Computational Geosciences

, Volume 22, Issue 5, pp 1251–1270 | Cite as

Conditioning reservoir models on rate data using ensemble smoothers

  • Geir EvensenEmail author
  • Kjersti Solberg Eikrem
Open Access
Original Paper


There are several issues to consider when we use ensemble smoothers to condition reservoir models on rate data. The values in a time series of rate data contain redundant information that may lead to poorly conditioned inversions and thereby influence the stability of the numerical computation of the update. A time series of rate data typically has correlated measurement errors in time, and negligence of the correlations leads to a too strong impact from conditioning on the rate data and possible ensemble collapse. The total number of rate data included in the smoother update will typically exceed the ensemble size, and special care needs to be taken to ensure numerically stable results. We force the reservoir model with production rate data derived from the observed production, and the further conditioning on the same rate data implies that we use the data twice. This paper discusses strategies for conditioning reservoir models on rate data using ensemble smoothers. In particular, a significant redundancy in the rate data makes it possible to subsample the rate data. The alternative to subsampling is to model the unknown measurement error correlations and specify the full measurement error covariance matrix. We demonstrate the proposed strategies using different ensemble smoothers with the Norne full-field reservoir model.


History matching Ensemble smoothers Iterative ensemble smoothers Production data 



Geir Evensen was supported by a research project funded by Statoil and for the finalization of the manuscript by the Research Council of Norway through the DIGIRES project with industry partners Statoil Petroleum AS, Eni Norge AS, VNG Norge AS, DEA Norge AS, Aker BP AS, and Petrobras.

Kjersti Solberg Eikrem acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, Aker BP ASA, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS of The National IOR Centre of Norway for support.

The study received access to Eclipse licenses from Schlumberger and benefited from the interaction and collaborations with the Nordforsk Nordic center of excellence in data assimilation, EMBLA. We are grateful to Joakim Hove and Statoil for making their Ensemble Reservoir Tool (ERT) available at Github. Finally, constructive comments from two anonymous reviewers have contributed to a clearer and more in-depth presentation and discussion.


  1. 1.
    Aanonsen, S.I., Naevdal, G., Oliver, D.S., Reynolds, A., Valles, B.: Ensemble Kalman filter in reservoir engineering – a review. SPE J 14(3), 393–412 (2009). CrossRefGoogle Scholar
  2. 2.
    Alturki, A., Baddourah, M., Pamuku, Y., Ravanelli, F., Hayder E.: An evaluation of assisted history matching methodologies for giant simulation models (2015).
  3. 3.
    Bennett, A.F.: Inverse methods in physical oceanography. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  4. 4.
    Chen, Y., Oliver, D.S.: Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math. Geosci. 44, 1–26 (2012)CrossRefGoogle Scholar
  5. 5.
    Chen, Y., Oliver, D.S.: Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Computat. Geosci. 17, 689–703 (2013)CrossRefGoogle Scholar
  6. 6.
    Chen, Y., Oliver, D.S.: History matching of the Norne full-field model using an iterative ensemble smoother. SPE Reserv. Eval. Eng. 17(2), 244–256 (2014)CrossRefGoogle Scholar
  7. 7.
    Eikrem, K.S., Nævdal, G., Jakobsen, M., Chen, Y.: Bayesian estimation of reservoir properties—effects of uncertainty quantification of 4d seismic data. Comput. Geosci. 20(6), 1211–1229 (2016)CrossRefGoogle Scholar
  8. 8.
    Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013)CrossRefGoogle Scholar
  9. 9.
    Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99(C5), 10,143–10,162 (1994)CrossRefGoogle Scholar
  10. 10.
    Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)CrossRefGoogle Scholar
  11. 11.
    Evensen, G.: Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn. 54, 539–560 (2004)CrossRefGoogle Scholar
  12. 12.
    Evensen, G.: Data assimilation: The ensemble Kalman filter, 2nd edn. Springer, Berlin (2009)CrossRefGoogle Scholar
  13. 13.
    Evensen, G.: The ensemble Kalman filter for combined state and parameter estimation. IEEE Control. Syst. Mag. 29(3), 83–104 (2009)CrossRefGoogle Scholar
  14. 14.
    Evensen, G.: Analysis of iterative ensemble smoothers for solving inverse problems. Computat Geosci (2018).
  15. 15.
    Evensen, G., van Leeuwen, P.J.: An ensemble Kalman smoother for nonlinear dynamics. Mon. Weather Rev. 128, 1852–1867 (2000)CrossRefGoogle Scholar
  16. 16.
    Hanea, R., Evensen, G., Hustoft, L., Ek, T., Chitu, A., Wilschut, F.: Reservoir management under geological uncertainty using Fast Model Update. 173305-MS SPE Conference Paper (2015)Google Scholar
  17. 17.
    Houtekamer, P.L., Zhang, F.: Review of the ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 144, 4489–4533 (2016)CrossRefGoogle Scholar
  18. 18.
    Iglesias, M.A.: Iterative regularization for ensemble data assimilation in reservoir models. Computat. Geosci. 19(1), 177–212 (2015)CrossRefGoogle Scholar
  19. 19.
    Iglesias, M.A.: A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems. Inverse Prob. 32(2), 025,002 (2016)CrossRefGoogle Scholar
  20. 20.
    Le, D.H., Emerick, A.A., Reynolds, A.C.: Iterative ensemble smoother as an approximate solution to a regularized minimum-average-cost problem: theory and applications. SPE J., SPE-173214-PA 21(6), 2195–2207 (2016)Google Scholar
  21. 21.
    van Leeuwen, P.J., Evensen, G.: Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Weather Rev. 124, 2898–2913 (1996)CrossRefGoogle Scholar
  22. 22.
    Luo, X., Stordal, A.S., Lorentzen, R.J.: Iterative ensemble smoother as an approximate solution to a regularized minimum-average-cost problem: Theory and applications. SPE J., SPE-176023-PA 20(5), 962–982 (2015)Google Scholar
  23. 23.
    Ma, X., Hetz, G., Wang, X., Bi, L., Stern, D., Hoda, N.: A robust iterative ensemble smoother method for efficient history matching and uncertainty quantification (2017)Google Scholar
  24. 24.
    Miyoshi, T., Kalnay, E., Li, H.: Estimating and including observation-error correlations in data assimilation. Inverse Prob. Sci. Eng. 21, 387–398 (2013)CrossRefGoogle Scholar
  25. 25.
    Nævdal, G., Johnsen, L.M., Aanonsen, S.I., Vefring, E.: Reservoir monitoring and continuous model updating using the ensemble Kalman filter. SPE Annual Technical Conference and Exhibition (SPE 84372) (2003)Google Scholar
  26. 26.
    Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Computat. Geosci. 15(1), 185–221 (2011). CrossRefGoogle Scholar
  27. 27.
    Seiler, A., Aanonsen, S.I., Evensen, G., Rivenæs, J.: Structural surface uncertainty modelling and updating using the ensemble Kalman filter. SPE J. (SPE-125352-PA) 15(4), 1062–1076 (2010). Google Scholar
  28. 28.
    Skjervheim, J.A., Evensen, G., Hove, J., Vabø, J.: An ensemble smoother for assisted history matching. SPE 141929 (2011)Google Scholar
  29. 29.
    Skjervheim, J.A., Hanea, R., Evensen, G.: Fast model update coupled to an ensemble based closed loop reservoir management. Petroleum Geostatistics (2015)Google Scholar
  30. 30.
    Stewart, L.M., Dance, S.L., Nichols, N.K.: Correlated observation errors in data assimilation. Int. J. Numer. Meth. Fluids 56, 1521–1527 (2008)CrossRefGoogle Scholar
  31. 31.
    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)CrossRefGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.International Research Institute of StavangerBergenNorway
  2. 2.Nansen Environmental and Remote Sensing CenterBergenNorway

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