Reservoir Simulation

March 30, 2018 | Author: daonguyencm10 | Category: Petroleum Reservoir, Partial Differential Equation, Fluid Dynamics, Numerical Analysis, Liquids
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FUNDAMENTALS OF RESERVOIRSIMULATION Dr. Mai Cao Lan, GEOPET, HCMUT, Vietnam Jan, 2014 ABOUT THE COURSE COURSE OBJECTIVE COURSE OUTLINE REFERENCES 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 2 Course Objective • To review the background of petroleum reservoir simulation with an intensive focus on what and how things are done in reservoir simulations • To provide guidelines for hands-on practices with Microsoft Excel 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 3 COURSE OUTLINE INTRODUCTION FLOW EQUATIONS FOR PETROLEUM RESERVOIRS FINITE DIFFERENCE METHOD & NUMERICAL SOLUTION FOR FLOW EQUATIONS SINGLE-PHASE FLOW SIMULATION MULTIPHASE FLOW SIMULATION . . Dalton. 1990. Abou-Kassem et al.. Reservoir Simulation. 2001.HCMUT 5 . Texas.H.Mattax & R. Texas  J.References  T. SPE. SPE. Eterkin et al. Texas.  C. 16-Jan-2014 Mai Cao Lân – Faculty of Geology & Petroleum Engineering . Petroleum Reservoir Simulation – A Basic Approach. Gulf Publishing Company. 2005. Basic Applied Reservoir Simulation. Houston. Faculty of Geology & Petroleum Engineering. HCMUT.INTRODUCTION NUMERICAL SIMULATION – AN OVERVIEW COMPONENTS OF A RESERVOIR SIMULATOR RESERVOIR SIMULATION BASICS 16-Jan-2014 Dr. Mai Cao Lan. Vietnam 6 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. HCMUT.Numerical Simulation – An Overview 16-Jan-2014 Dr. Vietnam 7 . Mathematical Formulation 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. HCMUT. Vietnam 8 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. HCMUT. Vietnam 9 .Numerical Methods for PDEs 16-Jan-2014 Dr. Vietnam 10 . HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan.Numerical Methods for Linear Equations 16-Jan-2014 Dr. HCMUT. Faculty of Geology & Petroleum Engineering. Vietnam 11 . Mai Cao Lan.Components of a Reservoir Simulator Computer Code Physical Model Reservoir Simulator Mathematical Model 16-Jan-2014 Numerical Model Dr. Faculty of Geology & Petroleum Engineering. Vietnam 12 . HCMUT.What is Reservoir Simulation? • A powerful tool for evaluating reservoir performance with the purpose of establishing a sound field development plan • A helpful tool for investigating problems associated with the petroleum recovery process and searching for appropriate solutions to the problems 16-Jan-2014 Dr. Mai Cao Lan. HCMUT. Vietnam 13 . 16-Jan-2014 Dr.Reservoir Simulation Basics • The reservoir is divided into a number of cells • Basic data is provided for each cell • Wells are positioned within the cells • The required well production rates are specified as a function of time • The equations are solved to give the pressure and saturations for each block as well as the production of each phase from each well. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 14 . Faculty of Geology & Petroleum Engineering.Simulating Flow in Reservoirs • Flow from one grid block to the next • Flow from a grid block to the well completion • Flow within the wells (and surface networks) Flow = Transmissibility * Mobility * Potential Difference Geometry & Properties 16-Jan-2014 Fluid Properties Well Production Dr. HCMUT. Mai Cao Lan. Faculty of Geology & Petroleum Engineering. Vietnam 15 . HCMUT.SINGLE-PHASE FLOW EQUATIONS ESSENTIAL PHYSICS CONTINUITY EQUATION MOMENTUM EQUATION CONSTITUTIVE EQUATION GENERAL 3D SINGLE-PHASE FLOW EQUATION BOUNDARY & INITIAL CONDITIONS 16-Jan-2014 Dr. Mai Cao Lan. Essential Physics The basic differential equations are derived from the following essential laws:  Mass conservation law  Momentum conservation law  Material behavior principles 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. HCMUT. Mai Cao Lan. Vietnam 16 . Conservation of Mass Mass conservation may be formulated across a control element with one fluid of density r. HCMUT. Vietnam 17 . Mai Cao Lan. flowing through it at a velocity u: u r Dx Mass into the  Mass out of the  Rate of change of mass     element at x element at x + Dx inside the element       16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. the continuity equation can be expressed as follow:     Ar u   A r  x t For constant cross section area. Faculty of Geology & Petroleum Engineering. HCMUT.Continuity Equation Based on the mass conservation law. Mai Cao Lan. one has:     r u   r  x t 16-Jan-2014 Dr. Vietnam 18 . horizontal flow is: k P u  x 16-Jan-2014 Dr.Conservation of Momentum Conservation of momentum for fluid flow in porous materials is governed by the semi-empirical Darcy's equation. Faculty of Geology & Petroleum Engineering. Vietnam 19 . Mai Cao Lan. which for one dimensional. HCMUT. Vietnam 20 . Mai Cao Lan. these equations express the relationships between rock & fluid properties with respect to the reservoir pressure.  In general. Faculty of Geology & Petroleum Engineering.Equation Governing Material Behaviors  The behaviors of rock and fluid during the production phase of a reservoir are governed by the constitutive equations or also known as the equations of state. 16-Jan-2014 Dr. HCMUT. Faculty of Geology & Petroleum Engineering. the constitutive equation of rock becomes d  c f dP 16-Jan-2014 Dr. Mai Cao Lan. HCMUT. Vietnam 21 .Constitutive Equation of Rock The behavior of reservoir rock corresponding to the pressure declines can be expressed by the definition of the formation compaction  1     cf         P T For isothermal processes. Faculty of Geology & Petroleum Engineering. w. l  o. Mai Cao Lan. Vietnam 22 . g V  P T For natural gas. HCMUT. the well-known equation of state is used: PV  nZRT 16-Jan-2014 Dr.Constitutive Equation of Fluids The behavior of reservoir fluids corresponding to the pressure declines can be expressed by the definition of fluid compressibility (for liquid) 1  V  cl     . Mai Cao Lan. HCMUT.Single-Phase Fluid System Normally. Faculty of Geology & Petroleum Engineering. we would deal with one of the following fluids: Fluid System One Phase Gas 16-Jan-2014 One Phase Water One Phase Oil Dr. in single-phase reservoir simulation. Vietnam 23 . HCMUT. Mai Cao Lan. Faculty of Geology & Petroleum Engineering. which means that crossing of the dew point line is not permitted in order to avoid condensate fall-out in the pores. Vietnam 24 .Single-Phase Gas The gas must be single phase in the reservoir. Gas behavior is governed by: r gs constant rg   Bg Bg 16-Jan-2014 Dr. Mai Cao Lan. Vietnam 25 . Faculty of Geology & Petroleum Engineering. which strictly speaking means that the reservoir pressure is higher than the saturation pressure of the water in case gas is dissolved in it. has a density described by: r ws constant rw   Bw Bw 16-Jan-2014 Dr. HCMUT.Single-Phase Water One phase water. Vietnam 26 . oil density is described by: ro  16-Jan-2014 r oS  r gS Rso Bo Dr. which means that the reservoir pressure is higher than the bubble point pressure. Mai Cao Lan. In the Black Oil fluid model. it must be undersaturated. Faculty of Geology & Petroleum Engineering. HCMUT.Single-Phase Oil In order for the oil to be single phase in the reservoir. Single-Phase Fluid Model For all three fluid systems, the one phase density or constitutive equation can be expressed as: constant r B 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 27 Single-Phase Flow Equation The continuity equation for a one phase, one-dimensional system of constant cross-sectional area is:    ru   r  x t The conservation of momentum for 1D, horizontal flow is: k P u  x The fluid model: constant r B Substituting the momentum equation and the fluid model into the continuity equation, and including a source/sink term, we obtain the single phase flow in a 1D porous medium:   k P  qsc          x   B x  Vb t  B  16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 28 Single-Phase Flow Equation for Slightly Compressible Fluids  c f d (1/ B)  P   k P  qsc       t x   B x  Vb B dP   Based on the fluid model, compressibility can now be defined in terms of the formation volume factor as: d (1/ B) cl  B , l  o, g , w dP Then, an alternative form of the flow equation is:   k P  qsc  P  ct P  c f  cl     x   B x  Vb B t B t 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 29 HCMUT.Single-Phase Flow Equation for Compressible Fluids   k P  qsc          x   B x  Vb t  B  16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 30 . • Neumann BCs: The values of the first derivative of the unknown are specified or given. there are two types of boundary conditions: • Dirichlet BCs: Values of the unknown at the boundaries are specified or given.Boundary Conditions (BCs) Mathematically. 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. HCMUT. Vietnam 31 . Boundary Conditions (BCs) From the reservoir engineering point of view:  Dirichlet BCs: Pressure values at the boundaries are specified as known constraints. HCMUT. Mai Cao Lan.  Neumann BCs: The flow rates are specified as the known constraints. Vietnam 32 . 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. 16-Jan-2014 Dr. Vietnam 33 . t  0  PL Px  L.Dirichlet Boundary Conditions For the one-dimension single phase flow. at some position of the reservoir. Mai Cao Lan. such as follows: Px  0. t  0  PR A pressure condition will normally be specified as a bottom-hole pressure of a production or injection well. the Dirichlet boundary conditions are the pressure the pressures at the reservoir boundaries. Faculty of Geology & Petroleum Engineering. HCMUT. Using Darcy's equation. the conditions become: kA  P  Q0       x  x 0 kA  P  QL       x  x  L For reservoir flow. at some position of the reservoir. Mai Cao Lan. a rate condition may be specified as a production or injection rate of a well. Faculty of Geology & Petroleum Engineering. 16-Jan-2014 Dr.Newmann Boundary Conditions In Neumann boundary conditions. Vietnam 34 . the flow rates at the end faces of the system are specified. HCMUT. or between non-communicating layers. or it is specified as a zero-rate across a sealed boundary or fault. HCMUT.General 3D Single-Phase Flow Equations The general equation for 3D single-phase flow in field units (customary units) is as follows:   Ax k x     Ay k y    Dy  c  Dx    c x   B x  y   B y  Vb       Az k z     c    Dz  qsc  z   B z   c t  B    p  Z   cr g 16-Jan-2014 Z: Elevation. c. Mai Cao Lan. c: Unit conversion factors Dr. Faculty of Geology & Petroleum Engineering. positive in downward direction c. Vietnam 35 . 3D Single-Phase Flow Equations for Horizontal Reservoirs The equation for 3D single-phase flow in field units for horizontal reservoir is as follow:   Ax k x p    Ay k y p   Dy  c  Dx    c x   B x  y   B y  Vb       Az k z p    c    Dz  qsc  z   B z   c t  B  16-Jan-2014 Dr. Vietnam 36 . HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Faculty of Geology & Petroleum Engineering. HCMUT. Mai Cao Lan. Vietnam 37 .1D Single-Phase Flow Equation with Depth Gradient Vb       Ax k x p    c Dx  qsc    x  B x   c t  B    Ax k x Z  Dx    c  x  B x  16-Jan-2014 Dr. HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 38 .Quantities in Flow Equations 16-Jan-2014 Dr. HCMUT. Mai Cao Lan. Vietnam 39 .Quantities in Flow Equations 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Faculty of Geology & Petroleum Engineering. Vietnam 40 . HCMUT.FINITE DIFFERENCE METHOD & NUMERICAL SOLUTION OF SINGLE-PHASE FLOW EQUATIONS FUNDAMENTALS OF FINITE DIFFERENCE METHOD FDM SOLUTION OF THE SINGLE-PHASE FLOW EQUATIONS 16-Jan-2014 Dr. Vietnam 41 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. HCMUT.Numerical Solution of Flow Equations  The equations describing flui flows in reservoirs are of partial differential equations (PDEs)  Finite difference method (FDM) is traditionally used for the numerical solution of the flow equations 16-Jan-2014 Dr. HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 42 . derivatives are replaced by a proper difference formula based on the Taylor series expansions of a function: (Dx)1 f (Dx) 2  2 f f ( x  Dx)  f ( x)   1! x x 2! x 2 (Dx)3  3 f  3 3!  x x (Dx) 4  4 f  4 4!  x x   x The first derivative can be written by re-arranging the terms: f f ( x  Dx)  f ( x) Dx  2 f   x x Dx 2! x 2 (Dx) 2  3 f  3 3!  x x   x Denoting all except the first terms by O (Dx) yields f f ( x  Dx)  f ( x)   O(Dx) x x Dx The difference formula above is of order 1 with the truncation error being proportional to Dx 16-Jan-2014 Dr.Fundamentals of FDM In FDM. Fundamentals of FDM (cont. Vietnam 43 . Taylor series expansion of the function is used from both side of x (Dx)1 f (Dx) 2  2 f f ( x  Dx)  f ( x)   1! x x 2! x 2 (Dx)1 f (Dx) 2  2 f f ( x  Dx)  f ( x)   1! x x 2! x 2 (Dx)3  3 f  3 3!  x x (Dx) 4  4 f  4 4!  x x (Dx)3  3 f  3 3!  x x   x (Dx) 4  4 f  4 4!  x x   x Subtracting the second from the first equation yields f f ( x  Dx)  f ( x  Dx) (Dx) 2  3 f   x x 2Dx 3! x3   x The difference formula above is of order 2 with the truncation error being proportional to (Dx)2 f f ( x  Dx)  f ( x  Dx)   O(Dx 2 ) x x 2Dx 16-Jan-2014 Dr.) To obtain higher order difference formula for the first derivative. Faculty of Geology & Petroleum Engineering. HCMUT. Mai Cao Lan. HCMUT. Vietnam 44 . Faculty of Geology & Petroleum Engineering. Mai Cao Lan.Typical Difference Formulas Forward difference for first derivatives (1D) f f ( x  Dx)  f ( x)   O(Dx) x x Dx or in space index form fi 1  fi f   O(Dx) x i Dx i-1 i i+1 Dx 16-Jan-2014 Dr. Mai Cao Lan.Typical Difference Formulas Backward difference for first derivatives (1D) f f ( x)  f ( x  Dx)   O(Dx) x x Dx or in space index form fi  fi 1 f   O(Dx) x i Dx i-1 i i+1 Dx 16-Jan-2014 Dr. Vietnam 45 . Faculty of Geology & Petroleum Engineering. HCMUT. Faculty of Geology & Petroleum Engineering. Vietnam 46 . Mai Cao Lan.Typical Difference Formulas Centered difference for first derivatives (1D) f f ( x  Dx)  f ( x  Dx)   O(Dx 2 ) x x 2Dx or in space index form f f f  i 1 i 1  O(Dx 2 ) x i 2Dx i-1 i i+1 Dx 16-Jan-2014 Dr. HCMUT. Vietnam 47 . Faculty of Geology & Petroleum Engineering. Mai Cao Lan. HCMUT.Typical Difference Formulas Centered difference for second derivatives (1D) 2 f x 2 x f ( x  Dx)  2 f ( x)  f ( x  Dx) 2   O ( D x ) 2 Dx or in space index form fi 1  2 fi  fi 1 2 f 2   O ( D x ) 2 2 x i Dx i-1 i i+1 Dx 16-Jan-2014 Dr. j i. HCMUT.j-1 16-Jan-2014 Dr.j i.j i+1.j+1 i-1. j 1  fi . y )   O(Dy ) y ( x . Faculty of Geology & Petroleum Engineering. y ) Dy or in space index form fi . j ) Dy i.Typical Difference Formulas Forward difference for first derivatives (2D) f f ( x. Mai Cao Lan. y  Dy )  f ( x. Vietnam 48 . j f   O(Dy ) y (i . j i.Typical Difference Formulas Backward difference for first derivatives (2D) f f ( x. y )  f ( x. j ) Dy i. j 1 f   O(Dy ) y (i .j i. Vietnam 49 . y  Dy )   O(Dy ) y ( x . Faculty of Geology & Petroleum Engineering.j i+1.j+1 i-1. j  fi . Mai Cao Lan. y ) Dy or in space index form fi . HCMUT.j-1 16-Jan-2014 Dr. j-1 16-Jan-2014 Dr.j i+1. j ) 2Dy i-1.Typical Difference Formulas Centered difference for first derivatives (2D) f y  ( x. j 1  fi . Faculty of Geology & Petroleum Engineering.j+1 fi . y  Dy )  f ( x. HCMUT.j i. y ) f ( x. y  Dy )  O(Dy 2 ) 2Dy or in space index form i.j i. Mai Cao Lan. j 1 f   O(Dy 2 ) y (i . Vietnam 50 . j i. j ) fi .j+1 2 f y 2  (i . y ) f ( x. Faculty of Geology & Petroleum Engineering.j i. j 1  2 fi . y )  f ( x. Vietnam 51 . Mai Cao Lan. j 1 Dy 2  O(Dy 2 ) i-1. HCMUT. y  Dy ) 2  O ( D y ) 2 Dy or in space index form i.Typical Difference Formulas Centered difference for second derivatives (2D) 2 f y 2  ( x.j i+1. y  Dy )  2 f ( x.j-1 16-Jan-2014 Dr. j  fi . Mai Cao Lan.Solving time-independent PDEs  Divide the computational domain into subdomains  Derive the difference formulation for the given PDE by replacing all derivatives with corresponding difference formulas  Apply boundary conditions to the points on the domain boundaries  Apply the difference formulation to every inner points of the computational domain  Solve the resulting algebraic system of equations 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. HCMUT. Vietnam 52 . Faculty of Geology & Petroleum Engineering. HCMUT. Mai Cao Lan.Exercise 1  Solve the following Poisson equation: 2 p 2   16  sin(4 x) 2 x 0  x 1 subject to the boundary conditions: p=2 at x=0 and x=1 16-Jan-2014 Dr. Vietnam 53 . Exercise 2  Solve the following Poisson equation:  2u  sin( x)sin( y ) 0  x  1. Mai Cao Lan. HCMUT. Faculty of Geology & Petroleum Engineering. y  1 16-Jan-2014 Dr. 0  y  1 subject to the boundary conditions: u  0 along the boundaries x  0. y  0. x  1. Vietnam 54 . HCMUT. Faculty of Geology & Petroleum Engineering.Boundary Condition Implementation Newmann BCs: p C x b p1  p0 p  C x 11/2 x1  x0 pnx 1  pnx p  C x nx 1/2 xnx 1  xnx p0  p1  C Dx1 pnx 1  pnx  C Dxnx 16-Jan-2014 Dr. Vietnam 55 . Mai Cao Lan. Vietnam 56 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. HCMUT.Boundary Condition Implementation Dirichlet BCs: pb  C 1    p1  p2  C Dx1  Dx1  Dx2 16-Jan-2014 1    pn  x  pnx 1  C Dxnx Dxnx  Dxnx 1 Dr. Faculty of Geology & Petroleum Engineering. 0  y  1. y  1 u   exp( x   y ). x  1 x 16-Jan-2014 Dr. y  0.Exercise 3  Solve the following Poisson equation:  2u  ( 2   2 ) exp( x   y ) 0  x  1. Vietnam 57 . HCMUT. x  0.   3 subject to the boundary conditions: u  exp( x   y).   2. Mai Cao Lan. HCMUT.Solving time-dependent PDEs  Divide the computational domain into subdomains  Derive the difference formulation for the given PDE by replacing all derivatives with corresponding difference formulas in both space and time dimensions  Apply the initial condition  Apply boundary conditions to the points on the domain boundaries  Apply the difference formulation to every inner points of the computational domain  Solve the resulting algebraic system of equations 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Vietnam 58 . Mai Cao Lan. 0. t )  u( x  1. t  0)  sin( x). 0  x  1. HCMUT. Faculty of Geology & Petroleum Engineering.0  x  1  Hints: Use explicit scheme for time discretization 16-Jan-2014 Dr.Exercise 4  Solve the following diffusion equation: u  2u  2 . t )  0. t  0 u( x. Mai Cao Lan. t  0 t x subject to the following initial and boundary conditions: u( x  0. Vietnam 59 . Explicit Scheme  The difference formulation of the original PDE in Exercise 4 is: uin1  uin uin1  2uin  uin1  Dt (Dx)2 where n=0. Faculty of Geology & Petroleum Engineering.NX: Grid point index 16-Jan-2014 Dr. Mai Cao Lan.NT: Time step i =1. HCMUT. Vietnam 60 . NT: Time step i =1. Mai Cao Lan.Implicit Scheme  The difference formulation for the original PDE in Exercise 4 n 1 i u n 1 i 1 u u  Dt n i n 1 i 2 n 1 i 1  2u  u (Dx) where n=0. Faculty of Geology & Petroleum Engineering.NX: Grid point index 16-Jan-2014 Dr. HCMUT. Vietnam 61 . NX: Grid point index When =0. HCMUT.NT: Time step i =1. Mai Cao Lan.Semi-Implicit Scheme Semi-Implicit Scheme for the Diffusion Equation in Exercise 4 is uin1  uin uin11  2uin1  uin11 uin1  2uin  uin1   (1   ) 2 Dt (Dx) (Dx)2 where 0≤≤1 n=0. Vietnam 62 . we have Crank-Nicolson scheme 16-Jan-2014 Dr.5. Faculty of Geology & Petroleum Engineering. HCMUT. Mai Cao Lan. Faculty of Geology & Petroleum Engineering.Discretization in Conservative Form    P f ( x)  x  x  i-1 i i+1 Dx P  P    f ( x )  f ( x )   P   x  i 1/2  x  i 1/2 2 f ( x )   O D x   x  x  i Dxi  Pi 1  Pi  P  1  O(Dx)    x ( D x  D x )  i 1/2 2 i i 1   P  f ( x)    x  x  i 16-Jan-2014 2 f ( x)i 1/2  Pi  Pi 1  P    O(Dx)   1  x i 1/2 2 (Dxi  Dxi 1 ) ( Pi 1  Pi ) ( Pi  Pi 1 )  2 f ( x)i 1/2 (Dxi 1  Dxi ) (Dxi  Dxi 1 )  O(Dx) Dxi Dr. Vietnam 64 . HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 65 .FDM for Flow Equations  FD Spatial Discretization  FD Temporal Discretization 16-Jan-2014 Dr. Single-Phase Flow Equations  For slightly compressible fluids (Oil) Vb ct p   Ax k x p   c  Dx  qsc  x   B x   c B t  For compressible fluids (Gas) Vb       Ax k x p   c  Dx  qsc  x   B x   c t  B  16-Jan-2014 Mai Cao Lân – Faculty of Geology & Petroleum Engineering .HCMUT 66 . HCMUT.FDM for Slightly Compressible Fluid Flow Equations  FD Spatial Discretization  FD Temporal Discretization 16-Jan-2014 Dr. Vietnam 67 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. Faculty of Geology & Petroleum Engineering. Vietnam 68 . HCMUT. Mai Cao Lan.FD Spatial Discretization of the LHS Discretization of the left side term   P  f ( x )    x  x  i where  P   P  f ( x )i  1    f ( x )i  1   2 2  x i  1  x i  1 Ak f ( x)   c x x B 2 2 Dxi  O(Dx) ( Pi 1  Pi ) ( Pi  Pi 1 )  P   P        1 (Dxi 1  Dxi ) / 2  x i  1 (Dxi 1  Dxi ) / 2  x i  2 2 The discretization of the left side term is then  Ax k x   Ax k x    Ax k x p   D x   ( P  P )   i 1 i  c  i  c   c  ( Pi  Pi 1 ) x   B x i   BDx i  12   BDx i  12 16-Jan-2014 Dr. Vietnam 69 .Transmissibility Define transmissibility as the coefficient in front of the pressure difference: Tx i 1 2  Ax k x   1     c   Dx i  1  B i  1  2 16-Jan-2014 2 Dr. HCMUT. Mai Cao Lan. Faculty of Geology & Petroleum Engineering. Faculty of Geology & Petroleum Engineering. Vietnam 70 . Mai Cao Lan. HCMUT.FD Spatial Discretization The left side term of the 1D single-phase flow equation is now discritized as follow:   Ax k x P   c  Dxi  Txi  12 ( Pi 1  Pi )  Txi  12 ( Pi 1  Pi ) x   B x i 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. HCMUT.Transmissibility Tx i 1 2  Ax k x   1    c   1  1 Dx i    B i   2 16-Jan-2014 2 Dr. Vietnam 71 . Mai Cao Lan. HCMUT 72 .Transmissibility (cont’d) 1  Ax k x   c  1 D x  i  2 1 1  1  Ax k x   Ax k x     c    c   2  Dx i 1  Dx i  or  Ax k x i 1  Ax k x i  Ax k x    c Dx  1  2 c A k Dx  A k  x x i i 1  x x i 1 Dxi  i  2 16-Jan-2014 Mai Cao Lân – Faculty of Geology & Petroleum Engineering . HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 73 .Weighted Average of Mobility i  1 2 1  B i  1 2 16-Jan-2014  Dxi 1i 1  Dxi i   Dxi 1  Dxi   Dxi 1i 1  Dxi i   Dxi 1  Dxi  Dr. Faculty of Geology & Petroleum Engineering. HCMUT. Vietnam 74 .Discretized Transmissibility Tx i 1 2  Ax k x   1     c   Dx i  1  B i  1  2 Tx i 1 2  Ax k x i 1  Ax k x i  2 c  Ax k x i Dxi 1   Ax k x i 1 Dxi 1  Dxi 1  Dxi 16-Jan-2014 2       1 1  Dxi 1       D x i       B  B  i 1  i   Dr. Mai Cao Lan. Faculty of Geology & Petroleum Engineering.FD Temporal Discretization Explicit Method Txni1/2  pin1  pin   Txni1/2  pin  pin1   qsc i Implicit Method n 1 n  Vb ct   pi  pi    Dt   c B i Txni1/21  pin11  pin 1   Txni1/21  pin 1  pin11   qsc i Semi-implicit Method  0    1 n 1 n p  p  Vb ct   i i    Dt   c B i qsc i   Txni1/21  pin11  pin 1   Txni1/21  pin 1  pin11   n 1 n p  p  Vb ct   i i  n n n n n n    1    Txi1/2  pi 1  pi   Txi1/2  pi  pi 1       B Dt  c i 16-Jan-2014 Dr. Vietnam 75 . HCMUT. Mai Cao Lan. c t =3. 15. The rock and fluid properties for this problem are: Dx  1000ft.5 10 6 psi -1. HCMUT. Vietnam 76 . determine the pressure distribution during the first year of production. Use time step sizes of =10. Mai Cao Lan. Dz  75ft B  1RB/STB.Exercise 5 For the 1D. 16-Jan-2014 Dr. Assume B is unchanged within the pressure range of interest. and 30 days. k x =15md. The initial reservoir pressure is 6000 psia.  =0. Dy  1000ft. =10cp.18. block-centered grid shown on the screen. Faculty of Geology & Petroleum Engineering. Faculty of Geology & Petroleum Engineering.Exercise 5 (cont’d) 1000 ft p 0 x qsc  150 STB/D p 0 x 75 ft 1 2 3 4 5 1000 ft 16-Jan-2014 Dr. Mai Cao Lan. Vietnam 77 . HCMUT. c t =3. and 30 days.Exercise 6 For the 1D. Use time step sizes of =10.  =10cp. Mai Cao Lan.5 106 psi -1. The rock and fluid properties for this problem are: Dx  1000ft. The initial reservoir pressure is 6000 psia. determine the pressure distribution during the first year of production. Assume B is unchanged within the pressure range of interest. Dz  75ft B  1RB/STB. 15. k x =15md. Vietnam 78 . Faculty of Geology & Petroleum Engineering. 16-Jan-2014 Dr. Dy  1000ft.  =0. HCMUT.18. block-centered grid shown on the screen. Exercise 6 (cont’d) 1000 ft 1 p  6000psia 16-Jan-2014 p 0 x qsc  150 STB/D 2 3 4 5 75 ft 1000 ft Dr. HCMUT. Vietnam 79 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. Vietnam 80 . HCMUT.FDM for Slightly Compressible Fluid Flow Equations  FD Spatial Discretization  FD Temporal Discretization 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 80 .FD Spatial Discretization of the LHS for Compressible Fluids Same as that for slightly compressible fluids   Ax k x p   c  Dxi  Txi  12 ( pi 1  pi )  Txi  12 ( pi 1  pi ) x   B x i 16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. HCMUT. Transmissibility Tx i 1 2  Ax k x   1     c   Dx i  1  B i  1  2 16-Jan-2014 2 Dr. HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 82 . Upstream Average of Mobility 1  B i  16-Jan-2014 1 2 i 1   i if pi 1  pi if pi 1  pi Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 82 . HCMUT. FD Spatial Discretization of the RHS for Compressible Fluids  Vb           c t  B   i n 1 n    Vb                 B      c Dt  B  i    ref 1  c f  p  p ref  16-Jan-2014 Dr. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 83 . HCMUT. 5 106 psi -1 Use time step sizes of =10 days.  =0.Exercise 7 For the 1D. The initial reservoir pressure is 5000 psia. Dy  1000ft.HCMUT 85 . block-centered grid shown on the screen. determine the pressure distribution during the first year of production. 16-Jan-2014 Mai Cao Lân – Faculty of Geology & Petroleum Engineering . The rock and fluid properties for this problem are: Dx  1000ft.18. ct =3. Dz  75ft k x =15md. 567 1.675 1.343 Mai Cao Lân – Faculty of Geology & Petroleum Engineering .339 1900 0.619 1.581 1.HCMUT 86 .637 1.306 3500 0.563 1.570 1.313 3000 0.600 1.557 1.292 4500 0.335 2100 0.Exercise 7 (cont’d) PVT data table: p (psia) 16-Jan-2014  (cp) B (bbl/STB) 5000 0.341 1800 0.321 2500 0.299 4000 0.656 1.330 2200 0.560 1.337 2000 0. Exercise 7 (cont’d) 1000 ft p 0 x qsc  150 STB/D p 0 x 1 2 3 4 5 75 ft 1000 ft 16-Jan-2014 Mai Cao Lân – Faculty of Geology & Petroleum Engineering .HCMUT 87 . Faculty of Geology & Petroleum Engineering.MULTIPHASE FLOW SIMULATION MULTIPHASE FLOW EQUATIONS FINITE DIFFERENCE APPROXIMATION TO MULTIPHASE FLOW EQUATIONS NUMERICAL SOLUTION OF THE MULTIPHASE FLOW EQUATIONS 16-Jan-2014 Dr. HCMUT. Mai Cao Lan. Vietnam 88 . g Dr. Mai Cao Lan. g 16-Jan-2014 Pcow  Po  Pw Pcog  Pg  Po S l 1 l  o. w. w. HCMUT. Vietnam 89 . Faculty of Geology & Petroleum Engineering. w.Multiphase Flow Equations  Continuity equation for each fluid flowing phase:     Arl ul   A rl Sl  x t  l  o. g Momentum equation for each fluid flowing phase: kkrl Pl ul    l x l  o. the flow equations for the two phases are as follows: k ro  Po Vb   So    Z     qosc o   Dx    c k x Ax x  o Bo  x x   c t  Bo  k rw  Pw Vb   S w    Z     qwsc w   Dx    c k x Ax x   w Bw  x x   c t  Bw  So  S w  1 16-Jan-2014 Pw  Po  Pcow Dr. Vietnam 90 . Mai Cao Lan. HCMUT.Oil-Water Flow Equations • Considering the fluid phases of oil and water only. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 91 . HCMUT. Faculty of Geology & Petroleum Engineering.Oil-Water Flow Equations k ro  Po Vb    1  S w     Z     qosc o   Dx    c k x Ax x  o Bo  x x   c t  Bo  k rw  Po Pcow Vb   S w    Z     qwsc  w   Dx    c k x Ax x   w Bw  x x x   c t  Bw  16-Jan-2014 Dr. Discretization of the Flow Equation Left side flow terms kro  Po   Z  o   Dxi  c k x Ax x  o Bo  x x  i  Txo i  1 ( Po i 1  Po i )  Txo i  1 ( Po i 1  Po i ) 2 2 krw  Po Pcow   Z   w   Dxi  c k x Ax x   w Bw  x x x  i  Txw i  1 ( Po i 1  Po i )  Txw i  1 ( Po i 1  Po i ) 2 16-Jan-2014 2 Dr. Mai Cao Lan. Vietnam 92 . HCMUT. Faculty of Geology & Petroleum Engineering. HCMUT. Mai Cao Lan. Vietnam 93 . Faculty of Geology & Petroleum Engineering.Phase Mobility k ro o   o Bo k rw w   w Bw 16-Jan-2014 Dr. Averaging of Phase Mobility Upstream: 1 2  i   i o 1 2 Qw o weighted average: o i  1 2  Dxi o i  Dxi 1o i 1   Dxi  Dxi 1  OIL Sw 1-Swir exact average upstream Swir x 16-Jan-2014 Dr. Vietnam 94 . Mai Cao Lan. Faculty of Geology & Petroleum Engineering. HCMUT. Faculty of Geology & Petroleum Engineering. Mai Cao Lan. Vietnam 95 .Upstream Average of Mobility wi  oi  16-Jan-2014 1 2 1 2 wi 1 if Pwi 1  Pwi   wi if Pwi 1  Pwi  oi 1 if Poi 1  Poi   oi if Poi 1  Poi Dr. HCMUT. Mai Cao Lan. HCMUT. Vietnam 96 .Discretization of Multiphase Flow Equation Left side flow terms kro  Po   Z   o   c k x Ax  Dxi   x  o Bo  x x   i  Txo 1 ( Po i1  Po i )  Txo 1 ( Po i1  Po i ) i 2 i 2 krw  Po Pcow   Z    w   c k x Ax  Dxi   x  w Bw  x x x   i  Txw 1 ( Po i1  Po i )  Txw 1 ( Po i1  Po i ) i 16-Jan-2014 2 i 2 Dr. Faculty of Geology & Petroleum Engineering. Vietnam 97 . Faculty of Geology & Petroleum Engineering.Discretization of the Oil-Phase Equation Right side flow terms   S o   S o       So   t  Bo  Bo t t  Bo  The second term:     i So So    t  Bo i Dt  cr d (1 / Bo)  n1 n  ( P  P o oi i )  Bo  dPo  i The first term: n 1 So 1  Sw 16-Jan-2014   S o     Bo t i  i Boi Dti ( S wni1  Swin ) Dr. Mai Cao Lan. HCMUT. Discretization of Oil-phase RHS   So     Cpooi ( Poni 1  Poin )  Cswoi ( Swin1  Swin ) t  Bo i Where: and 16-Jan-2014 Cpooi  i (1  Swi )  cr Cswoi   Dt d (1 / Bo)   Bo  dPo  i i Boi Dti Dr. Mai Cao Lan. Vietnam 98 . HCMUT. Faculty of Geology & Petroleum Engineering. Faculty of Geology & Petroleum Engineering. Vietnam 99 . Mai Cao Lan. HCMUT.Discretization of Water-Phase Equation Right side flow terms   S w   S w        S w   t  Bw  Bw t t  Bw         Pw     Po Pcow            t  Bw  Pw  Bw  t Pw  Bw  t t  Pcow dPcow S w  t dSw t 16-Jan-2014 Dr. Mai Cao Lan. Vietnam 100 . HCMUT. Faculty of Geology & Petroleum Engineering.Discretization of Water-phase RHS   S w     Cpowi ( Poni 1  Poin )  Cswwi ( Swin1  Swin ) t  Bw i Where: Cpowi i Swi  cr d (1 / Bw )      Dt  Bw dPw  i and Cswwi 16-Jan-2014 i  dPcow     Cpowi Bwi Dti  dSw i Dr. HCMUT. Vietnam 101 ... Faculty of Geology & Petroleum Engineering. N 16-Jan-2014 Dr.. Mai Cao Lan.Fully Discrete Oil-Water Flow Equations      S Txoi  1 Poni1 1  Poni  Txoi  1 Poni1 1  Poni  Cpooi Poni 1  Poin 2 2 Cswoi      n 1 wi  Swin   S  q osci  n n n 1 n n n    Txwi  1  Poni1 1  Poni  Pcow  P  T xw P  P  P  P 1 cow o o cow cow i 2  i 1 i  i 1 i i 1 i  2     C powi Poni 1  Poin  Cswwi n 1 wi   Swin  qwsci i  1.. Vietnam 102   . Faculty of Geology & Petroleum Engineering. Mai Cao Lan.IMPES Solution of Oil-Water Flow Equations First. the pressure is found by solving the following equation: T xo n i  12    iTxwin 1 2  P n 1 oi 1     Poni 1  Txo in 1   iTxwin 1  2  2  P n 1 oi 1 n n n n n   iTxwin 1 Pcow  P   T xw P  P cowi i cowi 1 cowi i 1 i 1  2  2  Cpooin   i Cswoin Poni 1  Poin  qosci   i qwsci Cswwin i   Cswoin 16-Jan-2014  Poni 1 Dr. HCMUT. Vietnam 103 .IMPES Pressure Solution Wi Wi n 1 n 1 T n 1 oi 1 P n xo 1 i 2 n 1 n 1 oi  Ci P  T  Ei C in 1   Txoin 1  Txoin 1  Cpooin  2 2 T   i Txwin 1  Txwin 1  Cpowin 2 2 n 1 oi 1  Ei P n 1 n xw i i  12 n 1 n xo 1 i 2 n 1  gi   iT n xw 1 i 2 n swwi n swoi C i   C  g in1  (Cpooin   i Cpowin ) Poin  qosci   i qwsci   iTxwin 1 ( Pcowin1  Pcowin )   iTxwin 1 ( Pcowin1  Pcowin ) 2 16-Jan-2014 2 Dr. HCMUT. Mai Cao Lan. Faculty of Geology & Petroleum Engineering. IMPES Water Saturation Once the oil pressures have been found, water saturations can be obtained by either the oil-phase equation or the water-phase equation.     n n 1 n 1 n n 1 n 1  1 Txoi  12 Poi1  Poi  Txoi  12 Poi1  Poi n 1 n  S wi  Swi  n Cswoi  qosc  Cpooin Pon1  Poin i i      i  1,..., N 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 104 Exercise 8 A homogeneous, 1D horizontal oil reservoir is 1,000 ft long with a cross-sectional area of 10,000 ft2. It is discretized into four equal gridblocks. The initial water saturation is 0.160 and the initial reservoir pressure is 5,000 psi everywhere. Water is injected at the center of cell 1 at a rate of 75 STB/d and oil is produced at the center of cell 4 at the same rate. Rock compressibility cr=3.5E-6 psi-1 . The viscosity and formation volume factor of water are given as w=0.8cp and Bw=1.02 bbl/STB. 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 105 Exercise 8 (cont’d) The gridblock dimensions and properties are: Dx=250ft, Dy=250ft, Dz=40ft, kx=300md, =0.20. PVT data including formation volume factor and viscosity of oil is given as in Table 1 as the functions of pressure. The saturation functions including relative permeabilities and capillary pressure. Using the IMPES solution method with Dt=1 day, find the pressure and saturation distribution after 100 days of production. 16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 106 Faculty of Geology & Petroleum Engineering.000 ft2 1 p 0 x Qo=-75 STB/d Qw=75 STB/d 2 3 4 250 ft p 0 x 16-Jan-2014 Dr. Mai Cao Lan. HCMUT.Exercise 8 (cont’d) Ax=10. Vietnam 107 . 5 0. Vietnam 108 .4 0.Exercise 8 (cont’d) The relative permeability data: Sw Krw 0.6 0.031 0. HCMUT.06 0.24 0. Faculty of Geology & Petroleum Engineering.045 0.7 0.8 16-Jan-2014 Kro 0 0.11 0.16 0.015 0 Dr.325 0.3 0.16 0.15 0.7 0.2 0. Mai Cao Lan.42 1 0.035 0.01 0. The End .
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