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7/27/2019 Discret Ization
1/20
DISCRETIZATION OF
THE FLOW EQUATIONS
SIG4042 Reservoir Simulation
7/27/2019 Discret Ization
2/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Home
Handouts(pdf file)
Contents
Explicit Formulation
Time Discretization
Crank-Nicholson Formulation
Implicit Formulation
Spatial Discretization
Questions
Constant grid block size
Variable grid block size
Boundary conditions
http://www.ipt.ntnu.no/~kleppe/SIG4042/note3.pdfhttp://www.ipt.ntnu.no/~kleppe/SIG4042/note3.pdfhttp://www.ipt.ntnu.no/~kleppe/SIG4042/note3.pdf7/27/2019 Discret Ization
3/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Spatial DiscretizationConstant grid block sizes
The approximation of the second derivative of pressure may be obtained by
forward and backward expansions of pressure:
...,'''!3
,''!2
,'!1
,,
32
txPx
txPx
txPx
txPtxxP
...,'''!3
,''!2
,'!1
,,
32
txPx
txPx
txPx
txPtxxP
2
2
11
2
2 2xO
x
PPP
x
P tit
i
t
i
t
i
it yields:
This approximation applies to the following grid system:
i-1 i i+1
x
Introduction
Finite difference approximations of the partial derivatives appearing in the
flow equations may be obtained from Taylor series expansions. We shallnow proceed to derive approximations for all terms needed in reservoirsimulation.
Continue
More
7/27/2019 Discret Ization
4/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Spatial DiscretizationVariable grid block sizes
A more realistic grid system is one of variable block lengths, which will be the
case in most simulations. Such a grid would enable finer description ofgeometry, and better accuracy in areas of rapid changes in pressures andsaturations, such as in the neighborhood of production and injection wells.For the simple one-dimensional system, a variable grid system would be:
it yields:
i-1 i i+1
x xi+1xi-1
...
!3
2/)(
!2
2/)(
!1
2/)(3
1
2
111 i
iii
iii
iiii P
xxP
xxP
xxPP
Continue
...
!3
2/)(
!2
2/)(
!1
2/)(3
1
2
111 i
iii
iii
iiii P
xxP
xxP
xxPP
The Taylor expansions become (dropping the time index):
An important difference is now that the error term is of only first order,due to the different block sizes.
+
7/27/2019 Discret Ization
5/20
7/27/2019 Discret Ization
6/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Spatial DiscretizationBoundary conditions
We have seen earlier that we have two types of boundary conditions,
Dirichlet, or pressure condition, and Neumann, or rate condition. If wefirst consider a pressure condition at the left side of our slab, as follows:
1 2
x1 x
2
PL
Morethen we will have to modify our approximation of the first derivative at the left
face, i=1/2, to become a forward difference instead of a central difference:
)(2/)( 1
1
2/1
xOx
PP
x
P L
The flow term approximation thus becomes:
)()(
)()(2
)(
)()(2
)(1
1
12/1
12
122/11
1
xOx
x
PPxf
xx
PPxf
x
Pxf
x
L
)()(
)()(2
)(
)()(2
)( 1
12/12/1
xOx
xx
PPxf
x
PPxf
x
Pxf
x N
NN
NNN
N
NRN
N
With a pressure PR specified atthe right hand face, we geta similar approximation forblockN:
7/27/2019 Discret Ization
7/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Spatial DiscretizationFor a flow rate specified at the left side (injection/production):
1 2
x1 x2
QL
More
we make use of Darcy's equation:
or:
Then, by substituting into the approximation, we get:
21
xP
BkAQL
kA
BQ
x
PL
2/1
)()(
)()(2
)(1
12
122/11
1
xOx
kA
BQ
xx
PPxf
x
Pxf
x
L
7/27/2019 Discret Ization
8/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Spatial DiscretizationWith a rate QRspecified at the right hand face, we get a similar
approximation for blockN:
More
For the case of a no-flow boundary between blocks iand i+1, the flow termsfor the two blocks become:
)()(
)()(2
)( 11
2/1
xOx
xxPPxf
kABQ
x
Pxf
x N
NN
NNNR
N
)()(
)()(2)(
1
12/1 xO
xxx
PPxf
x
Pxf
x iii
iii
i
)()(
)(
)(2)(121
12
2/111
xOxxx
PP
xfx
P
xfx iii
ii
ii
7/27/2019 Discret Ization
9/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Time discretizationTime discretization
We showed earlier that by expansion backward in time:
...),(!3
)(),(
!2
)(),(
!1),(),(
32
ttxPt
ttxPt
ttxPt
ttxPtxP
the following backward difference approximation with first order error term is obtained:
An expansion forward in time:
...),(
!3
)(),(
!2
)(),(
!1
),(),(32
txPt
txPt
txPt
txPttxP
yields a forward approximation, again with first order error term:
)( tOt
PP
t
P ti
tt
i
tt
i
)( tOt
PP
t
P titt
i
t
i
More
7/27/2019 Discret Ization
10/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Time discretizationFinally, expanding in both directions:
we get a central approximation, with a second order error term:
The time approximation used as great influence on the solutions of the equations.Using the simple case of the flow equation and constant grid size as example, wemay write the difference form of the equation for the three cases above.
...),(!3
)(),(
!2
)(),(
!1),(),(
2
3
2
2
2
2
2
2
2
tt
tt
tt
t txPtxPtxPtxPttxP
...),(!3
)(),(
!2
)(),(
!1),(),(
2
3
22
2
22
22
tt
tt
tt
t txPtxPtxPtxPtxP
22
tOt
PP
t
P titt
i
t
i
t
More
7/27/2019 Discret Ization
11/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Explicit FormulationExplicit formulation
Here, we use the forward approximation of the time derivative at
time level t. Hence, the left hand side is also at time level t, andwe can solve for pressures explicitly:
This formulation has limited stability,
and is therefore seldom used.
t
PP
k
c
x
PPP titt
i
t
i
t
i
t
i
2
11 2
7/27/2019 Discret Ization
12/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Implicit FormulationImplicit formulation
Here, we use the backward approximation of the time derivative at
time level tt, and thus left hand side is also at time level tt:
We have a set ofNequations with Nunknowns, which must be solvedsimultaneously, for instance using the Gaussian elimination method.
This formulation is unconditionally stable.
t
PP
k
c
x
PPP titt
i
tt
i
tt
i
tt
i
2
11 2
7/27/2019 Discret Ization
13/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Crank-Nicholson FormulationCrank-Nicholson formulation
Finally, by using the central approximation of the time derivative at time
level tt/2, and thus left hand side is also at time level tt/2:
The resulting set of linear equations may be solvedsimultaneously just as in the implicit case.
The formulation is unconditionally stable, but may
exhibit oscillatory behavior, and is seldom used.
tPP
k
c
x
PPP
x
PPP titt
i
tt
i
tt
i
tt
i
t
i
t
i
t
i
2
11
2
11 22
2
1
7/27/2019 Discret Ization
14/20FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Questions1) Use Taylor series to derive the following approximations
(include error terms):
a) Forward approximation of
b) Backward approximation of
c) Central approximation of (constant x)
d) Central approximation of (variable x)Next
2) Modify the approximation for grid block1, if the left side of the grid
block is maintained at a constant pressure, PL.
3) Modify the approximation for grid block1, if grid block is subjected to
a constant flow rate, QL.
4) Write the discretized equation on implicit and explicit forms.
t
P
2
2
x
P
t
P
2
2
x
P
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7/27/2019 Discret Ization
16/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
General information
About the author
Title: Discretization of the Flow Equations
Teacher(s): Jon Kleppe
Assistant(s): Szczepan Polak
Abstract: Derivation of finite difference approximations for all terms neededin reservoir simulation.
Keywords: discretization, spatial discretization, time discretization, explicitformulation, implicit formulation, Crank-Nicholson formulation
Topic discipline: Reservoir Engineering -> Reservoir Simulation
Level: 4Prerequisites: Good knowledge of reservoir engineering
Learning goals: Learn basic principles of Reservoir Simulation
Size in megabytes: 0.7
Software requirements: -
Estimated time to complete: 30 minutes
Copyright information: The author has copyright to the module
http://iptibm3.ipt.ntnu.no/~kleppe/http://polak.s.w.interia.pl/http://polak.s.w.interia.pl/http://iptibm3.ipt.ntnu.no/~kleppe/7/27/2019 Discret Ization
17/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
FAQ
7/27/2019 Discret Ization
18/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
References
Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied SciencePublishers LTD, London (1979)
Mattax, C.C. and Kyte, R.L.: Reservoir Simulation, Monograph Series, SPE,Richardson, TX (1990)
Skjveland, S.M. and Kleppe J.: Recent Advances in Improved Oil RecoveryMethods for North Sea Sandstone Reservoirs, SPOR Monograph, Norvegian
Petroleum Directoriate, Stavanger 1992
7/27/2019 Discret Ization
19/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
Summary
7/27/2019 Discret Ization
20/20
FAQ References SummaryInfo
Constant GridBlock Size
Boundary Conditions
Spatial Discretization
Variable Grid
Block Size
Questions
Time Discretization
Explicit Formulation
Implicit Formulation
Crank-NicholsonFormulation
Nomenclature
About the Author
Name
Jon Kleppe
Position
Professor at Department ofPetroleum Engineering and
Applied Geophysics at NTNU
Address: NTNU S.P. Andersensvei 15A
7491 Trondheim
E-mail:
kleppe@ipt.ntnu.no
Phone:
+47 73 59 49 33
Web:
http://iptibm3.ipt.ntnu.no/~kleppe/
mailto:kleppe@ipt.ntnu.no?subject=e-learning%20modulehttp://iptibm3.ipt.ntnu.no/~kleppe/http://iptibm3.ipt.ntnu.no/~kleppe/mailto:kleppe@ipt.ntnu.no?subject=e-learning%20moduleRecommended