Skip to main content
SpringerLink
Log in

Deterministic ensemble smoother with multiple data assimilation as an alternative for history-matching seismic data

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

This paper reports the results of an investigation on the use of a deterministic analysis scheme combined with the method ensemble smoother with multiple data assimilation (ES-MDA) for the problem of assimilating a large number of correlated data points. This is the typical case when history-matching time-lapse seismic data in petroleum reservoir models. The motivation for the use of the deterministic analysis is twofold. First, it tends to result in a smaller underestimation of the ensemble variance after data assimilation. This is particularly important for problems with a large number of measurements. Second, the deterministic analysis avoids the factorization of a large covariance matrix required in the standard implementation of ES-MDA with the perturbed observations scheme. The deterministic analysis is tested in a synthetic history-matching problem to assimilate production and seismic data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abadpour, A., Bergey, P., Piasecki, R.: 4D seismic history matching with ensemble Kalman filter-assimilation on hausdorff distance to saturation front. In: Proceedings of the SPE Reservoir Simulation Symposium. The Woodlands, Texas, USA, 18–20 February, number SPE-163635-MS. https://doi.org/10.2118/163635-MS (2013)

  2. Anderson, J.L.: An ensemble adjustment Kalman filter for data assimilation. Mon. Weather Rev. 129(12), 2884–2903 (2001). https://doi.org/10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2

    Article  Google Scholar 

  3. Avansi, G.D., Schiozer, D.J.: UNISIM-I: synthetic model for reservoir development and management applications. Int. J Model. Simul. Pet. Ind. 9(1), 21–30 (2015). http://www.ijmspi.org/ojs/index.php/ijmspi/article/view/152

    Google Scholar 

  4. Bishop, C.H., Etherton, B.J., Majumdar, S.J.: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Weather Rev. 129, 420–436 (2001). https://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2

    Article  Google Scholar 

  5. Burgers, G., van Leeuwen, P., Evensen, G.: Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126(6), 1719–1724 (1998). https://doi.org/10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2

    Article  Google Scholar 

  6. Chen, Y., Oliver, D.S.: Ensemble-based closed-loop optimization applied to Brugge field. SPE Reserv. Eval. Eng. 13(1), 56–71 (2010). https://doi.org/10.2118/118926-PA

    Article  Google Scholar 

  7. Chen, Y., Oliver, D.S.: History matching of the Norne full-field model with an iterative ensemble smoother. SPE Reserv. Eval. Eng. 17(2). https://doi.org/10.2118/164902-PA (2014)

  8. Deutsch, C.V., Journel, A.G.: GSLIB Geostatistical Software Library And User’s Guide, 2nd edition. Oxford University Press, New York (1998)

    Google Scholar 

  9. Emerick, A.A.: Analysis of the performance of ensemble-based assimilation of production and seismic data. J. Pet. Sci. Eng. 139, 219–239 (2016). https://doi.org/10.1016/j.petrol.2016.01.029

    Article  Google Scholar 

  10. Emerick, A.A., Reynolds, A.C.: Combining sensitivities and prior information for covariance localization in the ensemble Kalman filter for petroleum reservoir applications. Comput. Geosci. 15(2), 251–269 (2011). https://doi.org/10.1007/s10596-010-9198-y

    Article  Google Scholar 

  11. Emerick, A.A., Reynolds, A.C.: History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations. Comput. Geosci. 16(3), 639–659 (2012). https://doi.org/10.1007/s10596-012-9275-5

    Article  Google Scholar 

  12. Emerick, A.A., Reynolds, A.C.: Investigation on the sampling performance of ensemble-based methods with a simple reservoir model. Comput. Geosci. 17(2), 325–350 (2013a). https://doi.org/10.1007/s10596-012-9333-z

    Article  Google Scholar 

  13. Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013b). https://doi.org/10.1016/j.cageo.2012.03.011

    Article  Google Scholar 

  14. Emerick, A.A., Reynolds, A.C.: History matching of production and seismic data for a real field case using the ensemble smoother with multiple data assimilation. In: Proceedings of the SPE Reservoir Simulation Symposium. The Woodlands, Texas, USA, 18–20 February, number SPE-163675-MS. https://doi.org/10.2118/163675-MS (2013)

  15. Evensen, G.: Sampling strategies and square root analysis schemes for the enKF. Ocean Dyn. 54(6), 539–560 (2004). https://doi.org/10.1007/s10236-004-0099-2

    Article  Google Scholar 

  16. Evensen, G.: The ensemble Kalman filter for combined state and parameter estimation. IEEE Control. Syst. Mag., 83–104. https://doi.org/10.1109/MCS.2009.932223 (2009)

  17. Gaspari, G., Cohn, S.E.: Construction of correlation functions in two and three dimensions. Q. J. Roy. Meteorol. Soc. 125(554), 723–757 (1999). https://doi.org/10.1002/qj.49712555417

    Article  Google Scholar 

  18. Golub, G.H., van Loan, C.F.: Matrix Computations, 2nd edn. The Johns Hopkins University Press, Baltimore (1989)

  19. Hoteit, I., Pham, D.-T., Gharamti, M.E., Luo, X.: Mitigating observation perturbation sampling errors in the stochastic enKF. Mon. Weather. Rev. 143, 2918–2936 (2015). https://doi.org/10.1175/MWR-D-14-00088.1

    Article  Google Scholar 

  20. Houtekamer, P.L., Mitchell, H.L.: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 129(1), 123–137 (2001). https://doi.org/10.1175/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2

    Article  Google Scholar 

  21. Kalman, R.E.: A new approach to linear filtering prediction problems. Trans. ASME J. Basic Eng. 82, 35–45 (1960). https://doi.org/10.1115/1.3662552

    Article  Google Scholar 

  22. Le, D.H., Emerick, A.A., Reynolds, A.C.: An adaptive ensemble smoother with multiple data assimilation for assisted history matching. SPE J. 21(6), 2195–2207 (2016). https://doi.org/10.2118/173214-PA

    Article  Google Scholar 

  23. Leeuwenburgh, O., Arts, R.: Distance parameterization for efficient seismic history matching with the ensemble Kalman filter. Comput. Geosci. 18(3–4), 535–548 (2014). https://doi.org/10.1007/s10596-014-9434-y

    Article  Google Scholar 

  24. Maschio, C., Avansi, G.D., Santos, A.A., Schiozer, D.J.: UNISIM-I-H case study for history matching. Dataset. www.unisim.cepetro.unicamp.br/benchmarks/br/unisim-i/unisim-i-h (2013)

  25. Oliver, D.S.: Moving averages for Gaussian simulation in two and three dimensions. Math. Geol. 27(8), 939–960 (1995). https://doi.org/10.1007/BF02091660

    Article  Google Scholar 

  26. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  27. Rafiee, J., Reynolds, A.C.: Theoretical and Efficient Practical Procedures for the Generation of Inflation Factors for ES-MDA. Inverse Problems. https://doi.org/10.1088/1361-6420/aa8cb2 (2017)

  28. Sakov, P., Bertino, L.: Relation between two common localisation methods for the enKF. Comput. Geosci. 15(2), 225–237 (2011). https://doi.org/10.1007/s10596-010-9202-6

    Article  Google Scholar 

  29. Sakov, P., Oke, P.R.: Implications of the form of the ensemble transformation in the ensemble square root filters. Mon. Weather. Rev. 136, 1042–1053 (2008). https://doi.org/10.1175/2007MWR2021.1

    Article  Google Scholar 

  30. Sakov, P., Oke, P.R.: A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters. Tellus A 60(2), 361–371 (2008). https://doi.org/10.1111/j.1600-0870.2007.00299.x

    Article  Google Scholar 

  31. Skjervheim, J.-A., Evensen, G., Aanonsen, S.I., Ruud, B.O., Johansen, T.-A.: Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter. SPE J. 12(3), 282–292 (2007). https://doi.org/10.2118/95789-PA

    Article  Google Scholar 

  32. Sun, A.Y., Morris, A., Mohanty, S.: Comparison of deterministic ensemble Kalman filters for assimilating hydrogeological data. Adv. Water Resour. 32, 280–292 (2009). https://doi.org/10.1016/j.advwatres.2008.11.006

    Article  Google Scholar 

  33. Tarantola, A.: Inverse problem theory methods for model parameter estimation. SIAM, Philadelphia (2005)

    Book  Google Scholar 

  34. Tippett, M.K., Anderson, J.L., Bishop, C.H., Hamill, T.M., Whitaker, J.S.: Ensemble square-root filters. Mon. Weather Rev. 131, 1485–1490 (2003). https://doi.org/10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2

    Article  Google Scholar 

  35. Trani, M., Arts, R., Leeuwenburgh, O.: Seismic history matching of fluid fronts using the ensemble Kalman filter. SPE J. 18(1). https://doi.org/10.2118/163043-PA (2012)

  36. van Leeuwen, P.J.: An ensemble smoother with error estimates. Mon. Weather Rev. 129, 709–728 (2001). https://doi.org/10.1175/1520-0493(2001)129<0709:AESWEE>2.0.CO;2

    Article  Google Scholar 

  37. van Leeuwen, P.J.: Nonlinear data assimilation for high-dimensional systems—with geophysical applications, volume 2 of nonlinear data assimilation. In: Frontiers in Applied Dynamical Systems: Reviews and Tutorials, pp 1–73. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18347-3_1

  38. Whitaker, J.S., Hamill, T.M.: Ensemble data assimilation without perturbed observations. Mon. Weather Rev. 130(7), 1913–1924 (2002). https://doi.org/10.1175/1520-0493(2002)130<1913:EDAWPO>2.0.CO;2

    Article  Google Scholar 

  39. Zhang, Y., Liu, N., Oliver, D.S.: Ensemble filter methods with perturbed observations applied to nonlinear problems. Comput. Geosci. 14(2), 249–261 (2010). https://doi.org/10.1007/s10596-009-9149-7

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank Petrobras for supporting this research and for the permission to publish this paper. The author also thanks to two anonymous reviewers for the suggestions to improve the manuscript.

Author information

Authors and Affiliations

  1. Petrobras Research and Development Center – CENPES, Av. Horácio de Macedo 950, Cidade Universitária, Rio de Janeiro, RJ, 21941-915, Brazil

    Alexandre A. Emerick

Authors
  1. Alexandre A. Emerick
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre A. Emerick.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Emerick, A.A. Deterministic ensemble smoother with multiple data assimilation as an alternative for history-matching seismic data. Comput Geosci 22, 1175–1186 (2018). https://doi.org/10.1007/s10596-018-9745-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-018-9745-5

Keywords

Search

Navigation