Revisiting Brugge case study using a hierarchical ensemble Kalman filter

1. Revisiting Brugge case study using a hierarchical ensemble Kalman filter Geir Nævdal and Brice Vallès

2. Outline • Motivation for this study • Hierarchical ensemble Kalman filter • Examples: • PUNQS3 model • Brugge case

3. Motivation • Lorentzen et. al. (2005), SPE96375 • Problem with consistency between repeated runs • 10 independent initial ensembles • Generated with same distribution • Kolmogorov-Smirnov (KS) test on posterior distributions • “Posterior distributions are not coming from same distribution”

4. Reminder: SPE96375 • KS tests: 5 groups • 1 red, 1 blue and 3 independent blacks Pairwise comparisons of cdf for FOPT forecasts for PUNQ-S3 model

5. Motivation • Robustness can be improved • Inconsistency between repeated runs • Lorentzen et. al. (2005), SPE96375 • Due to spurious correlations • Update far from observation points • HEnKF • Address spurious correlations issue • Successful on simple models • Need testing on real size reservoir models

6. What is hierarchical ensemble Kalman filter (HEnKF) • Generic “localization” approach • Operates on Kalman gain matrix • Different from other localization techniques… • And from the hierarchical ensemble Kalman filter by Myrseth and Omre, SPE Journal, 2010 (available at online first).

7. Hierarchical filter Anderson, 2007 • Split the ensemble in Ng sub-ensembles of size Ne • At each time step n • Run each sub-ensemble using different Kalman gain matrices • Modify each Kalman gain matrices elements with factor • Update each state vector with “new” Kalman gain matrices • min,n acts as a damping factor!

8. (1) (2) • Forecast step: • Analysis step: • f : Simulator K : Kalman gain • d : Measurements H : Measurement matrix • y : state vector n : time index • Split the ensemble in Ng sub-ensembles of size Ne • Calculate αnbased on statistics between sub-ensembles • Automated generic “localization” approach • Minimize sampling errors HEnKF

9. HEnKF evaluation – PUNQ-S3 • Small-size synthetic 3-D reservoir engineering model • 19 x 28 x5 gridblocks • 1761 active gridblocks • Reservoir bounded by fault in east and south • Reservoir bounded by aquifer in west and north.

10. HEnKF evaluation – PUNQ-S3 • First 8 years: history matching phase • 1 year well testing, 3 years shut-in period, and 4 years production • Next 8.5 years: forecasting phase • During history matching phase • wells controlled by history oil target rates • During forecasting phase • wells controlled by oil target rate = 150 scm/day • Minimum bottom hole pressure = 120 bar • If GOR > 200, use cutback factor = 0.75

11. HEnKF evaluation – PUNQ-S3 • Description of experiment • Revisit Lorentzen et. al. (2005), SPE 96375 • PUNQ-S3 synthetic reservoir model • Use same 10 initial ensembles • Permeability and porosity are estimated • Comparisons of forecasts • Compare • EnKF 200 • Splitting 5x40 • HEnKF 5x40 • HEnKF 2x100

12. HEnKF evaluation – PUNQ-S3 Empirical cdf for FOPT obtained for the 10 runs Splitting 5x40 EnKF HEnKF 5x40 HEnKF 2x100

13. HEnKF evaluation – PUNQ-S3 Splitting 5x40 EnKF HEnKF 2x100 HEnKF 5x40

14. Brugge case • Comparative study prepared for SPE-ATW on “Closed loop reservoir management”, Bruges, June 2008 • Prepared by TNO • True case: 450 K gridblocks model (only known to TNO) • Participants: 104 upscaled realizations (60 K gridblocks) with 10 years production data + … • Request: • Solution of history matching for first ten years • Production strategy for next 20 years (2 x 10 years) aiming at maximizing NPV • Participants had a range of solutions • Top 3 groups (achieved NPV) used EnKF based methods for history matching • See SPE 119094 for report on comparative study

15. HEnKF evaluation – Brugge case • Brent type of reservoir • 9 layers • Two phases: • Oil & water • 10 injectors • 20 producers • PORO, PERMX, PERMY, PERMZ and NTG tuned • Measurements: WWCT, WOPR and WBHP

16. HEnKF evaluation – Brugge case • History matching first 10 years only • Compare EnKF vs. HEnKF • Initial ensemble size = 200 • Using variogram models (as Lorentzen et al., 2009 (SPE119101)) • Initial ensemble size = 210 • Uses all 104 original realizations • Modeled using different types of geostatistical modeling (both facies and variogram models)

17. HEnKF evaluation – Brugge case HEnKF 5x40 EnKF200 HEnKF 5x42 EnKF210 Mean PERMX layer 4

18. HEnKF evaluation – Brugge case HEnKF 5x40 EnKF200 HEnKF 5x42 EnKF210 Std log(PERMX) layer 4

19. HEnKF evaluation – Brugge case HEnKF 5x40 EnKF200 HEnKF 5x42 EnKF210 abs(last updated-init) log(PERMX) layer 4

20. HEnKF evaluation – Brugge case • Comments on parameters fields

21. HEnKF evaluation – Brugge case init200 EnKF200 HEnKF5x40 init210 EnKF210 HEnKF5x42 WOPR Prod. 5

22. HEnKF evaluation – Brugge case init200 EnKF200 HEnKF5x40 init210 EnKF210 HEnKF5x42 WOPR Prod. 14

23. Summary • HEnKF seems better than EnKF • Almost no spurious correlations with HEnKF • Applicable to real size reservoir models • But needs slightly larger ensembles (200) • Using 5 groups, 40 ensemble members • Brugge case: Better history match using all 104 original realizations

24. Acknowledgements