ll4 enpe470-2013 instability
DESCRIPTION
instability analysis in reservoir simulationTRANSCRIPT
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February 12, 2013 ENPE 470 RESERVOIR MODELING Dr. Ezeddin Shirif
INTRODUCTION TO RESERVOIR SIMULATION
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING2
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Home
Handouts(pdf file)
Contents
Neumanns Method
Karpluss Method
Implicit Scheme
Matrix Method
Instability
Explicit Scheme
Crank Nickolson
Fourier series Method
Explicit Method
Implicit Method
Crank Nickolson Method
Questions
Introduction
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING3
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
An unstable solution scheme will accumulate round-off error until the solution becomes meaning-less
Single precision, the truncation error is about 10-8
Double precision, the truncation error is about 10-16
Determination of stable time
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING4
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Determination of stable t:
1. Karpluss method 99 % of time gives correct results. This method does not include BCS
2. Fourier series (Newmanns method) 100 % of time gives correct results. This method does not include BCS
3. matrix method 100 % of time gives correct results. This method includes BCS and ICS
Determination of stable time
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING5
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Express the flow equation in the following form:
Neglect Gi and qi , then:
1. If all coefficients (a, b, c, ) are positive, then scheme is stable.
2. If one or more of the coefficients are negative, then for stability:
Karpluss method
Instability
0......)()()()( 11111 =++++ ++++ nininininininini ppdppcppbppa
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING6
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
For stability:
A + b + c + d + .. 0
Karpluss method
Instability
0......)()()()( 11111 =++++ ++++ nininininininini ppdppcppbppa
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING7
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
in
in
ibii
in
in
iin
in
ii GPPtcvqppTppT +
=
+++ )()()( 1'12/112/1
Karpluss method for explicit scheme
Rewrite the above equation and ignore q and G:
0)()()( 112/112/1 =+
++n
in
ibiin
in
iin
in
ii PPtcvppTppT
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING8
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
02/12/1 +
+ t
cvTT biiii
Karpluss method for explicit scheme
For stability:
or:
2/12/1 + +
ii
bii
TTcv
t
Calculate t for every i, then choose the smallest value
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING9
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction Karpluss method for explicit scheme
2-D:
2/1,2/1,,2/1,2/1
,,
++ +++
JiJiJiJi
jbijiTTTT
cvt
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING10
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
in
in
ibii
in
in
iin
in
ii GPPtcvqppTppT +
=
++
+
++++ )()()( 1'1112/11112/1
Karpluss method for implicit scheme
Rewrite the above equation and ignore q and G:
0)()()( 11112/11112/1 =++
+
++++
n
in
ibiin
in
iin
in
ii PPtcvppTppT
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING11
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction Karpluss method for implicit scheme
Rewrite the above equation
0)()()(
)()(11
12/11
2/1
12/1
112/1
=
+
++
+
++
+++
n
in
ibiin
in
iin
in
ii
n
in
iin
in
ii
PPt
cvppTppT
ppTppT
0)()(
)(11
11
2/1
1112/1
=
+
+
++
+
++++
n
in
ibiin
in
in
in
ii
n
in
in
in
ii
PPt
cvppppT
ppppT
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING12
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
02/12/12/12/1 +
++ t
cvTTTT biiiiii
Karpluss method for implicit scheme
For stability:
or:
t allfor true is this t
cvbii
0
This means, that the scheme is absolutely stable
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING13
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
in
in
ibii
in
in
iin
ii
n
in
iin
in
ii
GPPt
cv
qppTppT
ppTppT+
=
++
++
+
+
++++
)()]()([
21
)]()([21
1
'
12/112/1
11
12/1
1112/1
Karpluss method for Crank-Nickolson scheme
Rewrite the above equation and ignore q and G:
0)(2)()(
)()( 112/112/1
11
12/1
1112/1
=
+ +
++
+
+
++++ n
in
ibii
n
in
iin
in
ii
n
in
iin
in
ii PPt
cv
ppTppT
ppTppT
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING14
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction Karpluss method for Crank-Nickolson scheme
Rewrite the above equation
Instability
0)(2)()(
)()(1
12/112/1
11
12/1
1112/1
=
+++
+++
+
+
++++
n
in
ibiin
in
iin
in
ii
n
in
in
in
iin
in
in
in
ii
PPt
cvppTppT
ppppTppppT
0)(2)()(
)()()()(1
12/112/1
112/1
12/1
12/1
112/1
=
++
+++
+
+
++
+++
n
in
ibiin
in
iin
in
ii
n
in
iin
in
iin
in
iin
in
ii
PPt
cvppTppT
ppTppTppTppT
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING15
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
022/12/1 ++++ ++++ tcvTT biiii
Karpluss method for Crank-Nickolson scheme
For stability:
or:
TTcv
tii
bii
2/12/1
2++++ ++++
This means, that the scheme is conditionally stable
Karpluss criteria gives us a conservative answer
Instability
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING16
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Continue
)]()([(111,1
)(11,
])([(,1
)]([1,
])([,1
)(,
21
21
21
21
21
21
yyxxinnji
yxinnji
yxxinnji
yyxinnji
yxxinnji
yxinnji
eP
eP
eP
eP
eP
eP
+++++
+++
+
+++
+++
+
=
=
=
=
=
=
Instability
Stability analysis by Fourier Series method
The finite difference solution discrete value can be decomposed into a product of space and time dependent terms as:
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING17
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Continue
1max
1
max
=
+
n
n
AF
Instability
Stability analysis by Fourier Series method
The Fourier Series states that a scheme is stable as long as the amplification factor, AFmax, is less than one.
the amplification factor, AFmax describes how an error grows with time
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING18
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Consider the following flow equation
)()()(1
211
2
2
t
PPk
c
x
pppp
t
Pk
c
x
P
n
in
in
in
in
in
i
=
=
++
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING19
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
The finite difference solution discrete value can be decomposed into a product of space and time dependent terms as:
n
in
in
in
in
i PPpppr
xc
tkrlet
=+
=
++
111
2
)2(
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING20
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Using Fourier Series definitions:
][][1)]([
][)]([11
1
11 2xinxin
xxin
xinxxin
eee
eer
=
+
+
+
Dividing by and rearranging:xi
e 1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING21
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Using Fourier Series definitions:
nnxinnxin eer =+ + 1]2[ 11
Rearranging:
1]12)([ 11 + =++ nxixin reer
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING22
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Eulers identity:
sincos ie i =
Now, applying Eulers identity:
11111 ]12)sin(cos)sin(cos[ +=+++ nn rxixrxixr
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING23
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Rearrange:
11 ]12cos2[ +=+ nn rxr
11 )]cos1(21[ += nn xr
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING24
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Dividing both sides by n and rearranging:
)cos1(21 11
max xrAF nn
==+
For stability, AF max1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING25
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
1)cos1(21 1 xr
Consider the following three situation for the above argument:
pi
pi
=
=
=
x
x
x
1
1
1
)32
)2
0)1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING26
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
111)11(211)cos1(21
10cos0
1
1
==
r
xr
x
The above is always true but it provides no useful information.
Situation #1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING27
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Lets wait for our final decision.
Situation #2
121,1)01(21,1)cos1(21
,02
cos,2
1
1
==
rr
xr
x
pipi
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING28
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Lets wait for our final decision.
Situation #3
141,1)11(21,1)cos1(21
,1cos,
1
1
+
==
rr
xr
x
pipi
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING29
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Explicit scheme
Situation #3
21
,24,1)41(,
0,04,1)41(,141
+
rrr
ORrrrr
2xc
tkr Recall
=
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING30
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
210 r
Fourier Series method for Explicit scheme
Situation #3
kxc
t2
21
21
2
xc
tk
This means, this scheme is conditionally stable
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING31
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
Consider the following flow equation
)()()(1
2
11
1111
2
2
t
PPk
c
x
pppp
t
Pk
c
x
P
n
in
in
in
in
in
i
=
=
++
++++
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING32
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
The finite difference solution discrete value can be decomposed into a product of space and time dependent terms as:
n
in
in
in
in
i PPpppr
xc
tkrlet
=+
=
++
+++
111
111
2
)2(
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING33
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
Using Fourier Series definitions:
][][1)]([1
][1)]([111
1
11 2xinxin
xxin
xinxxin
eee
eer
=
+
+
+
+++
Dividing by and rearranging:xi
e 1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING34
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
Using Fourier Series definitions:
nnxinnxin eer =+ ++++ 1111 ]2[ 11
Rearranging:
nxixinreer =+ + ]12)([ 111
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING35
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
Eulers identity:
sincos ie i =
Now, applying Eulers identity:
nnrxixrxixr =+++ ]12)sin(cos)sin(cos[ 11111
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING36
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
Multiply by -1 and rearrange:
nnrxr =+ ]12cos2[ 11
nn xr =++ ]1)cos1(2[ 11
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING37
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
Dividing both sides by n and rearranging:
1)cos1(21
1
1
max +==
+
xrAF
n
n
For stability, AF max1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING38
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
11)cos1(2
11
+ xr
11)cos1(2 1 + xr
0)cos1(2 1 xr
11)cos1(2 1 + xr
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING39
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Implicit scheme
0)cos1(2 1 xr
0cos1 1 x
1cos 1 x
This is true for all values of , regardless of the value of r.
The scheme is unconditionally stable.
x1
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING40
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Crank-Nickolson scheme
Consider the following flow equation
)(2)()(
)()(1
211
2
11
1111
t
PPk
c
x
ppppx
ppppn
in
in
in
in
in
i
n
in
in
in
i
=
+
+
+
+
++++
t
Pk
c
x
P
=
2
2
-
Nomen References AuthorInfo
Derivation of the Flow Equations
REFERENCES ABOUT EXITHELPNomenclatureHome
ENPE 470 RESERVOIR MODELING41
Flow between
block
Transmissibility
Conservation of Mass
Constitutive Equations
Questions
Flow Equation
Single-phase
Flow
Non-horizontal
FlowMutlidimensional
FlowCoordinate Systems
Nomenclature
Introduction
Instability
Fourier Series method for Crank-Nickolson scheme
The finite difference solution discrete value can be decomposed into a product of space and time dependent terms as:
n
in
in
in
in
in
in
in
i PPppppppr
xc
tkrlet
=+++
=
++
+
+++
111
11
111
2
)22(2
2
-
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Derivation of the Flow Equations
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ENPE 470 RESERVOIR MODELING42
Flow between
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Fourier Series method for Crank-Nickolson scheme
Using Fourier Series definitions:
][][1
)]([][
)]([)]([1
][1)]([1
11
11
11
11
2
22 xinxin
xxinxin
xxinxxin
xinxxin
ee
ee
ee
ee
r
=
+
++
+
++
+++
Dividing by and rearranging:xi
e 1
-
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Derivation of the Flow Equations
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Flow between
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Fourier Series method for Crank-Nickolson scheme
Using Fourier Series definitions:
nn
xinnxin
xinnxin
ee
eer
=
+
++ +
+++1
111
11
11
22
2
Rearranging:
++
=+
+
]14)(2[]14)(2[
11
111
reer
reer
xixin
xixin
-
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Flow between
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Fourier Series method for Crank-Nickolson scheme
Eulers identity:
sincos ie i =
Now, applying Eulers identity:
+++=+++
]14)sin(cos2)sin(cos2[]14)sin(cos2)sin(cos2[
1111
11111
rxixrxixrrxixrxixr
n
n
-
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Flow between
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]14cos4[]14cos4[ 111 +=+ rxrrxr nn
Fourier Series method for Crank-Nickolson scheme
Multiply by -1 and rearrange:
]1)cos1(4[]1)cos1(4[ 111 =++ xrxr nn
-
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Derivation of the Flow Equations
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Flow between
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Flow Equation
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Flow
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Nomenclature
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Fourier Series method for Crank-Nickolson scheme
Dividing both sides by n and rearranging:
1)cos1(41)cos1(4
1
11
max ++
==
+
xr
xrAFn
n
For stability, AF max1
-
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Flow between
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11)cos1(41)cos1(4
1
1 ++
xr
xr
Fourier Series method for Crank-Nickolson scheme
1)cos1(41)cos1(4 11 ++ xrxr
1)cos 1 xThis is true for all values of , regardless of the value of r.
The scheme is unconditionally stable.
x1
-
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Derivation of the Flow Equations
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Flow between
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11)cos1(41)cos1(4
1
1 ++
xr
xr
Fourier Series method for Crank-Nickolson scheme
1)cos1(41)cos1(4 11 + xrxr
11
This is true for all values of , regardless of the value of r.
The scheme is unconditionally stable.
x1
-
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Derivation of the Flow Equations
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Flow between
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Flow Equation
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Nomenclature
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Incompressible fluid flow in 1-D
x3=150mx2=190m
Continue
Calculate pressure distribution as follows:
1. Explicit scheme at t1=t, t2=2t, and t100=100t
2. Implicit scheme at t2=2t, and t100=100t
3. Also calculate the MB
A2 A4
P3P2
K2=1m2 K4=0.1
P1
K1=.5A1=5000m
2
2=.25 3=.151=.2
x1=200m
T1/2=0
T7/2=0
Example: No flow boundary
All blocks have the same depth and area
qi=100m3/D
=50mPa.s, c=10-6kPa-1, Di=0, and initial conditions, Pio=10MPa
-
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Flow between
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Transmissibility
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Flow Equation
Single-phase
Flow
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Introduction
THE END