Well placement optimization

Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources

Well Placement Optimization
Zyed Bouzarkouna (IFP)
Didier Yu Ding (IFP)
Anne Auger (INRIA)
© 2010 - IFP Energies nouvelles, Rueil-Malmaison, France

ECMOR 2010 – 08/09/2010

Outline

 Well placement optimization

 Covariance Matrix Adaptation – ES (CMA-ES)

 Comparison with the genetic algorithm

 CMA-ES with meta-models

 Summary

2

Well Placement Optimization Problem

 multi-modal
 non-smooth
 non-convex
non-separable


 with a large dimension
 computationally expensive
 ... Onwunalu & Durlofsky (2010)

The use of a stochastic optimization algorithm
3

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

Initial population

Evaluating individuals

Next
generation x i( g 1)  m ( g )   ( g )N i (0, C( g ) )  i  1..

New population
4

5

CMA-ES (Cont’d)
Covariance Matrix Adaptation

CMA-ES (Cont’d)

 Moving the mean

m ( g 1)
  i x i(:λ1)
g

i 1

with 
i 1
i  1, 1  2  ...     0

 Adapting the covariance matrix rank-one update

rank-μ update


ccov 1
) i y i(:g 1) y i(:g 1)
T T
C( g 1)  (1  ccov )C( g )  p cg 1)p cg 1)  ccov (1 
( (
 
 cov cov i 1
( g 1)
m  m( g )
where p ( g 1)
 (1  cc )p (g)
 cc (2  cc )  eff
 (g)
c c

Step-size control p σg 1)
(
 c
 ( g 1)
 (g)
exp( (  1))
d E N (0, I )
1
(g)2 m ( g 1)  m ( g )
where p ( g 1)
 (1  c )p (g)
 c (2  c )  eff C
 (g)
σ σ

6

Handling Constraints with CMA-ES
 Adaptive penalization with rejection
 Adaptive penalization
 m = nbconstraints
2
1 m dj
f ( x)  f ( x)    j
m j 1  j

 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.

7

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)

8

Comparison with the Genetic Algorithm
Genetic Algorithm
Initial population

Evaluating individuals

Next
generation
Selection, Crossover, Mutation

New population

constraints handled with Genocop III (Emerick et al. 2009)
9

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

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

 1 unilateral producer
 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
   
10

11

14 runs
CMA-ES vs. GA

CMA-ES vs. GA (Cont’d)
Position of solution wells (Producers, Injectors)

CMA-ES GA

12

13

CMA-ES: Handling Constraints

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.

14

Meta-Models (MM)

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

 Local quadratic regression

point q to be evaluated
points used to evaluate q
other points from the training set

15

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 evaluate with f the best
best from Rank1 with Ranki

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dt

dt
dt
ot

ot
ot
he

he
he
Tra

Tra
Tra
ini

ini
ini
ng

ng
ng
Se

Se
Se
t

t
t

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

16

MM Acceptance Criteria (nlmm-CMA)
Bouzarkouna et al. (2010)

 The meta-model is accepted if it succeeds in keeping:

the best individual and the set of the best individuals unchanged



or
 the best individual unchanged, if more than one fourth of the
population is evaluated.

17

Well Placement with lmm-CMA
10 runs

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

18

Well Placement with lmm-CMA (Cont’d)
n layers
Map of    S o
k 1

 Engineer's proposed config.
INJ-2  Producers: Horizontal in layer 1;
INJ-1
Injectors: Horizontal in layer 5.


INJ-O

PROD-O  Optimized config.
 Wells: inclined in layer 3.

PROD-1/2

19

Well Placement with lmm-CMA (Cont’d)
n layers Production Curves
Map of    S o
k 1

INJ-2
INJ-1

INJ-O

PROD-O

PROD-1/2

20

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.

21

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.

22


Thank You for Your Attention

Zyed.Bouzarkouna@ifp.fr

ECMOR 2010 – 08/09/2010


Well Placement Optimization
Zyed Bouzarkouna (IFP)
Didier Yu Ding (IFP)
Anne Auger (INRIA)

Zyed.Bouzarkouna@ifp.fr

ECMOR 2010 – 08/09/2010

Well placement optimization

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