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 • In this work we have looked more closely at the effect of using Geir Evensen's resampling scheme (2004)
Outline • Evensen’s resampling scheme • Does resampling preserve the variogram? • Description of test case • Results • Conclusion
Evensen’s improved sampling scheme Aim 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).
Evensen’s improved sampling scheme To 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.
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
Does resampling preserve the variogram? • For the Gaussian model the elements in the covariance matrix are • Relationship between variogram and covariance
Evensen’s algorithm Evensen’s algorithm Effect of retaining only ½ of the singular values Gaussian Spherical Exponential Evensen’s algorithm Analytical model variogram
Effect of retaining 1/8 of the singular values Gaussian Spherical Exponential Evensen’s algorithm Evensen’s algorithm Evensen’s algorithm Analytical model variogram This shows that the variograms will be influenced by resampling if we truncate a large portion of the singular values, leading to smoother fields.
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%
Injector in the middle, producers in the upper left and right corners and lower left corner True static fields PERMX PORO
Test Case - Resampling setup Started with 500 ensemble members. Resampled down to 100 members • 100 random ensemble members • Used the 100 first of the 500 ensemble members • Generate 100 ensemble members using Evensen’s improved sampling scheme
Ensemble member generated from resampling Ordinary ensemble member Example of initial porosity fields I
Ensemble member generated from resampling Ordinary ensemble member Example of initial porosity fields II
Ensemble member generated from resampling Ordinary ensemble member Example of initial porosity fields III
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
Forecast Analyzed Measurement Injection pressure
Forecast Analyzed Measurement Data for PROD1
Forecast Analyzed Measurement Data for PROD2
Forecast Analyzed Measurement Data for PROD3
Compared results of 30 runs Performance measures • For a field m we use the formula
Effect on estimated water saturation
Effect on estimated pressure
Effect on estimated porosity
Effect on estimated permeability
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
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.
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
