Reservoir simulation (april 2017)

FUNDAMENTALS OF RESERVOIR
SIMULATION
Dr. Mai Cao Lan,
GEOPET, HCMUT, Vietnam
November, 2016

ABOUT THE COURSE
11/11/2019 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 2
COURSE OBJECTIVE
COURSE OUTLINE
REFERENCES

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

INTRODUCTION
FLOW EQUATIONS FOR PETROLEUM RESERVOIRS
FINITE DIFFERENCE METHOD & NUMERICAL SOLUTION FOR
FLOW EQUATIONS
SINGLE-PHASE FLOW SIMULATION
MULTIPHASE FLOW SIMULATION
COURSE OUTLINE

11/11/2019 5Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
 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.
References

INTRODUCTION
NUMERICAL SIMULATION – AN OVERVIEW
COMPONENTS OF A RESERVOIR SIMULATOR
RESERVOIR SIMULATION BASICS

Numerical Simulation – An Overview

Mathematical Formulation

Numerical Methods for PDEs

Numerical Methods for Linear Equations

Mathematical Model
Physical Model
Numerical Model
Computer Code
Reservoir
Simulator
Components of a Reservoir Simulator

• 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
What is Reservoir Simulation?

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.

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
Fluid
Properties
Well
Production

SINGLE-PHASE FLOW
EQUATIONS
ESSENTIAL PHYSICS
CONTINUITY EQUATION
MOMENTUM EQUATION
CONSTITUTIVE EQUATION
GENERAL 3D SINGLE-PHASE FLOW EQUATION
BOUNDARY & INITIAL CONDITIONS

Essential Physics
The basic differential equations are derived from the
following essential laws:
 Mass conservation law
 Momentum conservation law
 Material behavior principles

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:
Dx
u
r




















elementtheinside
massofchangeofRate
Dx+at xelement
theofoutMass
at xelement
theintoMass

Continuity Equation
Based on the mass conservation law, the continuity equation can be
expressed as follow:
   A u A
x t
r r
 
 
 
   u
x t
r r
 
 
 
For constant cross section area, one has:

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:
x
Pk
u





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.

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
f
T
c
P


   
   
  
For isothermal processes, the constitutive equation of rock
becomes
f
d
c
dP



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
, , ,l
T
V
c l o w g
V P
 
   
 
For natural gas, the well-known equation of state is used:
PV nZRT

Single-Phase Fluid System
Normally, in single-phase reservoir simulation, we would
deal with one of the following fluids:
One Phase Gas One Phase Water One Phase Oil
Fluid System

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:
rg 
rgs
Bg

constant
Bg

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:
rw 
rws
Bw

constant
Bw

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 
roS  rgSRso
Bo

Single-Phase Fluid Model
For all three fluid systems, the one phase density or
constitutive equation can be expressed as:
r 
constant
B

Single-Phase Flow Equation
The continuity equation for a one phase, one-dimensional system of
constant cross-sectional area is:
   rr
t
u
x 





The conservation of
momentum for 1D,
horizontal flow is: x
Pk
u




The fluid model:
r 
constant
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:
sc
b
qk P
x B x V t B


     
    
     

(1/ )
, , ,l
d B
c B l o g w
dP
 
sc t
f l
b
q ck P P P
c c
x B x V B t B t


    
           
Based on the fluid model, compressibility can now be defined in terms of
the formation volume factor as:
Then, an alternative form of the flow equation is:
(1/ )fsc
b
cqk P d B P
x B x V B dP t


    
          
Single-Phase Flow Equation for Slightly
Compressible Fluids

Single-Phase Flow Equation for Compressible
Fluids
sc
b
qk P
x B x V t B


     
    
     

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.

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.

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:
 
  R
L
PtLxP
PtxP


0,
0,0
A pressure condition will normally be specified as a bottom-hole
pressure of a production or injection well, at some position of the
reservoir.

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:
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.
0
0x
kA P
Q
x 
 
   
  Lx
L
x
PkA
Q












General 3D Single-Phase Flow Equations
The general equation for 3D single-phase flow in field units (customary
units) is as follows:
c
p Z
g

  r
    

Z: Elevation, positive in downward direction
c, c, c: Unit conversion factors
y yx x
c c
bz z
c sc
c
A kA k
x y
x B x y B y
VA k
z q
z B z t B
 
 


 
     
D  D  
      
     
 D     
     

3D Single-Phase Flow Equations for
Horizontal Reservoirs
The equation for 3D single-phase flow in field units for horizontal
reservoir is as follow:
y yx x
c c
bz z
c sc
c
A kA k p p
x y
x B x y B y
VA k p
z q
z B z t B
 
 


 
     
D  D  
      
     
 D     
     

x
x
Z
B
kA
x
Bt
V
qx
x
p
B
kA
x
xx
c
c
b
sc
xx
c
D


















D















1D Single-Phase Flow Equation with
Depth Gradient

Quantities in Flow Equations

FINITE DIFFERENCE METHOD &
NUMERICAL SOLUTION OF SINGLE-PHASE
FLOW EQUATIONS
FUNDAMENTALS OF FINITE DIFFERENCE METHOD
FDM SOLUTION OF THE SINGLE-PHASE FLOW EQUATIONS

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

Fundamentals of FDM
In FDM, derivatives are replaced by a proper difference formula based on
the Taylor series expansions of a function:
1 2 2 3 3 4 4
2 3 4
( ) ( ) ( ) ( )
( ) ( )
1! 2! 3! 4!x x x x
x f x f x f x f
f x x f x
x x x x
D  D  D  D 
 D      
   
2 2 3
2 3
( ) ( ) ( )
2! 3!x x x
f f x x f x x f x f
x x x x
  D  D  D 
   
 D  
The first derivative can be written by re-arranging the terms:
( ) ( )
( )
x
f f x x f x
O x
x x
  D 
  D
 D
Denoting all except the first terms by O (Dx) yields
The difference formula above is of order 1 with the truncation error being
proportional to Dx

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
2 3
3
( ) ( ) ( )
2 3!x x
f f x x f x x x f
x x x
  D   D D 
  
 D 
Subtracting the second from the first equation yields
2( ) ( )
( )
2x
f f x x f x x
O x
x x
  D   D
  D
 D
The difference formula above is of order 2 with the truncation error being
proportional to (Dx)2
1 2 2 3 3 4 4
2 3 4
( ) ( ) ( ) ( )
( ) ( )
1! 2! 3! 4!x x x x
x f x f x f x f
f x x f x
x x x x
D  D  D  D 
 D      
   
1 2 2 3 3 4 4
2 3 4
( ) ( ) ( ) ( )
( ) ( )
1! 2! 3! 4!x x x x
x f x f x f x f
f x x f x
x x x x
D  D  D  D 
 D      
   

Typical Difference Formulas
Forward difference for first derivatives (1D)
( ) ( )
( )
x
f f x x f x
O x
x x
  D 
  D
 D
1
( )i i
i
f ff
O x
x x
 
  D
 D
or in space index form
i-1 i i+1
Dx

Backward difference for first derivatives (1D)
( ) ( )
( )
x
f f x f x x
O x
x x
   D
  D
 D
1
( )i i
i
f ff
O x
x x

  D
 D
i-1 i i+1
Dx

Centered difference for first derivatives (1D)
2( ) ( )
( )
2x
f f x x f x x
O x
x x
  D   D
  D
 D
21 1
( )
2
i i
i
f ff
O x
x x
 
  D
 D
i-1 i i+1
Dx

Centered difference for second derivatives (1D)
2
2
2 2
( ) 2 ( ) ( )
( )
x
f f x x f x f x x
O x
x x
  D    D
  D
 D
2
21 1
2 2
2
( )i i i
i
f f ff
O x
x x
  
  D
 D
i-1 i i+1
Dx

Forward difference for first derivatives (2D)
( , )
( , ) ( , )
( )
x y
f f x y y f x y
O y
y y
  D 
  D
 D
, 1 ,
( , )
( )i j i j
i j
f ff
O y
y y
 
  D
 D
i-1,j i,j i+1,j
i,j+1
i,j-1

Backward difference for first derivatives (2D)
( , )
( , ) ( , )
( )
x y
f f x y f x y y
O y
y y
   D
  D
 D
, , 1
( , )
( )i j i j
i j
f ff
O y
y y

  D
 D
i-1,j i,j i+1,j
i,j+1
i,j-1

Centered difference for first derivatives (2D)
2
( , )
( , ) ( , )
( )
2x y
f f x y y f x y y
O y
y y
  D   D
  D
 D
, 1 , 1 2
( , )
( )
2
i j i j
i j
f ff
O y
y y
 
  D
 D
i-1,j i,j i+1,j
i,j+1
i,j-1

Centered difference for second derivatives (2D)
2
2
2 2
( , )
( , ) 2 ( , ) ( , )
( )
x y
f f x y y f x y f x y y
O y
y y
  D    D
  D
 D
2
, 1 , , 1 2
2 2
( , )
2
( )i j i j i j
i j
f f ff
O y
y y
  
  D
 D
i-1,j i,j i+1,j
i,j+1
i,j-1

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

Exercise 1
 Solve the following Poisson equation:
2
2
2
16 sin(4 )
p
x
x
 

 

subject to the boundary conditions:
p=2 at x=0 and x=1
10  x

Exercise 2
2
sin( )sin( )
0 1,0 1
u x y
x y
  
   
0 along the boundaries 0, 1, 0, 1u x x y y    

Boundary Condition Implementation
b
p
C
x



Newmann BCs:
1 0
1 1/2 1 0
0 1 1
p pp
C
x x x
p p C x


 
 
  D
1
1/2 1
1
x x
x x x
x x x
n n
n n n
n n n
p pp
C
x x x
p p C x

 


 
 
  D

Boundary Condition Implementation
Dirichlet BCs:
bp C
  1 2
1
1 2
1 p p C
x
x x
   
D
 
D  D
  1
1
1 x x
x
x x
n n
n
n n
p p C
x
x x


   
D
 
D  D

Exercise 3
2 2 2
( )exp( )
0 1,0 1, 2, 3
u x y
x y
   
 
   
     
exp( ); 0, 1u x y y y    
exp( ); 0, 1
u
x y x x
x
  

   


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

Exercise 4
 Solve the following diffusion equation:
2
2
,0 1.0, 0
u u
x t
t x
 
   
 
subject to the following initial and boundary conditions:
( 0, ) ( 1, ) 0, 0u x t u x t t    
( , 0) sin( ),0 1u x t x x   
 Hints: Use explicit scheme for time discretization

Explicit Scheme
 The difference formulation of the original PDE in Exercise 4 is:
1
1 1
2
2
( )
n n n n n
i i i i iu u u u u
t x

   

D D
where
n=0,NT: Time step
i =1,NX: Grid point index

Implicit Scheme
 The difference formulation for the original PDE in Exercise 4
1 1 1 1
1 1
2
2
( )
n n n n n
i i i i iu u u u u
t x
   
   

D D
where
n=0,NT: Time step

Semi-Implicit Scheme
Semi-Implicit Scheme for the Diffusion Equation in Exercise 4 is
1 1 1 1
1 1 1 1
2 2
2 2
(1 )
( ) ( )
n n n n n n n n
i i i i i i i iu u u u u u u u
t x x
 
   
       
  
D D D
where
0 ≤  ≤ 1
n=0,NT: Time step
When =0.5, we have Crank-Nicolson scheme

Discretization in Conservative Form
  21/2 1/2
( ) ( )
( ) i i
i i
P P
f x f x
P x x
f x O x
x x x
 
    
           
  D   D 
1
1
1/2 12
( )
( )
i i
i i i
P PP
O x
x x x

 
 
  D 
 D  D 
1
1
1/2 12
( )
( )
i i
i i i
P PP
O x
x x x

 
 
  D 
 D  D 
1 1
1/2 1/2
1 1
( ) ( )
2 ( ) 2 ( )
( ) ( )
( ) ( )
i i i i
i i
i i i i
i i
P P P P
f x f x
x x x xP
f x O x
x x x
 
 
 
 

D  D D  D  
  D   D 
( )
P
f x
x x
 
 
 
  
i-1 i i+1
Dx

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

 For slightly compressible fluids (Oil)
x x b t
c sc
c
A k V cp p
x q
x B x B t


 
   
D   
   
 For compressible fluids (Gas)
x x b
c sc
c
A k Vp
x q
x B x t B


 
     
D     
     
Single-Phase Flow Equations

FDM for Slightly Compressible Fluid Flow
Equations

Discretization of the left side term
The discretization of the left side term is then
1 1
2 21 1
2 2
( ) ( )
( ) ( )
i i
i i
i i
P P
f x f x
x xP
f x O x
x x x
 
 
    
   
      
  D   D 
where ( ) x x
c
A k
f x
B



1
1
1
2
( )
( ) / 2
i i
i i i
P PP
x x x

 
 
 
 D  D 
1
1
1
2
( )
( ) / 2
i i
i i i
P PP
x x x

 
 
 
 D  D 
FD Spatial Discretization of the LHS
1 1
2 2
1 1( ) ( )x x x x x x
c i c i i c i i
i i i
A k A k A kp
x P P P P
x B x B x B x
  
   
 
      
D         
  D D     

Define transmissibility as the coefficient in front of the
pressure difference:
2
1
2
1
1
2
1













D


ii
xx
cx
Bx
kA
T
i 

Transmissibility

FD Spatial Discretization
The left side term of the 1D single-phase flow equation is
now discritized as follow:
1 1
2 2
1 1( ) ( )x x
c i i i i ii i
i
A k P
x Tx P P Tx P P
x B x

   
  
D     
  

1
2 1 1
2 2
1
i
x x
x c
i i
A k
T
x B


 
  
   
D   
Transmissibility

   
   
1
1
1 1
2
2
x x x xx x i i
c c
i x x i x x ii i
A k A kA k
x A k x A k x
  
  
 
 
D D  D 
or
1 1 1
1
1
2
1
2
x x x x x x
c c c
i i i
A k A k A k
x x x
  
  
 
      
       D D D       
Transmissibility (cont’d)

 
 ii
iiii
i
xx
xx
DD
DD




1
11
2
1


 
 ii
iiii
i
xx
xx
DD
DD




1
11
2
1


B

1

Weighted Average of Mobility

2
1
2
1
1
2
1













D


ii
xx
cx
Bx
kA
T
i 

   
   














D





D
DD

DD







i
i
i
i
ii
iixxiixx
ixxixx
cx
B
x
B
x
xx
xkAxkA
kAkA
T
i


111
2
1
1
1
11
1
2
1
Discretized Transmissibility

FD Temporal Discretization
Explicit Method
   
 
1/2 1/2
1
1 1i i i
i
n n n
i in n n n n n n b t
x i i x i i sc
c
p pV c
T p p T p p q
B t

 

 
 
      
D 
Implicit Method
   
 
1/2 1/2
1 1
1 1 1 1 1 1 1
1 1i i i
i
n n n
i in n n n n n n b t
x i i x i i sc
c
p pV c
T p p T p p q
B t

 
 
      
 
 
      
D 
Semi-implicit Method
   
     
 
1/2 1/2
1/2 1/2
1 1 1 1 1 1 1
1 1
1 1
1 11
i ii
i i
i
n n n n n n n
sc x i i x i i
n n n
i in n n n n n b t
x i i x i i
c
q T p p T p p
p pV c
T p p T p p
B t




 
 
      
 
 
 
     
 
          D 
 0 1 

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:
6 -1
t
1000ft; 1000ft; 75ft
1RB/STB; =10cp;
k =15md; =0.18; c =3.5 10 psi ;
Use time step sizes of =10, 15, and 30 days.
Assume B is unchanged within the pressure range
of interest.
x
x y z
B 
 
D  D  D 


Exercise 5

-6 -1
6 -1
t
1000ft; 1000ft; 75ft
1RB/STB; =10cp; cf=1.0 10 psi
k =15md; =0.18 at p=3000psia; c =3.5 10 psi ;
Use time step sizes of =15 days.
Assume B and are unchanged within the pressure range
x
x y z
B 



D  D  D 
 

of interest. Also, the reservoir rock is considered as
a slightly compressible material.
Exercise 6

1 2 3 4 5
0
p
x



0
p
x



150 STB/Dscq  
1000 ft
75 ft
1000 ft
Exercise 5 (cont’d)

1 2 3 4 5
0
p
x



6000psiap 
1000
ft
75
ft
1000
ft

FDM for Slightly Compressible Fluid Flow
Equations

1 1
2 2
1 1( ) ( )x x
c i i i i ii i
i
A k p
x Tx p p Tx p p
x B x

   
  
D     
  
FD Spatial Discretization of the LHS for
Compressible Fluids
Same as that for slightly compressible fluids

2
1
2
1
1
2
1













D


ii
xx
cx
Bx
kA
T
i 

Transmissibility

1
2
1 1
1
if
if
i i i
i
i i i
p p
p p



 



 

1
B



Upstream Average of Mobility

1n n
b b
c ci i
V V
t B t B B
  
 

          
         
 D           
 expref ref
fc p p     
FD Spatial Discretization of the RHS for
Compressible Fluids

6 -1
t
1000ft; 1000ft; 75ft
k =15md; =0.18; c =3.5 10 psi
Use time step sizes of =10 days.
x
x y z
 
D  D  D 

Exercise 7

PVT data table:
p (psia)  (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

1 2 3 4 5
0
p
x



0
p
x



1000 ft
75 ft
1000 ft

MULTIPHASE FLOW
SIMULATION
MULTIPHASE FLOW EQUATIONS
FINITE DIFFERENCE APPROXIMATION TO MULTIPHASE FLOW EQUATIONS
NUMERICAL SOLUTION OF THE MULTIPHASE FLOW EQUATIONS

 Continuity equation for each fluid flowing phase:
   llll S
t
AuA
x
rr






x
Pkk
u l
l
rl
l




gwol ,,
wocow PPP 
ogcog PPP 
Sl
l  o, w, g
 1
gwol ,,
 Momentum equation for each fluid flowing phase:
11/11/2019 89Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Multiphase Flow Equations

• Considering the fluid phases of oil and water only, the
flow equations for the two phases are as follows:
scw
w
w
c
b
w
w
ww
rw
xxc q
B
S
t
V
x
x
Z
x
P
B
k
Ak
x








D

















 




sco
o
o
c
b
o
o
oo
ro
xxc q
B
S
t
V
x
x
Z
x
P
B
k
Ak
x








D

















 




cowow PPP 1 wo SS
Oil-Water Flow Equations

scw
w
w
c
b
w
cowo
ww
rw
xxc q
B
S
t
V
x
x
Z
x
P
x
P
B
k
Ak
x








D




















 




 
sco
o
w
c
b
o
o
oo
ro
xxc q
B
S
t
V
x
x
Z
x
P
B
k
Ak
x





 


D

















 1




Oil-Water Flow Equations

)()( 11 2
1
2
1 ioioixoioioixo
i
i
o
o
oo
ro
xxc
PPTPPT
x
x
Z
x
P
B
k
Ak
x

D






















)()( 11 2
1
2
1 ioioixwioioixw
i
i
w
cowo
ww
rw
xxc
PPTPPT
x
x
Z
x
P
x
P
B
k
Ak
x

D

























Left side flow terms
Discretization of the Flow Equation

o 
kro
oBo
ww
rw
w
B
k

 
Phase Mobility

1 2
ii oo
  2
1
 
 1
11
2
1



DD
DD

ii
ioiioi
io
xx
xx 

Upstream: weighted average:
x
Swir
Sw
1-Swir
Qw
average
upstream
exact
OIL
Averaging of Phase Mobility










ioioio
ioioio
i
o
PPif
PPif
1
11
2
1












iwiwiw
iwiwiw
i
w
PPif
PPif
1
11
2
1



Upstream Average of Mobility

1 1 1 1
2 2
( ) ( )i i i ii i
ro o
c x x o i
o o i
xo o o xo o o
k P Z
k A x
x B x x
T P P T P P
 

  
   
 D      
   
1 1 1 1
2 2
( ) ( )i i i ii i
rw o cow
c x x w i
w w i
xw o o xw o o
k P P Z
k A x
x B x x x
T P P T P P
 

  
    
  D       
   
Left side flow terms
Discretization of Multiphase Flow
Equation

Right side flow terms



















o
o
o
oo
o
Bt
S
t
S
BB
S
t

1 1
1(1/ )
( )i
n n
o n no r
oo o i
o o iio i
S c d B
S P P
t B t B dP

 
     
        D    
The second term:
1 1
1
( )i
n n
n no
ww i
oo ii
S
S S
B t B t

 
   
       
 D  
wo SS 1
The first term:
Discretization of the Oil-Phase Equation

and
11 1 1
( ) ( )n
ii i i
n n n n no
poo o swo w wo i i
o i
S
C P P C S S
t B
    
    
  
1 1
1 (1 ) (1/ )
i
i
n n
w on r
poo
o oi
S c d B
C
t B dP

 
    
    D   Where:
1
1
i
n
n
swo
o
i
C
B t


  
  
D 
Discretization of Oil-phase RHS

Right side flow terms



















w
w
w
ww
w
Bt
S
t
S
BB
S
t







































t
P
t
P
BPt
P
BPBt
cowo
ww
w
www

t
w
w
cow
t
cow S
dS
dPP





Discretization of Water-Phase Equation

and
Where:
11 1 1
( ) ( )n
ii i i
n n n n nw
pow o sww w wo i i
w i
S
C P P C S S
t B
    
    
  
11
1 (1/ )
i
i
nn
wn wr
pow
i w w
d BS c
C
t B dP


   
   
D   
1 1
1 1
i i
i
n n
n ncow
sww pow
ww i
dP
C C
B t dS

 
    
    D   
Discretization of Water-phase RHS

Ni ,...,1
     
 
1 1
1 12 2
1 1 1 1 1 1 1 1
1 1 1
i i i i i
i osci
n n n n n n n n n
xo xo poo oo o o o i o ii i
n n n n
swo wi w i
T P P T P P C P P
C S S q
 
       
 
  
    
  
       
   
1 1
1 1 1 12 2
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1
i i i i i i i i
i i wsci
n n n n n n n n n n
xw xwo o cow cow o o cow cowi i
n n n n n n n
o sww wpowi o i i w i
T P P P P T P P P P
C P P C S S q
   
         
 
    
         
   
    
Fully Discrete Oil-Water Flow Equations
(Fully Implicit)
Ni ,...,1

1
1
n
swoi
i n
swwi
C
C



 
       
   
  
1 1 1 1
1 12 2 2 2
1 1
1 12 2
1 1 1 1 1 1 1 1
1 1
1 1 1 1 1
i i i i
i i i i
i
n n n n n n n n
xo xw xo xwi o o i o oi i i i
n n n n n n
xw xwi cow cow i cow cowi i
n n n n n n
poo pow oi i i o i osci i wsci
T T P P T T P P
T P P T P P
C C P P q q
 
 
 
 
 
       
   
 
 
    
    
   
    
First, the pressure is found by solving the following equation:
IMPES Solution of Oil-Water Flow Equations

11
1
1111
1
1 




 nn
i
nn
i
nn
i
n
iiii
gPEPCPW ooo
1 1
2 2
1 1 1
i
n n n
xo xwii i
W T T  
 
 
1 1
2 2
1 1 1 1 1
1 1 1 1 1 1
1 1
( )
( ) ( )
i
n n n n n n
poo pow oi i i i osci i wsci
n n n n n n
xw cow cow xw cow cowi i i i i ii i
g C C P q q
T P P T P P
 
 
    
     
  
    
   
1
1
n
swoi
i n
swwi
C
C



 
 
 
1 1
2 2
1 1
2 2
1 1 1 1
1 1 1
i
n n n n
xo xo pooii i
n n n
xw xw powi ii i
C T T C
T T C
   
 
  
 
   
  
IMPES Pressure Solution
1 1
2 2
1 1 1
i
n n n
xo xwii i
E T T  
 
 

Once the oil pressures have been found, water saturations
can be obtained by either the oil-phase equation or the
water-phase equation.
   
 
1 1
1 12 2
1 1 1 1 1 1
1
1 1 1 1
1 i i i i
i
i
n n n n n n
xo xoo o o oi in n
ww i n n n n nswo poo oi osci i o i
T P P T P P
S S
C q C P P
 
     
 
   
   
  
    
Ni ,...,1
IMPES Water Saturation

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 constant and given as
w=0.8cp and Bw=1.02 bbl/STB during the period of interest.
Exercise 8

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 in Table 1 as the functions of pressure. The
saturation functions including relative permeabilities and
capillary pressure.
Using the IMPES solution method with Dt=10 days, find
the pressure and saturation distribution after 100 days of
production.

1 2 3 4
0
p
x



250 ft
Ax=10,000 ft2
0
p
x



Qo=-75 STB/dQw=75 STB/d

PVT data table:
p (psia)  (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

The relative permeability data:
Sw Krw Kro
0.16 0 1
0.2 0.01 0.7
0.3 0.035 0.325
0.4 0.06 0.15
0.5 0.11 0.045
0.6 0.16 0.031
0.7 0.24 0.015
0.8 0.42 0

DATA PREPARATION
INTRODUCTION
GROUPS OF DATA REQUIRED FOR A SIMULATION STUDY
SOURCES OF DATA FOR A SIMULATION STUDY

INTRODUCTION
The initial and often the most time consuming phase of a simulation study,
is the acquisition and interpretation of descriptive data for the reservoir
and reducing this data to a format acceptable to the simulation program.
DATA GATHERED
Seismic,
Cores, Logs
Total injection
Total production
Fluid properties (PVT)
Reservoir pressure (BHP)
Production by layer (PLT)
Fluid contacts (TDT, logs)
DATA USEAGE
Structure
Reservoir quality
Faulting
Continuity
Continuity
Depletion
Displacement
Fluid behavior
Residual oil
Sweep
MODELS +
DOCUMENTS
Development Drilling and
Production
Reservoir
Development
Strategy

Groups of Data Required for
a Simulation Study
Rock Data
permeability, relative permeability, capillary pressure,
porosity, saturations, thickness, depth, compressibility
Fluid Data
PVT, viscosity, density, formation volume factor,
compressibility, solution gas-oil ratio

Production Data
flow rate, pressure, PI, II
Mechanical and Operational Data
lifting capacity, operational constraints
Economic Data
product price, capex, opex, economic limit
Miscellaneous Data
Well stimulation, workover
Groups of Data Required for a Simulation Study

Data Required for a Simulation Study –
Sources of Data
Property Sources
Permeability Pressure transient testing,
Core analyses
Porosity, Rock
compressibility
Core analyses, Well logs
Relative permeability
and capillary
pressure
Laboratory core flow tests

Property Sources
Saturations Well logs, Core analyses,
Single-well tracer tests
Fluid property (PVT) data Laboratory analyses of reservoir
fluid samples
Faults, boundaries, fluid
contacts
Seismic, Pressure transient testing
Sources of Data

Property Sources
Aquifers Seismic, Regional exploration
studies
Fracture spacing,
orientation, connectivity
Core analyses, Well logs, Seismic,
Pressure transient tests,
Interference testing
Rate and pressure data,
completion and workover
data
Field performance history
Sources of Data

EXAMPLE OF RESERVOIR SIMULATION
WITH ECLIPSE100
PROBLEM DEFINITION
BRIEF INTRODUCTION ABOUT ECLIPSE
DATA SECTIONS IN ECLIPSE100
TYPICAL KEYWORDS IN SECTIONS

Problem Definition
Consider a 2-phase (oil,water) reservoir model having 5x5x3 cells (in X,Y,Z
directions, respectively). The cell sizes are 500ft x 500ft x 75ft, respectively and the
depth of reservoir top structure is 8,000ft. A production well (named as PROD) was
drilled at location (x,y)=(1,1) through the whole reservoir thickness. An injection
well (named as INJ) was drilled at location (x,y)=(5,5) through the whole reservoir
thickness. Both wells were completed by perforations in the entire reservoir
thickness, starting from the depth of 8,000ft.
The reservoir has 3 layers whose permeabilities in X,Y,Z directions are:
Layer Kx Ky Kz
1 200 150 20
2 1000 800 100
3 200 150 20
Create a data file to perform reservoir simulation by using ECLIPSE 100

Brief Introduction about Eclipse
• ECLIPSE 100 is a fully-implicit, three phase, three dimensional,
general purpose black oil simulator with gas condensate option.
• ECLIPSE 100 can be used to simulate 1, 2 or 3 phase systems. Two
phase options (oil/water, oil/gas, gas/water) are solved as two
component systems saving both computer storage and computer
time. In addition to gas dissolving in oil (variable bubble point
pressure or gas/oil ratio), ECLIPSE 100 may also be used to model
oil vaporizing in gas (variable dew point pressure or oil/gas ratio).
• Both corner-point and conventional block-center geometry
options are available in ECLIPSE. Radial and Cartesian block-
center options are available in 1, 2 or 3 dimensions. A 3D radial
option completes the circle allowing flow to take place across the
0/360 degree interface.

Data Sections in Eclipse100
RUNSPEC
SUMMARY
SOLUTION
REGIONS
PROPS
GRID
EDIT
Request output for line plots (optional section)
Initialization
Subdivision of the reservoir (optional section)
PVT & SCAL properties
Modification of the processed GRID data (optional section)
General model characteristics
Grid geometry and basic rock properties
SCHEDULE
Wells, completions, rate data, flow correlations, surface facilities
Simulator advance, control and termination

Typical Keywords in Sections
Runspec Section
Title, problem dimensions, switches, phases present, components etc.
TITLE title
DIMENS
OIL, WATER, GAS, VAPOIL, DISGAS
FIELD/METRIC/LAB
WELLDIMS
number of blocks in X,Y,Z directions
the active phases present
unit convention
well and group dimensions
1 The maximum number of wells in the model
2 The maximum number of connections per well
3 The maximum number of groups in the model
4 The maximum number of wells in any one group

data checking only, with no simulation
UNIFIN
UNIFOUT
START
NOSIM
indicates that input files are unified
indicates that output files are unified
start date of the simulation
Runspec Section

RUNSPEC
TITLE
3D 2-PHASE SIMULATION
-- Number of cells
-- NX NY NZ
DIMENS
5 5 3 /
-- Phases
OIL
WATER
-- Units
FIELD
-- Well dimensions
-- Maximum # connections # groups # wells
-- # wells per well per group
WELLDIMS
2 3 2 1 /
-- Unified output files
UNIFOUT
-- Simulation start date
START 16 MAR 2010 /
Runspec Section
Example

Grid Section
Cell properties such as PORO, PERMX,
PERMY, PERMZ, NTG are averages defined at
the centre
TOPS
depths of top faces of grid blocks for the
current box; data is taken from Structure
map, and geological model from IRAP
DX, DY, DZ
PORO
X,Y,Z-direction grid block sizes for the current box; data
is taken from Isopac map, and geological model from
IRAP
X,Y,Z-direction permeabilities for
the current box; data is taken from Isopac map, and
geological model from IRAP
grid block porosities for the current box; data is taken from
Isopac map, and geological model from IRAP
PERMX, PERMY, PERMZ

TYPICAL KEYWORDS IN SECTIONS
GRID SECTION
Example
GRID
Size of each cell in X,Y and Z directions
DX 75*500 /
DY 75*500 /
DZ 75*50 /
-- TVDSS of top layer only
-- X1 X2 Y1 Y2 Z1 Z2
BOX
1 5 1 5 1 1 /
TOPS
25*8000 /
ENDBOX
-- Permeability in X,Y and Z directions for each cell
PERMX 25*200 25*1000 25*200 /
PERMY 25*150 25*800 25*150 /
PERMZ 25*20 25*100 25*20 /
-- Porosity
PORO75*0.2 /
-- Output file with geometry and rock properties

props section
SWFN
water relative permeability and capillary
pressure as functions of Sw
PVT: Tables of properties of reservoir rock and fluids as functions of fluid
pressures, saturations and compositions
SCAL: Phase Relative Permeabilities
Column 1 The water saturation
Column 2 The corresponding water relative permeability
Column 3 The corresponding water-oil capillary pressure
SOF3
oil relative permeability as a function of So
in three phase system
Column 1 The oil saturation
Column 2 The corresponding oil relative permeability for regions
where only oil and water are present
Column 3 The corresponding oil relative permeability for
regions where only oil, gas and connate water are present.

props section
SGFN
gas relative permeability and capillary
pressure as functions of Sg
Column 1 The gas saturation.
Column 2 The corresponding gas relative permeability
Column 3 The corresponding oil-gas capillary pressure
SWOF Water / oil saturation functions versus water saturation
Column 1 The water saturation
Column 2 The corresponding water relative permeability
Column 3 The corresponding oil relative permeability when
only oil and water are present.
Column 4 The corresponding water-oil capillary pressure

DENSITY stock tank fluid densities
PVTG
PVTW
FVF and viscosity of wet gas as functions of
pressure and Rv
FVF, compressibility and viscosity of water
ROCK rock compressibility
props section
PVTO
FVF and viscosity of live oil as functions of
pressure and Rs
Item 2 The bubble point pressure (Pbub) for oil with dissolved
gas-oil ratio given by item 1.
Item 1 The dissolved gas-oil ratio (Rs)
Item 3 The oil formation volume factor for saturated oil at Pbub.
Item 4 The oil viscosity for saturated oil at Pbub.

Props Section
Example
PROPS
-- Densities in lb/ft3
-- Oil Water Gas
-- --- ---- ---
DENSITY
49 63 0.01 /
-- PVT data for dead oil
-- P Bo Vis
-- -- -- ---
PVDO
300 1.25 1.0
800 1.20 1.1
6000 1.15 2.0 /
-- PVT data for water
-- P BW CW VIS VISCOSIBILITY
-- -- -- -- --- -------------
PVTW
4500 1.20 3E-06 0.8 0.0 /

-- Rock compressibility
-- P Cr
-- -- --
ROCK
4500 4e-06 /
-- Water and oil relative perms and
capillary pressure
-- Sw Krw Kro Pc
-- -- --- --- --
SWOF
0.25 0.0 0.9 4.0
0.5 0.2 0.3 0.8
0.7 0.4 0.1 0.2
0.8 0.55 0.0 0.1 /
Props Section
Example

Splits computational grid into regions for calculation of:
- PVT properties (fluid densities and viscosities),
- saturation properties (relative permeabilities and capillary pressures)
- initial conditions, (equilibrium pressures and saturations)
- fluids in place (fluid in place and inter-region flows)
Regions Section
FIPNUM fluid-in-place region numbers
SATNUM saturation table regions
The region numbers should not be less than 1 or greater than
NTFIP (the maximum number of fluid-in-place regions)
The saturation function region number specifies which set of
saturation functions (input using SGFN, SOF3, etc. in the
PROPSsection)

Regions Section
EQLNUM
PVTNUM
Equilibration regions
PVT data regions
All blocks with the same equilibration region number must also
have the same PVT region number
The PVT region number specifies which set of PVT tables (input
using DENSITY, PVDG, PVDO, PVTG, PVTO, PVCO, PVTW and ROCK
in the PROPSsection) should be used to calculate PVT properties
of fluids in each grid block for a black oil model

Solution Section
The SOLUTION is used to define the initial state of every cell in the model
 Initial pressure and phase saturation
 Initial solution ratios
 Depth dependence of reservoir fluid properties
 Oil and gas re-solution rates
 Initial analytical aquifer conditions
EQUIL
fluid contact depths and other equilibration parameters;
data taken from well testing
RESTART
RPTSOL
name of the restart file
report switches for SOLUTION data
1 Datum depth
2 Pressure at the datum depth.
3 Depth of the water-oil contact

Solution Section
Example
SOLUTION
-- Initial equilibration conditions
-- Datum Pi WOC Pc@WOD
-- @datum
-- ---- ---- --- ------
EQUIL
8000 4500 8200 0.0 /
-- Output to restart file for t=0
(.UNRST)
-- Rst file Graphics
-- for ic only
-- ------- --------
RPTRST
BASIC=2 NORST=1 /

Summary Section
The SUMMARY section is used to specify variables that are to be
written to the Summary file(s) after each time step of the simulation
Well Oil Production Rate
FOPT Field Oil Production Total
FOPR
FGOR
FWIR
FOE
FPR
WBHP
FWCT
WOPR
Field Oil Production Rate
Field Gas-Oil Ratio
Field Water Injection Rate
Field Oil Efficiency
Field Pressure
Well Bottom Hole Pressure
Field Water CuT

Summary Section
Example
SUMMARY
-- Field average pressure
FPR
Bottomhole pressure of all wells
WBHP
/
-- Field oil production rate
FOPR
-- Field water production rate
FWPR
-- Field oil production total
FOPT
-- Field water production total
FWPT
-- Water cut in PROD
WWCT
PROD /
-- CPU usage
TCPU

Schedule Section
Specifies the operations to be simulated (production and injection controls and
constraints) and the times at which output reports are required.
Vertical flow performance curves and simulator tuning parameters may also be
specified in the SCHEDULE section.
RPTSCHED
TUNING
WELSPECS
report switches to select which simulation results are to be printed at
report times
time step and convergence controls
introduces a new well, defining its name, the position of the wellhead,
its bottom hole reference depth and other specification data
1 Well name
2 Name of the group to which the well belongs
3 I - location of well head or heel
4 J - location of well head or heel
5 Reference depth for bottom hole pressure
6 Preferred phase for the well
7 Drainage radius for productivity/injectivity index calculation

COMPDAT
specifies the position and properties of one or more well completions;
this must be entered after the WELSPECS
WCONPROD control data for production wells
1 Well name, well name template, well list or well list template
2 I - location of connecting grid block(s)
3 J - location of connecting grid block(s)
4 K - location of upper connecting block in this set of data
5 K - location of lower connecting block in this set of data
6 Open/shut flag of connection
2 Open/shut flag for the well
3 Control mode
4 Oil rate target or upper limit.
5 Water rate target or upper limit
Schedule Section

WCONHIST observed rates for history matching wells
WCONINJE control data for injection wells
TSTEP or DATE advances simulator to new report time(s) or specified report date(s)
2 Injector type
3 Open/shut flag for the well
4 Control mode
5 Surface flow rate target or upper limit
Schedule Section

Schedule Section
Example
SCHEDULE
-- Output to restart file for t>0 (.UNRST)
-- Restart File Graphics
-- every step only
-- ------------ -----------
RPTRST
BASIC=2 NORST=1 /
-- Location of well head and pressure gauge
-- Well Well Location BHP Pref.
-- name group I J datum phase
-- ---- ----- -- -- ----- -----
WELSPECS
PROD G1 1 1 8000 OIL /
INJ G2 5 5 8000 WATER /
/
-- Completion interval
-- Well Location Interval Status OTHER Well
-- name I J K1 K2 O or S PARAMS ID
-- ---- -- -- -- -- ------ ------ ----

COMPDAT
PROD 1 1 1 3 OPEN 2*
0.6667 /
INJ 5 5 1 3 OPEN 2*
0.6667 /
/
-- Production control
-- Well Status Control Oil Water Gas Liquid Resvr BHP
-- name mode rate rate rate rate
rate lim
-- ---- ------ ------ ---- ---- ---- ------ ----- -----
WCONPROD
PROD OPEN LRAT 3* 10000
1* 2000 /
/
-- Injection control
-- Well Fluid Status Control Surf Resvr Voidage BHP
-- name type mode rate rate frac flag lim
-- ---- ----- ----- ------ ----- ----- ---- ----
Schedule Section
Example

HISTORY MATCHING
OVERVIEW OF HISTORY MATCHING
WHAT IS MATCHED?
WHAT IS ADJUSTED?
ACTION STEPS IN HISTORY MATCHING
EXAMPLE OF ADJUSTMENT
PROBLEM DEFINITION

History Matching: Comprising the adjustment of reservoir parameters in the
model until the simulated performance matches the measured information
Mathematically: Inverse Problem That is, we know the Model and we know
the answer, but we do not know the input to the model. There are special techniques
for solving inverse problem, but these do not apply to reservoir simulation history
matching. Thus, we use trial and ERROR
Overview of History Matching

 Gas-Oil Ratio (GOR)
 Water-Oil Ratio (WOR)
 Temperature
 Individual Well History
 Shut-in Pressures (Build-ups)
What are matched?
 Rates
 Break Through (BT)
 Fluid Contact History
 Overall Reservoir Performance

 Thickness
What are adjusted?
Any parameters which describe the reservoir
 Permeability
 Porosity
 Net-to-Gross
 Uncertain Areas of the Structure

 Well Saturation
What are adjusted?
Any parameters which describe the reservoir
 Faults
 Shape and Endpoints of Saturation Functions
 Transmissibility

1. Assemble data on performance history.
2. Screen the data and evaluate their quality.
3. Define the specific objectives of the history matches.
Action Steps in History Matching
4. Develop a preliminary model based on the best
available data.
5. Simulate history with the preliminary model and
compared simulated performance with actual field history.
6. Decide whether the model is satisfactory.

7. Identify changes in model properties that are most likely to
improve agreement between observed and calculated
performance.
8. Decide whether an automatic matching program should be
used.
9. Make adjustments to the model. Consult with geologic,
drilling, production operations personnel to confirm
the realism of proposed changes.
10. Again, simulate part or all of the past performance data
to improve the match. Analyze results as in Step 6.
11. Repeat Step 6, 9, and 10 until a satisfactory match of
observed data is obtained.
Action Steps in History Matching

Simulation field pressure too high
Possible Changes
Pore Volume?
Aquifer?
Oil Initially in Place
(Contacts, So)
Energy?
Gas cap size?
Example of adjustment

Possible Changes
Krw / Kro ratio decrease
Aquifer size

Possible Changes
Effective end point Krw ?
Horiz. Permeability of well to aquifer layer?
Shale or barrier between wells and water?
Vertical permeability between wells and
water?
Numerical dispersion / grid effect?

Gas BT OK, After BT simulation slope in
error
Krg / Kro ratio increase?
Supply of gas?
Possible Changes

Well GOR simulation BT too early
Possible Changes
Shale or barrier between well and gas?
Vertical permeability between well and gas?

Well water simulation BT too early
Possible Changes
Shale or barrier?
Vertical permeability between well and
water?

HISTORY MATCHING – CASE STUDY
Given the history data in terms of oil, gas, water production rates, bottom
hole pressure, and reservoir pressure of a waterflooding project having
one injector and one producer as depicted in figure below, perform
history matching by adjusting the following unknown properties:
• Permeability in the horizontal direction
• Permeability in the vertical direction

Horizontal and vertical permeability
w
0 w
w
( )
( )
141.2 ln( ) 0.75
R f
f
e
o o
kh P P
Q P f k
r
B S
r


  
 
  
 
w
0 w
w
( )
( )
141.2 ln( ) 0.75
ro R f
f ro
e
o o
kk h P P
Q P f kk
r
B S
r


  
 
  
 
The well bottom hole pressure (WBHP) is the function of average permeability
when there is single phase flow ( See equation 1); when multi-phase flow occurs,
the WBHP is a function of relative permeability and average permeability (see
equation 2).
When the water breakthrough has not occurred, the WBHP depends on the
average permeability. Assume that the oil flows in the horizontal plane, so before
water breakthrough, WBHP depends on horizontal permeability (Kxx and Kyy).
After water breakthrough, the water flow up ward because of up dip water
injection. The WBHP mainly depend on vertical permeability (Kzz)
(1)
(2)

The permeability in the horizontal direction (Kxx = Kyy) was adjusted by
comparison of well bottom hole pressure of producer. Choose the first valve of
Kxx = Kyy = 250 md and Kzz = 0.1Kxx = 25md.
Figure 1 – The result of first trail of K = K = 250 md and K = 25md.
History
Simulation

Figure 1 shows that, the WBHP of producer in case of Kxx = 200 md is smaller
the base case. Based on equation 2, the horizontal permeability should be
increased. For the second trail, Kxx = Kyy = 315 md and Kzz = 0.1Kxx = 35md.
Figure 2 The result of first trail of Kxx = Kyy = 315 md and Kzz = 35md.
History
Simulation

From figure 2, the well bottom hole pressure is matched for the stage of before
water breakthrough.
Since this is updip water injection. In this matching work, well bottom hole
pressure of producer is a function of vertical permeability after water
breakthrough.
The bottom hole pressure is smaller than the base case when water
breakthrough. In this case, the vertical permeability should be increased and
Kzz was 73md in next trail

Figure 3 The result of first trail of Kxx = Kyy = 315 md and Kzz = 73md
History
Simulation
Thus, the horizontal Kxx = Kyy = 315md and Kzz = 73 md are matched with
the given data.

Reservoir simulation (april 2017)

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