1 / 28

Improved sampling

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

ceri
Download Presentation

Improved sampling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Geir Nævdal and Kristin M. Flornes* Improved sampling

  2. 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)‏

  3. Outline • Evensen’s resampling scheme • Does resampling preserve the variogram? • Description of test case • Results • Conclusion

  4. 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).

  5. 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.

  6. 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

  7. Does resampling preserve the variogram? • For the Gaussian model the elements in the covariance matrix are • Relationship between variogram and covariance

  8. Evensen’s algorithm Evensen’s algorithm Effect of retaining only ½ of the singular values Gaussian Spherical Exponential Evensen’s algorithm Analytical model variogram

  9. 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.

  10. 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%

  11. Injector in the middle, producers in the upper left and right corners and lower left corner True static fields PERMX PORO

  12. 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

  13. Ensemble member generated from resampling Ordinary ensemble member Example of initial porosity fields I

  14. Ensemble member generated from resampling Ordinary ensemble member Example of initial porosity fields II

  15. Ensemble member generated from resampling Ordinary ensemble member Example of initial porosity fields III

  16. 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

  17. Forecast Analyzed Measurement Injection pressure

  18. Forecast Analyzed Measurement Data for PROD1

  19. Forecast Analyzed Measurement Data for PROD2

  20. Forecast Analyzed Measurement Data for PROD3

  21. Compared results of 30 runs Performance measures • For a field m we use the formula

  22. Effect on estimated water saturation

  23. Effect on estimated pressure

  24. Effect on estimated porosity

  25. Effect on estimated permeability

  26. 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

  27. 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.

  28. 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

More Related