Petroleum Reservoir Simulations: The Engineering Approach

Cover
Petroleum Reservoir
Simulation:
The Engineering Approach
Copyright
Dedication
Preface
Introduction
Nomenclature
	Subscripts
	Superscripts
1
Introduction
	Chapter outline
	Background
	Milestones for the engineering approach
	Importance of the engineering and mathematical approaches
	Summary
	Exercises
2
Single-phase fluid flow equations in multidimensional domain
	Chapter outline
	Introduction
	Properties of single-phase fluid
	Properties of porous media
	Reservoir discretization
	Basic engineering concepts
	Multidimensional flow in Cartesian coordinates
		Block identification and block ordering
		Derivation of the one-dimensional flow equation in Cartesian coordinates
		Approximation of time integrals
		Flow equations in multidimensions using engineering notation
	Multidimensional flow in radial-cylindrical coordinates
		Reservoir discretization for single-well simulation
		Derivation of the multidimensional flow equation in radial-cylindrical coordinates
		Approximation of time integrals
	Summary
	Exercises
3
Flow equations using CVFD terminology
	Chapter outline
	Introduction
	Flow equations using CVFD terminology
		Flow equations using CVFD terminology and engineering notation
		Flow equations using CVFD terminology and the natural ordering scheme
	Flow equations in radial-cylindrical coordinates using CVFD terminology
	Flow equations using CVFD terminology in any block ordering scheme
	Summary
	Exercises
4
Simulation with a block-centered grid
	Chapter outline
	Introduction
	Reservoir discretization
	Flow equation for boundary gridblocks
	Treatment of boundary conditions
		Specified pressure gradient boundary condition
		Specified flow rate boundary condition
		No-flow boundary condition
		Specified boundary pressure condition
		Specified boundary block pressure
	Calculation of transmissibilities
	Symmetry and its use in solving practical problems
	Summary
	Exercises
5
Simulation with a point-distributed grid
	Chapter outline
	Introduction
	Reservoir discretization
	Flow equation for boundary gridpoints
	Treatment of boundary conditions
		Specified pressure gradient boundary condition
		Specified flow rate boundary condition
		No-flow boundary condition
		Specified boundary pressure condition
		Specified boundary gridpoint pressure
	Calculation of transmissibilities
	Symmetry and its use in solving practical problems
	Summary
	Exercises
6
Well representation in simulators
	Chapter outline
	Introduction
	Single-block wells
		Treatment of wells in 1-D linear flow
		Treatment of wells in 1-D radial flow
		Treatment of wells in 2-D areal flow
	Multiblock wells
		Vertical effects (flow within wellbore)
		Wellblock contribution to well rate
		Estimation of the wellblock geometric factor
		Estimation of well rate and FBHP
	Practical considerations dealing with modeling well operating conditions
	Summary
	Exercises
7
Single-phase flow equation for various fluids
	Chapter outline
	Introduction
	Pressure dependence of fluid and rock properties
		Incompressible fluid
		Slightly compressible fluid
		Compressible fluid
		Rock porosity
	General single-phase flow equation in multidimensions
		Incompressible fluid flow equation
			Algorithm for obtaining the pressure solution
			Material balance check for an incompressible fluid flow problem
		Slightly compressible fluid flow equation
			Formulations of the slightly compressible fluid flow equation
				Explicit formulation of the flow equation
				Implicit formulation of the flow equation
				Crank-Nicolson formulation of the flow equation
			Advancing the pressure solution in time
			Material balance check for a slightly compressible fluid flow problem
		Compressible fluid flow equation
			Formulations of compressible fluid flow equation
				Explicit formulation of the flow equation
				Implicit formulation of the flow equation
				Crank-Nicolson formulation of the flow equation
			Advancing the pressure solution in time
			Material balance check for a compressible fluid flow problem
	Summary
	Exercises
8
Linearization of flow equations
	Chapter outline
	Introduction
	Nonlinear terms in flow equations
	Nonlinearity of flow equations for various fluids
		Linearity of the incompressible fluid flow equation
		Nonlinearity of the slightly compressible fluid flow equation
		Nonlinearity of the compressible fluid flow equation
	Linearization of nonlinear terms
		Linearization of transmissibilities
			Linearization of fp in space
			Linearization of fp in time
		Linearization of well rates
		Linearization of fictitious well rates
		Linearization of coefficients in accumulation term
	Linearized flow equations in time
		Explicit transmissibility method
		Simple iteration on transmissibility method
		Fully implicit (Newtons iteration) method
	Summary
	Exercises
9
Methods of solution of linear equations
	Chapter outline
	Introduction
	Direct solution methods
		1-D rectangular or radial flow problems (Thomas algorithm)
			Forward solution
			Backward solution
		1-D tangential flow problem (Tangs algorithm)
			Forward solution
			Backward solution
		2-D and 3-D flow problems (sparse matrices)
			Initialization step
			Forward elimination step
			Back substitution step
	Iterative solution methods
		Point iterative methods
			Point Jacobi method
			Point Gauss-Seidel method
			Point SOR method
		Line and block SOR methods
			Line SOR method
			Block SOR method
		Alternating-direction implicit procedure
			Set of 1-D problems in the x-direction
			Set of 1-D problems in the y-direction
		Advanced iterative methods
	Summary
	Exercises
10
The engineering approach versus the mathematical approach in developing reservoir simulators
	Introduction
	Derivation of fluid flow equations in discretized form
		Basic principles
		Reservoir discretization
		The mathematical approach
			Derivation of PDE
			Discretization of PDE in space and time
				Space discretization
				Time discretization
					Forward-difference discretization
					Backward-difference discretization
					Central-difference discretization
			Observations on the derivation of the mathematical approach
		The engineering approach
			Derivation of the algebraic flow equations
			Approximation of time integrals
				Forward-difference equation
				Backward-difference equation
				Central-difference (Crank-Nicholson) equation
	Treatment of initial and boundary conditions
		Specified boundary pressure condition
			The mathematical approach
			The engineering approach
		Specified boundary pressure-gradient condition
			The mathematical approach
			The engineering approach
		Specified flow rate condition
			The mathematical approach
			The engineering approach
	Linearization of well flow rates
		The mathematical approach
		The engineering approach
	Summary
11
Introduction to modeling multiphase flow in petroleum reservoirs
	Chapter Outline
	Introduction
	Reservoir engineering concepts in multiphase flow
		Fluid properties
		Relative permeability
		Capillary pressure
		Darcys law in multiphase flow
	Multiphase flow models
		Flow equations for oil/water flow model
		Flow equations for gas/water flow model
		Flow equations for oil/gas flow model
		Flow equations for black-oil model
	Solution of multiphase flow equations
		Expansion of accumulation terms
		Well rate terms
			Production terms
			Injection terms
		Treatment of boundary conditions
			Specified pressure gradient boundary condition
			Specified flow rate boundary condition
			No-flow boundary condition
			Specified boundary pressure condition
		Treatment of nonlinearities
		Solution methods
			IMPES method
			SS method
	Material balance checks
	Advancing solution in time
	Summary
	Exercises
Glossary
Appendix A: Users manual for single-phase simulator
	Introduction
	Data file preparation
		Format procedure A
		Format procedure B
		Format procedure C
		Format procedure D
		Format procedure E
	Description of variables used in preparing a data file
	Instructions to run simulator
	Limitations imposed on the compiled version
	Example of a prepared data file
References
Author Index
Subject Index
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