Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying...
Transcript of Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying...
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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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Outline
O
• Smart Fields Consortium
• The History Matching Problem
• Challenges and Trends
• History Matching at Smart Fields
• Conclusions
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Outline
O
• Smart Fields Consortium
• The History Matching Problem
• Challenges and Trends
• History Matching at Smart Fields
• Conclusions
![Page 4: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/4.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
Field Development Optimization
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closing the Loop
Field Development Optimization
1
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Stanford Smart Fields
2
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Stanford Smart Fields
• Stanford University
2
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Stanford Smart Fields
• Stanford University
– Energy Resources Engineering
– Geophysics
– Management Sciences and Engineering
– CEES
– Geology, Electrical Engineering, …
2
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Stanford Smart Fields
• Stanford University
• Current consortium members
2
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Stanford Smart Fields
• Stanford University
• Current consortium members
– BP, Chevron, CMG, Conoco Phillips,
ExxonMobil, Landmark, NTNU, Petrobras,
Shell, Saudi Aramco, Statoil, Total
– in discussions with several others
2
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
Oilfield Review 2006
• Optimization techniques– well placement
– well type
– reservoir and production system operations
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
• Data filtering and integration
www.halliburton.com
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
• Data filtering and integration
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
• Data filtering and integration
• History matching
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
• Data filtering and integration
• History matching
• Fast reservoir modeling and proxies
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
• Data filtering and integration
• History matching
• Fast reservoir modeling and proxies
• Uncertainty propagation
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Scope of Work
• Optimization techniques
• Data filtering and integration
• History matching
• Fast reservoir modeling and proxies
• Uncertainty propagation
• Decision making
3
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
(from Cardoso et al., 2007)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
Injec. 1 Injec. 2
Prod. 1 Prod. 2
(from Cardoso et al., 2007)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
(from Cardoso et al., 2007)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
full model N = 800n = 1 + 7 = 8n = 2 + 7 = 9n = 8 + 26 = 34n = 23 + 47 = 61
(from Cardoso et al., 2007)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
full model N = 800n = 1 + 7 = 8n = 2 + 7 = 9n = 8 + 26 = 34n = 23 + 47 = 61
(from Cardoso et al., 2007)
![Page 35: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/35.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
full model N = 800n = 1 + 7 = 8n = 2 + 7 = 9n = 8 + 26 = 34n = 23 + 47 = 61
(from Cardoso et al., 2007)
![Page 36: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/36.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
full model N = 800n = 1 + 7 = 8n = 2 + 7 = 9n = 8 + 26 = 34n = 23 + 47 = 61
(from Cardoso et al., 2007)
![Page 37: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/37.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Model Order Reduction
4
• PCA, original model size N = 800
full model N = 800n = 1 + 7 = 8n = 2 + 7 = 9n = 8 + 26 = 34n = 23 + 47 = 61
(from Cardoso et al., 2007)
![Page 38: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/38.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Outline
O
• Smart Fields Consortium
• The History Matching Problem
• Challenges and Trends
• History Matching at Smart Fields
• Conclusions
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Smart Fields, Stanford UniversityIntegrated Operations 200707
History Matching
5
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Smart Fields, Stanford UniversityIntegrated Operations 200707
History Matching
m**
5
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Smart Fields, Stanford UniversityIntegrated Operations 200707
History Matching
m**observe
5
( )** mO
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Smart Fields, Stanford UniversityIntegrated Operations 200707
m
History Matching
m**observe
( )mO
5
( )** mO
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Smart Fields, Stanford UniversityIntegrated Operations 200707
m
History Matching
( ) ( ) ||mOmO|| minargm̂ **
Mm−=
∈
m**observe ( )** mO
( )mO
5
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
• Data filtering– solution quality ~– use observations to improve estimation– when noise/uncertainty is significant
||mm̂|| *−
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
• Data filtering
• Example: Kalman Filter
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
• Data filtering
• Example: Kalman Filter
• Original idea: linear systems only
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
• Data filtering
• Example: Kalman Filter
• Original idea: linear systems only
• Extended Kalman Filter– linearization around current estimate
– problematic with high nonlinearities
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
• Data filtering
• Example: Kalman Filter
• Original idea: linear systems only
• Extended Kalman Filter
• Ensemble Kalman Filter
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Smart Fields, Stanford UniversityIntegrated Operations 200707
A Different Perspective
6
• Data filtering
• Example: Kalman Filter
• Original idea: linear systems only
• Extended Kalman Filter
• Ensemble Kalman Filter– see next presentation!
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization– exact / approximate / no gradients
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization– exact / approximate / no gradients– gradients are faster
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization– exact / approximate / no gradients– gradients are faster– no gradients are robust
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization– exact / approximate / no gradients– gradients are faster– no gradients are robust– exact non trivial implementation
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Smart Fields, Stanford UniversityIntegrated Operations 200707
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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization– when gradients make no sense
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization– when gradients make no sense
(from Onwunalu et al., 2007)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization– when gradients make no sense
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization– when gradients make no sense– most of them stochastic
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization– when gradients make no sense– most of them stochastic– DIRECT, genetic, particle swarm,
differential evolution, simulated annealing
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization– when gradients make no sense– most of them stochastic– DIRECT, genetic, particle swarm,
differential evolution, simulated annealing– amenable to parallelization
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization
• Local faster than global
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Optimization
7
• Local optimization
• Global optimization
• Local faster than global
• Hybridization
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
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Smart Fields, Stanford UniversityIntegrated Operations 200707
m
Workflow
8
facies
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
rock properties
m
![Page 70: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/70.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
O1 (m)
O2 (m)
rock properties
m
![Page 71: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/71.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
O1 (m)
O2 (m)
m**
rock properties
m
![Page 72: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/72.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
O1 (m)
O2 (m)||.||
m**
rock properties
m
![Page 73: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/73.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
O1 (m)
O2 (m)||.||
m**
tooptimizerrock
propertiesm
![Page 74: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/74.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
flow
seis||.||
m**
tooptimizerrock
propertiesm
![Page 75: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/75.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
flow
seis||.||
m**
tooptimizer
manyparameters
rock properties
m
![Page 76: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/76.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
flow
seis||.||KPCA
m**
tooptimizer
manyparameters
fewerparameters
rock properties
mξ
![Page 77: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/77.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
flow
seis||.||KPCA
m**
tooptimizer
manyparameters
lessparameters
rock properties
mξ
gradients
![Page 78: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/78.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Outline
O
• Smart Fields Consortium
• The History Matching Problem
• Challenges and Trends
• History Matching at Smart Fields
• Conclusions
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
![Page 80: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/80.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization
![Page 81: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/81.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization
0.0
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization
0.0
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(from Caers et al., 2002)
![Page 83: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/83.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization
0.0
0.2
0.4
0.6
0.8
1.0
Wat
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0 200 400 600 800 1000tim e, days
a a a a a a a a a
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0.0 50.00.0
50.0
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50.0
0.0
200.0
400.0
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400.0
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(from Caers et al., 2002)
![Page 84: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/84.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization– select a solution that honors geology
– KPCA
![Page 85: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/85.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization– honors geology, KPCA
• Large-scale optimization– from thousands to millions of variables
– reduce number of variables
– KPCA
![Page 86: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/86.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization– honors geology, KPCA
• Large-scale optimization– reduce number of variables, KPCA
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Challenges and Trends
9
• Non-uniqueness in optimization– honors geology, KPCA
• Large-scale optimization– reduce number of variables, KPCA
• Cost functions expensive to compute– reduce number of function calls, gradients
![Page 88: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/88.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
![Page 89: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/89.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
• Global methods
![Page 90: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/90.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
• Global methods– dealing with uncertainty and noise
![Page 91: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/91.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
• Global methods– dealing with uncertainty and noise
– when a good initial guess is not available
![Page 92: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/92.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
• Global methods– dealing with uncertainty and noise
– when a good initial guess is not available
– combined with proxies
![Page 93: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/93.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
• Global methods– dealing with uncertainty and noise
– when a good initial guess is not available
– combined with proxies
– hybridize with local methods
![Page 94: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/94.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
More Trends
10
• Global methods– dealing with uncertainty and noise
– when a good initial guess is not available
– combined with proxies
– hybridize with local methods
• Data filtering
![Page 95: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/95.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Workflow
8
facies
flow
seis||.||KPCA
m**
tooptimizer
manyparameters
lessparameters
rock properties
mξ
gradients
![Page 96: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/96.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
12
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
⎪⎭
⎪⎬⎫
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+=
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212
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![Page 103: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/103.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
• Nonlinear is more flexible
![Page 104: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/104.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
• Nonlinear is more flexible
• PCA preserves 2nd order statistics
![Page 105: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/105.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
• Nonlinear is more flexible
• PCA preserves 2nd order statistics
requires more than 2nd order
![Page 106: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/106.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
• Nonlinear is more flexible
• PCA preserves 2nd order statistics
![Page 107: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/107.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
Kernel PCA
11
• PCA captures linear dependence
• Nonlinear is more flexible
• PCA preserves 2nd order statistics
• Higher dimensions preserve
higher order statistics
![Page 108: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/108.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
KPCA in Action(from Sarma et al., 2006)
12
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Smart Fields, Stanford UniversityIntegrated Operations 200707
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)
![Page 110: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/110.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
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)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
KPCA in Action
12
(from Sarma et al., 2006)
![Page 112: Semi-automatic history matching applying parameter ... · Semi-automatic history matching. applying . parameter estimation techniques. ... • History matching ... Workflow. 8. facies.](https://reader031.fdocuments.in/reader031/viewer/2022022602/5b53fad97f8b9add3a8c7902/html5/thumbnails/112.jpg)
Smart Fields, Stanford UniversityIntegrated Operations 200707
-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)
(from Sarma et al., 2006)
KPCA analysis (n = 30)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
-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)
(from Sarma et al., 2006)
KPCA analysis (n = 30)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Flow Gradients
13
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Flow Gradients
13
• Cost function = flow + seismic
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Flow Gradients
13
• Cost function = flow + seismic
• Flow cost function
( ) ( )∑−
=
=1N
0nn mLmJ m: model
( ) ( ) ||mOmO|| J(m) **11 −=
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Flow Gradients
13
• Cost function = flow + seismic
• Flow cost function
• Constraints (reservoir flow equations)
( ) 0m,x,xg n1nn =+ x: states
( ) ( )∑−
=
=1N
0nn mLmJ m: model
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Flow Gradients
13
• Cost function = flow + seismic
• 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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Flow Gradients
13
• Cost function = flow + seismic
• 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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Outline
O
• Smart Fields Consortium
• The History Matching Problem
• Challenges and Trends
• History Matching at Smart Fields
• Conclusions
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
random
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
random
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
random
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
high
low
cost function
curve optimumrandom
(from Caers et al., 2003)
15
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
(from Caers et al., 2003)
16
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Probability Perturbation
• Global Optimization
• 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...
(from Caers et al., 2003)
16
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closed-Loop Management
17
(from Sarma et al., 2006)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closed-Loop Management
• Model (sector)
– 32x46x8 (~12000) cells– 3 injectors and 4 producers, BHP control
(from Sarma et al., 2006)
17
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closed-Loop Management
• Control problem– permeability field unknown (model update)– maximize NPV in ~ 8 years (costly water)
(from Sarma et al., 2006)
17
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closed-Loop Management
• Constraints– bounds for controls– total injection ≤
20,000 STBD, watercut ≤
0.95
(from Sarma et al., 2006)
17
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Closed-Loop Management
• Reference model– producer BHP = 4500 psi– injection water distributed by kh
(from Sarma et al., 2006)
17
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Saturations(from Sarma et al., 2006)
18
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Saturations(from Sarma et al., 2006)
reference optimized(known permeability)
18
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Saturations(from Sarma et al., 2006)
optimized(known permeability)
optimized(unknown permeability)
18
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data(from Sarma et al., 2006)
19
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data(from Sarma et al., 2006)
base case
Cum
ulat
ives
(STB
)
optimized
reference
19
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data(from Sarma et al., 2006)
base case
Cum
ulat
ives
(STB
)
16% increase in oil production
optimized
reference
19
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data(from Sarma et al., 2006)
base case
Cum
ulat
ives
(STB
)
50% decrease in water production
optimized
reference
19
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data(from Sarma et al., 2006)
base case
Cum
ulat
ives
(STB
)
25% increase in NPV
optimized
reference
19
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir– three zones
zone 3
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir– three zones
– prograding fluvial channel
zone 3
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir– three zones
– prograding fluvial channel
– an asymmetric anticline
zone 3
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir– three zones
– prograding fluvial channel
– an asymmetric anticline
– 6 million cellszone 3
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir– three zones
– prograding fluvial channel
– an asymmetric anticline
– 6 million cells
– 4D seismic data set zone 3
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Integrating Data
• Production data and seismics
• Stanford VI reservoir
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Production data and seismics
• Stanford VI reservoir
• Tomographies, 4D
Integrating Data
20
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Production data and seismics
• Stanford VI reservoir
• Tomographies, 4D
• Flexible optimization– alternating matching production/seismics
– cumulative match from tn to tn+1
Integrating Data
20
(joint work with Mukerji and Santos)
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Preliminary Results
21
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
21
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
21
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
• k, φ: regression from well location– k from Kozeny-Carman’s
21
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
• k, φ: regression from well location
• VP and VS as seismics
21
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
• k, φ: regression from well location
• VP and VS as seismics– Gassmann fluid substitution
– Batzle-Wang fluid properties
21
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
• k, φ: regression from well location
• VP and VS as seismics
• PCA: 30 coefficients retained
21
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
• k, φ: regression from well location
• VP and VS as seismics
• PCA: 30 coefficients retained– 1000 realizations
21
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Preliminary Results
• 20x20x10 sector from zone 3
• m: facies
• k, φ: regression from well location
• VP and VS as seismics
• PCA: 30 coefficients retained
• Measured data only after 3 months
21
(joint work with Mukerji and Santos)
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22
Preliminary Results(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Alternating optimization strategy– production ( 2 iter)
– production + seismics ( 2 iter)
– seismics (10 iter)
22
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Alternating optimization strategy
22
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Alternating optimization strategy
• Optimizer: SQP + numerical gradients
22
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Alternating optimization strategy
• Optimizer: SQP + numerical gradients
• True model m*: original sector projected over truncated PCA basis
*m
22
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Smart Fields, Stanford UniversityIntegrated Operations 200707
• Alternating optimization strategy
• Optimizer: SQP + numerical gradients
• 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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Production Data
5 spot
injector
producer
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Production
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Production
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Production
23
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Oil Production
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Water Injection
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Water Injection
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Water Injection
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Water Injection
23
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Seismic Data
5 spot
injector
producer
24
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
5 spot
injector
producer
24
(joint work with Mukerji and Santos)
Seismic Data
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Velocities (VP )
true(projected)
initialguess
matchingproduction
(2 iter)
alternatingmatching
24
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Velocities (VP )
true(projected)
initialguess
matchingproduction
(2 iter)
alternatingmatching
24
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Velocities (VP )
initialguess
matchingproduction
(2 iter)
alternatingmatching
24
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Velocities (VP )
x
z
y
z
24
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Facies
25
(joint work with Mukerji and Santos)
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Facies
true(projected)
initialguess
matchingproduction
(2 iter)
alternatingmatching
25
(joint work with Mukerji and Santos)
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Facies
x
y
LAYER 125
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Facies
x
y
LAYER 125
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Facies
x
y
LAYER 425
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Facies
x
y
LAYER 425
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Facies
x
y
LAYER 625
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Facies
x
y
LAYER 625
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Facies
x
y
LAYER 1025
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Facies
x
y
LAYER 1025
true
after production
initial
alternating
(joint work with Mukerji and Santos)
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Outline
O
• Smart Fields Consortium
• The History Matching Problem
• Challenges and Trends
• History Matching at Smart Fields
• Conclusions
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Smart Fields, Stanford UniversityIntegrated Operations 200707
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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
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
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Smart Fields, Stanford UniversityIntegrated Operations 200707
Thank youfor
your attention!
October 3, 2007
D. EcheverrD. Echeverríía Ciaurria Ciaurri
Smart Fields ConsortiumStanford University
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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