Numerical Reservoir
Simulation
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Topic Overview
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
An introduction to standard numerical solution
techniques for reservoir flow equations.
Stability Analyses
Reservoir equations
Reservoir performance
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Introduction
Topic overview
Introduction
Differential equations
Gridding
Gridding
Difference approximation
Stability analyses
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
Differential
equations
for mass flow
Numerical
Modell
Reservoir
equations
Reservoir Performance
Difference Approximation
Discretization Error
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Discretization Techniques
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
General partial differential equations for reservoir
fluid flow must be discretized before they can be
treated computationally.
Reservoir equations
Reservoir performance
The most common techniques are:
- finite differences
- finite elements
We will in in this module learn about the finite
difference technique.
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Finite Differences
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Finite difference approximations are used in most
commercial reservoir simulation software to solve
fluid flow equations numerically.
Reservoir equations
Reservoir performance
Main steps in a discretization procedure:
- replace differential operators by algebraic
ciexpressions
- compute approximate solution at given points and
iiispecified times
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Differential Equations for Mass Flow
Topic overview
Introduction
Differential equations
Mass conservation equations for Black Oil models:
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
water :
[k ]krw
S

 [
(pw   wd )]  Qw  ( w )
 w Bw
t Bw
Where Ql are sink/source term
Reservoir performance
oil :
 [
[k ]k ro
S

(po   od )]  Qo  ( o )
 o Bo
t Bo
gas :
[k ]k rg
[k ]k ro
 [
(p g   g d )]    [
R (po   o d )]  Qg
 g Bg
 o Bo s

S
S R

( g   o s )
t
Bg
Bo
Discretization Techniques
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Reservoir Equations
Topic overview
Introduction
Differential equations
Gridding
Discrete equations for Black Oil models for
block i,j,k:
Difference approximation
Discretization error
water :
Stability Analyses
Reservoir equations
Reservoir performance
Vi , j ,k
 t (
Sw
)
Bw
 t (
So
)
Bo
Tg g  RsTo o  q g ,i , j ,k 
Vi , j ,k
Tw w  qw,i , j ,k 
t
oil :
To o  qo ,i , j ,k 
Vi , j ,k
t
gas :
t
t (
S g
Bg

Rs S o
)
Bo
For more information click on the equation you want to learn more about.
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Water Equation
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
The water equation consists of three parts; a flow
term, a well term and an accumulation term.
Stability Analyses
Reservoir equations
Reservoir performance
Flow term + well term = accumulation term
water :
Tw w  qw,i , j ,k
Vi , j ,k
Sw

 t (
)
t
Bw
For more information click on the term of the water equation you want to learn more about.
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Flow Term for Water
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The flow term for water consists of three terms,
one for each coordinate direction.
Reservoir equations
Reservoir performance
water :
Tw w   xTwx  x wx   yTwy  y wy   zTwz  z wz
For more information click on the term of the equation you want to learn more about.
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Flow Term for Water in x- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The x-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
water :
k x k rw
 xTwx  x w  (
) i  12 y j z k (w,i 1  w,i )
 w Bw x
k x k rw
(
) i  12 y j z k (w,i  w,i 1 )
 w Bw x
For information on block boundaries, click on the textbox.
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Flow Term for Water in y- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
The y-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir performance
water :
 yTwy  y w  (
(
k y k rw
 w Bw y
k y k rw
 w Bw y
) j  12 xi z k (w, j 1  w, j )
) j  12 xi zk (w, j  w, j 1 )
For information on block boundaries, click on the textbox.
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Flow Term for Water in z- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
The z-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir performance
water :
k z k rw
 zTwz  z w  (
) k  12 xi y j (w, z 1  w, z )
 w Bw z
k z k rw
(
) k  12 xi y j (w, z  w, z 1 )
 w Bw z
For information on block boundaries, click on the textbox.
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Well Term for Water
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Specification are different for production and
injection wells.
Reservoir equations
Reservoir performance
water
Click here to see how the production term for water is given.
Developers
References
Well Equations for Black Oil Model
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
water :
Reservoir equations
Reservoir performance
q w ,i
krw
 WI i (
)i ( pi  pwell )
 w Bw
WI i 
2kh
r
ln( e )  S
rw
Pwell = pressure in the well
Developers
References
Well Equations for Black Oil Model
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
oil :
Reservoir equations
Reservoir performance
qo , i
k ro
 WI i (
) i ( pi  pwell )
 o Bo
WI i 
2kh
r
ln( e )  S
rw
Pwell = pressure in the well
Developers
References
Well Equations for Black Oil Model
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
gass :
Reservoir equations
Reservoir performance
q g ,i
k rg
k ro Rs
 WI i (
) i ( pi  pwell )  WI i (
) i ( pi  pwell )
 g Bg
 o Bo
WI i 
2kh
r
ln( e )  S
rw
Pwell = pressure in the well
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Accumulation Term for Water
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The change of mass of water in block i,j,k during
time t between step n and n+1 is given by:
Reservoir equations
Reservoir performance
water :
Vi , j ,k S w n 1 S w n
Sw
 t ( ) 
[(
) (
) ]
t
Bw
t
Bw
Bw
Vi , j ,k
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Evaluation on Block Boundaries
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
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Oil Equation
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
The oil equation consists of three parts; a flow
term, a well term and an accumulation term.
Stability Analyses
Reservoir equations
Reservoir performance
Flow term + well term = accumulation term
oil :
To o  qo ,i , j , k
Vi , j , k
So

 t (
)
t
Bo
For more information click on the term of the oil equation you want to learn more about.
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Flow Term for Oil
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The flow term for oil consists of three
terms, one for each coordinate direction.
Reservoir equations
Reservoir performance
oil :
To o   xTox  x ox   yToy  y oy   zToz  z oz
For more information click on the term of the equation you want to learn more about.
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Flow Term for Oil in x- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The x-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
oil :
k x k ro
 xTox  x o  (
)i  12 y j z k (o ,i 1  o ,i )
 o Bo x
k x k ro
(
)i  12 y j z k (o ,i  o ,i 1 )
 o Bo x
For information on block boundaries, click on the textbox.
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Flow Term for Oil in y- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
The y-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir performance
oil :
 yToy  y o  (
(
k y k ro
 o Bo y
k y k ro
 o Bo y
) j  12 xi z k (o , j 1  o , j )
) j  12 xi z k (o , j  o , j 1 )
For information on block boundaries, click on the textbox.
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Flow Term for Oil in z- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
The z-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir performance
oil :
k z k ro
 zToz  z o  (
) k  12 xi y j (o , z 1  o , z )
 o Bo z
k z k ro
(
) k  12 xi y j (o , z  o , z 1 )
 o Bo z
For information on block boundaries, click on the textbox.
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Well Term for Oil
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Specification are different for production and
injection wells.
Reservoir equations
Reservoir performance
oil
Click here to see how the production term for oil is given.
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Accumulation Term for Oil
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The change of mass of water in block i,j,k during
time t between step n and n+1 is given by:
Reservoir equations
Reservoir performance
oil :
Vi , j ,k  S o n 1  S o n
So
 t ( ) 
[(
) (
) ]
t
Bo
t
Bo
Bo
Vi , j ,k
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Gas Equation
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The gas equation consists of a flow term for gas and
dissolved gas, a well term and an accumulation term for
gas and dissolved gas.
Reservoir equations
Reservoir performance
Flow terms + well term = accumulation terms
gas :
Tg g  RsTo o  q g ,i , j ,k
S g
Rs S o

t (

)
t
Bg
Bo
Vi , j ,k
For more information click on the term of the equation you want to learn more about.
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Flow Term for Gas
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The flow term for gas consists of three
terms, one for each coordinate direction.
Reservoir equations
Reservoir performance
gas :
Tg g   xTgx  x gx   yTgy  y gy   zTgz  z gz
For more information click on the term of the equation you want to learn more about.
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Flow Term for Gas in x- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The x-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
gas :
 xTgx  x g  (
(
k x k rg
 g Bg x
k x k rg
 g Bg x
)i  12 y j zk (g ,i 1  g ,i )
)i  12 y j zk (g ,i  g ,i 1 )
For information on block boundaries, click on the textbox.
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Flow Term for Gas in y- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The y-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
gas :
 yTgy  y g  (
(
k y k rg
 g Bg y
k y k rg
 g Bg y
) j  12 xi zk (g , j 1  g , j )
) j  12 xi zk (g , j  g , j 1 )
For information on block boundaries, click on the textbox. (not active yet)
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Flow Term for Gas in z- direction
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The x-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
gas :
 zTgz  z g  (
k z k rg
 g Bg z
(
) k  12 xi y j (g , z 1  g , z )
k z k rg
 g Bg z
) k  12 xi y j (g , z  g , z 1 )
For information on block boundaries, click on the textbox.
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Flow Term for Dissolved Gas
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The flow term for dissolved gas consists of three
terms, one for each coordinate direction.
Reservoir equations
Reservoir performance
dissolved gas :
RsTo o   x RsTox  x ox   y RsToy  y oy
  z RsToz  z oz
For more information click on the term of the equation you want to learn more about.
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Topic overview
Flow Term for Dissolved Gas in x- direction
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The x-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
dissolved gas :
k x k ro Rs
 x RsTox  x o  (
) i  y j z k (o ,i 1  o ,i )
 o Bo x
1
2
k x k ro Rs
(
) i  y j zk (o ,i  o ,i 1 )
 o Bo x
1
2
For information on block boundaries, click on the textbox.
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Topic overview
Flow Term for Dissolved Gas in y- direction
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The y-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
dissolved gas :
k y k ro Rs
 y RsToy  y o  (
) j  xi z k (o , j 1  o , j )
 o Bo y
1
2
k y k ro Rs
(
) j  xi z k (o , j  o , j 1 )
 o Bo y
1
2
For information on block boundaries, click on the textbox.
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Topic overview
Flow Term for Dissolved Gas in z- direction
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The z-part consists of two terms; one to compute
flow to neighbour block in the positive direction
and one for flow in the negative direction.
Reservoir equations
Reservoir performance
dissolved gas :
k z k ro Rs
 z RsToz  z o  (
) k  xi y j (o , z 1  o , z )
 o Bo z
1
2
k z k ro Rs
(
) k  xi y j (o , z  o , z 1 )
 o Bo z
1
2
For information on block boundaries, click on the textbox.
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Well Term for Gas
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Specification are different for production and
injection wells.
Reservoir equations
Reservoir performance
gas
Click here to see how the production term for gas is given.
Developers
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Accumulation Term for Gas and Dissolved Gas
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The change of mass of water in block i,j,k during
time t between step n and n+1 is given by:
Reservoir equations
Reservoir performance
gas :
Vi , j ,k
t
t (
 Sg
Bg

 Rs So
Bo
)
Vi , j ,k  S g  Rs S o n 1  S g  Rs S o n
[(

) (

) ]
t
Bg
Bo
Bg
Bo
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Definition of Symbols
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
l
s
p
ql,i,j,k
Ql,i,j,k

Sl
Bl
[k]
k
l
Vi,j,k
t
t
Rs
Rs
sTls
= o,w,g
= x,y,z
= i,j,k
=
=
=
=
=
=
=
=
=
=
=
=
=
=
sls
WIp
pi
pwell
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=
=
=
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Difference Approximations
Topic overview
Introduction
Differential equations
Gridding
Taylor series can be used to derive a difference
formula for single and double derivates.
Difference approximation
Discretization error
Stability Analyses
Taylor series of f(x+x) and f(x-x) are given by:
Reservoir equations
Reservoir performance
f ( x  x)  f ( x)  f ( x)x 
1
1
f ( x)x 2    f  k  ( x)x k
2!
k!
f ( x  x)  f ( x)  f ( x)x 
1
1
f ( x)x 2    f  k  ( x)x k
2!
k!
With these expansion we can deduce:
- first order approximation of f ’
- second order approximation of f ’
- second order approximation of f ’’
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First Order Approximation of f’
Topic overview
Introduction
From the expansion of f(x+Δx) we get an expression for f’(x):
Differential equations
Gridding
Difference approximation
Discretization error
f ( x) 
Stability Analyses
Reservoir equations
Reservoir performance
f ( x  x)  f ( x)
R
x
From the expansion of f(x-Δx) we get an expression for f’(x):
f ( x) 
f ( x)  f ( x  x)
R
x
This difference formula is used
for discretizing time derivative
in the mass equations
x
x
Click on the box to see how the approximation changes when the step size is halved.
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Difference Formula
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
A first order approximation of ut at the point
n+1 is given by:
Reservoir equations
ut 
Reservoir performance
n 1
u n1  u n

t
The time axis is divided into points at
distance Δt:
n
n 1
t
t  t
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First Order Approximation of f’
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
From the serie f(x+Δx):
Discretization error
Stability Analyses
Reservoir equations
f ( x) 
Reservoir performance
f ( x  x)  f ( x)
R
x
From the serie f(x-Δx):
f ( x) 
f ( x)  f ( x  x)
R
x
x
2
x
2
The step size reduction produces more accurate
approximations.
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Second Order Approximation of f’
Topic overview
Introduction
Differential equations
Gridding
Adding expansion of f(x+Δx) and f(x-Δx) results
in the approximations:
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
f ( x) 
f ( x  x)  f ( x  x)
R
2x
Reservoir performance
X
X
Click on the box to see how the approximation changes when the time step is halved.
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Second Order Approximation of f’
Topic overview
Introduction
Differential equations
The sum of f’(x) of the series f(x+Δx) and f(x-Δx):
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
f ( x) 
f ( x  x)  f ( x  x)
R
2x
Step size reduction
Reservoir performance
will produce
more accurate
approximations.
x
2
x
2
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Second Order Approximation of f’’
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
The sum of the Taylor series f(x+Δx) and
f(x-Δx) is used to deduced a second
order approximation of f’’:
Reservoir equations
Reservoir performance
f ( x) 
f ( x  x)  2 f ( x)  f ( x  x)
2





x
2
x
This approximation is frequently used and
the numerator is written:
2 f  f ( x  x)  2 f ( x)  f ( x  x)
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Difference Approximation
Topic overview
Introduction
Differential equations
Gridding
Uxx can be approximated at each point i by the
formula:
Difference approximation
u xx i  ui 1  2ui2  ui 1
Discretization error
Stability Analyses
x
Reservoir equations
Reservoir performance
i 1
i
i 1
x  x
x
x  x
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Discretization Error
Topic overview
Introduction
Differential equations
Gridding
The order of a difference approximation can by
analysed using Taylor expansions.
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
The discretization error approaches zero faster for
a high order approximation then for a low order
approximation.
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Gridding
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
A faulted reservoir
Click to the picture for sound (not
active yet)
Well locations
An imposed grid
Initial fluid distribution
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A Faulted Reservoir
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
(Not active yet)
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Well Locations
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
(Not active yet)
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An Imposed Grid
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
Main criteria for grid selection:
- The ability to identify saturations and pressures
ii at specific locations (existing and planned well i
iiiilocations).
- The ability to produce a solution with the i
iiiirequired accuracy (numerical dispersion and
iiiigrid orientation effects).
- The ability to represent geometry, geology and
iiiphysical properties of the reservoir (external
iiiboundaries, faults, permeability distribution
iiiincluding vertical layering).
- Keep the number of grid blocks small in order to
iiimeet requirements of limited money and time
iiiavailable for the study.
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Initial Fluid Distribution
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
(Not active yet)
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Stability Analyses
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
Stable
Unstable
(Not active yet)
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Stable
Topic overview
Introduction
Differential equations
 t

Animation of the stable solution  2  0,4 
 x

Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
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Unstable
Topic overview
Introduction
Differential equations
Gridding
 t

Animation of the unstable solution  2  0,6 
 x

Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
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Reservoir Performance
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
Sound not active yet
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Developers
Topic overview
Introduction
Differential equations
Gridding
Difference approximation
Discretization error
Stability Analyses
Reservoir equations
Reservoir performance
Informasjon på min web-side
http://www.ux.his.no/~hans-k
Made by students
Siril Strømme and Rune Simonsen
Stavanger university college
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