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FUNDAMENTALS OF RESERVOIR
SIMULATION
Dr. Mai Cao Lan,
GEOPET, HCMUT, Vietnam
Jan, 2014
ABOUT THE COURSE
16-Jan-2014 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 3
INTRODUCTION
FLOW EQUATIONS FOR PETROLEUM RESERVOIRS
FINITE DIFFERENCE METHOD & NUMERICAL SOLUTION FOR
FLOW EQUATIONS
SINGLE-PHASE FLOW SIMULATION
MULTIPHASE FLOW SIMULATION
COURSE OUTLINE
16-Jan-2014 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 6
Numerical Simulation – An Overview
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 7
Mathematical Formulation
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 8
Numerical Methods for PDEs
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 9
Numerical Methods for Linear Equations
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 10
Mathematical Model
Physical Model
Numerical Model
Computer Code
Reservoir Simulator
Components of a Reservoir Simulator
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 11
• 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?
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 12
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 13
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 14
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 15
Essential Physics
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 16
The basic differential equations are derived from the
following essential laws:
Mass conservation law
Momentum conservation law
Material behavior principles
Conservation of Mass
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 17
Mass conservation may be formulated across a control element with one fluid
of density r, flowing through it at a velocity u:
Dx
ur
element theinside
mass of change of Rate
Dx+at xelement
theofout Mass
at xelement
theinto Mass
Continuity Equation
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 18
Based on the mass conservation law, the continuity equation can be
expressed as follow:
A u Ax t
r r
ux tr r
For constant cross section area, one has:
Conservation of Momentum
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 19
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
Pku
Equation Governing Material Behaviors
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 20
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 21
The behavior of reservoir rock corresponding to the
pressure declines can be expressed by the definition of the
formation compaction
1f
T
cP
For isothermal processes, the constitutive equation of rock becomes
f
dc
dP
Constitutive Equation of Fluids
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 22
The behavior of reservoir fluids corresponding to the
pressure declines can be expressed by the definition of fluid
compressibility (for liquid)
1, , ,l
T
Vc l o w g
V P
For natural gas, the well-known equation of state is used:
PV nZRT
Single-Phase Fluid System
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 23
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 24
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 25
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 rwsBw
constant
Bw
Single-Phase Oil
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 26
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 27
For all three fluid systems, the one phase density or
constitutive equation can be expressed as:
r
constant
B
Single-Phase Flow Equation
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 28
The continuity equation for a one phase, one-dimensional system of
constant cross-sectional area is:
rrt
ux
The conservation ofmomentum for 1D,horizontal flow is: x
Pku
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 Bc B l o g w
dP
sc tf l
b
q ck P P Pc 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 29
Single-Phase Flow Equation for Compressible Fluids
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 30
sc
b
qk P
x B x V t B
Boundary Conditions (BCs)
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 31
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)
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 32
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 33
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 34
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 PQ
x
Lx
Lx
PkAQ
General 3D Single-Phase Flow Equations
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 35
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 xc c
bz zc sc
c
A kA kx y
x B x y B y
VA kz q
z B z t B
D D
D
3D Single-Phase Flow Equations for Horizontal Reservoirs
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 36
The equation for 3D single-phase flow in field units for horizontal
reservoir is as follow:
y yx xc c
bz zc sc
c
A kA k p px y
x B x y B y
VA k pz q
z B z t B
D D
D
xx
Z
B
kA
x
Bt
Vqx
x
p
B
kA
x
xxc
c
bsc
xxc
D
D
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 37
1D Single-Phase Flow Equation with Depth Gradient
Quantities in Flow Equations
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 38
Quantities in Flow Equations
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 39
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 40
Numerical Solution of Flow Equations
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 41
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 ff 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 xO 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 42
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 xO 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 ff 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 ff x x f x
x x x x
D D D D D
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 43
Typical Difference Formulas
Forward difference for first derivatives (1D)
( ) ( )( )
x
f f x x f xO x
x x
D D
D
1 ( )i i
i
f ffO x
x x
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 44
i-1 i i+1
Dx
Typical Difference Formulas
Backward difference for first derivatives (1D)
( ) ( )( )
x
f f x f x xO x
x x
D D
D
1 ( )i i
i
f ffO x
x x
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 45
i-1 i i+1
Dx
Typical Difference Formulas
Centered difference for first derivatives (1D)
2( ) ( )( )
2x
f f x x f x xO x
x x
D D D
D
21 1 ( )2
i i
i
f ffO x
x x
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 46
i-1 i i+1
Dx
Typical Difference Formulas
Centered difference for second derivatives (1D)
22
2 2
( ) 2 ( ) ( )( )
x
f f x x f x f x xO x
x x
D D D
D
221 1
2 2
2( )i i i
i
f f ffO x
x x
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 47
i-1 i i+1
Dx
Typical Difference Formulas
Forward difference for first derivatives (2D)
( , )
( , ) ( , )( )
x y
f f x y y f x yO y
y y
D D
D
, 1 ,
( , )
( )i j i j
i j
f ffO y
y y
D
D
or in space index form
i-1,j i,j i+1,j
i,j+1
i,j-1
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 48
Typical Difference Formulas
Backward difference for first derivatives (2D)
( , )
( , ) ( , )( )
x y
f f x y f x y yO y
y y
D D
D
, , 1
( , )
( )i j i j
i j
f ffO y
y y
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 49
i-1,j i,j i+1,j
i,j+1
i,j-1
Typical Difference Formulas
Centered difference for first derivatives (2D)
2
( , )
( , ) ( , )( )
2x y
f f x y y f x y yO y
y y
D D D
D
, 1 , 1 2
( , )
( )2
i j i j
i j
f ffO y
y y
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 50
i-1,j i,j i+1,j
i,j+1
i,j-1
Typical Difference Formulas
Centered difference for second derivatives (2D)
22
2 2
( , )
( , ) 2 ( , ) ( , )( )
x y
f f x y y f x y f x y yO y
y y
D D D
D
2, 1 , , 1 2
2 2
( , )
2( )
i j i j i j
i j
f f ffO y
y y
D
D
or in space index form
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 51
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 52
Exercise 1
Solve the following Poisson equation:
22
216 sin(4 )
px
x
subject to the boundary conditions:
p=2 at x=0 and x=1
10 x
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 53
Exercise 2
Solve the following Poisson equation:
2 sin( )sin( )
0 1,0 1
u x y
x y
subject to the boundary conditions:
0 along the boundaries 0, 1, 0, 1u x x y y
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 54
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 55
Boundary Condition Implementation
b
pC
x
Newmann BCs:
1 0
1 1/2 1 0
0 1 1
p ppC
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 ppC
x x x
p p C x
D
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 56
Boundary Condition Implementation
Dirichlet BCs:
bp C
1 2
1
1 2
1 p p C
x
x x
D
D D
1
1
1x x
x
x x
n n
n
n n
p p C
x
x x
D
D D
Exercise 3
Solve the following Poisson equation:
2 2 2( )exp( )
0 1,0 1, 2, 3
u x y
x y
subject to the boundary conditions:
exp( ); 0, 1u x y y y
exp( ); 0, 1u
x y x xx
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 58
Exercise 4
Solve the following diffusion equation:
2
2,0 1.0, 0
u ux 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 59
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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 60
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
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
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
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
21/2 1/2
( ) ( )
( ) i i
i i
P Pf x f x
P x xf x O x
x x x
D D
1
11/2 12
( )( )
i i
i i i
P PPO x
x x x
D
D D
1
11/2 12
( )( )
i i
i i i
P PPO x
x x x
D
D D
1 11/2 1/2
1 1
( ) ( )2 ( ) 2 ( )
( ) ( )( ) ( )
i i i ii i
i i i i
i i
P P P Pf x f x
x x x xPf x O x
x x x
D D D D D D
( )P
f xx x
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 64
i-1 i i+1
Dx
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
16-Jan-2014 66Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
For slightly compressible fluids (Oil)
x x b tc sc
c
A k V cp px q
x B x B t
D
For compressible fluids (Gas)
x x bc sc
c
A k Vpx q
x B x t B
D
Single-Phase Flow Equations
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
Discretization of the left side term
The discretization of the left side term is then
1 12 21 1
2 2
( ) ( )
( ) ( )
i i
i i
i i
P Pf x f x
x xPf x O x
x x x
D D
where ( ) x xc
A kf x
B
1
11
2
( )
( ) / 2
i i
i i i
P PP
x x x
D D
1
11
2
( )
( ) / 2
i i
i i i
P PP
x x x
D D
FD Spatial Discretization of the LHS
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 68
1 12 2
1 1( ) ( )x x x x x xc i c i i c i i
i i i
A k A k A kpx P P P P
x B x B x B x
D
D D
Define transmissibility as the coefficient in front of thepressure difference:
2
1
2
1
1
21
D
ii
xxcx
Bx
kAT
i
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 69
Transmissibility
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 70
FD Spatial Discretization
The left side term of the 1D single-phase flow equation is now discritized as follow:
1 12 2
1 1( ) ( )x xc i i i i ii i
i
A k Px Tx P P Tx P P
x B x
D
12 1 1
2 2
1
i
x xx c
i i
A kT
x B
D
Transmissibility
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 71
16-Jan-2014 72Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
1
11 1
2
2x 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
11
2
1
2
x x x x x xc c c
i i i
A k A k A k
x x x
D D D
Transmissibility (cont’d)
ii
iiii
ixx
xx
DD
DD
1
11
21
ii
iiii
ixx
xx
DD
DD
1
11
21
B
1
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 73
Weighted Average of Mobility
2
1
2
1
1
21
D
ii
xxcx
Bx
kAT
i
D
D
DD
DD
i
i
i
i
ii
iixxiixx
ixxixx
cx
Bx
Bx
xx
xkAxkA
kAkAT
i
111
2
1
1
1
11
1
2
1
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 74
Discretized Transmissibility
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 75
FD Temporal Discretization
Explicit Method
1/2 1/2
1
1 1i i i
n n
i in n n n n n b tx i i x i i sc
c i
p pV cT p p T p p q
B t
D Implicit Method
1/2 1/2
1
1 1 1 1 1 1
1 1i i i
n n
i in n n n n n b tx i i x i i sc
c i
p pV cT 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 11
i i i
i i
n n n n n n
sc x i i x i i
n n
i in n n n n n b tx i i x i i
c i
q T p p T p p
p pV cT 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 76
1 2 3 4 5
0p
x
0p
x
150 STB/Dscq
1000 ft
75 ft
1000 ft
Exercise 5 (cont’d)
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 77
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 6
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 78
1 2 3 4 5
0p
x
6000psiap
150 STB/Dscq
1000 ft
75 ft
1000 ft
Exercise 6 (cont’d)
16-Jan-2014 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
1 12 2
1 1( ) ( )x xc i i i i ii i
i
A k px 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
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 80
2
1
2
1
1
21
D
ii
xxcx
Bx
kAT
i
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 82
Transmissibility
12
1 1
1
if
if
i i i
i
i i i
p p
p p
1
B
Upstream Average of Mobility
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 82
1n n
b b
c ci i
V V
t B t B B
D
1ref ref
fc p p
FD Spatial Discretization of the RHS for Compressible Fluids
16-Jan-2014 Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam 83
16-Jan-2014 85Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
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:
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
16-Jan-2014 86Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
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
Exercise 7 (cont’d)
16-Jan-2014 87Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT
1 2 3 4 5
0p
x
0p
x
150 STB/Dscq
1000 ft
75 ft
1000 ft
Exercise 7 (cont’d)
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
Continuity equation for each fluid flowing phase:
llll St
AuAx
rr
x
Pkku l
l
rll
gwol ,,
wocow PPP
ogcog PPP
Sll o,w, g
1gwol ,,
Momentum equation for each fluid flowing phase:
16-Jan-2014 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
bw
w
ww
rwxxc q
B
S
t
Vx
x
Z
x
P
B
kAk
x
D
sco
o
o
c
bo
o
oo
roxxc q
B
S
t
Vx
x
Z
x
P
B
kAk
x
D
cowow PPP 1 wo SS
16-Jan-2014 90Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Oil-Water Flow Equations
scw
w
w
c
bw
cowo
ww
rwxxc q
B
S
t
Vx
x
Z
x
P
x
P
B
kAk
x
D
sco
o
w
c
bo
o
oo
roxxc q
B
S
t
Vx
x
Z
x
P
B
kAk
x
D
1
16-Jan-2014 91Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Oil-Water Flow Equations
)()(11
21
21 ioioixoioioixo
i
i
oo
oo
roxxc
PPTPPT
xx
Z
x
P
B
kAk
x
D
)()(11
21
21 ioioixwioioixw
i
i
wcowo
ww
rwxxc
PPTPPT
xx
Z
x
P
x
P
B
kAk
x
D
Left side flow terms
16-Jan-2014 92Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Discretization of the Flow Equation
o krooBo
ww
rww
B
k
16-Jan-2014 93Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Phase Mobility
1 2
ii oo
21
1
11
21
DD
DD
ii
ioiioi
ioxx
xx
Upstream: weighted average:
x
Swir
Sw
1-Swir
Qw
average
upstream
exact
OIL
16-Jan-2014 94Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Averaging of Phase Mobility
ioioio
ioioio
io
PPif
PPif
1
11
21
iwiwiw
iwiwiw
iw
PPif
PPif
1
11
21
16-Jan-2014 95Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Upstream Average of Mobility
1 1 1 12 2
( ) ( )i i i ii i
ro oc x x o i
o o i
xo o o xo o o
k P Zk A x
x B x x
T P P T P P
D
1 1 1 12 2
( ) ( )i i i ii i
rw o cowc x x w i
w w i
xw o o xw o o
k P P Zk A x
x B x x x
T P P T P P
D
Left side flow terms
16-Jan-2014 96Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Discretization of Multiphase Flow Equation
Right side flow terms
o
oo
oo
o
BtS
t
S
BB
S
t
)()/1( 1 n
ion
o
io
o
o
roi
io
o PPdP
Bd
B
c
t
S
BtS
i
D
The second term:
)( 1
1
n
iwn
w
iio
i
n
i
o
o
SStBt
S
B i
D
wo SS 1
The first term:
16-Jan-2014 97Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Discretization of the Oil-Phase Equation
and
)()( 11 n
iwwiswon
ion
oipoo
io
o SSCPPCB
S
t
n
ii
io
o
o
riwiipoo
dP
Bd
B
c
t
SC
D
)/1()1(Where:
iio
iiswo
tBC
D
16-Jan-2014 98Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Discretization of Oil-phase RHS
Right side flow terms
w
ww
ww
w
BtS
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
16-Jan-2014 99Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Discretization of Water-Phase Equation
and
Where:
)()( 11 n
iwwiswwn
ion
oipow
iw
w SSCPPCB
S
t
n
ii
iw
w
w
riwiipow
dP
Bd
B
c
t
SC
D
)/1(
ipow
iw
cow
iwi
iisww C
dS
dP
tBC
D
16-Jan-2014 100Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Discretization of Water-phase RHS
Ni ,...,1
1 11 12 2
1 1 1
1
i i i i i
i i
n n n n n nxo xo poo oo o o o i o ii i
n nswo wi w i osc
T P P T P P C P P
C S S q
1 11 1 1 12 2
1 1
1 1
i i i i i i i i
i i i
n n n n n n n nxw xwo o cow cow o o cow cowi i
n n n no sww wpowi o i i w i wsc
T P P P P T P P P P
C P P C S S q
16-Jan-2014 101Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Fully Discrete Oil-Water Flow Equations
n
iswo
n
isww
iC
C
iii
iiii
iiii
wsciosc
n
ion
o
n
iswoi
n
ipoo
n
cow
n
cow
n
ixwi
n
cow
n
cow
n
ixwi
n
o
n
o
n
ixwi
n
ixo
n
o
n
o
n
ixwi
n
ixo
qqPPCC
PPTPPT
PPTTPPTT
1
1111
121
121
121
21
121
21
First, the pressure is found by solving the following equation:
16-Jan-2014 102Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
IMPES Solution of Oil-Water Flow Equations
11
1
1111
1
1
nn
i
nn
i
nn
i
n
iiiigPEPCPW ooo
16-Jan-2014 103Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
n
ixwi
n
ixo
n TTWi 2
121
1
)()(
)(
11
1
21
21
n
icown
icown
ixwi
n
icown
icown
ixwi
wsciosc
n
ion
ipowi
n
ipoon
PPTPPT
qqPCCgiii
n
iswo
n
isww
iC
C
n
ixwi
n
ixo
n TTEi 2
121
1
1 12 2
1 12 2
1
i
n n n nxo xo pooii i
n n nxw xw powi ii i
C T T C
T T C
IMPES Pressure Solution
16-Jan-2014 104Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
Once the oil pressures have been found, water saturationscan be obtained by either the oil-phase equation or thewater-phase equation.
n
ion
o
n
ipooosc
n
o
n
o
n
ixo
n
o
n
o
n
ixo
n
iswo
n
iwn
w
PPCq
PPTPPT
CSS
ii
iiii
i 1
1111
1 121
1211
Ni ,...,1
IMPES Water Saturation
16-Jan-2014 105Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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.
Exercise 8
16-Jan-2014 106Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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.
Exercise 8 (cont’d)
1 2 3 4
0p
x
250 ft
Ax=10,000 ft2
0p
x
Qo=-75 STB/dQw=75 STB/d
Exercise 8 (cont’d)
16-Jan-2014 107Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
16-Jan-2014 108Dr. Mai Cao Lan, Faculty of Geology & Petroleum Engineering, HCMUT, Vietnam
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
Exercise 8 (cont’d)
The End