Reservoir Simulation

July 26, 2017 | Author: daonguyencm10 | Category: Petroleum Reservoir, Partial Differential Equation, Fluid Dynamics, Numerical Analysis, Liquids
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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

16-Jan-2014

Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam

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Numerical Methods for Linear Equations

16-Jan-2014

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

16-Jan-2014

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

16-Jan-2014

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

16-Jan-2014

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.

16-Jan-2014

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 

16-Jan-2014

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.

16-Jan-2014

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.

16-Jan-2014

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:

Px  0, t  0  PL

Px  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 

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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

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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

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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 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

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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

16-Jan-2014

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

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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:

uin1  uin uin1  2uin  uin1  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

61

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,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

62

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

64

FDM for Flow Equations  FD Spatial Discretization  FD Temporal Discretization

16-Jan-2014

Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam

65

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

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

67

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

68

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. 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

16-Jan-2014

Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam

70

Transmissibility

Tx

i 1 2

 Ax k x   1    c   1  1 Dx i    B i   2

16-Jan-2014

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

16-Jan-2014

Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT

72

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. 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

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. 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 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

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

79

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

16-Jan-2014

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 106 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 1o 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 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. 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)  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, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam

97

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, 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 ( 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. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam

100

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,..., 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 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. 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 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

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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