A1258690735sfRiu

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 AM)

(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

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

8
Adaptive Local-Global Upscaling (ALG)
Well-driven global coarse flow
y
x

• Thresholding Local calculations only in
  high-flow regions (computational efficiency)

9
Thresholding in ALG

• Regions for
• Local calculations

Permeability
Streamlines
Coarse blocks

• Identify high-flow region, gt ? (? ?
  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

• Channelized layer (59) from 10th SPE CSP

Upscaling 220 ? 60 ? 22 ? 6

• 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

14
Another Channelized System
100 realizations 120 ? 120 ? 24 ? 24
ALG T
T NWSU
k only
15
Results Multiple Realizations

• Fine scale

• BHP (PSIA)

• Time (days)

• 100 realizations conditioned to seismic and well
  data
• Oil-water flow, M5
• 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
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
1.0
0.5
0.0
Fine scale Sw (220 ? 60)
Reconstructed Sw from extended local T (22 ? 6)

19
Results for 3D Systems (SPE 10)
Typical layers

• 50 channelized layers, 3 wells
• pinj1, pprod0

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

• 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)

• Coarse scale pressure equation

(modified form for total mobility, ?Sc dependence)
26
Calculation of GCD Functions

• D and W2 computed over purely local domain

(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

• 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
0
100x100 ? 10x10, Side Flow
30
Results for Two-Point Geostatistics (Contd)

• Permeability with longer correlation length

?x 0.5, ? y 0.05, ?logk 2.0
10
5
0
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
lx 0.25, lz 0.01, s 2
0 ? t ? 0.8 PVI
Oil Cut
p 1 S 1
t gt 0.8 PVI
PVI
32
Corner to Corner Flow (M 5)
lx 0.2, lz 0.02, s 1.5

• 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

• 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

• 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 lt 0.6 PVI
t ? 0.6 PVI
flow rate
oil cut
37
Unstructured Modeling - Workflow
coarse model
fine model
upscaling
gridding
info. maps
Gocad interface
flow simulation
flow simulation
38
Numerical Discretization Technique
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
p1
p0
fitted extended regions
ltqijgt
Tij -
ltpjgt
ltpigt
-
cell j
cell i
40
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)

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 M1
• Compute errors in Q/Dp and L1 norm of Fw

43
Flow-Based Upscaling Results
6 x 6 x 13 468 nodes
8 x 8 x 18 1152 nodes
1
1
Reference (fine)
0.8
TPFA
0.8
MPFA
Fw
Fw
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
PVI
PVI
(from M. Prevost, 2003)
44
Layered Reservoir Flow Rate Adaptation

• Grid density from flow rate

• Flow results

(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