Reservoir Multiscale Data Assimilation Using the Ensemble Kalman Filter

Santha R. Akella

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DOI: 10.4236/am.2011.22019                

Abstract

In this paper we propose a way to integrate data at different spatial scales using the ensemble Kalman filter (EnKF), such that the finest scale data is sequentially estimated, subject to the available data at the coarse scale (s), as an additional constraint. Relationship between various scales has been modeled via upscaling techniques. The proposed coarse-scale EnKF algorithm is recursive and easily implementable. Our numerical results with the coarse-scale data provide improved fine-scale field estimates when compared to the results with regular EnKF (which did not incorporate the coarse-scale data). We also tested our algorithm with various precisions of the coarse-scale data to account for the inexact relationship between the fine and coarse scale data. As expected, the results show that higher precision in the coarse-scale data, yielded improved estimates.

Keywords

Kalman Filter, Reservoir Engineering, Uncertainty Quantification, Multiscale Data

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. M. Lewis, S. Lakshmivarahan and S. Dhall, “Dynamic Data Assimilation: A Least Squares Approach (Encyclopedia of Mathematics and its Applications),” Cambridge University Press, Cambridge, 2006. doi:10.1017/CBO 9780511526480
[2]	G. Evensen, “Data Assimilation: The Ensemble Kalman Filter,” Springer, Berlin, 2006.
[3]	Y. Liu and H. V. Gupta, “Uncertainty in Hydrologic Modeling: Toward and Integrated Data Assimilation Framework,” Water Resources Research, Vol. 43, 2007, W07401. doi:10.1029/2006WR005756
[4]	D. E. Lumley, “Time-Lapse Seismic Reservoir Monitoring,” Geophysics, Vol. 66, No. 1, 2001, pp. 50-53. doi:10.1190/1.1444921
[5]	G. N?vdal, L. Johnson, S. Aanonsen and E. Vefring, “Reservoir Monitoring and Continuous Model Updating Using Ensemble Kalman Filter,” SPE Journal, Vol. 10, No. 1, 2005, pp. 66-74.
[6]	X.-H Wen and W. H. Chen, “Real-Time Reservoir Model Updating Using Ensemble Kalman Filter,” Proceedings of SPE Reservoir Simulation Symposium 2005, SPE 92991, 31 January-2 February 2005, Texas.
[7]	Y. Q. Gu and D. S. Oliver, “The Ensemble Kalman Filter for Continuous Updating of Reservoir Simulation Models,” Journal of Energy Resources Technology-Transac- tions of the ASME, Vol. 128, 2006, pp.79-87. doi:10.1115/1.2134735
[8]	S. H. Lee, A. Malallah, A. Datta-Gupta and D. Higdon, “Multiscale Data Integration Using Markov Random Fields,” SPE Reservoir Evaluation Engineering, Vol. 5, 2002, pp. 68-78. doi:10.2118/76905-PA
[9]	Y. Efendiev, A. Datta-Gupta, V. Ginting, X. Ma and B. Mallick, “An Efficient Two-Stage Markov Chain Monte Carlo Method for Dynamic Data Integration,” Water Resources Research, Vol. 41, 2005, doi:10.1029/2004WR003764
[10]	M. Christie, “Upscaling for Reservoir Simulation,” Journal of Petroleum Technology, Vol. 48, No. 11, 1996, pp. 1004-1010. doi:10.2118/37324-MS
[11]	J. W. Barker and S. Thibeau, “A Critical Review of the Use of Pseudo-Relative Permeabilities for Upscaling,” SPE Reservoir Engineering, Vol. 12, 1997, pp.138-143. doi:10.2118/35491-PA
[12]	Y. Efendiev, A. Datta-Gupta, I. Osako and B. Mallick, “Multiscale Data Integration Using Coarse-Scale Models,” Advances in Water Resources, Vol. 28, No. 3, 2005, pp. 303-314. doi:10.1016/j.advwatres.2004.10.013
[13]	D. Devegowda, E. Arroyo-Negrete and A. Datta-Gupta, “Efficient and Robust Reservoir Model Updating Using Ensemble Kalman Filter with Sensitivity-Based Covariance Localization,” Proceedings of SPE Reservoir Simulation Symposium, Houston, February 2007. doi:10.2118/106144-MS
[14]	G. Evensen, “Sequential Data Assimilation with a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods to Forecast Error Statistics,” Journal of Geophysical Research, Vol. 99, 1994, pp. 143-162. doi:10.1029/94JC00572
[15]	G. Burgers, P. J. van Leeuwen and G. Evensen, “Analysis Scheme in the Ensemble Kalman Filter,” Monthly Weather Review, Vol. 126, No. 6, 1998, pp. 1719-1724. doi:10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2
[16]	Y. Chen and D. Zhang, “Data Assimilation for Transient Flow in Geologic Formations Via Ensemble Kalman Filter,” Advances in Water Resources, Vol. 29, 2006, pp. 1107-1122. doi:10.1016/j.advwatres.2005.09.007
[17]	L. J. Durlofsky, “Numerical Calculation of Equivalent Grid Block Permeability Tensors for Heterogeneous Porous Media,” Water Resources Research, Vol. 27, No. 5, 1991, pp. 699-708. doi:10.1029/91WR00107
[18]	L. J. Durlofsky, “Coarse Scale Models of Two Phase Flow in Heterogeneous Reservoirs: Volume Averaged Equations and Their Relationship to the Existing Upscaling Techniques,” Computational Geosciences, Vol. 2, 1998, pp. 73-92. doi:10.1023/A:1011593901771
[19]	X. H. Wu, Y. R. Efendiev and T. Y. Hou, “Analysis of Upscaling Absolute Permeability,” Discrete and Continuous Dynamical Systems, Series B, Vol. 2, 2002, pp. 185-204. doi:10.3934/dcdsb.2002.2.185
[20]	S. Yoon, A. H. Malallah, A. Datta-Gupta, D. W. Vasco and R. A. Behrens, “A Multiscale Approach to Production-Data Integration Using Streamline Models,” SPE Journal, Vol. 6, No. 2, 2001, pp. 182-192.
[21]	Y. Gu and D. S. Oliver, “History Matching of the PUNQ-S3 Reservoir Model Using the Ensemble Kalman Filter,” SPE Journal, Vol. 10, No. 2, 2005, pp. 217-224. doi:10.2118/89942-PA
[22]	C. V. Deutsch, “Journel A. GSLIB: Geostatistical Software Library and User’s Guide,” Oxford Univiversity Press, Oxford, 1992.
[23]	Y. Dong Y, Y. Gu Y and D. S. Oliver, “Sequential Assimilation of 4D Seismic Data for Reservoir Description Using the Ensemble Kalman Filter,” Journal of Petroleum Science and Engineering, Vol. 53, 2006, pp. 83-99. doi:10.1016/j.petrol.2006.03.028
[24]	J.-A. Skjervheim, G. Evensen, S. Aanonsen, B. Ruud and T. Johansen, “Incorporating 4D Seismic Data in Reservoir Simulation Models Using Ensemble Kalman Filter,” SPE Journal, Vol. 12, No. 3, 2007, pp. 282-292. doi:10.2118/95789-PA