well placement optimization (with a reduced number of reservoir simualtions)
DESCRIPTION
EAGE Conference & Exhibition incorporating SPE EUROPEC, Vienna, Austria, May 25, 2011.TRANSCRIPT
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Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources
SPE EUROPEC 2011©20
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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 !!
Well Placement Problem
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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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: approximate function (MM)
f̂f : 'true' objectivefunction
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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: approximate function (MM)
f̂f : 'true' objectivefunction
Building the meta-model
Locally weighted regression
A training set containing m points with their objective function values
mjfy jjj ...1)),(,( xx
CMA-ES with Meta-models (Cont'd)
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: approximate function (MM)
f̂f : 'true' objectivefunction
Building the meta-model
Locally weighted regression
We select the k nearest neighbor data points to q according to the Mahalanobis distance with respect to the current covariance matrix C.
CMA-ES with Meta-models (Cont'd)
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: approximate function (MM)
f̂f : 'true' objectivefunction
Building the meta-model
Locally weighted regression
Building the full quadratic meta-model on q
f̂
CMA-ES with Meta-models (Cont'd)
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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
add to the training set
Training Set(n + 2 ) elements
evaluate with
rank with (Ranki)
If (NO criteria) evaluate with the best
with Rank2.
^f
^f
f
add to the training set
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.
Well Placement 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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: approximate function (MM)
f̂f : 'true' objectivefunction
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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Locally weighted regression
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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Building the p-sep Meta-Model
Locally weighted regression
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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Locally weighted regression
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 p-sep Meta-Model
)(qi
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Locally weighted regression
Building the p-sep Meta-Model
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-S310 runs
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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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Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources
SPE EUROPEC 2011©20
10 -
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Thank you for Your Attention
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Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources
SPE EUROPEC 2011©20
10 -
IFP
Ener
gies
nou
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Mal
mai
son,
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Zyed [email protected]
Joint work withDidier Yu DingAnne Auger
Partially Separated Meta-Models with Evolution Strategies for
Well Placement Problem