Computational Geosciences

, Volume 19, Issue 4, pp 821–844 | Cite as

Assessment of ordered sequential data assimilation

  • Kristian FossumEmail author
  • Trond Mannseth


Ensemble based data assimilation methods, such as the (sequential) ensemble Kalman Filter (EnKF) and the (non-sequential) ensemble smoother (ES), are widely used for history matching petroleum reservoir models. In a recent study (Fossum and Mannseth, Inverse Probl. 30(11):114002-3, [2014]), investigating the difference between sequential and non-sequential assimilation, it was shown that, for a series of weakly non-linear data, the sequential assimilation strategy outperformed the non-sequential approach, especially if the data were ordered according to ascending degree of non-linearity. In this paper, we assess, numerically, various assimilation strategies. Here, we consider numerous data types representing a large variation in the degree of non-linearity, and we consider both simple and complex forward models. The numerical study is divided into two parts. Firstly, the assimilation methods are assessed for problems that allow a controllable variation in the degree of data non-linearity. This investigation is conducted by toy models to ensure that a sufficiently large range of non-linear data is tested. Secondly, considering a 2D synthetic reservoir case, the assimilation methods are assessed for different production strategies and reference models. The numerical experiments show that for most models, considering data with a suitable degree of non-linearity, assimilating the data ordered after ascending degree of non-linearity produce the lowest approximation error. Two counter examples illustrate that the optimal assimilation strategy cannot be determined for all cases, especially if the degree of non-linearity depends greatly on the position in parameter space.


Sequential estimation Data assimilation Ensemble methods Numerical assessment 

Mathematics Subjects Classification (2010)

62L12 65C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aziz, K., Settari, A.: Petroleum reservoir simulation. Elsevier Appl. Sci., New York (1979)Google Scholar
  2. 2.
    Barker, J., Cuypers, M., Holden, L.: Quantifying uncertainty in production forecasts: another look at the PUNQ-S3 problem. SPE J. 6(4), 433–441 (2001). doi: 10.2118/74707-PA CrossRefGoogle Scholar
  3. 3.
    Bates, D.M., Watts, D.G.: Relative curvature measures of nonlinearity. J. Roy. Stat. Soc. B Met. 42(1), 1–25 (1980)Google Scholar
  4. 4.
    Berlinet, A., Thomas-Agnan, C.: Springer, Boston (2004)Google Scholar
  5. 5.
    Brooks, S., Gelman, A., Jones, G.L., Meng, X.L. (eds.): Handbook of Markov Chain Monte Carlo. Chapman and Hall / CRC (2011)Google Scholar
  6. 6.
    Burgers, G., van Leeuwen, P.J., Evensen, G.: Analysis scheme in the ensemble Kalman Filter. Mon. Weather Rev. 126(6), 1719–24 (1998)CrossRefGoogle Scholar
  7. 7.
    Chen, Z., Huan, G., Ma, Y.: Computational methods for multiphase flows in porous media. computational science and engineering. society for industrial and applied mathematics, Philadelphia (2006)Google Scholar
  8. 8.
    Chilès, J.P., Delfiner, P.: Geostatistics, modeling spatial uncertainty: Wiley Series in Probability and Statistics, 2nd edn. Wiley, New Jersey (2012)Google Scholar
  9. 9.
    Emerick, A.A., Reynolds, A.C.: Combining the ensemble Kalman Filter with Markov-Chain monte carlo for improved history matching and uncertainty characterization. SPE J. 17(2), 418–40 (2012). doi: 10.2118/141336-PA CrossRefGoogle Scholar
  10. 10.
    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), 639–59 (2012). doi: 10.1007/s10596-012-9275-5 CrossRefGoogle Scholar
  11. 11.
    Emerick, A.A., Reynolds, A.C.: Investigation of the sampling performance of ensemble-based methods with a simple reservoir model. Comput. Geosci. 17(2), 325–50 (2013). doi: 10.1007/s10596-012-9333-z CrossRefGoogle Scholar
  12. 12.
    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 (1994). doi: 10.1029/94JC00572 CrossRefGoogle Scholar
  13. 13.
    Evensen, G.: Advanced data assimilation for strongly nonlinear dynamics. Mon. Weather Rev. 125(6), 1342–54 (1997). doi: 10.1175/1520-0493(1997)125<1342:ADAFSN>2.0.CO;2 CrossRefGoogle Scholar
  14. 14.
    Evensen, G.: Data assimilation. Springer Berlin Heidelberg, Berlin, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Evensen, G., van Leeuwen, P.J.: An ensemble Kalman Smoother for nonlinear dynamics. Mon. Weather Rev. 128(6), 1852–67 (2000)CrossRefGoogle Scholar
  16. 16.
    Floris, F., Bush, M., Cuypers, M., Roggero, F., Syversveen, A.R.: Methods for quantifying the uncertainty of production forecasts: a comparative study. Pet. Geosci. 7, 87–96 (2001). doi: 10.1144/?petgeo.7.S.S87 CrossRefGoogle Scholar
  17. 17.
    Fossum, K., Mannseth, T.: Parameter sampling capabilities of sequential and simultaneous data assimilation: I. Analytical comparison. Inverse Probl. 30(11), 114,002 (2014). doi: 10.1088/0266-5611/30/11/114002 CrossRefGoogle Scholar
  18. 18.
    Fossum, K., Mannseth, T.: Parameter sampling capabilities of sequential and simultaneous data assimilation: II. Statistical analysis of numerical results. Inverse Probl. 30(11), 114,003 (2014). doi: 10.1088/0266-5611/30/11/114003 CrossRefGoogle Scholar
  19. 19.
    Fossum, K., Mannseth, T., Oliver, D.S., Skaug, H.J.: Numerical comparison of ensemble Kalman filter and randomized maximum likelihood. In: 13th Eur. Conf. Math. Oil Recover. (ECMORXIII). Biarritz, France (2012)Google Scholar
  20. 20.
    Gao, G., Zafari, M., Reynolds, A.C.: Quantifying uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF. SPE J. 11(4), 506–15 (2006). doi: 10.2118/93324-PA CrossRefGoogle Scholar
  21. 21.
    Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian data analysis, 2edn. Chapman and Hall/CRC (2003)Google Scholar
  22. 22.
    Gelman, A., Rubin, D.B.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7(4), 457–72 (1992)CrossRefGoogle Scholar
  23. 23.
    Gretton, A., Borgwardt, K.M., Rasch, M.J., Schölkopf, B., Smola, A.J.: A kernel two-sample test. J. Mach. Learn. Res. 13, 723–73 (2012)Google Scholar
  24. 24.
    Gretton, A., Borgwardt, K.M., Schölkopf, B., Smola, A.J. In: Schölkopf, B., Platt, J., Hoffman, T. (eds.) : A kernel method for the two sample problem, pp. 513–20. MIT Press (2007). Adv. Neural Inf. Process. Syst. 19, 157Google Scholar
  25. 25.
    Grimstad, A.A., Kolltveit, K., Mannseth, T., Nordtvedt, J.E.: Assessing the validity of a linearized accuracy measure for a nonlinear parameter estimation problem. Inverse Probl. 17(5), 1373–90 (2001). doi: 10.1088/0266-5611/17/5/309 CrossRefGoogle Scholar
  26. 26.
    Gu, Y., Oliver, D.S.: An iterative ensemble Kalman Filter for multiphase fluid flow data assimilation. SPE J. 12(4), 438–46 (2007). doi: 10.2118/108438-PA CrossRefGoogle Scholar
  27. 27.
    Hvidevold, H.K., Alendal, G., Johannessen, T., Mannseth, T.: Assessing model parameter uncertainties for rising velocity of CO 2 droplets through experimental design. Int. J. Greenh. Gas Control 11, 283–89 (2012). doi: 10.1016/j.ijggc.2012.09.008 CrossRefGoogle Scholar
  28. 28.
    Iglesias, M.A., Law, K.J.H., Stuart, A.M.: Ensemble Kalman methods for inverse problems. Inverse Probl. 29(4), 045,001 (2013). doi: 10.1088/0266-5611/29/4/045001 CrossRefGoogle Scholar
  29. 29.
    Iglesias, M.A., Law, K.J.H., Stuart, A.M.: Evaluation of Gaussian approximations for data assimilation in reservoir models. Comput. Geosci. 17(5), 851–85 (2013). doi: 10.1007/s10596-013-9359-x CrossRefGoogle Scholar
  30. 30.
    Jazwinski, A.H.: Stochastic processes and filtering theory. Academic Press, New York (1970)Google Scholar
  31. 31.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. J. Basic Eng 82(1), 35–45 (1960)CrossRefGoogle Scholar
  32. 32.
    Kullback, S., Leibler, R.: On information and sufficiency. Ann. Math. Stat 22(1), 79–86 (1951)CrossRefGoogle Scholar
  33. 33.
    Law, K.J.H., Stuart, A.M.: Evaluating data assimilation algorithms. Mon. Weather Rev. 140(11), 3757–82 (2012). doi: 10.1175/MWR-D-11-00257.1 CrossRefGoogle Scholar
  34. 34.
    van Leeuwen, P.J., Evensen, G.: Data assimilation and inverse methods in terms of a probabilistic formulation. Mon. Weather Rev. 124(12), 2898–913 (1996). doi: 10.1175/1520-0493(1996)124<2898:DAAIMI>2.0.CO;2 CrossRefGoogle Scholar
  35. 35.
    Li, X.R.: Measure of nonlinearity for stochastic systems. In: 15th Int. Conf. Inf. Fusion, c, pp. 1073–80 (2012)Google Scholar
  36. 36.
    Lie, K., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16(2), 297–322 (2011). doi: 10.1007/s10596-011-9244-4 CrossRefGoogle Scholar
  37. 37.
    Lorentzen, R.J., Nævdal, G., Vallès, B., Berg, A., Grimstad, A.A.: Analysis of the ensemble Kalman Filter for estimation of permeability and porosity in reservoir models. In: Proc. SPE Annu. Tech. Conf. Exhib. Society of Petroleum Engineers (2005), doi: 10.2118/96375-MS
  38. 38.
    Mandel, J., Cobb, L., Beezley, J.D.: On the convergence of the ensemble Kalman Filter. Appl. Math. 56(6), 533–41 (2011). doi: 10.1007/s10492-011-0031-2 CrossRefGoogle Scholar
  39. 39.
    Mannseth, T.: Permeability identification from pressure observations: some foundations for multiscale regularization. Multiscale Model. Simul. 5(1), 21–44 (2006). doi: 10.1137/050630167 CrossRefGoogle Scholar
  40. 40.
    Nævdal, G., Thulin, K., Skaug, H.J., Aanonsen, S.I.: Quantifying monte Carlo uncertainty in the ensemble Kalman Filter. SPE J. 16(1), 172–82 (2011). doi: 10.2118/123611-PA CrossRefGoogle Scholar
  41. 41.
    Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2010). doi: 10.1007/s10596-010-9194-2 CrossRefGoogle Scholar
  42. 42.
    Oliver, D.S., Cunha, L.B., Reynolds, A.C.: Markov chain Monte Carlo methods for conditioning a permeability field to pressure data. Math. Geol. 29(1), 61–91 (1997). doi: 10.1007/BF02769620 CrossRefGoogle Scholar
  43. 43.
    Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–67 (2009). doi: 10.1198/jcgs.2009.06134 CrossRefGoogle Scholar
  44. 44.
    Simon, D.: Optimal state estimation:Kalman,H infinity, and nonlinear approaches. Wiley-Interscience (2006)Google Scholar
  45. 45.
    Skjervheim, J.A., Evensen, G.: An Ensemble Smoother for assisted History Matching. In: Proc. SPE Reserv. Simul. Symp., 2003, pp. 1–15. Society of Petroleum Engineers (2011), doi: 10.2118/141929-MS
  46. 46.
    Smola, A., Gretton, A., Song, L., Schölkopf, B.: A Hilbert space embedding for distributions. In: Hutter, M., Serverdio, R.A., Takimoto, E. (eds.) : 18th Int. Conf. Algorithmic Learn. Theory, pp. 13–31. Springer, Sendai (2007)CrossRefGoogle Scholar
  47. 47.
    Stordal, A.S., Karlsen, H.A., Nævdal, G., Skaug, H.J., Vallès, B.: Bridging the ensemble Kalman Filter and particle filters: the adaptive gaussian mixture filter. Comput. Geosci. 15(2), 293–305 (2010). doi: 10.1007/s10596-010-9207-1 CrossRefGoogle Scholar
  48. 48.
    Thulin, K., Li, G., Aanonsen, S.I., Reynolds, A.C.: Estimation of initial fluid contacts by assimilation of production data with EnKF. In: Proc. SPE Annu. Tech. Conf. Exhib., i. Society of Petroleum Engineers (2007). doi: 10.2118/109975-MS
  49. 49.
    Wang, Y., Li, G., Reynolds, A.C.: Estimation of depths of fluid contacts and relative permeability curves by history matching using iterative ensemble-kalman smoothers. SPE J. 15(2) (2010). doi: 10.2118/119056-PA

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Uni Research CIPRBergenNorway
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations