Direct Construction and History Matching of Ensembles of Coarse-Scale Reservoir

Models

Céline Scheidt and Jef CaersStanford University

Yuguang ChenChevron Energy Technology Company

1

Motivations

Ensemble modeling becoming increasingly “popular”

CPU limitations prevent the construction of multiple history-matched models

Usually, only a single model is history matchedOften without great regard to the intended geological spatial continuity

Availability of multiple models is critical for uncertainty quantification

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Objective

Develop a workflow addressing the issue of multiple history matched models

Done through ensemble level reservoir modeling

History match to be performed on coarse-scale models where simulation is feasible

Construct new HM realizations consistent with fine-scale data

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Proposed Workflow

Use of ensemble level upscaling or other rigorous upscaling methods to make flow simulations on all models feasible

Use of distance-based techniques to define a parameterization of the ensemble

Definition of an application-tailored distance

Use of KL-expansion and kernel methods to generate new realizations which honor the geological and historical production data

Refer to previous presentations of Caers and Scheidt4 SCRF Affiliate Meeting, May 1st 2009

ϕ-1

Reconstruction of real.

. ..

L

Fine-Scale perm.

Definition of Distance δ

Tailored to application

ϕMDS

WI*

Upscaling

kx* ky*

0 200 400 600 800500

1000

1500

2000

2500

3000

3500

Time (days)

Pre

ssur

e at

01

Coarse scale Simulations

Fine-Scale perm. kx* ky*WI*

…

Workflow

Feature Space Metric Space

K-L Exp. in F:newnew Φbx =)( ϕ xXx

5

xPost-image

Test Case 1

P1

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Objective: History match pressure at observation we

UpscalingUse of extended local upscaling and near-well upscaling on all L fine-scale permeabilities (100 x 100)

Coarse-scale permeabilities (kx*, ky*): dimension of 20 x 20Well indices (WI*)

Flow simulations are performed on the coarse-scale models

Distances are defined from the simulation of the response of interest

Note: Wells are located at the center of the coarse-id

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Example of some realizations…

P1

O1

P1

O1

P1

O1

P1

O1

Fine-Scale perm.

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

kx* ky* Pressure - Eclipse

Pressure (P10, P50 and P90)Eclipse

8

Definition of a Metric SpaceDefinition of distance:

Difference of pressure at O1 for each time step (from coarse simulation)

Map reservoir models xi in Metric space (using MDS):

Find configuration of points xi such that: δ(xi,xj) ~ dEucl(xd,i,xd,j)

1 2 3 ... data

1 δ1,1 δ1,2 δ1,3 ... δdata,1

2 δ2,1 δ2,2 δ2,3 ... δdata,2

3 δ3,1 δ3,2 δ3,3 ... δdata,3

... ... ... ... ... …

data δdata,1 δdata,2 δdata,3 … δdata,data0 200 400 600 800

500

1000

1500

2000

2500

3000

3500

Time (days)

Pre

ssur

e at

01

Coarse scale Simulations

x

True dataTrue data

True dataTrue data

2D proj.of modelsfrom metric space

Distance Matrix D

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Model Expansion in Feature Space

featurespace

ector)Gaussian v standard a is (L

1 with )(

:expansionLoeve-Karhunen

y

ybbx KVΦ ==ϕ

ΦX)( ii aa ⇒xφx

L)(L Matrix Gramor Kernel

)()(

),(

×⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−==

K

expkK jiT

j,dijiij σ

xxxxxx

TKKK VVK Λ=

x

ϕ

Recap from Cae

)( xnewϕ

2D proj.of modelsin feature space

2D proj.of modelsFrom metric space

10

x

True dataTrue data

SCRF Affiliate Meeting, May 1st 2009

The Post-image problemLocation of new realizations in Metric space (M) is KNOWN

Where location of HM model is found: xd,trueGiven by definition of distance

Definition of the problem in M (Caers):

Definition of the problem in F:

d

MDS

trueddd xxxxx a with 0),( :such that find , =

xxd,true

xX ?

M

F

2D proj.of modelsin feature space

2D proj.of modelsFrom metric space

11

X ?

ϕ(xd,true)d

MDS

trued xxxφxφxx

a with )()(minarg: such that find2

, −

newKVL yΦ1

SCRF Affiliate Meeting, May 1st 2009

The Post-image problem

Definition of the problem in F:

Optimization on ynew in the K-L expansionUse of Gradual Deformation (GDM) on

)sin()cos( 21 ππ aanew yyy +=

)1,0(~);1,0(~ 21 NN yy

xxd,true

xxxxx

ϕ(xd,true)

M

F

2D proj.of modelsin feature space

2D proj.of modelsFrom metric space

12

newy

X ?

2

,new 1)(minarg: such that find new

Ktrued VLnew

yΦxφyy

−

SCRF Affiliate Meeting, May 1st 2009

Reconstruction of new realizationsPost image-problem gives: andClassical pre-image solution:

Apply the same weights to construct new realizations of:

coarse perms:

WI*:

fine perms:

Transmissibilities, etc.

newK

new VL yb 1=newy

∑=

=L

ii

optinew

1

** kyky β∑=

=L

ii

optinew

1

** kxkx β

∑=

=L

ii

optinew WIWI

1

** β

∑=

=L

i

finei

opti

finenew

1kk β

13

∑∑∑

=

=+ =′

′=

L

iid

opti

idnnew

di

L

i ididnnew

dinnewd kb

kb

1,

,,

1 ,,,

1,

),ˆ(),ˆ(

ˆ xxxxxx

x β

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SCRF Affiliate Meeting, May 1st 2009

Construction of 10 HM models (1/2)Fine-Scale perm. kx* ky*

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

P1

O1

Reference Model

Example of 3 New HM Models

14

Construction of 10 HM models (2/2)

Metric Space

All HM realizations mapped in the same location

Flow simulation on coarse-scale perm.kx*

ky*

WI*

k

Flow simulation on fine-scale perm.

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Test Case 2Case similar to the 1st test case, but

Heterogeneity with a angle of 45 deg.

Upscaling: 100x100 10x10Compute Tx*, Ty* and WI* using extended local upscaling + near-well upscaling

Wells are not at the center of the coarse grid

P1

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Construction of 10 HM models

Metric Space

All HM realizations mapped in the same location

Flow simulation on coarse-scale trans.

Flow simulation on fine-scale perm.

10 HM realizations constructed from coarse-simulations on trans.

Creation of new Tx*, Ty*, WI*, k_fine

17

Developed a workflow constructing multiple HM realizations

Distance is constructed by simulations of response of interest

Simulations on coarse-scale model needed (much faster than fine-scale simulations)No further simulations are required to find HM models

Can reconstruct simultaneously the fine grid and associated coarse grids from post-image

18

Conclusions

SCRF Affiliate Meeting, May 1st 2009

Future WorkApply the method to more complex cases

E.g. Channelized models where the upscaling is more difficult as well as the post-image optimization

Use of proxy distances in cases where one cannot run flow simulation on the entire ensemble

How to incorporate potential errors in the upscaling and be able to generate history-matched fine-scale models ?

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AcknowledgmentsMany thanks to Chevron for their financial support

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The Post-image

21)(minarg)( newT

RN αyΦxφx −=ρ { }newTTnewnewTT ΦbΦbΦbxφxφxφ −−= )(2)()(minarg

{ }newTnewnewk KbbbXx −−= ),(21minarg

K‐L Exp.• Pre-image problem:

21)(minarg)( newT

Rnew N αyΦxφy −=ρ

• Post-image problem:

newR

new N αyb 1= To be optimized

To be optimized

{ }newTnewnewk KbbbXx −−= ),(21minarg

21 SCRF Affiliate Meeting, May 1st 2009