1887

Abstract

Summary

For data assimilation problems there are different ways in using available observations. While certain data assimilation algorithms, for instance, the ensemble Kalman filter (EnKF, see, for example, ) assimilate the observations sequentially in time, other data assimilation algorithms may instead collect the observations at different time instants and assimilate them simultaneously. In general such algorithms can be classified as smoothers. In this aspect, the ensemble smoother (ES, see, for example, ) can be considered as an smoother counterpart of the EnKF.

The EnKF has been widely used for reservoir data assimilation problems since its introduction to the community of petroleum engineering ( ). The applications of the ES to reservoir data assimilation problems are also investigated recently. Compared to the EnKF, the ES has certain technical advantages, including, for instance, avoiding the restarts associated with each update step in the EnKF and also having fewer variables to update, which may result in a significant reduction in simulation time, while providing similar assimilation results to those obtained by the EnKF ( ).

To further improve the performance of the ES, some iterative ensemble smoothers are suggested in the literature, in which the iterations are carried out in the forms of certain iterative optimization algorithms, e. g., the Gaussian-Newton ( ) or the Levenberg-Marquardt method ( ), or in the context of adaptive Gaussian mixture (AGM, see Stordal and Lorentzen, 2013).

In this contribution we show that the iteration formulae used in can also be derived from the regularized Levenberg-Marquardt (RLM) algorithm in inverse problems theory ( ), with certain linearization approximations introduced to the RLM. This does not only lead to an alternative theoretical tool in understanding and analyzing the behaviour of the aforementioned iterative ES, but also provide insights and guidelines for further developments of the iterative ES algorithm. As an example, we show that an alternative implementation of the iterative ES can be derived based on the RLM algorithm. For illustration, we apply this alternative algorithm to a facies estimation problem previously investigated in , and compare its performance to that of the (approximate) iterative ES used in .

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2014-09-08
2020-01-19
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References

  1. Aanonsen, S., Nævdal, G., Oliver, D., Reynolds, A. and Vallès, B.
    [2009] The ensemble Kalman filter in reservoir engineering: a review. SPE Journal, 14(3), 393–412.
    [Google Scholar]
  2. Chen, Y. and Oliver, D.
    [2013] Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Computational Geosciences, 17, 703–.
    [Google Scholar]
  3. Chen, Y. and Oliver, D.S.
    [2014] History matching of the Norne full-field model with an iterative ensemble smoother. SPE Reservoir Evaluation & Engineering, in press, SPE-164902-PA.
    [Google Scholar]
  4. [2010] Cross-covariances and localization for EnKF in multiphase flow data assimilation. Computational Geosciences, 14, 601–.
    [Google Scholar]
  5. [2012] Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Mathematical Geosciences, 44(1), 1–26.
    [Google Scholar]
  6. Emerick, A.A. and Reynolds, A.C.
    [2012] Ensemble smoother with multiple data assimilation. Computers & Geosciences, 55, 15–.
    [Google Scholar]
  7. Engl, H.W., Hanke, M. and Neubauer, A.
    [2000] Regularization of Inverse Problems. Springer.
    [Google Scholar]
  8. Evensen, G.
    [2006] Data Assimilation: The Ensemble Kalman Filter. Springer.
    [Google Scholar]
  9. Evensen, G. and van Leeuwen, P.J.
    [2000] An ensemble Kalman smoother for nonlinear dynamics. Mon. Wea. Rev., 128, 1867–.
    [Google Scholar]
  10. Gu, Y. and Oliver, D.
    [2007] An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE Journal, 12, 146–.
    [Google Scholar]
  11. Hanke, M.
    [1997] A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse problems, 13(1), 79–95.
    [Google Scholar]
  12. Iglesias, M. and Dawson, C.
    [2013] The regularizing Levenberg-Marquardt scheme for history matching of petroleum reservoirs. Computational Geosciences, 17(6), 1033–1053, ISSN 1420–0597, doi:10.1007/s10596‑013‑9373‑z.
    https://doi.org/10.1007/s10596-013-9373-z [Google Scholar]
  13. Jin, Q.
    [2010] On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems. Numerisohe Mathematik, 115(2), 229–259.
    [Google Scholar]
  14. Kaltenbacher, B., Neubauer, A. and Scherzer, O.
    [2008] Iterative regularization methods for nonlinear ill-posed problems. Walter de Gruyter.
    [Google Scholar]
  15. Li, G. and Reynolds, A.
    [2009] Iterative ensemble Kalman filters for data assimilation. SPE Journal, 14, 505–.
    [Google Scholar]
  16. Lorentzen, R. and Nævdal, G.
    [2011] An iterative ensemble Kalman filter. IEEE Transactions on Automatic Control, 56, 1995–.
    [Google Scholar]
  17. Lorentzen, R., Flornes, K. and Nævdal, G.
    [2012] History matching channelized reservoirs using the ensemble Kalman filter. SPE Journal, 17, 151–.
    [Google Scholar]
  18. Lorenz, E.N. and Emanuel, K.A.
    [1998] Optimal sites for supplementary weather observations: Simulation with a small model. J. Atmos. Sci., 55, 414–.
    [Google Scholar]
  19. Luo, X. and Hoteit, H.
    [2014a] Ensemble Kalman filtering with residual nudging: an extension to the state estimation problems with nonlinear observations. Mon. Wea. Rev., in press.
    [Google Scholar]
  20. Luo, X. and Hoteit, I.
    [2012] Ensemble Kalman filtering with residual nudging. Tellus A, 64, 17130, open access, doi:10.3402/tellusa.v64i0.17130.
    https://doi.org/10.3402/tellusa.v64i0.17130 [Google Scholar]
  21. [2013] Covariance inflation in the ensemble Kalman filter: a residual nudging perspective and some implications. Mon. Wea. Rev., 141, 3360–3368, doi:10.1175/MWR‑D‑13‑00067.1.
    https://doi.org/10.1175/MWR-D-13-00067.1 [Google Scholar]
  22. [2014b] Efficient particle filtering through residual nudging. Quart. J. Roy. Meteor. Soc., 140, 557–572, doi:10.1002/qj.2152.
    https://doi.org/10.1002/qj.2152 [Google Scholar]
  23. Marquardt, D.W.
    [1963] An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial & Applied Mathematics, 11(2), 431–441.
    [Google Scholar]
  24. Mitchell, I.M.
    [2008] The flexible, extensible and efficient toolbox of level set methods. Journal of Stientific Computing, 35, 329–.
    [Google Scholar]
  25. Nævdal, G., Mannseth, T. and Vefring, E.
    [2002] Near-well reservoir monitoring through ensemble Kalman filter. DOE Improved Oil Recovery Symposium, 13–17 April, Tulsa, Oklahoma, USA, SPE-75235-MS.
    [Google Scholar]
  26. Nævdal, G., Johnsen, L.M., Aanonsen, S.I., Vefring, E.H. et al.
    [2005] Reservoir monitoring and continuous model updating using ensemble Kalman filter. SPE journal, 10, 74–.
    [Google Scholar]
  27. Nocedal, J. and Wright, S.J.
    [2006] Numerical optimization. Springer, 2nd edn.
    [Google Scholar]
  28. Oliver, D.S., Reynolds, A.C. and Liu, N.
    [2008] Inverse Theory for Petroleum Reservoir Characterization and History Matching,. Cambridge University Press.
    [Google Scholar]
  29. Simon, D.
    [2006] Optimal State Estimation: Kalman, H-Infinity, and Nonlinear Approaches. Wiley-Interscience.
    [Google Scholar]
  30. Skjervheim, J.A. and Evensen, G.
    [2011] An ensemble smoother for assisted history matching. SPE Reservoir Simulation Symposium, 21–23 February, The Woodlands, Texas, USA, SPE-141929-MS.
    [Google Scholar]
  31. Stordal, A.S. and Lorentzen, R.J.
    [2014] An iterative version of the adaptive Gaussian mixture filter. Computational Geosciences, in press, doi:10.1007/s10596‑014‑9402‑6.
    https://doi.org/10.1007/s10596-014-9402-6 [Google Scholar]
  32. Strebelle, S.
    [2002] Conditional simulation of complex geological structures using multiple-point statistics. Mathematical Geology, 34, 21–.
    [Google Scholar]
  33. Wolpert, D.H. and Macready, W.G.
    [1997] No free lunch theorems for optimization. Evolutionary Computation, IEEE Transactions on, 1, 82–.
    [Google Scholar]
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