Correction of ensemble mean

1. Correction of ensemble mean Leonid Grigoriev Nick Kazakov

2. Outline • Ensemble mean in History matching • Deficiency of traditional mean calculation • Algorithm of mean correction • Example of mean correction (Brugge case) • Mean correction as additional step in EnKF • Summary

3. Ensemble mean in History matching Final ensemble mean is a result of history matching Final ensemble mean is an initial model for Close-loop exercises Optimization based on mean model Quality of history matching is defined by comparing real observation and synthetic from final ens. mean forecast .

4. 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

5. Brugge case • History matching of first 5 years • Measurements observed each 1 month • Initial ensemble size = 205 • 104 original members by TNO • 1 member from 104 is a TRUE model • Ensemble includes 2*103 – 1=205(to be divided by 5) • Initial ensemble for HEnKF has 5 groups with 41 member in each

6. Forecast of final ensemble from 0 to 1828 day (Brugge case) WOPR WWCT

7. Deficiency of mean calculation Ensemble mean is arithmetic average of all members Each member of ensemble including completely wrong members influences the mean equally

8. Algorithm of mean correction 1) Make forecast of ensemble from time zero by simulator 2) Calculate for each well for each ensemble member correlation between synthetic and real observation

9. Algorithm of mean correction 3)Calculate critical value for each well 4) According to eject bad members for each well which have less then critical value W1 W2 W3 W4 Wn ……. Ens. size

10. Algorithm of mean correction 5) Generate one ensemble without members which are bad for every well. 6) Corrected mean is average of new ensemble W1 W2 W3 W4 Wn ……. Ens. size Corrected ens.

11. Example of mean correction (Brugge case) final ens. corrected final ens. true data mean of corrected final ens. mean of final ens.

12. Example of mean correction (Brugge case) true data mean of corrected final ens. mean of final ens.

13. Time update or “Forecast step” (1) Measurement update or “Analyze step” (2) (1) (2) Part of ensemble mean in EnKF algorithm Time update: Measurement update: f : Simulator K : Kalman gain d : Measurements H : Measurement matrix y : state vector n : time index

14. Part of ensemble mean in EnKF algorithm Kalman gain matrix depends on ensemble mean through the covariance matrix

15. Measurement update or “Analyze step” (3) (corrected mean is used to calculate covariance matrix ) Time update or “Forecast step” (1) (simulator run ensemble from current to next time step) Mean correction as additional step in EnKF Mean correction (2) (simulator run ensemble from start point)

16. Mean correction as additional step in EnKF Time update: Mean correction: Measurement update: (1) (2) (3)

17. Summary • Sensitivity of result from the way of mean calculation is pointed • Mean can be defined more exactly using correlation analysis • Algorithm of avoiding wrong members and correcting the mean is presented • Good prospects of algorithm application in EnKF • Calculation time increases in 1.5-2 times