×

Conditioning reservoir models on rate data using ensemble smoothers. (English) Zbl 1406.86031

Summary: 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.

MSC:

86A32 Geostatistics
86A22 Inverse problems in geophysics
76S05 Flows in porous media; filtration; seepage
86A05 Hydrology, hydrography, oceanography

Software:

EnKF
PDFBibTeX XMLCite
Full Text: DOI

References:

[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). https://doi.org/10.21188/117274-PA · doi:10.21188/117274-PA
[2] Alturki, A., Baddourah, M., Pamuku, Y., Ravanelli, F., Hayder E.: An evaluation of assisted history matching methodologies for giant simulation models (2015). https://doi.org/10.2118/172796-MS
[3] Bennett, A.F.: Inverse methods in physical oceanography. Cambridge University Press, Cambridge (1992) · Zbl 0782.76002 · doi:10.1017/CBO9780511600807
[4] Chen, Y., Oliver, D.S.: Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math. Geosci. 44, 1-26 (2012) · doi:10.1007/s11004-011-9376-z
[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) · Zbl 1382.65031 · doi:10.1007/s10596-013-9351-5
[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) · doi:10.2118/164902-PA
[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) · Zbl 1386.86017 · doi:10.1007/s10596-016-9585-0
[8] Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3-15 (2013) · doi:10.1016/j.cageo.2012.03.011
[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) · doi:10.1029/94JC00572
[10] Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343-367 (2003) · doi:10.1007/s10236-003-0036-9
[11] Evensen, G.: Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn. 54, 539-560 (2004) · doi:10.1007/s10236-004-0099-2
[12] Evensen, G.: Data assimilation: The ensemble Kalman filter, 2nd edn. Springer, Berlin (2009) · Zbl 1395.93534 · doi:10.1007/978-3-642-03711-5
[13] Evensen, G.: The ensemble Kalman filter for combined state and parameter estimation. IEEE Control. Syst. Mag. 29(3), 83-104 (2009) · Zbl 1395.93534 · doi:10.1109/MCS.2009.932223
[14] Evensen, G.: Analysis of iterative ensemble smoothers for solving inverse problems. Computat Geosci (2018). https://doi.org/10.1007/s10596-018-9731-y · Zbl 1405.86018
[15] Evensen, G., van Leeuwen, P.J.: An ensemble Kalman smoother for nonlinear dynamics. Mon. Weather Rev. 128, 1852-1867 (2000) · doi:10.1175/1520-0493(2000)128<1852:AEKSFN>2.0.CO;2
[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)
[17] Houtekamer, P.L., Zhang, F.: Review of the ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 144, 4489-4533 (2016) · doi:10.1175/MWR-D-15-0440.1
[18] Iglesias, M.A.: Iterative regularization for ensemble data assimilation in reservoir models. Computat. Geosci. 19(1), 177-212 (2015) · Zbl 1330.86015 · doi:10.1007/s10596-014-9456-5
[19] Iglesias, M.A.: A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems. Inverse Prob. 32(2), 025,002 (2016) · Zbl 1334.65110 · doi:10.1088/0266-5611/32/2/025002
[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)
[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) · doi:10.1175/1520-0493(1996)124<2898:DAAIMI>2.0.CO;2
[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)
[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)
[24] Miyoshi, T., Kalnay, E., Li, H.: Estimating and including observation-error correlations in data assimilation. Inverse Prob. Sci. Eng. 21, 387-398 (2013) · Zbl 1278.93250 · doi:10.1080/17415977.2012.712527
[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)
[26] Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Computat. Geosci. 15(1), 185-221 (2011). https://doi.org/10.1007/s10596-010-9194-2 · Zbl 1209.86001 · doi:10.1007/s10596-010-9194-2
[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). https://doi.org/10.21188/125352-MS · doi:10.21188/125352-MS
[28] Skjervheim, J.A., Evensen, G., Hove, J., Vabø, J.: An ensemble smoother for assisted history matching. SPE 141929 (2011)
[29] Skjervheim, J.A., Hanea, R., Evensen, G.: Fast model update coupled to an ensemble based closed loop reservoir management. Petroleum Geostatistics (2015)
[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) · Zbl 1133.86001 · doi:10.1002/fld.1636
[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) · doi:10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.