Well placement optimization

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)

2

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

Outline

Well placement optimization

Covariance Matrix Adaptation – ES (CMA-ES)

Comparison with the genetic algorithm

CMA-ES with meta-models

Summary

3

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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)

4

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

CMA-ESCovariance Matrix Adaptation – Evolution Strategy (Hansen & Ostermeier, 2001)

New population

Initial population

Evaluating individuals

Nextgeneration ..1 ),0( )()()()1( ig

iggg

i Cmx N

5

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

CMA-ES (Cont’d)Covariance Matrix Adaptation

6

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

7

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

8

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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)

9

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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)

10

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

11

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

CMA-ES vs. GA14 runs

12

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

Position of solution wells (Producers, Injectors)

CMA-ES GA

CMA-ES vs. GA (Cont’d)

13

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

CMA-ES: Handling Constraints

14

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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.

15

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

16

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

17

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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)

18

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

Well Placement with lmm-CMA

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

10 runs

19

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

20

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

Well Placement with lmm-CMA (Cont’d)

INJ-1INJ-2

INJ-O

PROD-O

PROD-1/2

Map of

layersn

koS

1

Production Curves

21

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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.

22

©20

10 -

IFP

Ener

gies

nou

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

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

velle

s, R

ueil-

Mal

mai

son,

Fra

nce

Thank You for Your Attention

Zyed.Bouzarkouna@ifp.fr

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)

Zyed.Bouzarkouna@ifp.fr