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Partially Separated Meta-Models
with Evolution Strategies for
Well Placement Problem
Zyed Bouzarkouna
IFP-EN (French Institute of Petroleum)
INRIA
© 2010 - IFP Energies nouvelles, Rueil-Malmaison, France

Joint work with
Didier Yu Ding (IFP-EN)
Anne Auger (INRIA)
SPE EUROPEC 2011

2


Onwunalu & Durlofsky (2010)


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3

Outline

 Optimization Approach: CMA-ES

 CMA-ES with meta-models

 Exploiting the partial Separability of the objective function

 Results and Discussions

4

CMA-ES
Covariance Matrix Adaptation – Evolution Strategy
Hansen & Ostermeier (2001)

Initializing

Sampling: x i  m   N i (0, C)  i  1..

Next
generation Evaluating individuals

Adapting the distribution parameters

5

6

CMA-ES (Cont'd)

CMA-ES with Meta-Models

f : 'true' objective ˆ
f : approximate
function function (MM)

simulated well configuration
non-simulated well configuration : approximated with ˆ
f
7

CMA-ES with Meta-models (Cont'd)
Building the meta-model

f : approximate

 Locally weighted regression

q : point to evaluate
n

^
f (q) : full quadratic meta-model on q

8


f : approximate


A training set containing m points with their
objective function values
(x j , y j  f (x j )), j  1...m

9


f : approximate


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

10


f : approximate


Building the full quadratic meta-model ˆ
f
on q

11

Approximate Ranking Procedure

^ ^ ^
 evaluate with f  evaluate with f  evaluate with f
^ ^ ^
 rank with f (Rank0)  rank with f (Rank1)  rank with f (Ranki)
Training Set ...
n elements  evaluate with f the  If (NO criteria)  If (NO criteria)
best from Rank0.  evaluate with f the best  evaluate with f the best
from Rank2. with Rank2.

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n

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Training Set Training Set Training Set
(n + 1 ) elements (n + 2 ) elements (n + 1 + i ) elements

12

MM Acceptance Criteria: nlmm-CMA
Bouzarkouna et al. (2010a)
 The meta-model is accepted if it succeeds in keeping:
 the best individual and the ensemble of the μ best individuals
unchanged
or
 the best individual unchanged, if more than one fourth of the
population is evaluated.

13

Test Case Di
me
ns
ion
=
 PUNQ S-3: 19 x 28 x 5. 12

 2 wells to be placed:
 1 unilateral producer vertical, horizontal or deviated.

 1 unilateral injector Lmax = 1000 m.

 NPV = the objective function
T
 Qo   Co 
Q  C  )  C
Y
1
NPV   ( n  g  g
n 1 (1  APR )
d

Qw  n Cw  n
   

14

CMA-ES with meta-models: Performance
10 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 this work

 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
simulations
Reduce the dimension of the approximate model

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W3 W4
W1 W2 W5
Objective function: Net Present Value
(NPV)
NPV (field)   NPV (well )
wells
i i

 When evaluating the NPV, we have

Reservoir Simulation access to all the NPVi

W1
 Each NPVi can be approximated
using only a few variables instead of
W2
all the variables of the problem.
Production
W3
curves for
each well
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Partial Separability of the Objective Function

  
N
f ( x)  f i  i ( x)
i 1

 Two Conditions
  i must be explicit ;

  i must define a number of variables < dimension;
 well placement problem:
 f i : The NPV for each well
  i: defines the variables for each fi

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f : approximate

  
N

  
N
f ( x)  f i  ( x)
i ˆ
f ( x)  ˆ  i ( x)
fi

i 1 i 1

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


qn: point to evaluate on fˆ

f i ( i (q)) ???
i (q) ni : point to evaluate on ˆ
fi
^
f i : full quadratic meta-model on  i
(q)

20



A training set containing mi points with their
 i (q) true element function values

  (x ), f ( (x ))  ,
i
j i
i
j j  1,..., mi

21



We select the ki nearest neighbor data points to
 i (q) 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.

22



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

 i (q) ni ( ni  3)

 
ki
min  ˆi  i i  j 
 f  i (x ),   f  i (x ) 2   , w.r.t.    1
2

j 1 
j j

i

23

Test Case Di m
ens
ion
=1
8
 PUNQ S-3: 19 x 28 x 5.

 1 injector already drilled
I-1
 3 unilateral producers to be placed

 NPV = the objective function
T
 Qo   Co 
Q  C  )  C
Y
1
NPV   ( n  g  g
n 1 (1  APR )
d

Qw  n Cw  n
   

24

Problem Modeling Di m
ens
ion
=1
8
 Meta-models to approximate the NPV of each well
NPV(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

We build 4 meta-models
For wells to be drilled, each meta-model depends on 8 parameters
For wells already drilled, the meta-model depends on 2 parameters
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10 runs
Performance on PUNQ-S3

Performance on PUNQ-S3 (Cont'd)

Map of HPhiSo
I-1
Position of P-1
solution wells

P-2

P-3

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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

zyed.bouzarkouna@ifpen.fr

SPE EUROPEC 2011


with Evolution Strategies for
Zyed Bouzarkouna
zyed.bouzarkouna@ifpen.fr

Joint work with
Didier Yu Ding
Anne Auger
SPE EUROPEC 2011

�ݺ�ߣ

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

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