Well Placement Optimization (with a reduced number of reservoir simualtions)

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Partially Separated Meta-Models with Evolution Strategies for

Well Placement ProblemZyed Bouzarkouna

IFP-EN (French Institute of Petroleum)INRIA

Joint work withDidier Yu Ding (IFP-EN)Anne Auger (INRIA)

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Onwunalu & Durlofsky (2010)

Well Placement Problem

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Onwunalu & Durlofsky (2010)

several minutes to several hours !!


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Outline

Optimization Approach: CMA-ES

CMA-ES with meta-models

Exploiting the partial Separability of the objective function

Results and Discussions

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Evaluating individuals

Initializing

Adapting the distribution parameters

Sampling:

Nextgeneration

..1 ),0( iii Cmx N

CMA-ESCovariance Matrix Adaptation – Evolution StrategyHansen & Ostermeier (2001)

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CMA-ES (Cont'd)

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CMA-ES with Meta-Models

: approximate function (MM)

f̂f : 'true' objectivefunction

simulated well configuration non-simulated well configuration : approximated with f̂

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Building the meta-model

Locally weighted regression

nq : point to evaluate

)(^

qf : full quadratic meta-model on q

CMA-ES with Meta-models (Cont'd)

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A training set containing m points with their objective function values

mjfy jjj ...1)),(,( xx


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We select the k nearest neighbor data points to q according to the Mahalanobis distance with respect to the current covariance matrix C.


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Building the full quadratic meta-model on q

f̂


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Training Setn elements

add to the training set

evaluate with

rank with (Rank0)

evaluate with the best from Rank0.

^f

^f

f

Training Set(n + 1 ) elements

evaluate with

rank with (Rank1)

If (NO criteria) evaluate with the best

from Rank2.

^f

^f

f


Training Set(n + 2 ) elements

evaluate with

rank with (Ranki)

If (NO criteria) evaluate with the best

with Rank2.

^f

^f

f


Training Set(n + 1 + i ) elements

...

CMA-ES with Meta-models (Cont'd)Approximate Ranking Procedure

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MM Acceptance Criteria: nlmm-CMA

The meta-model is accepted if it succeeds in keeping: the best individual and the ensemble of the μ best individuals

unchangedor the best individual unchanged, if more than one fourth of the

population is evaluated.

Bouzarkouna et al. (2010a)

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PUNQ S-3: 19 x 28 x 5.

2 wells to be placed: 1 unilateral producer 1 unilateral injector

NPV = the objective function

vertical, horizontal or deviated.

Lmax = 1000 m.

d

nw

g

oT

nw

g

oY

nn C

CCC

QQQ

APRNPV

))1(

1(1

Dimension = 12

Test Case

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CMA-ES with meta-models: Performance10 runs on the PUNQ-S3 reservoir case Bouzarkouna et al. (ECMOR 2010)

The number of reservoir simulations is reduced by 19 - 25%

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Why ? The well placement problem is still demanding in reducing the

number of reservoir simulations

Idea Building a more accurate approximate model

How ? Exploit the problem structure to reduce more the number of

simulationsReduce the dimension of the approximate model

Why this work

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W1 W2 W4W5

W3

Reservoir Simulation

Productioncurves foreach well

wells

)(well NPV (field) NPV ii

W1

W2

W3

Objective function: Net Present Value (NPV)

When evaluating the NPV, we have access to all the NPVi

Each NPVi can be approximated using only a few variables instead of all the variables of the problem.


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Partial Separability of the Objective Function

Two Conditions

must be explicit ; must define a number of variables < dimension;

well placement problem: : The NPV for each well : defines the variables for each

N

i

iiff

1

)()( xx

i

ifif

i

i

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Partially Separated Meta-Models

N

i

iiff

1

)()( xx

N

i

iiff

1

)(ˆ)(ˆ xx

Building N meta-models (1 for each element function)instead of 1 meta-model for the whole objective function.

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Building the p-sep Meta-Model

nq : point to evaluate on

^

if : full quadratic meta-model on )(qi

ii n )(q : point to evaluate on

f̂

if̂( ( ))???i

if q

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A training set containing mi points with their true element function values

( ), ( ( )) , 1,...,i ij i j if j m x x

)(qi

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We select the ki nearest neighbor data points to Φi (q) according to the Mahalanobis distance with respect to a matrix Ci.

Ci is an ni ni matrix adapted to the local shape of the landscape of fi.


)(qi

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Building the full quadratic meta-model on Φi(q)

if̂

( 3)2 1

2

1

ˆmin ( ), ( ) , w.r.t. i ii n nk

i ii j i i j j i

j

f f

x x)(qi

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PUNQ S-3: 19 x 28 x 5.

1 injector already drilled

3 unilateral producers to be placed

NPV = the objective function

d

nw

g

oT

nw

g

oY

nn C

CCC

QQQ

APRNPV

))1(

1(1

Test Case Dimension = 18

I-1

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Meta-models to approximate the NPV of each wellNPV(field) = NPV(P1) + NPV(P2) + NPV(P3) + NPV(I1)

Each sub-objective function will be approximated with a few parameters the coordinates of the considered well the minimum distance to other producers the minimum distance to the injector

Problem Modeling Dimension = 18

We build 4 meta-modelsFor wells to be drilled, each meta-model depends on 8 parametersFor wells already drilled, the meta-model depends on 2 parameters

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Performance on PUNQ-S3 (Cont'd)

I-1

P-1

P-2

P-3

Map of HPhiSo

Position of solution wells

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Summary

New approach based on exploiting the partial separability of the objective function

The approach can be combined with any other stochastic optimizer

Promising results on the PUNQ-S3: It reduces the number of simulations by: 60% compared to CMA-ES; 28% compared to CMA-ES with meta-models;



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Thank you for Your Attention

[email protected]



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Zyed [email protected]

Joint work withDidier Yu DingAnne Auger

Partially Separated Meta-Models with Evolution Strategies for


Well Placement Optimization (with a reduced number of reservoir simualtions)

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