
Treatment of Nonlinearities
Weighting of Transmissibilities
Approximation to Transmissibilities in Time
Nonlinearity due to P c Function
Gas Percolation
The Sequential Solution Method (SEQ)
SEQ Method for Two-Phase Flow
Other Forms and Derivations
Numerical Results
SEQ Method for Three-Phase Flow
Discussion
Treatment of Production Terms
Differential Form of Boundary Conditions
Discretisation of Boundary Conditions
Exercises

SOLUTION OF BLOCK TRIDIAGONAL EQUATIONS
Introduction
Methods of Solution
Extension of Thomas’ Algorithm
Use of Methods for Band Matrices

FLOW OF A SINGLE FLUID IN TWO DIMENSIONS
Introduction
Classification of 2-D Problems
Areal Problems ( x–y )
Cross-Sectional Problems ( x–z )
Single-Well Problems ( r–z )
Comments on Two-Dimensional Models
Discretisation of Flow Equations
Difference Approximations
Stability of Difference Schemes
Boundary Conditions
No-Flow or Closed Boundaries
Flow Boundaries
Discretisation of Boundary Conditions
Initial Conditions
Treatment of Nonlinearities
Treatment of Individual Wells
Equations in Matrix Form
Special Methods for 2-D Problems
Alternating Direction Explicit (ADE) Methods
Alternating Direction Implicit (ADI) and Related Methods Comparison of Methods
Methods of Grid Construction
Irregular Grid in 2-D
Use of a Curvilinear Grid
Concluding Remarks
Exercises

SOLUTION OF PENTADIAGONAL MATRIX EQUATIONS
Introduction
Direct Methods of Solution
LU Factorisation
Ordering of Equations
Sparse Matrix Techniques
Iterative Methods
Point Jacobi Method
Point Gauss-Seidel Method
Point Successive Over Relaxation (SOR) Method
Line and Block SOR Methods
Additive Correction Methods
Iterative Alternating Direction Implicit (ADI) Methods Strongly Implicit Method
Other Methods
Comparison of Iterative Methods
Practical Considerations in the Use of Iterative Methods Comparison of Iterative and Direct Methods
Concluding Remarks
Exercises

MULTIPHASE FLOW IN TWO DIMENSIONS
Introduction
Classification of 2-D Problems
Areal Problems ( x– y )
Cross-Sectional Problems ( x–z )
Single-Well Problems ( r–z )
General Comments
Methods of Solution and Their Comparison
Discretisation in 2-D
Stability of SS and IMPES in 2-D
Comparison of Various Solution Methods and Computer Requirements
Boundary Conditions
Differential Formulation
Compatibility Conditions and Constraints
Finite-Difference Formulation
Initial Conditions
Simulation of Aquifers
Simulation of Areal and Cross-Sectional Problems
Use of Curvilinear Grid
Treatment of Individual Wells
Grid Orientation Phenomenon
Simulation of Single-Well Problems
Treatment of the Production Terms (the Well Model) Comparison of Stability and Efficiency of Various Treatments of Transmissibilities
Practical Considerations
Concluding Remarks
Exercise

SOLUTION OF BLOCK PENTADIAGONAL EQUATIONS
Introduction
Direct Methods
Iterative Methods
BSOR Method
Iterative ADI Method
The SIP Method
Comparison of Iterative Methods
Comparison of Iterative and Direct Methods
Concluding Remarks
Exercise

THREE-DIMENTIONAL PROBLEMS AND SOLUTION TECHNIQUES
Introduction
Single-Phase Flow
  Basic Equation and Discretisation
  Special Methods for 3-D Problems  
  Direct Methods of Solution
  Iterative Methods
  Comparison of Methods
Multiphase Flow
  Basic Solution Methods and Their Work Requirements
  Methods for Solving the Matrix Equations
Concluding Remarks
Exercises

SPECIAL TOPICS
Introduction
Pseudo-Functions
  Coats et al. (1971a) Vertical Equilibrium Model
  Other Pseudo-Functions
Stream Tube and Related Models
Simulation of Variable Bubble-Point Problems
  Simulation of Miscible Displacement
  Simulation of Compositional Effects
History-Dependent Saturation Functions  
  Physical Model of Hystersis
  Numerical Treatment of Hysteresis
Simulation of Naturally Fractured Reservoirs
Automatic Time-Step Control
Concluding Remarks
Exercise

PRACTICAL CONSIDERATIONS
Program Development  
  Development of the Mathematical Model
  Development of the Numerical Model
  Developmetn of the Computer Model (program)
Program Usage
  Steps Involved in a Simulation Study
  Selection and Design of the Model
  History Matching
Concluding Remarks
APPENDIX A
APPENDIX B
BIBLIOGRAPHY
INDEX