Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying...

Smart Fields, Stanford UniversityIntegrated Operations 200707

Semi-automatic history matchingapplying

parameter estimation techniques

October 3, 2007

D. EcheverrD. Echeverríía Ciaurria Ciaurri

Smart Fields ConsortiumStanford University


Outline

O

• Smart Fields Consortium

• The History Matching Problem

• Challenges and Trends

• History Matching at Smart Fields

• Conclusions


Closing the Loop

1


Closing the Loop

Field Development Optimization

1


Stanford Smart Fields

2



• Stanford University

2




– Energy Resources Engineering

– Geophysics

– Management Sciences and Engineering

– CEES

– Geology, Electrical Engineering, …

2




• Current consortium members

2




• Current consortium members

– BP, Chevron, CMG, Conoco Phillips,

ExxonMobil, Landmark, NTNU, Petrobras,

Shell, Saudi Aramco, Statoil, Total

– in discussions with several others

2


Scope of Work

3


Scope of Work

• Optimization techniques

3


Scope of Work

Oilfield Review 2006

• Optimization techniques– well placement

– well type

– reservoir and production system operations

3


Scope of Work


3


Scope of Work


• Data filtering and integration

www.halliburton.com

3


Scope of Work



3


Scope of Work



• History matching

3


Scope of Work




• Fast reservoir modeling and proxies

3


Scope of Work





• Uncertainty propagation

3


Scope of Work





• Uncertainty propagation

• Decision making

3


Model Order Reduction

4

(from Cardoso et al., 2007)



4

• PCA, original model size N = 800

Injec. 1 Injec. 2

Prod. 1 Prod. 2




4





4


full model N = 800n = 1 + 7 = 8n = 2 + 7 = 9n = 8 + 26 = 34n = 23 + 47 = 61



Outline

O





• Conclusions


History Matching

5


History Matching

m**

5


History Matching

m**observe

5

( )** mO


m

History Matching

m**observe

( )mO

5

( )** mO


m

History Matching

( ) ( ) ||mOmO|| minargm̂ **

Mm−=

∈

m**observe ( )** mO

( )mO

5


A Different Perspective

6



6

• Data filtering– solution quality ~– use observations to improve estimation– when noise/uncertainty is significant

||mm̂|| *−



6

• Data filtering

• Example: Kalman Filter



6

• Data filtering


• Original idea: linear systems only



6

• Data filtering



• Extended Kalman Filter– linearization around current estimate

– problematic with high nonlinearities



6

• Data filtering



• Extended Kalman Filter

• Ensemble Kalman Filter



6

• Data filtering



• Extended Kalman Filter

• Ensemble Kalman Filter– see next presentation!


Optimization

7


Optimization

7

• Local optimization


Optimization

7

• Local optimization– exact / approximate / no gradients


Optimization

7

• Local optimization– exact / approximate / no gradients– gradients are faster


Optimization

7

• Local optimization– exact / approximate / no gradients– gradients are faster– no gradients are robust


Optimization

7

• Local optimization– exact / approximate / no gradients– gradients are faster– no gradients are robust– exact non trivial implementation


Optimization

7

• Local optimization– exact / approximate / no gradients– gradients are faster– no gradients are robust– exact non trivial implementation– direct methods easier but much slower


Optimization

7


• Global optimization


Optimization

7


• Global optimization– when gradients make no sense


Optimization

7



(from Onwunalu et al., 2007)


Optimization

7




Optimization

7


• Global optimization– when gradients make no sense– most of them stochastic


Optimization

7


• Global optimization– when gradients make no sense– most of them stochastic– DIRECT, genetic, particle swarm,

differential evolution, simulated annealing


Optimization

7


• Global optimization– when gradients make no sense– most of them stochastic– DIRECT, genetic, particle swarm,

differential evolution, simulated annealing– amenable to parallelization


Optimization

7



• Local faster than global


Optimization

7



• Local faster than global

• Hybridization


Workflow

8


m

Workflow

8

facies


Workflow

8

facies

rock properties

m


Workflow

8

facies

O1 (m)

O2 (m)

rock properties

m


Workflow

8

facies

O1 (m)

O2 (m)

m**

rock properties

m


Workflow

8

facies

O1 (m)

O2 (m)||.||

m**

rock properties

m


Workflow

8

facies

O1 (m)

O2 (m)||.||

m**

tooptimizerrock

propertiesm


Workflow

8

facies

flow

seis||.||

m**

tooptimizerrock

propertiesm


Workflow

8

facies

flow

seis||.||

m**

tooptimizer

manyparameters

rock properties

m


Workflow

8

facies

flow

seis||.||KPCA

m**

tooptimizer

manyparameters

fewerparameters

rock properties

mξ


Workflow

8

facies

flow

seis||.||KPCA

m**

tooptimizer

manyparameters

lessparameters

rock properties

mξ

gradients


Outline

O





• Conclusions


Challenges and Trends

9



9

• Non-uniqueness in optimization



9


0.0

0.2

0.4

0.6

0.8

1.0

Wat

erC

ut

0 200 400 600 800 1000tim e, days

a a a a a a a a a

a

a

a

a

a

a

a

a

a

a

a

a

aa

a

a

a

aa

aa a a a a a a a a a a a a a

a a a a a a a a a

a

a

a

a

a

a

a

a

a

a

a

a

a

aa

aa

aa

aa

aa

aa a a a a a a a a a

a

a

In itia l Perm eab

Wat

er c

ut

t (days)

(from Caers et al., 2002)



9


0.0

0.2

0.4

0.6

0.8

1.0

Wat

erC

ut

0 200 400 600 800 1000tim e, days

a a a a a a a a a

a

a

a

a

a

a

a

a

a

a

a

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aa

a

a

a

aa

aa a a a a a a a a a a a a a

a a a a a a a a a

a

a

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a

a

a

a

a

a

a

a

a

a

aa

aa

aa

aa

aa

aa a a a a a a a a a

a

a

In itia l Perm eab

Wat

er c

ut

t (days)

East

Nor

th

0.0 50.00.0

50.0

0.0

200.0

400.0

600.0

800.0

1000.0

East

Nor

th

0.0 50.00.0

50.0

0.0

200.0

400.0

600.0

800.0

1000.0

East

Nor

th

0.0 50.00.0

50.0

0.0

200.0

400.0

600.0

800.0

1000.0

East

Nor

th

0.0 50.00.0

50.0

0.0

200.0

400.0

600.0

800.0

1000.0




9

• Non-uniqueness in optimization– select a solution that honors geology

– KPCA



9

• Non-uniqueness in optimization– honors geology, KPCA

• Large-scale optimization– from thousands to millions of variables

– reduce number of variables

– KPCA



9


• Large-scale optimization– reduce number of variables, KPCA



9


• Large-scale optimization– reduce number of variables, KPCA

• Cost functions expensive to compute– reduce number of function calls, gradients


More Trends

10


More Trends

10

• Global methods


More Trends

10

• Global methods– dealing with uncertainty and noise


More Trends

10


– when a good initial guess is not available


More Trends

10



– combined with proxies


More Trends

10




– hybridize with local methods


More Trends

10




– hybridize with local methods

• Data filtering


Workflow

8

facies

flow

seis||.||KPCA

m**

tooptimizer

manyparameters

lessparameters

rock properties

mξ

gradients


Kernel PCA

12


Kernel PCA

11

• PCA captures linear dependence


Kernel PCA

11


⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+=

=

21

212

1

12

yxyxyarctanx


Kernel PCA

11


• Nonlinear is more flexible


Kernel PCA

11



• PCA preserves 2nd order statistics


Kernel PCA

11




requires more than 2nd order


Kernel PCA

11





Kernel PCA

11




• Higher dimensions preserve

higher order statistics


KPCA in Action(from Sarma et al., 2006)

12


KPCA in Action

12

-1 0 1 20

0.5

1Approximate pdf

-2 -1 0 1 2 30

0.2

0.4

0.6

0.8Approximate pdf

-0.5

0

0.5

1

1.5 Perm Field from KLE

10 20 30 40

10

20

30

40

0

1

2Perm Field from KPCA

10 20 30 40

10

20

30

40

PCA analysis (n = 30)

(from Sarma et al., 2006)

KPCA analysis (n = 30)


KPCA in Action

12



-1 0 1 20

0.5

1Approximate pdf

-2 -1 0 1 2 30

0.2

0.4

0.6

0.8Approximate pdf

-0.5

0

0.5

1

Perm Field from KLE

10 20 30 40

10

20

30

40

-0.5

0

0.5

1

1.5

Perm Field from KPCA

10 20 30 40

10

20

30

40

KPCA in Action

12

PCA analysis (n = 30)


KPCA analysis (n = 30)


Flow Gradients

13


Flow Gradients

13

• Cost function = flow + seismic


Flow Gradients

13


• Flow cost function

( ) ( )∑−

=

=1N

0nn mLmJ m: model

( ) ( ) ||mOmO|| J(m) **11 −=


Flow Gradients

13


• Flow cost function

• Constraints (reservoir flow equations)

( ) 0m,x,xg n1nn =+ x: states

( ) ( )∑−

=

=1N

0nn mLmJ m: model


Flow Gradients

13


• Cost function and constraints

• Optimality

( ) 0m,x,xg ,0mJ

n1nnA ==

∂∂

+

( ) ( ) ( )m,x,xg m,xL,mJ n1nnT

1n

1N

0n1nnA ++

−

=+ λ+=λ ∑

( ) ( )λ=λλ

,mJ minargˆ,m̂ A,m

ff


Adjoint Equations

• Adjoint equations directly obtained from reservoir simulator

• Store Jacobians: and

• Nontrivial implementation

• Other nonlinear constraints can be considered

n

1n

xg∂∂ −

n

n

xg∂∂

14


Outline

O





• Conclusions


Probability Perturbation(from Caers et al., 2003)

15


Probability Perturbation

• Global Optimization


15




high

low

cost function


15




high

low

cost function

random


15




high

low

cost function

curve optimumrandom


15





16




• Related projects (Jef Caers)– History matching based on multiple alternative

geological scenarios

– Modeling of hydrogeological deposits constrained to pressure data

– Integration of 4D seismics and production data

– just to name a few...


16


Closed-Loop Management

17




• Model (sector)

– 32x46x8 (~12000) cells– 3 injectors and 4 producers, BHP control


17



• Control problem– permeability field unknown (model update)– maximize NPV in ~ 8 years (costly water)


17



• Constraints– bounds for controls– total injection ≤

20,000 STBD, watercut ≤

0.95


17



• Reference model– producer BHP = 4500 psi– injection water distributed by kh


17


Oil Saturations(from Sarma et al., 2006)

18



reference optimized(known permeability)

18



optimized(known permeability)

optimized(unknown permeability)

18


Production Data(from Sarma et al., 2006)

19



base case

Cum

ulat

ives

(STB

)

optimized

reference

19



base case

Cum

ulat

ives

(STB

)

16% increase in oil production

optimized

reference

19



base case

Cum

ulat

ives

(STB

)

50% decrease in water production

optimized

reference

19



base case

Cum

ulat

ives

(STB

)

25% increase in NPV

optimized

reference

19


Integrating Data

20

(joint work with Mukerji and Santos)


Integrating Data

• Production data and seismics

20



Integrating Data


• Stanford VI reservoir

20



Integrating Data


• Stanford VI reservoir– three zones

zone 3

20



Integrating Data



– prograding fluvial channel

zone 3

20



Integrating Data




– an asymmetric anticline

zone 3

20



Integrating Data





– 6 million cellszone 3

20



Integrating Data





– 6 million cells

– 4D seismic data set zone 3

20



Integrating Data



20





• Tomographies, 4D

Integrating Data

20





• Tomographies, 4D

• Flexible optimization– alternating matching production/seismics

– cumulative match from tn to tn+1

Integrating Data

20



Preliminary Results

21



Preliminary Results

• 20x20x10 sector from zone 3

21



Preliminary Results


• m: facies

21



Preliminary Results


• m: facies

• k, φ: regression from well location– k from Kozeny-Carman’s

21



Preliminary Results


• m: facies

• k, φ: regression from well location

• VP and VS as seismics

21



Preliminary Results


• m: facies


• VP and VS as seismics– Gassmann fluid substitution

– Batzle-Wang fluid properties

21



Preliminary Results


• m: facies



• PCA: 30 coefficients retained

21



Preliminary Results


• m: facies



• PCA: 30 coefficients retained– 1000 realizations

21



Preliminary Results


• m: facies



• PCA: 30 coefficients retained

• Measured data only after 3 months

21



22

Preliminary Results(joint work with Mukerji and Santos)


• Alternating optimization strategy– production ( 2 iter)

– production + seismics ( 2 iter)

– seismics (10 iter)

22



• Alternating optimization strategy

22




• Optimizer: SQP + numerical gradients

22





• True model m*: original sector projected over truncated PCA basis

*m

22





• True model m*: original sector projected over truncated PCA basis

• Initial guess m0*: random realization

taken from the 1000 for the PCA

*m

m̂

22



Production Data

23



Production Data

5 spot

injector

producer

23



Oil Production

23



Water Injection

23



Seismic Data

5 spot

injector

producer

24



5 spot

injector

producer

24


Seismic Data


Velocities (VP )

true(projected)

initialguess

matchingproduction

(2 iter)

alternatingmatching

24



Velocities (VP )

initialguess

matchingproduction

(2 iter)

alternatingmatching

24



Velocities (VP )

x

z

y

z

24

true

after production

initial

alternating



Facies

25



Facies

true(projected)

initialguess

matchingproduction

(2 iter)

alternatingmatching

25



Facies

x

y

LAYER 125

true

after production

initial

alternating



Facies

x

y

LAYER 425

true

after production

initial

alternating



Facies

x

y

LAYER 625

true

after production

initial

alternating



Facies

x

y

LAYER 1025

true

after production

initial

alternating



Outline

O





• Conclusions


Conclusions

• Semi-automatic history matching

• Seen as an optimization problem

• Local vs. global optimization

• KPCA reduces number of optimization variables and honors geology

• Different types of data integrated

C


Acknowledgement

• Khalid Aziz, Jef Caers, Lou Durlofsky, Roland Horne, Tapan Mukerji and Eduardo Santos

• Marco Cardoso, Jerome Onwunalu, Danny Rojas and Pallav Sarma

• Others at the Smart Field Consortium

A


Thank youfor

your attention!

October 3, 2007




Semi-automatic history matchingapplying

parameter estimation techniques

October 3, 2007



Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying...

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