An Introduction to Multiphase, Multicomponent Reservoir Simulation

1. Review of Rock and Fluid Properties

1.1 Introduction 　

1.2 Definitions

1.2.1 Phases and Components in Subsurface Porous Media

1.2.2 Density, Saturation, and Concentrations

1.3 Phase Behavior

1.4 Rock and Fluid Properties

1.4.1 Formation properties

1.4.2 Gaseous phase properties

1.4.3 Oleic phase properties

1.4.4 Aqueous phase properties

1.5 Petrophysical properties

1.5.1 Darcy’s Law

1.5.2 Relative Permeability

1.5.3 Capillary Pressure 　

1.6 Reservoir Initialization 　
References 　

Chapter 2. Single Phase Flow in Porous Media

　

2.1 Introduction　

2.2 Continuity Equation 　

2.2.1 Cartesian coordinates 　

2.2.2 Cylindrical coordinates 　

2.2.3 General form 　

2.3 Specific cases of　single phase　flow 　

2.3.1 Flow of a single aqueous phase 　

2.3.2 Flow of a single oleic phase in the presence of immobile water 　

2.3.3 Flow of a single gaseous phase in the presence of immobile water (primary recovery) 　

2.4 Diffusivity equation 　

2.4.1 Slightly compressible flow 　

2.4.2 Compressible flow 　

2.4.3 Cylindrical Coordinates 　

2.5 analytical solutions 　

2.5.1 1D Heat Equation in Finite Medium 　

2.5.2 1D Heat equation in a semi-infinite medium 　

2.5.3 1D Diffusivity equation in cylindrical coordinates (radial flow in infinite-acting and finite domains) 　

2.5.4 Numerical solutions to the diffusivity equation 　
References 　

Chapter 3. Finite Difference Solutions to the 1D Diffusivity Equation 　

3.1 Introduction 　

3.2 Taylor Series and Finite Differences 　

3.2.1 First-Order Forward Difference Approximation 　

3.2.2 First-Order Backward Difference Approximation 　

3.2.3 Second-Order Centered Difference Approximation 　

3.2.4 Approximations to the Second Derivative 　

3.2.5 Higher-Order Approximations 　

3.3 Discretization of the Parabolic Diffusivity (Heat) Equation 　

3.3.1 Explicit Solution to the Diffusivity Equation 　

3.3.2 Implicit Solution to the Diffusivity Equation 　

3.4 Boundary and Initial Conditions 　

3.4.1 Dirichlet Boundary Condition 　

3.4.2 Neumann Boundary Condition 　

3.4.3 Other Boundary Conditions 　

3.5 Stability and Convergence 　

3.6 Matrix Equations in Terms of Flow Units 　

3.7 Mixed Methods and Crank-Nicholson 　

3.8 Pseudocode 　

3.9 Point Distributed Grids 　
References 　

Chapter 4. Control Volume Approach, Heterogeneities, Gravity, and Nonlinearities

　

4.1 Introduction 　

4.2 Control Volume Approach 　

4.2.1 Flow in and out of blocks 　

4.2.2 Accumulation 　

4.2.3 Sources and Sinks (Wells) 　

4.2.4 Balance Equations 　

4.3 Reservoir heterogeneities 　

4.3.1 Permeability 　

4.3.2 Variable grid blocks 　

4.3.3 Other Heterogeneities 　

4.3.4 Mass Balance Equations 　

4.3.5 Boundary Conditions 　

4.4 Gravity 　

4.5 Non-Linearities in Reservoir Simulation 　

4.5.1 Explicit Update of Fluid and Reservoir Properties 　

4.5.2 Fixed Point Iteration 　

4.5.3 Newton’s method 　

4.5.3.1 1D Newton’s method 　

4.5.3.2 Multidimensional Newton’s method 　

4.6 Pseudocode 　

4.7 Examples of Nonlinearities in Single-Phase Flow Problems 　

4.7.1 Gas Flow 　

4.7.1.3 General Numerical Solution to Gas Flow 　

4.7.2 Non-Newtonian Flow 　

4.7.3　Forchheimer　Flow 　
References 　

CHAPTER 5. 2D AND 3D SINGLE-PHASE FLOW

　

5.1 Introduction 　

5.2 Grid Block Numbering in 2D 　

5.3 Mass Balance Equations 　

5.3 Special Issues in 2D Flow 　

5.3.1　Interblock　Transmissibility 　

5.3.4 Irregular Geometry and Inactive Grids 　

5.4 Initial and Boundary Conditions 　

5.4.1 Neumann Conditions 　

5.5.2 Dirichlet Conditions 　

5.5.3 Corner blocks 　

5.6 3D Flow 　

5.7 Pseudocode for 2D and 3D 　
References 　

CHAPTER 6. WELLS, WELL MODELS, AND RADIAL FLOW

　

6.1 Introduction 　

6.2 Radial Flow Equations and Analytical Solutions 　

6.3 Numerical Solutions to the Radial Diffusivity Equation 　

6.3.1 Direct Discretization 　

6.3.2 Variable Transformation 　

6.3.3 2D and 3D flow problems in cylindrical coordinates 　

6.4 Wells and Well Models in Cartesian Grids 　

6.4.1 Constant Rate Wells 　

6.4.2 Constant Bottom Hole Pressure (BHP) Wells 　

6.4.2.1 Steady-state radial flow around a well 　

6.4.2.2 Mass Balance on the well-containing grid block 　

6.4.2.3 Alternatives to　Peaceman’s　correction 　

6.5 Inclusion of the well model into the matrix equations 　

6.6 Horizontal Wells 　
References 　

Chapter 7. Component Transport in Porous Media

　

7.1 Introduction 　

7.2 Transport Mechanisms 　

7.2.1 Advection 　

7.2.2 Diffusion and Dispersion 　

7.2.3 Reaction 　

7.3 Component Balance Equations 　

7.3.1 Cylindrical coordinates 　

7.3.2 General form 　

7.4 Analytical solutions to the ADE 　

7.5 Finite Difference Solution to the ADE 　

7.5.1　 Implicit Pressure, Explicit Concentration (IMPEC) 　

7.5.2 Sequential Implicit 　

7.5.3 Fully Implicit 　

7.6 Boundary Conditions and wells 　

7.6.1 Constant concentration (Dirichlet) 　

7.6.2 Mixed (specified flux) 　

7.6.3 No flux boundary 　

7.6.4 Wells 　

7.7 Upwinding 　

7.8 Stability 　

7.9 Numerical Dispersion 　

7.10 Viscous Fingering (TBD) 　

7.11 Multicomponent Transport (TBD) 　

7.12　 Reactive Transport (TBD) 　
References 　

Chapter 8. Numerical Solution of the Black Oil Model 　

8.1 Introduction 　

8.2 Derivation of 3-phase black oil equations 　

8.3 Analytical Solutions –　Bulkley-Leverett (Note: Taken partially from Gary Pope’s (2008) Reservoir II notes) 　

8.4 Numerical solution to 1d, 2-phase undersaturated oil 　

8.4.1 Finite Differences (1D, 2-phase flow, no capillary pressure or gravity) 　

8.4.2 Implicit Pressure, Explicit Saturation (IMPES) 　

8.4.3 SS Method 　

8.4.4　Interblock　transmissibilities　and　upwinding 　

8.4.5 Wells and Well Models 　

8.5 2-phase, 2D and 3D flow with capillary pressure and gravity 　

8.5.1 IMPES solution 　

8.5.2 Simultaneous Solution (SS) 　

8.5.3 Simultaneous Solution (SS) #2 Method 　

8.5.4 Fully Implicit Method 　

8.6 Numerical solutions to 3-phase flow (black oil model) 　
REFERENCES 　

Chapter 9. Multiphase Compositional Modeling

　

9.1 Introduction 　

9.2 Multiphase Flow Compositional Equations 　

9.3 Finite Difference Equations 　

9.4 Flash Calculations 　

9.4.1 K-values 　

9.4.2 Flash Calculation 　

9.5 Equations of State 　

9.5.1 Cubic EOS 　

9.5.2 Mixing Rules 　

9.5.3. Solution to the cubic EOS 　

9.5.3. Analytical solution to cubic polynomials 　

9.5.4 K-value calculation 　

9.6 Phase Viscosity 　

9.7 Phase Saturation, Relative Permeability, and Transmissibility 　

9.9 Moles and Accumulation 　

9.10 Pressure and Composition Solution 　
REFERENCES