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

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*

• 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

9

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

10

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 Channelized Layers

Extended local T * Adaptive local-global T *

10th SPE CSP

14

Another Channelized System

100 realizations 120 120 24 24

ALG T *T * + NWSU k * only

15

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.

16

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)

17

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

19

Results for 3D Systems (SPE 10)

50 channelized layers, 3 wellspinj=1, pprod=0

Typical layers

Upscale from 6022050 124410

using different methods

20

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

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

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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

32

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

33

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

34

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

38

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

39

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

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velocity

grid density

Upscaled k*

Unstructured Gridding and Upscaling

(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/p and L1 norm of Fw

p=0p=1 1 0. 5

0.25

43

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)

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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