Upscaling of Geocellular Models for Flow Simulation Louis J. Durlofsky Department of Petroleum...
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Transcript of Upscaling of Geocellular Models for Flow Simulation Louis J. Durlofsky Department of Petroleum...
Upscaling of Geocellular Models for Flow Simulation
Louis J. Durlofsky
Department of Petroleum Engineering, Stanford University
ChevronTexaco ETC, San Ramon, CA
2
Acknowledgments
• Yuguang Chen (Stanford University)
• Mathieu Prevost (now at Total)
• Xian-Huan Wen (ChevronTexaco)
• Yalchin Efendiev (Texas A&M)
(photo by Eric Flodin)
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• Issues and existing techniques
• Adaptive local-global upscaling
• Velocity reconstruction and multiscale solution
• Generalized convection-diffusion transport model
• Upscaling and flow-based grids (3D unstructured)
• Outstanding issues and summary
Outline
4
Requirements/Challenges for Upscaling
• Accuracy & Robustness
– Retain geological realism in flow simulation
– Valid for different types of reservoir heterogeneity
– Applicable for varying flow scenarios (well conditions)
• EfficiencyInjector
Producer
Injector
Producer
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Existing Upscaling Techniques
• Single-phase upscaling: flow (Q /p)
– Local and global techniques (k k* or T *)
• Multiphase upscaling: transport (oil cut)
– Pseudo relative permeability model (krj krj*)
• “Multiscale” modeling
– Upscaling of flow (pressure equation)
– Fine scale solution of transport (saturation equation)
6
Local Upscaling to Calculate k*
• Local BCs assumed: constant pressure difference
• Insufficient for capturing large-scale connectivity in highly heterogeneous reservoirs
or
Local Extended Local Solve (kp)=0 over local region
for coarse scale k * or T *
Global domain
7
A New Approach
• Standard local upscaling methods unsuitable for
highly heterogeneous reservoirs
• Global upscaling methods exist, but require global
fine scale solutions (single-phase) and optimization
• New approach uses global coarse scale solutions to determine appropriate boundary conditions for local k* or T * calculations
– Efficiently captures effects of large scale flow
– Avoids global fine scale simulation
Adaptive Local-Global Upscaling
8
Adaptive Local-Global Upscaling (ALG)
Well-driven global coarse flow
• Thresholding: Local calculations only in high-flow regions (computational efficiency)
y
x
Coarse scale properties
k* or T * and upscaled well index
Local fine scale calculation
Interpolated pressure
gives Local BCs
Coarse pressure
Local fine scale calculation
Interpolated pressure
gives local BCs
Coarse pressure
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Thresholding in ALG
Permeability Streamlines Coarse blocks
Regions for
Local calculations
• Avoids nonphysical coarse scale properties (T *=q c/p c)• Coarse scale properties efficiently adapted to a given
flow scenario
• Identify high-flow region, > ( 0.1)|q c||q c|max
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Multiscale Modeling
0 cp*k 0 )(
St
Sv
• Solve flow on coarse scale, reconstruct fine
scale v, solve transport on fine scale
• Active research area in reservoir simulation:– Dual mesh method (FD): Ramè & Killough (1991),
Guérillot & Verdière (1995), Gautier et al. (1999)
– Multiscale FEM: Hou & Wu (1997)
– Multiscale FVM: Jenny, Lee & Tchelepi (2003, 2004)
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Reconstruction of Fine Scale Velocity
0 cp*k 0 )(
St
Sv
Upscaling, global
coarse scale flow
Solve local fine scale(kp)=0
Partition coarse
flux to fine scale
Reconstructed fine scale v
(downscaling)
• Readily performed in upscaling framework
12
Results: Performance of ALG
Averaged fine
Pressure Distribution
Coarse: extended local
Coarse: Adaptive local-global
Channelized layer (59) from 10th SPE CSP
Flow rate for specified
pressure
• Fine scale: Q = 20.86
• Extended T *: Q = 7.17
• ALG upscaling: Q = 20.010.0
5.0
10.0
15.0
20.0
25.0
0 1 2 3 4Iteration
Q
Q (Fine scale) = 20.86
ALG, Error: 4%
Extended local,
Error: 67%
Upscaling 220 60 22 6
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Results: Multiple Realizations
• 100 realizations conditioned to seismic and well data
• Oil-water flow, M=5
• Injector: injection rate constraint, Producer: bottom hole pressure constraint
• Upscaling: 100 100 10 10
100 realizations
Time (days)
BH
P (
PS
IA)
Fine scale
mean
90% conf. int.
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Results: Multiple Realizations
Coarse: Purely local upscaling Coarse: Adaptive local-global
Time (days)
BH
P (
PS
IA)
Mean (coarse scale)
90% conf. int. (coarse scale)
Time (days) B
HP
(P
SIA
)
Mean (fine scale)
90% conf. int. (fine scale)
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Results (Fo): Channelized System
220 60 22 6
Fractional Flow Curve
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1PVI
Fo
Fine scale
ALG T *
Extended local T * Flow rates
• Fine scale: Q = 6.30
• Extended T *: Q = 1.17
• ALG upscaling: Q = 6.26
Oil cut from reconstruction
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Results (Sw): Channelized System
Fine scale Sw (220 60)
Reconstructed Sw from
extended local T * (22 6)
0.5
1.0
0.0
Reconstructed Sw from
ALG T * (22 6)
Fine scale streamlines
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Results for 3D Systems (SPE 10)
50 channelized layers, 3 wellspinj=1, pprod=0
Typical layers
Upscale from 6022050 124410
using different methods
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Results for Well Flow Rates - 3D
Average errors
• k* only: 43%
• Extended T* + NWSU: 27%
• Adaptive local-global: 3.5%
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Results for Transport (Multiscale) - 3D
fine scale
ALG T *
local T * w/nw
standard k*
Producer 1
Fo
PVI
fine scale
ALG T *
standard k*
Producer 2
local T * w/nwFo
PVI
• Quality of transport calculation depends on the accuracy of the upscaling
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Velocity Reconstruction versus Subgrid Modeling
• Multiscale methods carry fine and coarse grid information over the entire simulation
• Subgrid modeling methods capture effects of fine grid velocity via upscaled transport functions:
- Pseudoization techniques
- Modeling of higher moments
- Generalized convection-diffusion model
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• Coarse scale pressure and saturation equations of same form as fine scale equations
• Pseudo functions may vary in each block and may be directional (even for single set of krj in fine scale model)
Pseudo Relative Permeability Models
, 0),( ** cc pS kx 0),(
* c
c
StS
xF
)()(
)()(
),( ,=),(
**
*c*
c****
c*
oirowirw
wirwi
icii
o
ro
w
rw
μk+μkμk
Sf
SfFμkk
S
xvx
* upscaled function
c coarse scale p, S
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Generalized Convection-Diffusion Subgrid Model for Two-Phase Flow
• Pseudo relative permeability description is convenient but incomplete, additional functionality required in general
• Generalized convection-diffusion model introduces new coarse scale terms
- Form derives from volume averaging and homogenization procedures
- Method is local, avoids need to approximate
- Shares some similarities with earlier stochastic approaches of Lenormand & coworkers (1998, 1999)
)()( yx ji vv
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• Coarse scale saturation equation:
Generalized Convection-Diffusion Model
cccc
SSStS
),(),(
xDxG
),()(),( cccc SSfS xmvxG
• Coarse scale pressure equation:
cccc SSSWS ),(),()( 21* xWx
(modified convection m and diffusion D terms)
(modified form for total
mobility, Sc dependence)
“primitive” termGCD term
0),( ** cc pS kx
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• D and W2 computed over purely local domain:
Calculation of GCD Functions
p = 1 S = 1
p = 0)()()( SfSfSS vvD
• m and W1 computed using extended local domain:
(D and W2 account for local subgrid effects)
SSSfSfS )()()( )( Dvvm
(m and W1 - subgrid effects due to longer range interactions)
target coarse block
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Solution Procedure
• Generate fine model (100 100) of prescribed parameters
• Form uniform coarse grid (10 10) and compute k* and directional GCD functions for each coarse block
• Compute directional pseudo relative permeabilities via total mobility (Stone-type) method for each block
• Solve saturation equation using second order TVD scheme, first order method for simulations with pseudo krj
fine grid: lx lz
Lx = Lz
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Oil Cuts for M =1 Simulations
• GCD and pseudo models agree closely with fine scale (pseudoization technique selected on this basis)
lx = 0.25, lz= 0.01, =2, side to side flow
100 x 100
10 x 10 (GCD)
10 x 10 (primitive)
10 x 10 (pseudo)
Oil
Cu
t
PVI
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Results for Two-Point Geostatistics
x =0.05, y = 0.01, logk = 2.0
100x100 10x10, Side Flow
10
0
5
• Diffusive effects only
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Results for Two-Point Geostatistics (Cont’d)
x =0.5, y = 0.05, logk = 2.0
100x100 10x10, Side Flow
10
0
5
• Permeability with longer correlation length
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Effect of Varying Global BCs (M =1)
p = 1 S = 1
p = 0
0 t 0.8 PVI
p = 1 S = 1 t > 0.8 PVI
p = 0
lx = 0.25, lz= 0.01, =2
Oil
Cu
t
PVI
100 x 100
10 x 10 (GCD)
10 x 10 (primitive)
10 x 10 (pseudo)
lx = 0.25, lz= 0.01, =2
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Corner to Corner Flow (M = 5)
100 x 100
10 x 10 (GCD)
10 x 10 (pseudo)
Oil
Cu
t
PVIT
ota
l Ra
tePVI
lx = 0.2, lz= 0.02, =1.5
• Pseudo model shows considerable error, GCD model provides comparable agreement as in side to side flow
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Effect of Varying Global BCs (M = 5)
100 x 100
10 x 10 (GCD)
10 x 10 (pseudo)
Oil
Cu
t
PVIT
ota
l Ra
tePVI
lx = 0.2, lz= 0.02, =1.5
• Pseudo model overpredicts oil recovery, GCD model in close agreement
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Effect of Varying Global BCs (M = 5)
lx = 0.5, lz= 0.02, =1.5
• GCD model underpredicts peak in oil cut, otherwise tracks fine grid solution
100 x 100
10 x 10 (GCD)
10 x 10 (pseudo)
Oil
Cu
t
PVIT
ota
l Ra
tePVI
35
),( * cc SS
0 ** cpkCoarse scale flow:
Pseudo functions:
GCD model:
T * from ALG, dependent on global flow
*, m(S c) and D(S c)
• Consistency between T * and * important for highly
heterogeneous systems
Combine GCD with ALG T* Upscaling
)( * cS
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ALG + Subgrid Model for Transport (GCD)
t < 0.6 PVI t 0.6 PVI
• Stanford V model (layer 1)
• Upscaling: 100130 1013
• Transport solved on coarse scale
flow rate oil cut
37
flow simulation flow simulation
upscaling
gridding
diagnostic
GocadGocadinterface
coarse model
info. maps
fine model
Unstructured Modeling - Workflow
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Numerical Discretization Technique
• CVFE method: – Locally conservative; flux on a face expressed as linear
combination of pressures
– Multiple point and two point flux approximations
• Different upscaling techniques for MPFA and TPFA
i j
k
qij = a pi + b pj + c pk + ... or qij = Tij ( pi - pj )
Primal and dual grids
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3D Transmissibility Upscaling (TPFA)
Dual cells Primal grid connectionp=1
p=0fitted extended regions
cell j cell i
Tij*= -<qij>
<pj> <pi>-
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Grid Generation: Parameters
• Specify flow-diagnostic
• Grid aspect ratio
• Grid resolution constraint:
– Information map (flow rate, tb)
– Pa and Pb , sa and sb
– N (number of nodes)
min max
1
property
cumulative frequency
a b
Pa
Pb
min max
Sa
Sb
property
resolution constraint
a b
42
Flow-Based Upscaling: Layered System
• Layered system: 200 x 100 x 50 cells
• Upscale permeability and transmissibility
• Run k*-MPFA and T*-TPFA for M=1
• Compute errors in Q/p and L1 norm of Fw
p=0p=1 1 0. 5
0.25
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Flow-Based Upscaling: Results
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
PVI
Reference (fine)TPFAMPFA
8 x 8 x 18 = 1152 nodes 6 x 6 x 13 = 468 nodes
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Fw
error in Fw error in Q/p
TPFA 7.6% -1.2%
MPFA 17.9% -25.2%
error in Fw error in Q/p
TPFA 16.8% -5.9%
MPFA 21.3% -31.7%
PVI
Fw
(from M. Prevost, 2003)
44
Layered Reservoir: Flow Rate Adaptation
• Grid density from flow rate
log |V| grid size
sb
sa
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
PVI
Fw
referenceuniform coarse (N=21x11x11=2541)
flow-rate adapted (N=1394)
Qc=0.82
Qc=0.99
PVI
Fw
• Flow results
(from Prevost, 2003)
(Qf = 1.0)
45
Summary
• Upscaling is required to generate realistic coarse scale models for reservoir simulation
• Described and applied a new adaptive local-global method for computing T *
• Illustrated use of ALG upscaling in conjunction with multiscale modeling
• Described GCD method for upscaling of transport
• Presented approaches for flow-based gridding and upscaling for 3D unstructured systems
46
Future Directions
• Hybridization of various upscaling techniques (e.g., flow-based gridding + ALG upscaling)
• Further development for 3D unstructured systems
• Linkage of single-phase gridding and upscaling approaches with two-phase upscaling methods
• Dynamic updating of grid and coarse properties
• Error modeling