Abstract
The ensemble Kalman filter (EnKF) is a sequential data assimilation method that has been demonstrated to be effective for history matching reservoir production data and seismic data. To avoid, however, the expense of repeatedly updating variables and restarting simulation runs, an ensemble smoother (ES) has recently been proposed. Like the EnKF, the ES obtains all information necessary to compute a correction to model variables directly from an ensemble of models without the need of an adjoint code. The success of both methods for history matching reservoir data without iteration is somewhat surprising since traditional gradient-based methods for history matching typically require 10 to 30 iterations to converge to an acceptable minimum. In this manuscript we describe a new iterative ensemble smoother (batch-EnRML) that assimilates all data simultaneously and compare the performance of the iterative smoother with the two non-iterative methods and the previously proposed sequential iterative ensemble filter (seq-EnRML). We discuss some aspects of the use of the ensemble estimate of sensitivity, and show that by sequentially assimilating data, the nonlinearity of the assimilation problem is substantially reduced. Although reasonably good data matches can be obtained using a non-iterative ensemble smoother, iteration was necessary to achieve results comparable to the EnKF for nonlinear problems.
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References
Aanonsen SI, Nævdal G, Oliver DS, Reynolds AC, Vallès B (2009) Ensemble Kalman filter in reservoir engineering—a review. SPE J 14(3):393–412
Anderson JL (2007) Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D, Nonlinear Phenom 230(1–2):99–111
Annan JD, Hargreaves JC (2004) Efficient parameter estimation for a highly chaotic system. Tellus A 56(5):520–526
Burgers G, van Leeuwen PJ, Evensen G (1998) Analysis scheme in the ensemble Kalman filter. Mon Weather Rev 126(6):1719–1724
Caya A, Sun J, Snyder C (2005) A comparison between the 4DVAR and the ensemble Kalman filter techniques for radar data assimilation. Mon Weather Rev 133(11):3081–3094
Chen Y, Oliver DS (2010a) Ensemble-based closed-loop optimization applied to Brugge Field. SPE Reserv Eval Eng 13(1):56–71
Chen Y, Oliver DS (2010b) Parameterization techniques to improve mass conservation and data assimilation for ensemble Kalman filter (SPE 133560). In: SPE Western regional meeting, Anaheim, California, USA, 27–29 May 2010
Evensen G (1994a) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophys Res 99(C5):10143–10162
Evensen G (1994b) Advanced data assimilation for strongly nonlinear dynamics. Mon Weather Rev 125(6):1342–1354
Evensen G (2003) The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn 53:43–367
Evensen G (2009) Data assimilation: the ensemble Kalman filter, 2nd edn. Springer, Berlin
Gao G, Reynolds AC (2006) An improved implementation of the LBFGS algorithm for automatic history matching. SPE J 11(1):5–17
Gu Y, Oliver DS (2007) An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE J 12(4):438–446
Houtekamer PL, Mitchell HL (1998) Data assimilation using an ensemble Kalman filter technique. Mon Weather Rev 126(3):796–811
Kalnay E, Li H, Miyoshi T, Yang SC, Ballabrera-Poy J (2007) 4-D-Var or ensemble Kalman filter? Tellus A 59(5):758–773
Li G, Reynolds AC (2009) Iterative ensemble Kalman filters for data assimilation. SPE J 14(3):496–505
Li R, Reynolds AC, Oliver DS (2003) History matching of three-phase flow production data. SPE J 8(4):328–340
Liu N, Oliver DS (2005) Critical evaluation of the ensemble Kalman filter on history matching of geologic facies. SPE Reserv Eval Eng 8(6):470–477
Lorentzen RJ, Nævdal G (2011) An iterative ensemble Kalman filter. IEEE Trans Autom Control 56(8):1990–1995
Oliver DS, Chen Y (2011) Recent progress on reservoir history matching: a review. Comput Geosci 15(1):185–221
Oliver DS, Zhang Y, Phale HA, Chen Y (2011) Distributed parameter and state estimation in petroleum reservoirs. Comput Fluids 46(1):70–77
Peters L, Arts R, Brouwer G, Geel C, Cullick S, Lorentzen R, Chen Y, Dunlop K, Vossepoel F, Xu R, Sarma P, Alhutali A, Reynolds A (2010) Results of the Brugge benchmark study for flooding optimization and history matching. SPE Reserv Eval Eng 13(3):391–405
Sakov P, Evensen G, Bertino L (2010) Asynchronous data assimilation with the ensemble Kalman filter. Tellus A 62(1):24–29
Sakov P, Oliver DS, Bertino L (2011) An iterative EnKF for strongly nonlinear systems. Mon Weather Rev (submitted)
Skjervheim JA, Evensen G (2011) An ensemble smoother for assisted history matching (SPE–141929). In: SPE reservoir simulation symposium, The Woodlands, Texas, 21–23 February
van Leeuwen PJ, Evensen G (1996) Data assimilation and inverse methods in terms of a probabilistic formulation. Mon Weather Rev 124(12):2898–2913
Wang Y, Li G, Reynolds AC (2010) Estimation of depths of fluid contacts and relative permeability curves by history matching using iterative ensemble Kalman smoothers (SPE 119056). SPE J 15(2):509–525
Wen XH, Chen WH (2007) Some practical issues on real-time reservoir model updating using ensemble Kalman filter. SPE J 12(2):156–166
Zhang Y, Oliver DS (2010) Improving the ensemble estimate of the Kalman gain by bootstrap sampling. Math Geosci 42(3):327–345
Zhao Y, Reynolds AC, Li G (2008) Generating facies maps by assimilating production data and seismic data with the ensemble Kalman filter, SPE-113990. In: Proceedings of the 2008 SPE improved oil recovery symposium, Tulsa, Oklahoma, 21–23 April
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Chen, Y., Oliver, D.S. Ensemble Randomized Maximum Likelihood Method as an Iterative Ensemble Smoother. Math Geosci 44, 1–26 (2012). https://doi.org/10.1007/s11004-011-9376-z
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DOI: https://doi.org/10.1007/s11004-011-9376-z