improved sampling
DESCRIPTION
Geir Nævdal and Kristin M. Flornes*. Improved sampling. EnKF with improved sampling of the initial ensemble. Goal: Improve the performance of the EnKF without increasing the ensemble size Different resampling techniques for the initial ensemble have been proposed - PowerPoint PPT PresentationTRANSCRIPT
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Improved sampling
Geir Nævdal and Kristin M. Flornes*
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EnKF with improved sampling of the initial ensembleGoal: Improve the performance of the EnKF
without increasing the ensemble size
• Different resampling techniques for the initial ensemble have been proposed
• In this work we have looked more closely at the effect of using Geir Evensen's resampling scheme (2004)
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Outline• Evensen’s resampling scheme
• Does resampling preserve the variogram?
• Description of test case
• Results
• Conclusion
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Evensen’s improved sampling schemeAim
Introduce a maximum rank and conditioning of the ensemble matrix for a given ensemble size.
Based on ideas from Singular Evolution Interpolated Kalman (SEIK) Filters
(Pham, 2001).
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Evensen’s improved sampling schemeTo generate an ensemble of size N do the following: Generate a large ensemble of size β*N. Do a SVD of the ensemble matrix A and retain only
the N largest singular values.
Create a new ensemble of size N based on these singular values.
TVUA
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Does Evensen’s resampling scheme preserve the variogram?• Is the variogram the same for a resampled
ensemble of N members as for the large initial ensemble with β*N members?
• We used the analytical covariance matrix in 1-D for Gaussian, spherical and exponential variogram to study theeffect of removing
singular values
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Does resampling preserve the
variogram? • For the Gaussian model the elements in the covariance matrix
are
• Relationship between variogram and covariance
2
23exp
a
ipi
ii pp 0
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Effect of retaining only ½ of the singular values
Gaussian Spherical Exponential
Analytical model variogram
Evensen’s algorithm
Evensen’s algorithm
Evensen’s algorithm
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Effect of retaining 1/8 of the singular values
Gaussian Spherical Exponential
This shows that the variograms will be influenced by resampling if we truncate a large portion of the singular values, leading to smoother fields.
Evensen’s algorithm
Analytical model variogram
Evensen’s algorithm Evensen’s algorithm
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Test Case - Description• Synthetic 2D case (50 X 50)• 3 producers (corners),1 injector (in the
middle)• Fields are generated using sgsim2
– Spherical variogram– Variogram range: 10 grid cells
• Static variables: PORO and PERMX• PERMY=PERMX• Measurement errors
– OPR, WPR, GPR: 10 % – BHP: 1%
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True static fields
Injector in the middle, producers in the upper left and right corners and lower left corner
PERMX PORO
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Test Case - Resampling setupStarted with 500 ensemble members.
Resampled down to 100 members
1. 100 random ensemble members • Used the 100 first of the 500 ensemble
members
2. Generate 100 ensemble members using Evensen’s improved sampling scheme
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Example of initial porosity fields I
Ordinary ensemble member
Ensemble member generated fromresampling
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Example of initial porosity fields II
Ordinary ensemble member
Ensemble member generated fromresampling
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Example of initial porosity fields III
Ordinary ensemble member
Ensemble member generated fromresampling
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Effect of resampling on initial
fields • The resampled ensemble members are smoother
• This is an effect of removing singular values
• Permeability fields are generated independently from porosity fields, and also resampled independently
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Injection pressure Forecast AnalyzedMeasurement
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Data for PROD1Forecast AnalyzedMeasurement
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Data for PROD2Forecast AnalyzedMeasurement
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Data for PROD3Forecast AnalyzedMeasurement
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Performance measures• For a field m we use the formula
Compared results of 30 runs
e mN
i
N
j
truej
ij
me
mmNN
mR1 1
21)(
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Effect on estimated water saturation
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Effect on estimated pressure
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Effect on estimated porosity
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Effect on estimated permeability
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Summary and conclusion• The dynamic variables are better
estimated using ordinary ensembles compared to resampled– Holds in particular for water saturation
• Small effects on static variables
• No argument for resampling
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Acknowledgement This work was done with financial
support from the ROAW project, funded by the Research Council of Norway (PETROMAKS) and industrial sponsors. Licenses to the Eclipse simulator were provided by Schlumberger.
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Literature• Evensen (Ocean Dynamics 2004)
“Sampling strategies and square root analysis schemes for the EnKF”
• Zafari et. al., SPE95750 (RMS measure)
• G. Nævdal, K.M. Flornes SPE118729 “Using ensemble Kalman filter with improved sampling of the initial ensemble” Submitted