Coupled EnKF

1. Geir Nævdal and Brice Vallès Coupled EnKF

2. Outline • Motivation • Background: Alternative implementation • Examples • Simple 1-D linear model • PUNQS3 • Further work

3. Coupled EnKF – motivation • Lorentzen et. al., 2005, SPE96375 • Problem with consistency between repeated runs • Thulin et. al., in preparation, previous talk + ECMOR XI • Set of independent EnKFs to estimate Monte-Carlo uncertainty • Anderson, Physica D, vol. 230, 2007 • “Hierarchical ensemble filter” to calculate localization • Use several independent EnKF, calculate a reduction in Kalman gain based on statistics

4. Reminder: SPE96375 • 10 initial ensembles • Generated with same distribution • Kolmogorov-Smirnov test on posterior distributions • Posterior distributions are not coming from same distribution Example: FOPT

5. Hierarchical ensemble filter (by Anderson) • Split the ensemble in several sub-ensembles • Run each sub-ensemble using different Kalman gain matrices • Modify each of the Kalman gain matrices • multiplied with factor

6. Localization by Hierarchical EnKF – simple 1-D example • Initial guess: • Zero mean • Gaussian variogram • Correlation length: 5 • Standard deviation: 1 • Measurement: y=2 at x=26 • Measurement uncertainty: 2 • Kalman filter gives updated mean (and covariance)

7. Localization by Hierarchical EnKF – simple 1-D example • Initial guess: • Zero mean • Gaussian variogram • Correlation length: 5 • Standard deviation: 1 • Measurement: y=2 at x=26 • Measurement uncertainty: 2 • Kalman filter gives updated mean (and covariance) • Compare • EnKF with 100 ens. members • Hierarchical ensemble filter with 5x20 members

8. EnKF Localization by Hierarchical ensemble filter – simple example (40 runs) Results from 40 runs (ens. mean) Average of 40 runs Hierarchical True True

9. EnKF Localization by Hierarchical ensemble filter – simple example (40 runs) Results from 40 runs (ens. mean) Standard deviation of mean of 40 runs Hierarchical True True

10. PUNQS3 • The PUNQ-S3 is a small-size synthetic 3-D reservoir engineering model. • The reservoir consists of 19 x 28 x5 gridblocks, where 1761 are active. • Equal 180 meter sides in x- and y-directions. • Reservoir is bounded by a fault in east and south. • Reservoir is bounded by an aquifer in west and north. • New webpage: http://www3.imperial.ac.uk/earthscienceandengineering/research/perm/punq-s3model

11. PUNQS3 – production history and forecasting • First 8 years: history matching phase. • 1 year of well testing, • 3 year shut-in period, and • 4 years of production. • Next 8.5 years: forecasting phase. • During history matching phase: • wells are controlled by using history target rates for oil. • During forecasting phase: • wells are controlled using target oil rate of 150 scm/day. • Minimum bottom hole pressure of 120 bar. • If gas/oil ratio is greater than 200, a cutback factor of 0.75 is used.

12. Investigation • Initial ensemble generated based on description on old PUNQS3 webpage • Permeability and porosity are estimated • Comparing result of forecasts • Ordinary EnKF versus hierarchical ensemble filter (200 members vs. 5 x 40 members) • Arguing for 40 members in each batch: • For PUNQS3 Gu & Oliver found reasonable history match with 40 members • For field case, Bianco et. al. found reasonable history match with 50 members

13. Comparison: Ordinary EnKF compared to 5x40 members with hierarchical ensemble filter – Forecasted FOPT • 10 initial ensembles used in both cases • Compare forecasted FOPT from final estimates • Figure shows maximum, mean, and minimum of cdfs for FOPT • There is generally less deviation in the results from hierarchical ensemble filter EnKF Hierarchical

14. EnKF Hierarchical Comparison: Ordinary EnKF compared to 5x40 members with hierarchical ensemble filter – Forecasted FOPT

15. Quality of the solutions: History matching • Evaluate the estimated fields by rerunning from time zero • 117 measurements • Objective function: EnKF Hierarchical

16. Quality of solutions: Estimated porosity • Compare quality of solutions with following measure: • Hierarchical more robust EnKF Hierarchical

17. Quality of solutions: Estimated log-perm Log-permZ Log-permX EnKF Hierarchical

18. Quality of solutions: Estimated dynamic quantities Pressure Gas-oil ratio EnKF Hierarchical

19. Quality of solutions: Estimated saturations Water saturation Gas saturation EnKF Hierarchical

20. EnKF Comparison of mean of final estimates – the concept: Results from 40 runs (ens. mean) Standard deviation of mean of 40 runs Hierarchical True True

21. Comparison of std. deviation of the mean estimate for the 10 runs: Porosity layer 1 – final time Hierarchical filter Ordinary EnKF

22. Porosity layer 5 – 10 runs – final time Hierarchical filter Ordinary EnKF

23. Log-Permx - layer 1 – final time Hierarchical filter Ordinary EnKF

24. Water saturation - layer 2 – final time Hierarchical filter Ordinary EnKF

25. Gas saturation - layer 3 – final time Hierarchical filter Ordinary EnKF

26. Pressure - layer 2 – final time Hierarchical filter Ordinary EnKF

27. Conclusion of PUNQS3 study • Slightly better history matches with EnKF compared to hierarchical ensemble filter • Hierarchical ensemble filter seems to be more robust and have less variations in repeated runs • Computation time is of same order for the two approaches • PUNQS3 forecasts do not differ to much

28. Conclusions & suggestions for further work • Hierarchical ensemble filter • Gives the opportunity to estimate Monte-Carlo uncertainty • Seems to be more robust • Have computation time as ordinary EnKF • Other approaches for localization could be evaluated • Datta-Gupta and coworkers based on streamlines • Approaches based on Schur product • … • Evaluate hierarchical ensemble filter on more challenging examples • Evaluate different partitions than 5 x 40 members

29. Acknowledgment • This work has been done with financial support from Research Council of Norway (PETROMAKS) and industrial partners • Licenses for Eclipse have been provided by Schlumberger