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Development of
a Reservoir Simulator with
Unique Grid-Block System
Master Division
Student Paper Contest 2004
Harold Vance Department of Petroleum Engineering
Presentation Outline
Motivation
Problem Definition
Objectives
Approach
Results
Conclusions
Motivation
Not to scale
Diagonal Parallel
“Homogeneous Reservoir”
Homogeneous Reservoir
0
5000
10000
0 500 1000 1500 2000
Time (days)
Cu
m. O
il R
ec
ov
ery
SC
(b
bl) Diagonal Grid Parallel Grid
0
0.1
0.2
0.3
0.4
0.5
0.6
0 500 1000 1500 2000Time (days)
Wa
ter
Cu
t S
C
Diagonal Grid Parallel Grid
Motivation
Not to scale
Diagonal Parallel
K1 K2K1 >> K2
“Heterogeneous Reservoir”
Heterogeneous Reservoir
0
5000
10000
0 500 1000 1500 2000Time (days)
Cu
m. O
il R
ec
ov
ery
SC
(b
bl)
Diagonal Grid Parallel Grid
0
0.1
0.2
0.3
0.4
0.5
0.6
0 500 1000 1500 2000Time (days)
Wa
ter
Cu
t S
C
Diagonal Grid Parallel Grid
Motivation
Brand, Heinemann, and Aziz (1992) –
“In general, Grid Orientation Effect
cannot be overcome with
grid refinement.”
(SPE 21228)
Motivation
Todd et.al. (1972)
Yanosik & McCracken
(1979)
Pruess & Bodvarsson
(1983)
Shiralkar & Stephenson
(1987)
Shiralkar (1990)Brand et.al. (1991)
Sammon (1991)Chen & Durlofsky
(1991)
Ostebo & Kazemi (1992)
Mattax & Dalton (1990)Wolcott et.al.
(1996)
Fractures with Multiple Joint Sets
Courtesy of Imperial College
Fractures create high permeability anisotropy in rock masses!
Motivation
Motivation
General Darcy’s Law:
Permeability Tensor:
Simplified:
k
u~
00633.0Φ
μ
k0.00633u
~
zzzyzx
yzyyyx
xzxyxx
kkk
kkk
kkk
k~
zz
yy
xx
k00
0k0
00k
k~
Grid orientation and heterogeneity significantly affects the results of reservoir simulation
Problem Definition
Problem Definition
Problem Definition
We need a grid model that can We need a grid model that can incorporate incorporate permeability
anisotropy in multiple directions – in multiple directions –
a full tensor representation must a full tensor representation must be considered!be considered!
Objectives
Developing a 2-D, 3-Phase reservoir simulator using finite difference formulation
Reducing the grid orientation effects in a grid model
Creating a grid model that can be used to simulate multiple permeability directions
Approach
2-D, 3-Phase IMPES finite difference simulator using VBA
with unique grid model
2D,3-Phase
•Initial Condition
•Rock/Fluid Properties
Well ModelHGB Model
IMPES
Transmissibility Terms
Grid Numbering
Matrix Form
Matrix Solver
Pn+1, Son+1, Swn+1, Sgn+1
Program Validation
Well Constraints
3 4 5
1
2
3
4
I
J
1 21 2I
J
I
J
W E
N
S
NE
SESW
NW
Hybrid Grid Block (HGB) System
Hybrid Grid Block (HGB) System
Example Grid:
5 x 4
Total Number
of Grid Blocks = 61
I
J
1 2 3 4 5
1
2
3
4
IMPES MethodFinite Difference Equations
OilOil
WaterWater
GasGas
n
w
wp1n
w
wp1nnw B
SV
B
SV
t1
pa
n
g
gp
n
g
gpnng B
SV
B
SV
tpa
1
1 1
n
o
op
n
o
opnno B
SV
B
SV
tpa
1
1 1
x x x x x x xx x x x x x x x
x x x x x x xx x x x x x
x x x x x x x xx x x x x x x
x x x x xx x x x x
x x xx x x
x x xx x x
x x xx x x
x xx x
x xx x
Grid Numbering #1 & Matrix Form
1 of 3
1 2 3
4 5 6
7 8
15 9 10 16
13 14
17 11 12 18
Example: 3x2
2 4 6
9 11 13
10 12
1 3 5
8 14
15 16 17 18
x xx x x x x x x
x x xx x x x x x x x
x x xx x x x x x x
x xx x xx x x x x x xx x x x x
x x x x x x x xx x x x x
x x x x x x xx x x
x xx x x
x x xx x
2 of 3
7
Example: 3x2
Grid Numbering #2 & Matrix Form
x xx x x
x x x x x x xx x x
x x xx x x x x x x x
x x x x xx x x x x x x x
x x xx x
x x x x x x xx x x x xx x x x x x x x x
x x x x xx x x x x x x
x xx x xx x x x x x x x
x x x x xx x x x x x x x
x x xx x xx x x x x x x
x x xx x
3 of 3
1
2
3
4
5
6
1
8
10
7
9
11
12
18
19
14
13
15
16
17
20
21
22
24
25
23
1
2
3
4
5
6
8
10
7
9
11
12
18
19
14
13
15
16
17
20
21
22
24
25
23
Example: 3x3
Grid Numbering #3 & Matrix Form
Well Model
ooj r
m
kh
qBPP ln
2
Peaceman Well Model Well Model
(1983):(1983):
For square gridblock,
Δm
mro 208.0where,ro = effective wellbore radius
Well Model
Well Model for regular
polygon (after Palagi,1992):
j = neighbor of wellblock i
bij = side of polygon
dij = distance between gridpoints
Θij= angle open to flow
ijj
ij
ijj
o
db
θdb
expr
bij
dij
Ɵij
i
j
Results: Case#1
Model Dimension: Model Dimension: 640 ft x 640 ft x 10 ft640 ft x 640 ft x 10 ft
Permeability: 100mDPermeability: 100mDPorosity: 20%Porosity: 20%
Well Constraints:-Well Constraints:- Const. QConst. Qinjinj
Const. QConst. Qoo
Inj Prod
Maximum Material Balance Error = 4.2602E-05%
Results: Case#1
10 days 20 days
Contour Map
40 days
Results: Case #2
Same dataset, Same dataset, except:except:1 permeability permeability
directiondirectionKKmaxmax = 500 mD = 500 mD
KKminmin = 100 mD = 100 mD
Inj
Prod2
Prod1
Maximum Material Balance Error = 2.587E-03%
Results: Case #2
10 days 20 days 40 days
Contour Map
Results: Case#2
Results: Case #3
Inj
Prod2
Prod1
Maximum Material Balance Error = 5.2654E-03%
Same dataset, Same dataset, except:except:3 permeability permeability
directionsdirectionsKKmaxmax = 500 mD = 500 mD
KKminmin = 100 mD = 100 mD
Results: Case #3
10 days 20 days 40 days
Contour Map
Results: Case #4
Maximum Material Balance Error = 2.9696E-03%
Homogeneous Homogeneous reservoirreservoir
1 injector1 injector4 producers4 producers
Results: Case #4
Pressure Distribution Chart
Conclusions
Grid orientation and and heterogeneity affects significantly the results of affects significantly the results of reservoir simulation (ie. water reservoir simulation (ie. water breakthrough times & recovery)breakthrough times & recovery)
A A full tensor representation must representation must be considered if reservoir flow be considered if reservoir flow performance is to be predicted performance is to be predicted accuratelyaccurately
Conclusions
Proposed HGB model is able to
reduce the grid orientation the grid orientation effectseffects
model model different sets of of permeability anisotropypermeability anisotropy
Future Application
Local Grid Refinement
Dr. David Schechter
Dr. Erwin Putra
U.S Department of Energy
Acknowledgement
THANK THANK YOUYOU
Development of
a Reservoir Simulator with
Unique Grid-Block System
Master Division
Student Paper Contest 2004
Harold Vance Department of Petroleum Engineering
