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Transcript of NTNU
DISCRETIZATION AND GRID BLOCKS
NTNU
Author: Professor Jon Kleppe
Assistant producers:
Farrokh Shoaei
Khayyam Farzullayev
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Discrete system: Continuous system:
Contineous and discrete systems
?t
p
K
C
x
p
2
2
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
For i=2 to N-1:
Summation of forward and backward equations give us:
Constant grid block sizes
...,'''!3
,''!2
,'!1
,,32
txPx
txPx
txPx
txPtxxP
...,'''!3
,''!2
,'!1
,,32
txPx
txPx
txPx
txPtxxP
2
211
2
2 2xO
x
PPP
x
P ti
ti
ti
t
i
i-1 i i+1
x
x
i-1 i i+1i-1 i i+1
Backward expansions of pressure:
Forward expansion of pressure:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Solving backward equation and Eq. II give us:
For i=N:
For i=1:
...,'''!3
2/,''
!2
2/,'
!1
2/,,0
32
txPx
txPx
txPx
txPtxP
22
12
12
2
)(4
323
xOx
PPP
x
P Lttt
...,'''!3
2/,''
!2
2/,'
!1
2/,,
32
txPx
txPx
txPx
txPtNxP
Eq. I
Solving forward equation and Eq. I give us:
21
x
PL
N-1 N
x
PR
22
12
2
)(4
323
xOx
PPP
x
P RtN
tN
t
N
Eq. II
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Variable grid block sizes
Finer description of geometry More realistic grid system
Better accuracy in areas of rapid changes in pressures and saturations
Specially useful in the neighborhood of production and injection wells
i-1 i i+1
x xi+1xi-1
The Taylor expansions:
...
!3
2/)(
!2
2/)(
!1
2/)( 31
211
1 iii
iii
iii
ii Pxx
Pxx
Pxx
PP
...
!3
2/)(
!2
2/)(
!1
2/)( 31
211
1 iii
iii
iii
ii Pxx
Pxx
Pxx
PP
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
The flow term :
22/12/1
)()(
)( xOx
xP
xfxP
xf
x
Pxf
x i
ii
i
Where :
)()( 12
1
1
2/1
xOxx
PP
x
P
ii
ii
i
)()( 12
1
1
2/1
xOxx
PP
x
P
ii
ii
i
For i=2 to N-1 :
)()()(
)(2)()(
)(2
)( 1
12/1
1
12/1
xOx
xxPP
xfxxPP
xf
x
Pxf
x i
ii
iii
ii
iii
i
Due to the different block sizes, the error terms are of first order only.
Flow equation for i=1 and i=N depends to boundary conditions.
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
x1 x2
1 2PL
Boundary conditions
Pressure condition at the sides of slab:
)(2/)( 1
1
2/1
xOx
PP
x
P L
)()(
)()(2
)(
)()(2
)(1
1
12/11
12
122/11
1
xOx
x
PPxf
xx
PPxf
x
Pxf
x
L
)()()(
)(2)()(
)(2
)( 1
12/12/1
xOx
xxPP
xfxPP
xf
x
Pxf
x N
NN
NNN
N
NRN
N
Same for pressure at the right hand side:
For i=1:
For i=N:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Flow rate specified at the sides of slab:
QL1 2
x1 x2
21
x
P
B
kAQL kA
BQ
x
PL
2/1
)()(
)()(2
)(1
12
122/11
1
xOx
kA
BQ
xx
PPxf
x
Pxf
x
L
)()(
)()(2
)( 1
12/1
xOx
xx
PPxf
kA
BQ
x
Pxf
x N
NN
NNNR
N
Same for flow rate at the right hand side:
For flow rate at the left hand side:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Time discretization
...),(!3
)(),(
!2
)(),(
!1),(),(
32
ttxP
tttxP
tttxP
tttxPtxP
Expansion forward:
...),(!3
)(),(
!2
)(),(
!1),(),(
32
txP
ttxP
ttxP
ttxPttxP
Expansion backward:
)( tOt
PP
t
P ti
tti
tt
i
)( tOt
PP
t
P ti
tti
t
i
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
All parameters except pi at (t+Δt) are in time t and are known , so simply by solving equation you can find pi at (t+Δt).
Use the forward approximation of the time derivative at time level t.
The left hand side is also at time level t.
Solve for pressures explicitly.
Numerical formulations
t
PP
k
c
x
PPP ti
tti
ti
ti
ti
2
11 2
x
i-1 i i+1t
This formulation has limited stability, and is therefore seldom used.
i-1 i i+1t + Δt
Explicit method:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Use the backward approximation of the time derivative at time level t+Δt.
The left hand side is also at time level t+Δt.
Solve for pressures implicitly.
t
PP
k
c
x
PPP ti
tti
tti
tti
tti
2
11 2
x
i-1 i i+1t
i-1 i i+1t + Δt
A set of N equations with N unknowns, which must be solved simultaneously. For instance using the Gaussian elimination method.
This formulation is unconditionally stable.
Implicit method:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Use the central approximation of the time derivative at time level t+Δt / 2.
The left hand side is also at time level t+Δt / 2.
t
PP
k
c
x
PPP
x
PPP ti
tti
tti
tti
tti
ti
ti
ti
211
211 22
2
1
The resulting set of linear equations may be solved simultaneously just as in the implicit case.
The formulation is unconditionally stable, but may exhibit oscillatory behavior, and is seldom used.
Crank-Nicholson method:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
SWir
100 grid blocks:
40 grid blocks:
20 grid blocks:
10 grid blocks:
Sensitivity to number of grid blocks1 i 10
1 i 20
1 i 40
1 i 100
1020
40
100
SW
X / L
The more grid blocks we have, the smaller are the blocks sizes (Δx), smaller is the numerical dispersion because the discretization error is proportional to Δx2 .
1-Sor
0
1
0 1
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 10 20 30 40 50 60 70 80 90 100
Sensitivity to time step
Δt = 20 sec
Δt = 10 sec
Δt = 1 sec
The smaller are the time steps (Δt), the smaller is the numerical dispersion due to the discretization where the error is proportional to Δt.
SWir
SW
X / L
1-Sor
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Capillary and viscous forces pressure difference across a grid block will be directly proportional to the size of this
( in the flow direction ).
The direct effect of capillary pressure will therefore often be dominating in a core-sized grid block while it is negligible in a full field simulation formation scale grid block.
Core plug:
Simulation grid block:
Capillary forces (capillary endpoint pressure)
Viscous forces (viscous pressure drop)10 cm
0.01
bar
production injection
0.7
bar
150 m
production injection
21 bar
0.7
bar
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Geological models may contain millions of grid blocks representing geologically interpolated data (geostatistical realizations).
Numerical simulators cannot handle this level of detail due to cost limitations (applicable with less than millions of grid blocks).
The magnitude of the difference between fine and coarse scales is very significant.
The key problem is how to obtain effective input for the numerical flow simulator from data on finer scales.
This process is called upscaling.
Upscaling
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Laminar sclae
grid blocks: 1.5 * 0.5 m
core plug
Formation scale
grid blocks: 60 * 5 m
Formation scalegrid blocks: 12 * 2.5 m
Formation scalegrid blocks: 1.5 * 0.5 m
High permiability Low permiability
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Fine grids and coarse grids:
Fine grids:
Coarse grids:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Cij
Fij
The permeability tensor of a porous medium is specified on each fine-scale cell Fij, and must be upscaled or homogenized over each coarse-scale or computational cell Cij
Permeability tensor:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Recovery
Time
Fine grid model with original relative permeability
Coarse grid model with upscaled relative permeability
Every single upscaling step is quality controlled during the upscaling by comparing recovery from the fine model with the recovery from the upscaled coarse gridded model incorporating the pseudo curves.
Quality control:
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Questions
1. Use Taylor series to derive the following approximations (include error terms):
4. Write the discretized equation on implicit and explicit forms.
t
P
2
2
x
P
t
P
2
2
x
P
a) Forward approximation of
d) Central approximation of
b) Backward approximation of
c) Central approximation of (constant x)
(variable x)
2. Modify the approximation for grid block 1, if the left side of the grid block is
maintained at a constant pressure, PL.
3. Modify the approximation for grid block 1, if grid block is subjected to a constant flow
rate, QL.
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
References
Kleppe J.: Reservoir Simulation, Lecture note 3
EPS
Discretization and Grid Blocks
REFERENCES ABOUT EXIT
Contineous and discrete systems
Constant grid block sizes
Variable grid block sizes
Time discretization
Numerical formulations
Sensitivity to number of grid blocks
Sensitivity to Time step
Capillary and viscous forces
Upscaling
QUESTIONS
Title: DISCRETIZATION AND GRID BLOCKS (PDF)
Author: Name: Prof. Jon Kleppe
Address:NTNU
S.P. Andersensvei 15A
7491 Trondheim
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