Use of Model Error With the EnKF

1. Use of Model Error With the EnKF S. I. Aanonsen Voss Workshop 18 – 20 June 2008 Centre for Integrated Petroleum Research University of Bergen, Norway

2. Motivation • An unkown model error is present in all real problems • Originally, the Kalman filter and the EnKF was developed to update state variables for models with error • Within the petroleum applications focus has mainly been on EnKF as a history-matching tool. Model error is normally not included(?) • However, real-time reservoir management also requires short term predictions based on a very accurate data match Centre for Integrated Petroleum Research University of Bergen, Norway

3. Questions • Is it possible to obtain good short term predictions and better estimates of unknown parameters by including a model error term? • Can a model error be estimated as a part of the EnKF updating process? Centre for Integrated Petroleum Research University of Bergen, Norway

4. Model Problem • Single-phase flow: Centre for Integrated Petroleum Research University of Bergen, Norway

5. True permeability field Fine grid (20x20) Centre for Integrated Petroleum Research University of Bergen, Norway

6. Estimated mean permeabilityFine Scale EnKF. No model error Log-normal Prior model: Mean log(K) = 2 (100mD) SD [log(K)] = 0.7 Spherical variogram with correlation length = L/4 Ensemble size: 40 Centre for Integrated Petroleum Research University of Bergen, Norway

7. EnKF update and ensemble predictions Well pressures vs time Centre for Integrated Petroleum Research University of Bergen, Norway Measurement error: 0.3

8. Rerun from time zero Well pressures vs time Centre for Integrated Petroleum Research University of Bergen, Norway

9. EnKF update and ensemble predictionsCoarse grid (10x10). No model error Same prior model as in fine-scale EnKF, except that the correlation length is L/2 Centre for Integrated Petroleum Research University of Bergen, Norway

10. Mean estimated permeability coarse modelNo model error Centre for Integrated Petroleum Research University of Bergen, Norway

11. Including model error • Alternative 1. Uncorrelated error (in time): where Cw has the same correlation structure as the prior permeability field and same variance as the measurement error. c = 1. No error added to the permeability. Centre for Integrated Petroleum Research University of Bergen, Norway

12. Including model error • Alternative 2. Correlated error (from Geir Evensen’s book). Model error is added to the state vector and updated “as usual”. Forecast step: where Centre for Integrated Petroleum Research University of Bergen, Norway

13. EnKF update and ensemble predictionsCoarse grid (10x10). Uncorrelated model error Centre for Integrated Petroleum Research University of Bergen, Norway

14. EnKF update and ensemble predictionsCoarse grid (10x10). Correlated model error, r= 1 Centre for Integrated Petroleum Research University of Bergen, Norway

15. Mean estimated permeability coarse modelWith model error Uncorrelated error Correlated error, r= 1 Centre for Integrated Petroleum Research University of Bergen, Norway

16. Rerun from time zeroCoarse model. No model error Centre for Integrated Petroleum Research University of Bergen, Norway

17. Rerun from time zeroCorrelated model error Centre for Integrated Petroleum Research University of Bergen, Norway

18. Rerun from time zeroUncorrelated model error Centre for Integrated Petroleum Research University of Bergen, Norway

19. Conclusions • A typical model error may result in EnKF not being able to match data, ensemble collapse and poor predictions and parameter estimates. • Adding an “unknown” model error may improve all of this and provide reasonably good predictions, at least short term. Centre for Integrated Petroleum Research University of Bergen, Norway