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Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources
ECMOR 2010 – 08/09/2010©20
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Well Placement Optimization
Zyed Bouzarkouna (IFP)Didier Yu Ding (IFP)Anne Auger (INRIA)
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Outline
Well placement optimization
Covariance Matrix Adaptation – ES (CMA-ES)
Comparison with the genetic algorithm
CMA-ES with meta-models
Summary
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Well Placement Optimization Problem
multi-modal non-smooth non-convex non-separable with a large dimension computationally expensive ...
The use of a stochastic optimization algorithm
Onwunalu & Durlofsky (2010)
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CMA-ESCovariance Matrix Adaptation – Evolution Strategy (Hansen & Ostermeier, 2001)
New population
Initial population
Evaluating individuals
Nextgeneration ..1 ),0( )()()()1( ig
iggg
i Cmx N
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CMA-ES (Cont’d)Covariance Matrix Adaptation
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CMA-ES (Cont’d)
Moving the mean
Adapting the covariance matrix
Step-size control
)(
)()1()()1(
1
)1(:
)1(:
covcov
)1()1(
cov
cov)(cov
)1(
)2()1( where
)11()1(
g
gg
effccg
cg
i
Tgi
gii
Tgggg
ccc
ccc
mmpp
yyppCC
cc
cc
0... ,1with 211
1
)1()1(
ii
i
gi
g:λixm
rank-one update rank-μ update
)(
)()1(21
)()()1(
)1()()1(
)2()1( where
))1),0(
(exp(
g
ggg
effgg
ggg
ccc
Edc
mmCpp
Ip
σσ
σ
N
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Handling Constraints with CMA-ES
Adaptive penalization with rejection Adaptive penalization
m = nbconstraints
where j are weights increased if the distribution mean moved away from the feasible domain.
Rejecting and resampling If an individual is far away from the feasible domain.
m
j j
jj
dm
ff1
21)()(
xx
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Why CMA-ES ?
A problem difficult to solve multimodal; non-smooth; non-separable; with a high dimension; an expensive objective function; ....
CMA-ES is one of the most powerful continuous optimization algorithms (Hansen et al. 2010)
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Comparison with the Genetic Algorithm
New population
Initial population
Evaluating individuals
Selection, Crossover, MutationNextgeneration
Genetic Algorithm
constraints handled with Genocop III (Emerick et al. 2009)
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Test Case
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
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CMA-ES vs. GA14 runs
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Position of solution wells (Producers, Injectors)
CMA-ES GA
CMA-ES vs. GA (Cont’d)
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CMA-ES: Handling Constraints
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First Conclusions
CMA-ES outperforms GA: Higher NPV with less simulations.
CMA-ES proposes solutions in a well-defined zone.
Well configurations generated by CMA-ES are, in general, either feasible or close to feasible domain.
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Meta-Models (MM)
: approximate function (MM)
f̂f : 'true' objectivefunction
point q to be evaluatedpoints used to evaluate qother points from the training set
Local quadratic regression
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Approximate Ranking Procedure
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 Rank1
^f
^f
f
add to the Training Set
Training Set(n + 2 ) elements
evaluate with
rank with (Ranki)
if (NO criteria)evaluate with the best with Ranki
^f
^f...
f
add to the Training Set
Training Set(n + i ) elements
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MM Acceptance Criteria (nlmm-CMA)
The meta-model is accepted if it succeeds in keeping:
the best individual and the set of the best individuals unchangedor
the best individual unchanged, if more than one fourth of the population is evaluated.
Bouzarkouna et al. (2010)
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Well Placement with lmm-CMA
The number of reservoir simulations is reduced by 19 - 25%
10 runs
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Well Placement with lmm-CMA (Cont’d)
Engineer's proposed config. Producers: Horizontal in layer 1; Injectors: Horizontal in layer 5.
Optimized config. Wells: inclined in layer 3.
INJ-1INJ-2
INJ-O
PROD-O
PROD-1/2
Map of
layersn
koS
1
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Well Placement with lmm-CMA (Cont’d)
INJ-1INJ-2
INJ-O
PROD-O
PROD-1/2
Map of
layersn
koS
1
Production Curves
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Meta-Models: Conclusions
Using Meta-Models reduces the number of simulations by ≈ 20%.
The methodology adds ≈ 60% to engineer's proposed well configurations' cumulative oil production.
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Summary
A successful application of CMA-ES in well placement optimization.
Constraints handled using an adaptive penalization with rejection technique.
Meta-Models coupled to CMA-ES to reduce the number of simulations.
Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources
ECMOR 2010 – 08/09/2010©20
10 -
IFP
Ener
gies
nou
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s, R
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Mal
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son,
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nce
Thank You for Your Attention
Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources
ECMOR 2010 – 08/09/2010©20
10 -
IFP
Ener
gies
nou
velle
s, R
ueil-
Mal
mai
son,
Fra
nce
Well Placement Optimization
Zyed Bouzarkouna (IFP)Didier Yu Ding (IFP)Anne Auger (INRIA)
