An example of flow based covariance localisation in the EnKF

1. An example of flow based covariance localisation in the EnKF Dan Kuznetsov, Dick KachumaGRC, Total E&P UK

2. Talk • Simple 2D example • Streamline-based localisation • Real field example • Conclusions

3. Covariance in the EnKF • Update equation • is the key element • is estimated from the ensemble • Estimation from the ensemble of a practical size may lead to a spurious correlation between model parameters and predictions

4. 2D synthetic example • Single well • 51x51x1 • Permeability is a Gaussian field • Conditioned at the well location • Other parameters are constant • BHP is the observation data • Ensemble of 100 realisations • 5 EnKF assimilation steps

5. Initial ensemble

6. BHP

7. Consider data along the line

8. Correlation coefficient and pressure along the line Pressure Corr Coef Step 1, 100 members

9. Correlation coefficient for bigger ensemble Step 1, 1000 members

10. Correlation coefficient and pressure along the line Step 1, 100 members

11. Covariance localisation • Covariance matrix estimated from the ensemble is modified in order to make it decreasing at some distance from the observation point • Multiplied by 1 within some zone and by 0 outside • or multiplied by some coefficients derived, for example from streamline sensitivities • Distance based localisation • Houtekamer and Mitchell 1998, Hamill et al 2001, Skjervheim et al 2006, … • Truncation of the covariance for production data may not be physically based since the well influence zone may have non trivial shape • Streamline based localisation • Aroyyo-Negrete et al 2006, Devegowda et al 2007 • Covariance is truncated with respect to the flow path and/or streamline based sensitivities

12. SL based localisation for production data • Limit covariance to the zone that has been swept by the produced liquid • Reverse problem: if a producer had been an injector the swept zone would be limited by a front of the injected liquid • Propagation of the liquid front along streamlines can be described in terms of Time Of Flight • To find the localisation zone: • Trace the streamlines based on the velocity field • Calculate time of flight • Limit drainage and injection zones accordingly to the current model time

13. SL based localisation Trace streamline -> Select a zone -> Update within the zone

14. Pressure TOF • Asymptotic solution to the pressure diffusivity equation • Kulkarni, Datta-Gupta and Vasco, 2000 • is the velocity of the pressure front • Pressure Time Of Flight • Pressure TOF is related to the observed time as in 2D and in 3D • Track the pressure front and set covariance to 0 outside the influence zone

15. Localisation within the drainage zone. Step 1

16. Localisation within the drainage zone. Step 5

17. Correlation coefficient. Step 1 Pressure Corr Coef Pressure TOF

18. Correlation coefficient. Step 1 after truncation Pressure Corr Coef Pressure TOF

19. Correlation coefficient. Step 5

20. Correlation coefficient. Step 5 after truncation

21. No localisation SL TOF localisation Match with and without localisation Initial

22. Permeability along the line. Step 5 Initial 1000 realisation No localisation SL TOF localisation

23. Real field case example • 70x49x34, 69290 active cells • Modified parameters:porosity,permeability,aquifer strength • NTG is different for each realisation but not updated • Initial ensemble for the fault permeability was generated separately and included into Kx and Ky • 3 producers, 2 water injectors • 900 days of production • Data to match: WBHP, WOPR, WWCT

24. Algorithm for reservoir simulation • Forward step with a conventional finite-difference reservoir simulator • Streamline tracing using separate routine (Texas A&M) • Compute TOF and Pressure TOF • Practically, for a full field reservoir simulation a full drainage zone is used for localisation of the covariance with bottom hole pressure • Stack influence zones for each well and observation for the current time • EnKF update with modified covariance

25. Perm X, realisation 1, Lay 8

26. Drainage zones, lay 8

27. Drainage zones, lay 8

28. Drainage zones, lay 8

29. Drainage zones, lay 8

30. Drainage zones, lay 8

31. Drainage zones, lay 8

32. Drainage zones, lay 8

33. Drainage zones, lay 8

34. Drainage zones, lay 8

35. Drainage zones, lay 8

36. Covariance: perm X, lay 8 and OPR in P1

37. Covariance: perm X, lay 8 and OPR in P1

38. Covariance: perm X, lay 8 and OPR in P1

39. Covariance: perm X, lay 8 and OPR in P1

40. Covariance: perm X, lay 8 and OPR in P1

41. Covariance: perm X, lay 8 and OPR in P1

42. Covariance: perm X, lay 8 and OPR in P1

43. Covariance: perm X, lay 8 and OPR in P1

44. Covariance: perm X, lay 8 and OPR in P1

45. Covariance: perm X, lay 8 and OPR in P1

46. BHP Well 1 Initial No Localisation SL TOF Localisation

47. BHP Well 2 Initial No Localisation SL TOF Localisation

48. BHP Well 3 Initial No Localisation SL TOF Localisation

49. WOPR Well 1 Initial No Localisation SL TOF Localisation

50. WOPR Well 2 Initial No Localisation SL TOF Localisation