Advances in Particle Swarm Optimization and application to history Matching: Stanford VI

Juan Luis Fernández MartínezStanford University.

UC Berkeley‐Lawrence Berkeley Lab.Oviedo University Spain.

In collaboration withTapan Mukerji, Amit Suman

and Esperanza García‐Gonzalo (Oviedo University,Spain).

Stanford Center for Reservoir forecastingStanford Center for Reservoir forecasting

Annual Meeting 2010

SCRF 2010 2

INDEX

• Advances in PSO design• Application of PSO to the History Matching

Problem (Uncertainty analysis)• (TIP) Preliminary results on Differential

Evolution

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I. Advances in PSO design

Work done in collaboration with Esperanza García-Gonzalo (University of Oviedo)

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The spring-mass analogy

( ) ( ) ( ) ( ) ( ) ( ) ( )φ φ φ φ+ − ⋅ + + ⋅ = ⋅ + ⋅1 2 1 2

'' 1 ' .i i i ix t w x t x t l t g t

gk

φ1

xik

lik

m=11-w

φ2

lik-xikxi

k-gk

1 2(1 (1 ) ) ( ( ) ( ),( 1) ( ) ( 1)( 1) ( ) ( ) ( ) ( )

.i i i i i

i i i

v w v x g k x lk x

tk

t tv k t

k k k k kx

φ φ= − − + + −

+ = +

∆ ∆ ∆

∆

+ −+

DISCRETIZATION

in ( ) ( )'' ', .i ix t x t

GPSO

(Fernández Martínez et al, 2008)

(Fernández Martínez and García Gonzalo, 2008)

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PSO Analysis & Design

Based on this mechanical analogy we have1. Shown that PSO BELONGS TO A FAMILY:

• Design and stochastic stability analysis of a whole family of PSO optimizers: PSO, CC-PSO, CP-PSO (Fernández Martínez and García Gonzalo, Swarm Int., 2009), PP-PSO, RR-PSO (García Gonzalo and Fernández Martínez,2010).

2. Shown that PSO IS NOT HEURISTIC: • Full stochastic stability of the PSO family (Fernández Martínez and García

Gonzalo, 2010).

3. Designed a PSO Cloud Algorithm with variable time step (cooling and exploration) (Fernández Martínez et al, 2009, 2010).

• Avoids tuning of the PSO parameters (automatic)

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Parameter tuning: the cloud of particles

ω

¹

PSO ROSENBROCK

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

-1

0

1

2

3

4

5

6

7

ω

¹

CC ROSENBROCK

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

-1

0

1

2

3

4

5

6

7

ω

¹

CP ROSENBROCK

-1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

1

2

3

4

5

6

7

PP ROSENBROCK

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

4

1

2

3

4

5

6

7

φ φ

φ φ

ωω

ωω

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RR-PSO is very different

_

14 / 3( 1)φ ω= −

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The ∆t parameter

∆t>=1 INITIAL BIG EXPLORATION

Stability region shrinks.

∆t<1 FINAL TUNING

Stability region expands.

13

1 2(1 (1 ) ) ( ( ) ( ),( 1) ( ) ( 1)( 1) ( ) ( ) ( ) ( )

.tv w v x g k x lt t

k x k tv kk k k k k

xφ φ= − − + + −

+ = +

∆ ∆ ∆

∆

+ −+

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II. HISTORY MATCHING, TIME LAPSE SEISMICS AND

UNCERTAINTY ANALYSISWith the collaboration of Tapan Mukerji and Amit Suman

Acknowledgments: David Echeverría, Eduardo Santos and Grégoire Mariethoz

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Optimization Workflow (Echeverría and Mukerji, 2009)

facies

m**

tooptimizer

manyparameters

Few PCAparameters

mξ

PSO

DE

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WHY UNCERTAINTY ANALYSIS IS NEEDED IN THE HISTORY MATCHING PROBLEM?

1. MINIMA ARE LOCATED ALONG FLAT ELONGATED VALLEYS.

2. NOISE IN DATA INTRODUCES LOCAL MINIMA.

3. NOISE HAS ALSO A REGULARIZATION EFFECT (MAKES THE SAMPLING EASIER).

4. THE MODEL REDUCTION INTRODUCES SINGULARITIES IN THE COST FUNCTION TOPOGRAPHY (potential danger for local methods).

SCRF 2010 12First PCA

Sec

ond

PC

A

Case 2-Stanford VI reservoir

5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

First PCA

Sec

ond

PC

A

Case 1-Stanford VI reservoir. 10% Gaussian Noise

9 10 11 12 13 14-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

First PCA

Sec

ond

PC

A

Case 1-Stanford VI reservoir

9 10 11 12 13 14-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

First PCA

Case 2-Stanford VI reservoir. 10% Gaussian Noise

Seco

ndPC

A

5 6 7 8 9 101

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

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PSO Results: Swarm size 20

0 10 20 30 40 50 60 70 800

0.003

0.005

0.01

0.015

0.02

0.025SWARM SIZE=20. 10 simulations

iterations

Erro

r

PSO medianPP medianCP median dt=1CP median dt=0.8CC median dt=0.8

0 10 20 25 30 40 50 60 70 800.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022SEQUENTIAL: 10PCA-20PCA

Iterations

Erro

r

PSOPPCPCCCP dt=1,0.8

Similar results are obtained for swarm sizes of 50 and 100 particles

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PSO as a posterior sampler(In collaboration with Gregoire Mariethoz, Stanford University)

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Computing uncertainty from samples

Median sample

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Noise Free 10%Gaussian Error

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Data Match: ProductionCumulated oil Injected water

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Data Match: Tomograms

Reference

Median for theInitial swarm

Median of lowmisfit samples

Section 1 Section 2

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III. DIFFERENTIAL EVOLUTIONWith the collaboration ofEsperanza García-Gonzalo (University of Oviedo)

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Differential Evolution(Storn and Price , 1997)

( ) ( )1( 1) ( ) ( ) ( ) ( ) ,( 1) ( ) ( 1),

i l n r s

i j i

k k k k kk k k+ = − + −

+ = + +2v x x x x

m x vF F1. MUTATION

: Crossover probabilityrC2. CROSSOVER

3. SELECTION

Rand-1, Best-1, Target-to-best, Rand-2, Best-2

GA like-mechanisms

PSO like-mechanism

3 parameters to tune: 1 2 rF , F , C

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RosenbrockGriewank

F F

F

Rastrigin Sphere

F

Cr

Cr

Cr

Cr

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DE PerformanceConvergence rate

Exploration capabilities

Iterations

Iterations

Rel

ativ

e m

isfit

Med

ian

dist

ance

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CONCLUSIONS • PSO

– All the PSO family members are able to provide facies models from the low misfit region, and can be used with small number of particles.

– Sequential inversion allows to increase dynamically the number of PCA parameters as needed.

– The topography of the cost function corresponds to flat valleys. The seismic data helps to partially constraint the space of possible solutions.

– PSO samples can be used to provide an approximate measure of model uncertainty.

A paper has been submitted to Computational Geosciences.

• DE – Very promising results: good balance between exploration

and exploitation.

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Acknowledgments

• Smart Fields and SCRF Consortia.• Schlumberger-EMI.• University of California-Berkeley and Lawrence Berkeley

Lab.

• University of Oviedo and Spanish Ministry of Innovation.

• Eduardo Santos (formerly Stanford University) and David Echeverría for providing the forward programs to model the HM problem (Stanford VI), and Grégoire Mariethozfor collaboration in the posterior sampling in hydrogeology.

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ARE THERE ANY QUESTIONS?

THANK YOU FORYOUR ATTENTION

When you see the face of the anger,look behind it ,and it will suddenly change to the face of the pride.

Jalaluddin Rumi (1207-1273)