DISCRETIZATION AND GRID BLOCKS
Author: Professor Jon Kleppe
Assistant producers:
Farrokh Shoaei
Khayyam Farzullayev
NTNU
ENTER
Discretization and Grid Blocks
Contineous and discrete systems
Contineous and
discrete systems
Constant grid block
sizes
Variable grid block
sizes

Time discretization
Continuous system:

Discrete system:
Numerical
formulations
Sensitivity to
number of grid
blocks
 2 p  C  p


x 2  K  t
?
Sensitivity to Time
step
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
Constant grid block sizes

Forward expansion of pressure:
i-1
Constant grid block
sizes
Px  x, t   Px, t  
Time discretization

x  P' ' x, t   x  P' ' ' x, t   ...
x
P '  x, t  
1!
2!
3!
2
3
Backward expansions of pressure:
i-1
Sensitivity to
number of grid
blocks
i
i+1
x
2
3



 x 
 x 
 x 
Px  x, t   Px, t  
P '  x, t  
P ' '  x, t  
P' ' ' x, t   ...
Sensitivity to Time
step
Capillary and viscous
forces
i+1
x
Variable grid block
sizes
Numerical
formulations
i
1!

2!
3!
For i=2 to N-1:
Summation of forward and backward equations give us:
t
QUESTIONS

  2 P  Pi t 1  2 Pi t  Pi t 1
2
 2  
 O x 
2
x 
 x i
Upscaling
REFERENCES
ABOUT

EXIT
Discretization and Grid Blocks
Contineous and
discrete systems

For i=1:
PL
1
Constant grid block
2
x
sizes
 x / 2 P' ' x, t    x / 2 P' ' ' x, t   ...
 x / 2
Px  0, t   Px, t  
P '  x, t  
1!
2!
3!
2
Variable grid block
sizes

Time discretization
Numerical
formulations

3
Solving forward equation and Eq. I give us:
Eq. I

t
 2P 
P2t  3P1t  2 PL
2
 2  
 O x 
3
 x 1
(x) 2
4

For i=N:
N-1
Sensitivity to
number of grid
blocks
N
PR
x
Sensitivity to Time
step
x / 2 P' ' x, t   x / 2 P' ' ' x, t   ...
x / 2
P  x  N , t   P  x, t  
P '  x, t  
1!
2!
3!
2
3
Eq. II
Capillary and viscous
forces

Upscaling
QUESTIONS
REFERENCES
Solving backward equation and Eq. II give us:
t
 2P 
PNt 1  3PNt  2 PR
2
 2  
 O x 
3
 x  N
(x) 2
4

ABOUT
EXIT

Discretization and Grid Blocks
Variable grid block sizes
Contineous and
discrete systems
Constant grid block
sizes
Variable grid block
sizes
 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
Numerical
formulations

Upscaling
QUESTIONS
i-1
i
xi-1
x
i+1
xi+1
The Taylor expansions:
Sensitivity to Time
step
Capillary and viscous
forces
Finer description of geometry

Time discretization
Sensitivity to
number of grid
blocks

(xi  xi 1 ) / 2 P (xi  xi 1 ) / 2 P...
(xi  xi 1 ) / 2
Pi 1  Pi 
Pi
i
i
1!
2!
3!
2
3
 (xi  xi 1 ) / 2 P  (xi  xi 1 ) / 2 P...
 (xi  xi 1 ) / 2
Pi 1  Pi 
Pi
i
i
1!
2!
3!
2
REFERENCES
ABOUT
3
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems

The flow term :
P 
P 


f
(
x
)

f
(
x
)
 
P  
x  i 1/ 2 
x  i 1/ 2
2
f ( x)  
 O x 

x 
x  i
xi

Constant grid block
sizes

Where :
Variable grid block
sizes
P  Pi
 P 
 1 i 1
 O(x)
 
 x  i 1 / 2 2 (xi  xi 1 )
P P
 P 
 1 i i 1  O(x)
 
 x i 1/ 2 2 (xi  xi 1 )
Time discretization
Numerical
formulations

For i=2 to N-1 :
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
 
P 
f ( x)  

x 
x  i
2 f ( x)i 1/ 2
( Pi 1  Pi )
( Pi  Pi 1 )
 2 f ( x)i 1/ 2
(xi 1  xi )
(xi  xi 1 )
 O(x)
xi

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.
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems

Boundary conditions
 Pressure condition at the sides of slab:
Constant grid block
sizes
Variable grid block
sizes
PL
1
2
x1
x2
P P
 P 
   1 L  O(x)
 x 1/ 2 (x1 ) / 2
Time discretization
 For i=1:
Numerical
formulations
 
P 
f
(
x
)

x 
x  1
Sensitivity to
number of grid
blocks
( P2  P1 )
( P  PL )
 2 f ( x)11 / 2 1
(x 2  x1 )
(x1 )
 O(x)
x1
 For i=N:
Sensitivity to Time
step

Capillary and viscous
forces
Same for pressure at the right hand side:
 
P 
f
(
x
)

x 
x  N
Upscaling
QUESTIONS
2 f ( x)11 / 2
REFERENCES
ABOUT
2 f ( x) N 1/ 2
( PR  PN )
( PN  PN 1 )
 2 f ( x) N 1/ 2
(xN )
(xN  xN 1 )
 O(x)
xN
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
 Flow rate specified at the sides of slab:
 For flow rate at the left hand side:
Constant grid block
sizes
Variable grid block
sizes
kA  P 
QL  
 
B  x  12
QL
1
B
 P 
   QL
kA
 x 1/ 2
2
x1
x2
Time discretization
Numerical
formulations
 
P 
f
(
x
)

x 
x  1
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
( P2  P1 )
B
 QL
(x 2  x1 )
kA
 O(x)
x1
 Same for flow rate at the right hand side:
Capillary and viscous
forces
 
P 
f
(
x
)

x 
x  N
Upscaling
QUESTIONS
2 f ( x)11 / 2
REFERENCES
ABOUT
 QR
B
kA
 2 f ( x) N 1 / 2
x N
( PN  PN 1 )
(x N  x N 1 )
 O(x)
EXIT
Discretization and Grid Blocks
Time discretization
Contineous and
discrete systems
Constant grid block
sizes

Expansion forward:
P( x, t  t )  P( x, t ) 
Variable grid block
sizes
t
(t ) 2
(t )3
P( x, t ) 
P( x, t ) 
P( x, t )  ...
1!
2!
3!
t  t
 P 
 
 t i
Time discretization
Pi t  t  Pi t

 O(t )
t
Numerical
formulations
Sensitivity to
number of grid
blocks

Expansion backward:
P( x, t )  P( x, t  t ) 
Sensitivity to Time
step
 t
(t ) 2
(t )3
P( x, t  t ) 
P( x, t  t ) 
P( x, t  t )  ...
1!
2!
3!
t  t
t
 P  Pi  Pi
 O(t )
  

t

t
 i
t
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Numerical formulations
Contineous and
discrete systems
Constant grid block
sizes
Variable grid block
sizes

Explicit method:

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.
Time discretization
Numerical
formulations
Sensitivity to
number of grid
blocks
Pi t 1  2 Pi t  Pi t 1
x 2

t
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).
t + Δt
Sensitivity to Time
step
t
Capillary and viscous
forces
i-1
i
i+1
i-1
i
i+1
x
Upscaling
QUESTIONS
t  t
 Pi
 c  Pi


t
 k 
This formulation has limited stability, and is therefore seldom used.
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
Constant grid block
sizes
Variable grid block
sizes

Implicit method:

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.
Pi t 1t  2 Pi t  t  Pi t 1t  c  Pi t  t  Pi t


t
x 2
 k 
Time discretization
Numerical
formulations
t + Δt
Sensitivity to
number of grid
blocks
t


i+1
i-1
i
i+1
A set of N equations with N unknowns, which must be solved simultaneously.
For instance using the Gaussian elimination method.
Upscaling
QUESTIONS
i
x
Sensitivity to Time
step
Capillary and viscous
forces
i-1
This formulation is unconditionally stable.
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
Constant grid block
sizes

Crank-Nicholson method:

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.
Variable grid block
sizes
1  Pi t 1  2 Pi t  Pi t 1 Pi t 1t  2 Pi t  t  Pi t 1t   c  Pi t  t  Pi t



 
2
k
t
x 2
x 2



Time discretization
Numerical
formulations
Sensitivity to
number of grid
blocks

The resulting set of linear equations may be solved simultaneously just as in the
implicit case.
Sensitivity to Time
step

The formulation is unconditionally stable, but may exhibit oscillatory behavior, and is
seldom used.
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Sensitivity to number of grid blocks
Contineous and
discrete systems

10 grid blocks:

20 grid blocks:

40 grid blocks:

100 grid blocks:
1
i
10
Constant grid block
sizes
Variable grid block
sizes
Time discretization
Numerical
formulations
1
i
1
20
i
1
40
i
100
1
10
1-Sor
Sensitivity to
number of grid
blocks
20
40
SW
100
Sensitivity to Time
step
SWir
Capillary and viscous
forces
0
0
Upscaling

QUESTIONS
X/L
1
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 .
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Sensitivity to time step
Contineous and
discrete systems
1
Constant grid block
sizes
1-Sor
Variable grid block
sizes
0,9
Δt = 20 sec
0,8
Δt = 10 sec
Δt = 1 sec
0,7
0,6
Time discretization
SW
Numerical
formulations
0,5
0,4
0,3
Sensitivity to
number of grid
blocks
SWir
0,2
0,1
Sensitivity to Time
step
0
0
10

QUESTIONS
30
40
50
60
70
80
90
100
X/L
Capillary and viscous
forces
Upscaling
20
The smaller are the time steps (Δt), the smaller is the numerical dispersion due to the
discretization where the error is proportional to Δt.
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
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.
Constant grid block
sizes
Time discretization
Numerical
formulations
production
injection
Capillary forces (capillary endpoint pressure)
0.7 bar
Variable grid block
sizes
 Core plug:
Viscous forces (viscous pressure drop)
10 cm
Sensitivity to
number of grid
blocks
injection
production
Sensitivity to Time
step

Simulation grid block:
0.7 bar
Capillary and viscous
forces
Upscaling
150 m
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
Upscaling

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.
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
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems
High permiability
Constant grid block
Low permiability
sizes

Laminar sclae
grid blocks: 1.5 * 0.5 m
core plug

Formation scale
grid blocks: 1.5 * 0.5 m

Formation scale
grid blocks: 12 * 2.5 m

Formation scale
Variable grid block
sizes
Time discretization
Numerical
formulations
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
Capillary and viscous
forces
grid blocks: 60 * 5 m
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems

Fine grids and coarse grids:
 Fine grids:
Constant grid block
sizes
Variable grid block
sizes
Time discretization
Numerical
formulations
Sensitivity to
number of grid
blocks
 Coarse grids:
Sensitivity to Time
step
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems

Permeability tensor:
Constant grid block
sizes
Variable grid block
sizes
Fij
Time discretization
Cij
Numerical
formulations
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
Capillary and viscous
forces
Upscaling
QUESTIONS
 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
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Contineous and
discrete systems

Quality control:
Constant grid block
sizes
Variable grid block
sizes
Recovery
Time discretization
Fine grid model with original relative permeability
Coarse grid model with upscaled relative permeability
Numerical
formulations
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
Capillary and viscous
forces
Time
 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.
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
Questions
Contineous and
discrete systems
1.
Constant grid block
Use Taylor series to derive the following approximations (include error terms):
a) Forward approximation of
sizes
b) Backward approximation of
Variable grid block
sizes
c) Central approximation of
2P
x 2
(constant x)
d) Central approximation of
2P
x 2
(variable x)
Time discretization
Numerical
formulations
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
Capillary and viscous
forces
2.
P
t
P
t
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.
4.
Write the discretized equation on implicit and explicit forms.
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
References
Contineous and
discrete systems
Constant grid block
sizes
Variable grid block
sizes

Kleppe J.: Reservoir Simulation, Lecture note 3

EPS
Time discretization
Numerical
formulations
Sensitivity to
number of grid
blocks
Sensitivity to Time
step
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT
Discretization and Grid Blocks
About this module
Contineous and
discrete systems

Title: DISCRETIZATION AND GRID BLOCKS (PDF)

Author:
Constant grid block
sizes
Variable grid block
sizes

Time discretization
 Address:
Name: Prof. Jon Kleppe
NTNU
S.P. Andersensvei 15A
7491 Trondheim
Numerical
formulations
 Website
Sensitivity to
number of grid
blocks
 Email
Sensitivity to Time
step

Size: 660 KB
Capillary and viscous
forces
Upscaling
QUESTIONS
REFERENCES
ABOUT
EXIT