Abstract
Ensemble-based data assimilation methods have recently become popular for solving reservoir history matching problems, but because of the practical limitation on ensemble size, using localization is necessary to reduce the effect of sampling error and to increase the degrees of freedom for incorporating large amounts of data. Local analysis in the ensemble Kalman filter has been used extensively for very large models in numerical weather prediction. It scales well with the model size and the number of data and is easily parallelized. In the petroleum literature, however, iterative ensemble smoothers with localization of the Kalman gain matrix have become the state-of-the-art approach for ensemble-based history matching. By forming the Kalman gain matrix row-by-row, the analysis step can also be parallelized. Localization regularizes updates to model parameters and state variables using information on the distance between the these variables and the observations. The truncation of small singular values in truncated singular value decomposition (TSVD) at the analysis step provides another type of regularization by projecting updates to dominant directions spanned by the simulated data ensemble. Typically, the combined use of localization and TSVD is necessary for problems with large amounts of data. In this paper, we compare the performance of Kalman gain localization to two forms of local analysis for parameter estimation problems with nonlocal data. The effect of TSVD with different localization methods and with the use of iteration is also analyzed. With several examples, we show that good results can be obtained for all localization methods if the localization range is chosen appropriately, but the optimal localization range differs for the various methods. In general, for local analysis with observation taper, the optimal range is somewhat shorter than the optimal range for other localization methods. Although all methods gave equivalent results when used in an iterative ensemble smoother, the local analysis methods generally converged more quickly than Kalman gain localization when the amount of data is large compared to ensemble size.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Annan, J. D., Hargreaves, J. C., Edwards, N. R., Marsh, R.: Parameter estimation in an intermediate complexity earth system model using an ensemble Kalman filter. Ocean Model. 8(1–2), 135–154 (2005). doi:10.1016/j.ocemod.2003.12.004. ISSN 1463– 5003
Chen, Y., Oliver, D. S.: Ensemble-based closed-loop optimization applied to Brugge Field. SPE Reserv. Eval. Eng. 13(1), 56–71 (2010a). doi:10.2118/118926-PA
Chen, Y., Oliver, D. S.: Cross-covariances and localization for EnKF in multiphase flow data assimilation. Comput. Geosci. 14, 579–601 (2010b). doi:10.1007/s10596-009-9174-6. ISSN 1420-0597
Chen, Y., Oliver, D. S.: Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math. Geosci. 44(1), 1–26 (2012). doi:10.1007/s11004-011-9376-z. ISSN 1874-8961
Chen, Y., Oliver, D. S.: Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Comput. Geosci. 17(4), 689–703 (2013). doi:10.1007/s10596-013-9351-5. ISSN 1420–0597
Chen, Y., Oliver, D. S.: History matching of the Norne full-field model with an iterative ensemble smoother. SPE Reserv. Eval. Eng. 17(2), 244–256 (2014). doi:10.2118/164902-PA
Emerick, A., Reynolds, A.: Combining sensitivities and prior information for covariance localization in the ensemble Kalman filter for petroleum reservoir applications. Comput. Geosci. 15(2), 251–269 (2011). doi:10.1007/s10596-010-9198-y. ISSN 1420-0597
Emerick, A. A.: Analysis of the performance of ensemble-based assimilation of production and seismic data. J. Petrol. Sci. Eng. Online first (2016)
Emerick, A. A., Reynolds, A. C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013a). doi:10.1016/j.cageo.2012.03.011. ISSN 0098-3004
Emerick, A. A., Reynolds, A. C.: History-matching production and seismic data in a real field case using the ensemble smoother with multiple data assimilation, SPE-164902. In: Proc of SPE RSS. The Woodlands (2013b)
Evensen, G: Data Assimilation: The Ensemble Kalman Filter, 2n edn.. Springer Verlag (2009)
Fahimuddin, A., Aanonsen, S. I., Skjervheim, J.-A.: 4D seismic history matching of a real field case with EnKF: Use of local analysis for model updating. In: SPE Annual Technical Conference and Exhibition, 19-22 September 2010. Florence (2010)
Fertig, E. J., Hunt, B. R., Ott, E., Szunyogh, I.: Assimilating non-local observations with a local ensemble Kalman filter. Tellus A 59(5), 719–730 (2007). doi:10.1111/j.1600-0870.2007.00260.x
Furrer, R., Bengtsson, T.: Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivar. Anal. 98(2), 227–255 (2007). doi:10.1016/j.jmva.2006.08.003. ISSN 0047-259X
Gaspari, G., Cohn, S. E.: Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc. 125(554), 723–757 (1999)
Greybush, S. J., Kalnay, E., Miyoshi, T., Ide, K., Hunt, B. R.: Balance and ensemble Kalman filter localization techniques. Mon. Weather Rev. 139(2), 511–522 (2011). doi:10.1175/2010MWR3328.1. ISSN 0027-0644
Hamill, T. M., Whitaker, J. S., Snyder, C.: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Weather Rev. 129(11), 2776–2790 (2001)
Houtekamer, P. L., Mitchell, H. L.: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 129(1), 123–137 (2001)
Houtekamer, P. L., Mitchell, H. L.: Data assimilation using an ensemble Kalman filter technique. Mon. Weather Rev. 126(3), 796–811 (1998)
Hunt, B. R., Kostelich, E. J., Szunyogh, I.: Efficient data assimilation for spatiotemporal chaos: a local ensemble transform Kalman filter. Physica D: Nonlin. Phenom. 230(1–2), 112–126 (2007)
Li, R., Reynolds, A. C., Oliver, D. S.: History matching of three-phase flow production data. SPE J. 8 (4), 328–340 (2003). doi:10.2118/87336-PA
Miyoshi, T., Yamane, S.: Local ensemble transform Kalman filtering with an AGCM at a T159/L48 resolution. Mon. Weather Rev. 135(11), 3841–3861 (2007). doi:10.1175/2007MWR1873.1
Nerger, L., Janjić, T., Schrȯter, J., Hiller, W.: A regulated localization scheme for ensemble-based Kalman filters. Q. J. R. Meteorol. Soc. 138(664), 802–812 (2012). doi:10.1002/qj.945
Oliver, D. S.: Calculation of the inverse of the covariance. Math. Geol. 30(7), 911–933 (1998)
Oliver, D. S., Reynolds, A. C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching, 1st edn. Cambridge University Press, Cambridge (2008)
Ott, E., Hunt, B. R., Szunyogh, I., Zimin, A. V., Kostelich, E. J., Corazza, M., Kalnay, E., Patil, D. J., Yorke, J. A.: A local ensemble Kalman filter for atmospheric data assimilation. Tellus A 56(5), 415–428 (2004)
Peters, L., Arts, R. J., Brouwer, G. K., Geel, C. R., Cullick, S., Lorentzen, R. J., Chen, Y., Dunlop, K. N. B., Vossepoel, F. C., Xu, R., Sarma, P., Alhutali, A. H., Reynolds, A. C.: Results of the Brugge benchmark study for flooding optimization and history matching. SPE Reserv. Evalu. Eng. 13(3), 391–405 (2010). doi:10.2118/119094-PA
Sætrom, J., Hove, J., Skjervheim, J.-A., Vabø, J. G.: Improved uncertainty quantification in the ensemble Kalman filter using statistical model-selection techniques. SPE J. 17, 152–162 (2012). doi:10.2118/145192-PA
Sakov, P., Bertino, L.: Relation between two common localisation methods for the EnKF. Comput. Geosci. 15(2), 225–237 (2011). doi:10.1007/s10596-010-9202-6. ISSN 1420-0597
Skjervheim, J.-A., Evensen, G.: An ensemble smoother for assisted history matching. In: SPE Reservoir Simulation Symposium, 21–23 February. The Woodlands, 10.2118/141929-MS (2011)
Weaver, A. T., Mirouze, I.: On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation. Q. J. R. Meteorol. Soc. 139(670), 242–260 (2013). doi:10.1002/qj.1955. ISSN 1477-870X
Zhao, Y., Reynolds, A. C., Li, G.: Generating facies maps by assimilating production data and seismic data with the ensemble Kalman filter, SPE-113990. In: Proc of SPE IOR Symp. Tulsa, doi:10.2118/113990-MS(2008)
Acknowledgments
The authors would like to thank Total for the permission to publish this work. Primary support for the second author has been provided by the CIPR/IRIS cooperative research project “4D Seismic History Matching” which is funded by industry partners Eni, Petrobras, and Total, as well as the Research Council of Norway (PETROMAKS).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, Y., Oliver, D.S. Localization and regularization for iterative ensemble smoothers. Comput Geosci 21, 13–30 (2017). https://doi.org/10.1007/s10596-016-9599-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-016-9599-7