1. 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)

3. Outline 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

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)
Efficiency
Injector
Producer
Injector
Producer

5. 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*or
Local
Extended Local
Solve (kp)=0 over local region
for coarse scale k * or T *
Global domain
Local BCs assumed: constant pressure difference
Insufficient for capturing large-scale connectivity in highly heterogeneous reservoirs

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
Local fine scale calculation
Interpolated pressure gives Local BCs
Coarse pressure
Local fine scale calculation
Interpolated pressure gives local BCs
Coarse pressure y
Coarse scale properties
k* or T * and upscaled well index x
Thresholding: Local calculations only in high-flow regions (computational efficiency)

9. Thresholding in ALG |q c| Identify high-flow region, >  (  0.1)
Regions for
Local calculations
Permeability
Streamlines
Coarse blocks
|q c|
|q c|max
Identify high-flow region, >  (  0.1)
Avoids nonphysical coarse scale properties (T *=q c/p c)
Coarse scale properties efficiently adapted to a given flow scenario

10. Multiscale Modeling
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)

11. Reconstruction of Fine Scale Velocity
Partition coarse
flux to fine scale
Solve local fine scale
(kp)=0
Upscaling, global
coarse scale flow
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
Upscaling 220  60  22  6
Q (Fine scale) = 20.86
ALG, Error: 4%
Extended local,
Error: 67%
Flow rate for specified pressure
Fine scale: Q = 20.86
Extended T *: Q = 7.17
ALG upscaling: Q = 20.01

13. Results: Multiple Channelized Layers
10th SPE CSP
Extended local T *
Adaptive local-global T *

14. Another Channelized System
100 realizations
120  120  24  24
k * only
T * + NWSU
ALG T *

15. Results: Multiple Realizations
Fine scale
mean
90% conf. int.
100 realizations
BHP (PSIA)
Time (days)
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

16. Results: Multiple Realizations
Coarse: Purely local upscaling
Coarse: Adaptive local-global
Time (days)
BHP (PSIA)
Time (days)
BHP (PSIA)
Mean (coarse scale)
Mean (fine scale)
90% conf. int. (coarse scale)
90% conf. int. (fine scale)

17. Results (Fo): Channelized System
Oil cut from reconstruction
220  60  22  6
ALG T *
Flow rates
Fine scale: Q = 6.30
Extended T *: Q = 1.17
ALG upscaling: Q = 6.26
Extended local T *
Fine scale

18. Results (Sw): Channelized System
Fine scale streamlines
1.0
0.5
0.0
Fine scale Sw (220  60)
Reconstructed Sw from
ALG T * (22  6)
Reconstructed Sw from
extended local T * (22  6)

19. Results for 3D Systems (SPE 10)
Typical layers
50 channelized layers, 3 wells
pinj=1, pprod=0
Upscale from
6022050  124410
using different methods

20. Results for Well Flow Rates - 3D
Average errors
k* only: 43%
Extended T* + NWSU: 27%
Adaptive local-global: 3.5%

21. Results for Transport (Multiscale) - 3D
fine scale
ALG T *
local T * w/nw
standard k*
Producer 1 Fo
PVI
Producer 2
Quality of transport calculation depends on the accuracy of the upscaling

22. 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

23. Pseudo Relative Permeability Models
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)
*  upscaled function
c  coarse scale p, S

24. 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)

25. Generalized Convection-Diffusion Model
Coarse scale saturation equation:
(modified convection m and diffusion D terms)
“primitive” term
GCD term
Coarse scale pressure equation:
(modified form for total mobility, Sc dependence)

26. Calculation of GCD Functions
D and W2 computed over purely local domain:
p = 1
S = 1
p = 0
(D and W2 account for local subgrid effects)
m and W1 computed using extended local domain:
(m and W1 - subgrid effects due to longer range interactions)
target coarse block

27. 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

28. Oil Cuts for M =1 Simulations
lx = 0.25, lz= 0.01, s =2, side to side flow
 100 x 100
 10 x 10 (GCD)
 10 x 10 (primitive)
 10 x 10 (pseudo)
Oil Cut
PVI
GCD and pseudo models agree closely with fine scale (pseudoization technique selected on this basis)

29. Results for Two-Point Geostatistics
Diffusive effects only
x =0.05,  y = 0.01, logk = 2.0 10 5
100x100  10x10, Side Flow

30. Results for Two-Point Geostatistics (Cont’d)
Permeability with longer correlation length
x =0.5,  y = 0.05, logk = 2.0 10 5
100x100  10x10, Side Flow

31. Effect of Varying Global BCs (M =1)
lx = 0.25, lz= 0.01, s =2
 100 x 100
 10 x 10 (GCD)
 10 x 10 (primitive)
 10 x 10 (pseudo)
p = 1
S = 1
p = 0
lx = 0.25, lz= 0.01, s =2
0  t  0.8 PVI
Oil Cut
p = 0
p = 1
S = 1
t > 0.8 PVI
PVI

32. Corner to Corner Flow (M = 5)
lx = 0.2, lz= 0.02, s =1.5
 100 x 100
 10 x 10 (GCD)
 10 x 10 (pseudo)
Oil Cut
PVI
Total Rate
Pseudo model shows considerable error, GCD model provides comparable agreement as in side to side flow

33. Effect of Varying Global BCs (M = 5)
lx = 0.2, lz= 0.02, s =1.5
 100 x 100
 10 x 10 (GCD)
 10 x 10 (pseudo)
Oil Cut
PVI
Total Rate
Pseudo model overpredicts oil recovery, GCD model in close agreement

34. Effect of Varying Global BCs (M = 5)
lx = 0.5, lz= 0.02, s =1.5
 100 x 100
 10 x 10 (GCD)
 10 x 10 (pseudo)
Oil Cut
PVI
Total Rate
GCD model underpredicts peak in oil cut, otherwise tracks fine grid solution

35. Combine GCD with ALG T* Upscaling
Coarse 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

36. ALG + Subgrid Model for Transport (GCD)
Stanford V model (layer 1)
Upscaling: 100130  1013
Transport solved on coarse scale
t < 0.6 PVI
t  0.6 PVI
flow rate
oil cut

37. Unstructured Modeling - Workflow
fine model
coarse model
upscaling
gridding
info. maps
Gocad
interface
flow simulation
flow simulation
diagnostic

38. Numerical Discretization Techniquej k
Primal and dual grids
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
qij = a pi + b pj + c pk + ...
or qij = Tij ( pi - pj )

39. 3D Transmissibility Upscaling (TPFA)
Dual cells
Primal grid connection
p=1
fitted extended regions
p=0
<qij>
Tij*= -
<pj>
<pi> -
cell j
cell i

40. Grid Generation: Parameters
min
max 1
property
cumulative frequency a b Pa Pb Sa Sb
resolution constraint
Specify flow-diagnostic
Grid aspect ratio
Grid resolution constraint:
Information map (flow rate, tb)
Pa and Pb , sa and sb
N (number of nodes)

41. Unstructured Gridding and Upscaling
velocity
grid density
Upscaled k*
(from Prevost, 2003)

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/Dp and L1 norm of Fw
p=0
p=1 1
0. 5
0.25

43. Flow-Based Upscaling: Results
8 x 8 x 18 = 1152 nodes
6 x 6 x 13 = 468 nodes 1 1
Reference (fine)
0.8
TPFA
0.8 Fw
MPFA Fw
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.4
0.6
0.8 1
0.2
0.4
0.6
0.8 1
PVI
PVI
(from M. Prevost, 2003)

44. Layered Reservoir: Flow Rate Adaptation
log |V|
grid size sb sa
Grid density from flow rate
Flow results
0.2
0.4
0.6
0.8 1
PVI F w
reference
uniform coarse (N=21x11x11=2541)
flow-rate adapted (N=1394)
Qc=0.82
Qc=0.99 Fw
(Qf = 1.0)
(from Prevost, 2003)

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