Reservoir Simulation
Short Description
Reservoir Simulation...
Description
FUNDAMENTALS OF RESERVOIR SIMULATION
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
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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
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COURSE OUTLINE INTRODUCTION FLOW EQUATIONS FOR PETROLEUM RESERVOIRS FINITE DIFFERENCE METHOD & NUMERICAL SOLUTION FOR
FLOW EQUATIONS SINGLE-PHASE FLOW SIMULATION MULTIPHASE FLOW SIMULATION
References
T. Eterkin et al., 2001. Basic Applied Reservoir Simulation, SPE, Texas
J.H. Abou-Kassem et al., 2005. Petroleum Reservoir
Simulation – A Basic Approach, Gulf Publishing Company, Houston, Texas.
C.Mattax & R. Dalton, 1990. Reservoir Simulation, SPE, Texas.
16-Jan-2014
Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
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INTRODUCTION NUMERICAL SIMULATION – AN OVERVIEW
COMPONENTS OF A RESERVOIR SIMULATOR RESERVOIR SIMULATION BASICS
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Numerical Simulation – An Overview
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Mathematical Formulation
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Numerical Methods for PDEs
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Numerical Methods for Linear Equations
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Components of a Reservoir Simulator Computer Code
Physical Model
Reservoir Simulator
Mathematical Model
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Numerical Model
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Essential Physics The basic differential equations are derived from the following essential laws:
Mass conservation law
Momentum conservation law
Material behavior principles
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Conservation of Mass Mass conservation may be formulated across a control element with one fluid of density r, 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Continuity Equation Based on the mass conservation law, the continuity equation can be
expressed as follow:
Ar u A r x t For constant cross section area, one has:
r u r x t 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Conservation of Momentum Conservation of momentum for fluid flow in porous materials is governed by the semi-empirical Darcy's equation, which for one dimensional, horizontal flow is:
k P u x
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. In general, these equations express the relationships between rock & fluid properties with respect to the reservoir pressure.
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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, the constitutive equation of rock becomes
d c f dP 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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 , l o, w, g V P T For natural gas, the well-known equation of state is used:
PV nZRT 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Single-Phase Fluid System Normally, in single-phase reservoir simulation, we would deal with one of the following fluids: Fluid System
One Phase Gas
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One Phase Water
One Phase Oil
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Single-Phase Gas The gas must be single phase in the reservoir, which means that crossing of the dew point line is not permitted in order to avoid condensate fall-out in the pores. Gas behavior is
governed by:
r gs
constant rg Bg Bg 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Single-Phase Water One phase water, 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
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Single-Phase Oil In order for the oil to be single phase in the reservoir, it must be undersaturated, which means that the reservoir pressure is higher than the bubble point pressure. In the Black Oil fluid model, oil density is described by:
ro 16-Jan-2014
r oS r gS Rso Bo
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Single-Phase Fluid Model For all three fluid systems, the one phase density or constitutive equation can be expressed as:
constant r B
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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
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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
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Single-Phase Flow Equation for Compressible Fluids
k P qsc x B x Vb t B
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Boundary Conditions (BCs) Mathematically, there are two types of boundary conditions:
• Dirichlet BCs: Values of the unknown at the boundaries are specified or given. • Neumann BCs: The values of the first derivative of the unknown are specified or given.
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Boundary Conditions (BCs) From the reservoir engineering point of view:
Dirichlet BCs: Pressure values at the boundaries are specified as known constraints. Neumann BCs: The flow rates are specified as the known constraints.
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Dirichlet Boundary Conditions For the one-dimension single phase flow, the Dirichlet boundary conditions are the pressure the pressures at the reservoir boundaries, such as follows:
Px 0, t 0 PL
Px L, t 0 PR
A pressure condition will normally be specified as a bottom-hole pressure of a production or injection well, at some position of the reservoir. 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Newmann Boundary Conditions In Neumann boundary conditions, the flow rates at the end faces of the system are specified. Using Darcy's equation, the conditions become:
kA P Q0 x x 0
kA P QL x x L
For reservoir flow, a rate condition may be specified as a production or injection rate of a well, at some position of the reservoir, or it is specified as a zero-rate across a sealed boundary or fault, or between
non-communicating layers. 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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, positive in downward direction c, c, c: Unit conversion factors
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Quantities in Flow Equations
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Quantities in Flow Equations
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Fundamentals of FDM In FDM, 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Fundamentals of FDM (cont.) To obtain higher order difference formula for the first derivative, 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Typical Difference Formulas Forward difference for first derivatives (2D)
f f ( x, y Dy ) f ( x, y ) O(Dy ) y ( x , y ) Dy or in space index form
fi , j 1 fi , j f O(Dy ) y (i , j ) Dy
i,j+1 i-1,j
i,j
i+1,j
i,j-1
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Typical Difference Formulas Backward difference for first derivatives (2D)
f f ( x, y ) f ( x, y Dy ) O(Dy ) y ( x , y ) Dy or in space index form
fi , j fi , j 1 f O(Dy ) y (i , j ) Dy
i,j+1 i-1,j
i,j
i+1,j
i,j-1
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Typical Difference Formulas Centered difference for first derivatives (2D)
f y
( x, y )
f ( x, y Dy ) f ( x, y Dy ) O(Dy 2 ) 2Dy
or in space index form i,j+1
fi , j 1 fi , j 1 f O(Dy 2 ) y (i , j ) 2Dy
i-1,j
i,j
i+1,j
i,j-1
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Typical Difference Formulas Centered difference for second derivatives (2D)
2 f y 2
( x, y )
f ( x, y Dy ) 2 f ( x, y ) f ( x, y Dy ) 2 O ( D y ) 2 Dy
or in space index form i,j+1
2 f y 2
(i , j )
fi , j 1 2 fi , j fi , j 1 Dy
2
O(Dy 2 )
i-1,j
i,j
i+1,j
i,j-1
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Exercise 2 Solve the following Poisson equation:
2u sin( x)sin( y ) 0 x 1, 0 y 1 subject to the boundary conditions:
u 0 along the boundaries x 0, x 1, y 0, y 1
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Boundary Condition Implementation Newmann BCs:
p C x b
p1 p0 p C x 11/2 x1 x0
pnx 1 pnx p C x nx 1/2 xnx 1 xnx
p0 p1 C Dx1
pnx 1 pnx C Dxnx
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Boundary Condition Implementation Dirichlet BCs:
pb C
1 p1 p2 C Dx1 Dx1 Dx2
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1 pn
x
pnx 1 C
Dxnx Dxnx Dxnx 1
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Exercise 3 Solve the following Poisson equation:
2u ( 2 2 ) exp( x y ) 0 x 1, 0 y 1, 2, 3 subject to the boundary conditions:
u exp( x y); y 0, y 1 u exp( x y ); x 0, x 1 x 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
57
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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Exercise 4 Solve the following diffusion equation:
u 2u 2 , 0 x 1.0, t 0 t x subject to the following initial and boundary conditions:
u( x 0, t ) u( x 1, t ) 0, t 0
u( x, t 0) sin( x),0 x 1 Hints: Use explicit scheme for time discretization 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Explicit Scheme The difference formulation of the original PDE in Exercise 4 is:
uin1 uin uin1 2uin uin1 Dt (Dx)2 where n=0,NT: Time step i =1,NX: Grid point index
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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,NT: Time step i =1,NX: Grid point index
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Semi-Implicit Scheme Semi-Implicit Scheme for the Diffusion Equation in Exercise 4 is
uin1 uin uin11 2uin1 uin11 uin1 2uin uin1 (1 ) 2 Dt (Dx) (Dx)2 where 0≤≤1 n=0,NT: Time step i =1,NX: Grid point index When =0.5, we have Crank-Nicolson scheme 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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FDM for Flow Equations FD Spatial Discretization FD Temporal Discretization
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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
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FDM for Slightly Compressible Fluid Flow Equations FD Spatial Discretization FD Temporal Discretization
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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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
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2
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
69
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
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Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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Transmissibility
Tx
i 1 2
Ax k x 1 c 1 1 Dx i B i 2
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2
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
71
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
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Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
72
Weighted Average of Mobility
i
1 2
1 B
i 1
2
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Dxi 1i 1 Dxi i Dxi 1 Dxi
Dxi 1i 1 Dxi i Dxi 1 Dxi
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
73
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, HCMUT, Vietnam
74
FD Temporal Discretization Explicit Method
Txni1/2 pin1 pin Txni1/2 pin pin1 qsc i Implicit Method
n 1 n Vb ct pi pi Dt c B i
Txni1/21 pin11 pin 1 Txni1/21 pin 1 pin11 qsc i Semi-implicit Method
0 1
n 1 n p p Vb ct i i Dt c B i
qsc i Txni1/21 pin11 pin 1 Txni1/21 pin 1 pin11
n 1 n p p Vb ct i i n n n n n n 1 Txi1/2 pi 1 pi Txi1/2 pi pi 1 B Dt c i
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
75
Exercise 5 For the 1D, block-centered grid shown on the screen,
determine the pressure distribution during the first year of production. The initial reservoir pressure is 6000 psia. The rock and fluid properties for this problem are:
Dx 1000ft; Dy 1000ft; Dz 75ft B 1RB/STB; =10cp; k x =15md; =0.18; c t =3.5 10 6 psi -1; Use time step sizes of =10, 15, and 30 days. Assume B is unchanged within the pressure range of interest. 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
76
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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
77
Exercise 6 For the 1D, block-centered grid shown on the screen, determine the pressure distribution during the first year of production. The initial reservoir pressure is 6000 psia. The rock and fluid properties for this problem are:
Dx 1000ft; Dy 1000ft; Dz 75ft B 1RB/STB; =10cp; k x =15md; =0.18; c t =3.5 106 psi -1; Use time step sizes of =10, 15, and 30 days. Assume B is unchanged within the pressure range of interest. 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
78
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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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FDM for Slightly Compressible Fluid Flow Equations FD Spatial Discretization FD Temporal Discretization
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
80
Transmissibility
Tx
i 1 2
Ax k x 1 c Dx i 1 B i 1 2
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2
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
82
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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
83
Exercise 7 For the 1D, block-centered grid shown on the screen, determine the pressure distribution during the first year of
production. The initial reservoir pressure is 5000 psia. The rock and fluid properties for this problem are:
Dx 1000ft; Dy 1000ft; Dz 75ft k x =15md; =0.18; ct =3.5 106 psi -1 Use time step sizes of =10 days.
16-Jan-2014
Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
85
Exercise 7 (cont’d) PVT data table: p (psia)
16-Jan-2014
(cp)
B (bbl/STB)
5000
0.675
1.292
4500
0.656
1.299
4000
0.637
1.306
3500
0.619
1.313
3000
0.600
1.321
2500
0.581
1.330
2200
0.570
1.335
2100
0.567
1.337
2000
0.563
1.339
1900
0.560
1.341
1800
0.557
1.343
Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
86
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
MULTIPHASE FLOW SIMULATION MULTIPHASE FLOW EQUATIONS FINITE DIFFERENCE APPROXIMATION TO MULTIPHASE FLOW EQUATIONS NUMERICAL SOLUTION OF THE MULTIPHASE FLOW EQUATIONS
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
88
Multiphase Flow Equations Continuity equation for each fluid flowing phase:
Arl ul A rl Sl x t
l o, w, g
Momentum equation for each fluid flowing phase:
kkrl Pl ul l x l o, w, g 16-Jan-2014
Pcow Po Pw
Pcog Pg Po
S
l
1
l o, w, g
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
89
Oil-Water Flow Equations • Considering the fluid phases of oil and water only, 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
90
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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
91
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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
92
Phase Mobility
k ro o o Bo
k rw w w Bw 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
93
Averaging of Phase Mobility Upstream:
1
2
i i o
1 2
Qw
o
weighted average:
o i
1 2
Dxi o i Dxi 1o i 1 Dxi Dxi 1
OIL Sw 1-Swir exact average upstream
Swir x 16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
94
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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
95
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 i1 Po i ) Txo 1 ( Po i1 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 i1 Po i ) Txw 1 ( Po i1 Po i ) i
16-Jan-2014
2
i
2
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
96
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) n1 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 wni1 Swin )
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
97
Discretization of Oil-phase RHS So Cpooi ( Poni 1 Poin ) Cswoi ( Swin1 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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
98
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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
99
Discretization of Water-phase RHS S w Cpowi ( Poni 1 Poin ) Cswwi ( Swin1 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
100
Fully Discrete Oil-Water Flow Equations
S
Txoi 1 Poni1 1 Poni Txoi 1 Poni1 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 Poni1 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,..., N
16-Jan-2014
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
101
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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
102
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 in1 (Cpooin i Cpowin ) Poin qosci i qwsci iTxwin 1 ( Pcowin1 Pcowin ) iTxwin 1 ( Pcowin1 Pcowin ) 2
16-Jan-2014
2
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
103
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 Poi1 Poi Txoi 12 Poi1 Poi n 1 n S wi Swi n Cswoi qosc Cpooin Pon1 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
Exercise 8 (cont’d) Ax=10,000 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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
107
Exercise 8 (cont’d) The relative permeability data:
Sw
Krw 0.16 0.2 0.3 0.4 0.5 0.6 0.7 0.8
16-Jan-2014
Kro 0 0.01 0.035 0.06 0.11 0.16 0.24 0.42
1 0.7 0.325 0.15 0.045 0.031 0.015 0
Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
108
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