An Introduction to Multiphase, Multicomponent Reservoir Simulation

Front Cover
DEVELOPMENTS IN PETROLEUM SCIENCE 75
An Introduction to Multiphase, Multicomponent Reservoir Simulation
Copyright
Dedication
Contents
Preface
Acknowledgments
Nomenclature
1 - Review of reservoir rock and fluid properties
1.1 Introduction
1.2 Overview of reservoir engineering principles
1.3 Definitions
1.3.1 Phases and components in subsurface porous media
1.3.2 Porosity, saturation, density, and concentrations
1.4 Phase behavior
1.5 Rock and Fluid Properties
1.5.1 Formation properties
1.5.2 Gaseous phase properties
1.5.3 Oleic phase properties
1.5.4 Aqueous phase properties
1.6 Petrophysical properties
1.6.1 Darcy's law
1.6.2 Relative permeability
1.6.2.1 Definition and relationship with phase saturation
1.6.2.2 Empirical models
1.6.3 Capillary pressure
1.6.3.1 Definition and relationship with phase saturation
1.6.3.2 Capillary pressure models
1.6.4 Capillary pressure scanning curves
1.7 Reservoir initialization
1.8 Pseudocode
1.8.1 Relative permeability
1.8.2 Capillary pressure
1.8.3 Initialization
1.8.4 Preprocess
1.9 Exercises
References
2 - Phase mass balances and the diffusivity equation
2.1 Introduction
2.2 Phase mass balances
2.2.1 Mass balance of a phase in Cartesian coordinates
2.3 The continuity equation
2.4 The diffusivity equation
2.4.1 General multiphase flow
2.4.2 Single-phase flow
2.4.2.1 Slightly compressible liquids
2.4.2.2 Compressible gases
2.5 Analytical solutions
2.5.1 1D heat equation in a finite medium
2.5.2 1D heat equation in a semi-infinite medium
2.5.3 Solution in cylindrical coordinates (around a wellbore)
2.5.3.1 Pseudo-steady-state (pss) flow
2.5.3.2 Infinite-acting flow
2.6 Exercises
References
3 - Finite difference solutions to PDEs
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 Generalization to higher-order approximations
3.3 Discretization of the parabolic diffusivity (heat) equation
3.4 Boundary and initial conditions
3.4.1 Dirichlet boundary condition
3.4.2 Neumann boundary condition
3.4.3 Robin boundary conditions
3.5 Solution methods
3.5.1 Explicit solution to the diffusivity equation
3.5.2 Implicit solution to the diffusivity equation
3.5.3 Mixed methods and Crank–Nicolson
3.5.4 Linear systems of equations
3.6 Stability and convergence
3.7 Higher-order approximations
3.8 Pseudocode for 1D, single-phase flow
3.9 Exercises
References
4 - Multidimensional reservoir domains, the control volume approach, and heterogeneities
4.1 Introduction
4.2 Gridding and block numbering in multidimensions
4.2.1 Grid block indexing in 2D and 3D
4.2.2 Grid dimensions
4.2.3 Irregular geometry and inactive grids
4.3 Single-phase flow in multidimensions and the control volume approach
4.3.1 Accumulation
4.3.2 Flux terms
4.3.3 Sources and sinks (wells)
4.3.4 Single-phase flow
4.4 Wells, boundary conditions, and initial conditions
4.4.1 Constant rate wells
4.4.2 Neumann boundary conditions
4.4.3 Dirichlet conditions
4.4.4 Corner blocks
4.4.5 Initial conditions
4.5 Reservoir heterogeneities
4.5.1 Fluid properties
4.5.2 Geometric properties
4.5.3 Accumulation terms
4.6 Matrix arrays
4.6.1 Accumulation and compressibility
4.6.2 Transmissibility
4.6.3 Source terms
4.6.4 Gravity
4.7 Pseudocode for single-phase flow in multidimensions
4.7.1 Preprocessing
4.7.2 Interblock transmissibility
4.7.3 Well Arrays
4.7.4 Grid Arrays
4.7.5 Main code
4.7.6 Postprocessing
4.8 Exercises
References
5 - Radial flow, wells, and well models
5.1 Introduction
5.2 Radial flow equations and analytical solutions
5.3 Numerical solutions to the radial diffusivity equation
5.3.1 Gridding
5.3.2 Discretization
5.4 Wells and well models in Cartesian grids
5.4.1 Well constraints
5.4.2 Steady-state radial flow around a well
5.4.3 Mass balance on the well-residing grid block
5.4.4 Extension to horizontal wells and anisotropy
5.5 Inclusion of the well model into the matrix equations
5.6 Practical considerations
5.7 Pseudocode for single-phase flow with constant BHP wells
5.8 Exercises
References
6 - Nonlinearities in single-phase flow through subsurface porous media
6.1 Introduction
6.2 Examples of nonlinearities in single-phase flow problems
6.2.1 Gas flow
6.2.1.1 Compressibility
6.2.1.2 Viscosity
6.2.1.3 Permeability and the Klinkenberg effect
6.2.2 Non-Newtonian flow
6.2.3 Forchheimer flow
6.3 Numerical methods for nonlinear problems
6.3.1 Explicit update of fluid and reservoir properties
6.3.2 Picard iteration
6.3.3 Newton's method
6.3.3.1 1D Newton's method
6.3.3.2 Multidimensional Newton's method
6.4 Pseudocode for Newton's method
6.5 Exercises
References
7 - Component transport in porous media
7.1 Introduction
7.2 Transport mechanisms
7.2.1 Advection
7.2.2 Hydrodynamic dispersion
7.2.2.1 Diffusive transport
7.2.2.2 Mechanical dispersion
7.2.2.3 Combined effects and hydrodynamic dispersion
7.2.3 Reactive transport and other source terms
7.3 Component mass balance equations
7.3.1 Single-phase flow
7.3.2 Overall compositional equations
7.4 Analytical solutions
7.4.1 1D Cartesian ADE in a semi-infinite domain
7.4.1.1 Constant concentration at xD=0
7.4.1.2 Mixed boundary condition at xD = 0
7.4.2 Semianalytical solution to two-phase flow
7.4.2.1 Fractional flow
7.4.2.2 Buckley–Leverett solution
7.4.2.3 Shock fronts
7.4.2.4 Breakthrough time and oil recovery
7.4.2.5 Capillary pressure and smearing of the shock front
7.4.2.6 Pseudocode for semianalytical Buckley–Leverett solution
7.5 Exercises
References
8 - Numerical solution to single-phase component transport
8.1 Introduction
8.2 Finite difference solution to the ADE in 1D for a single component
8.3 Discretization of advective terms
8.3.1 Cell-centered
8.3.2 Upwinding
8.3.3 Matrices
8.4 Wells and boundary conditions
8.4.1 Wells
8.4.2 No flux boundary condition
8.4.3 Constant concentration (Dirichlet)
8.5 Solution methods
8.5.1 Implicit pressure, explicit concentration (IMPEC)
8.5.2 Implicit pressure, implicit concentration
8.5.3 Fully implicit
8.6 Stability
8.7 Numerical dispersion
8.8 Channeling and viscous fingering
8.9 Multicomponents, multidimensions, and additional forms
8.10 Pseudocode for component transport
8.11 Exercises
References
9 - Numerical solution to the black oil model
9.1 Introduction
9.2 The black oil model
9.3 Finite difference equations for multiphase flow
9.4 Solution methods
9.4.1 Implicit pressure, explicit saturation
9.4.2 Simultaneous solution method
9.4.3 Fully implicit method
9.5 Interblock transmissibilities and upwinding
9.6 Stability
9.7 Wells and well models
9.7.1 Constant rate injector wells
9.7.2 Constant rate producer wells
9.7.3 Constant BHP injector wells
9.7.4 Constant BHP producer wells
9.7.5 Time-dependent well constraints
9.8 Pseudocode for multiphase flow
9.8.1 Preprocessing
9.8.2 Block properties
9.8.3 Interblock properties
9.8.4 Well productivity index
9.8.5 Well arrays
9.8.6 Grid arrays
9.8.7 Main code
9.8.8 Postprocessing
9.9 Exercises
References
10 - Numerical solution to multiphase, multicomponent transport
10.1 Introduction
10.2 Compositional equations for multiphase flow
10.3 Finite difference equations
10.4 Solution method
10.4.1 Flash calculations
10.4.1.1 K-values
10.4.1.2 Rachford–Rice flash
10.4.2 Equations of state
10.4.2.1 Cubic EOS
10.4.2.2 Mixing rules
10.4.2.3 Solution to the cubic EOS
10.4.2.4 K-value calculation
10.4.3 Phase saturation
10.4.4 Two-phase compressibility
10.4.5 Phase viscosity
10.4.6 Relative permeability and transmissibility
10.4.7 Wells and source terms
10.4.8 Pressure and composition solution
10.5 Oleic–aqueous bipartitioning components
10.6 Pseudocode for multiphase, multicomponent transport
10.7 Exercises
References
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
R
S
T
U
V
W
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