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)

Onwunalu & Durlofsky (2010)

Well Placement Problem

Onwunalu & Durlofsky (2010)

several minutes to several hours !!

Outline

Optimization Approach: CMA-ES

CMA-ES with meta-models

Exploiting the partial Separability of the objective function

Results and Discussions

Evaluating individuals

Initializing

Adapting the distribution parameters

Sampling:

Nextgeneration

..1 ),0( iii Cmx N

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

CMA-ES (Cont'd)

CMA-ES with Meta-Models

: approximate function (MM)

f̂f : 'true' objectivefunction

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

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)

A training set containing m points with their objective function values

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

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

Building the full quadratic meta-model on q

Training Setn elements

add to the training set

evaluate with

rank with (Rank0)

evaluate with the best from Rank0.

Training Set(n + 1 ) elements

evaluate with

rank with (Rank1)

If (NO criteria) evaluate with the best

from Rank2.

Training Set(n + 2 ) elements

evaluate with

rank with (Ranki)

If (NO criteria) evaluate with the best

with Rank2.

Training Set(n + 1 + i ) elements

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

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)

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.

APRNPV

Dimension = 12

Test Case

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%

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

W1 W2 W4W5

Reservoir Simulation

Productioncurves foreach well

)(well NPV (field) NPV ii

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.

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

)()( xx

Partially Separated Meta-Models

)()( xx

)(ˆ)(ˆ xx

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

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

if̂( ( ))???i

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

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

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.

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

( 3)2 1

ˆmin ( ), ( ) , w.r.t. i ii n nk

i ii j i i j j i

x x)(qi

PUNQ S-3: 19 x 28 x 5.

1 injector already drilled

3 unilateral producers to be placed

NPV = the objective function

APRNPV

Test Case Dimension = 18

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

Performance on PUNQ-S310 runs

Performance on PUNQ-S3 (Cont'd)

Map of HPhiSo

Position of solution wells

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;

Thank you for Your Attention

zyed.bouzarkouna@ifpen.fr

Zyed Bouzarkounazyed.bouzarkouna@ifpen.fr

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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