Skip to main content
SpringerLink
Log in

Integration of Markov mesh models and data assimilation techniques in complex reservoirs

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

We present a methodology that allows conditioning the spatial distribution of geological and petrophysical properties of reservoir model realizations on available production data. The approach is fully consistent with modern concepts depicting natural reservoirs as composite media where the distribution of both lithological units (or facies) and associated attributes are modeled as stochastic processes of space. We represent the uncertain spatial distribution of the facies through a Markov mesh (MM) model, which allows describing complex and detailed facies geometries in a rigorous Bayesian framework. The latter is then embedded within a history matching workflow based on an iterative form of the ensemble Kalman filter (EnKF). We test the proposed methodology by way of a synthetic study characterized by the presence of two distinct facies. We analyze the accuracy and computational efficiency of our algorithm and its ability with respect to the standard EnKF to properly estimate model parameters and assess future reservoir production. We show the feasibility of integrating MM in a data assimilation scheme. Our methodology is conducive to a set of updated model realizations characterized by a realistic spatial distribution of facies and their log permeabilities. Model realizations updated through our proposed algorithm correctly capture the production dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vallès, B.: The ensemble Kalman filter in reservoir engineering—a review. SPE J. 14(3), 393–412 (2009). doi:10.2118/117274-PA

    Article  Google Scholar 

  2. Abend, K., Harley, T.J., Kanal, L.N.: Classification of binary random patterns. IEEE Trans. Inf. Theory 11(4), 538–544 (1965). doi:10.1109/TIT.1965.1053827

    Article  Google Scholar 

  3. Arpat, G.B., Caers, J.: Conditional simulation with patterns. Math. Geol 39(2), 177–203 (2007). doi:10.1007/s11004-006-9075-3

    Article  Google Scholar 

  4. Astrakova, A., Oliver, D.S.: Conditioning truncated pluri-Gaussian models to facies observations in ensemble-Kalman-based data assimilation. Math. Geosci. 47(3), 345–367 (2014). doi:10.1007/s11004-014-9532-3

    Article  Google Scholar 

  5. Bridge, J.S., Leeder, M.R.: A simulation model of alluvial stratigraphy. Sedimentology 26(5), 617–644 (1979). doi:10.1111/j.1365-3091.1979.tb00935.x

    Article  Google Scholar 

  6. Burgers, G., van Leeuwen, P.J., Evensen, G.: Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126(6), 1719–1724 (1998). doi:10.1175/1520-0493(1998)126<1719:ASITEK>2.0.CO;2

    Article  Google Scholar 

  7. Daly, C. In: Leuangthong, O., Deutsch, C.V. (eds.) : Higher order models using entropy, Markov random fields and sequential simulation, pp 215–224. Springer, Berlin (2005)

  8. Deutsch, C.V., Wang, L.: Hierarchical object-based stochastic modeling of fluvial reservoirs. Math. Geol. 28(7), 857–880 (1996). doi:10.1007/BF02066005

    Article  Google Scholar 

  9. Deutsch, C.V., Gringarten, E.: Accounting for multiple-point continuity in geostatistical modeling, p 2000. Geostatistical Congress, South Africa (2000)

    Google Scholar 

  10. Deutsch, C.V., Tran, T.T.: FLUVSIM: a program for object-based stochastic modeling of fluvial depositional systems. Comput. Geosci. 28(4), 525–535 (2002). doi:10.1016/S0098-3004(01)00075-9

    Article  Google Scholar 

  11. Endres, D.M., Schindelin, J.E.: A new metric for probability distributions. IEEE Trans. Inf. Theory 49(7), 1858–1860 (2003). doi:10.1109/TIT.2003.813506

    Article  Google Scholar 

  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), 10143–10162 (1994). doi:10.1029/94JC00572

    Article  Google Scholar 

  13. Gaspari, G., Cohn, S.E.: Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc 125(554), 723–757 (1999). doi:10.1002/qj.49712555417

    Article  Google Scholar 

  14. Gross, L.J., Small, M.J.: River and floodplain process simulation for subsurface characterization. Water Resour. Res. 34(9), 2365–2376 (1998). doi:10.1029/98WR00777

    Article  Google Scholar 

  15. Hu, L.Y., Zhao, Y., Liu, Y., Scheepens, C., Bouchard, A.: Updating multipoint simulations using the ensemble Kalman filter. Comput. Geosci. 51, 7–15 (2013). doi:10.1016/j.cageo.2012.08.020

    Article  Google Scholar 

  16. Jafarpour, B., McLaughlin, D.B.: History matching with an ensemble Kalman filter and discrete cosine parameterization. Comput. Geosci. 12(2), 227–244 (2008). doi:10.1007/s10596-008-9080-3

    Article  Google Scholar 

  17. Jafarpour, B., Khodabakhshi, M.: A probability conditioning method (PCM) for nonlinear flow data integration into multipoint statistical facies simulation. Math. Geosci. 43(2), 133–164 (2011). doi:10.1007/s11004-011-9316-y

    Article  Google Scholar 

  18. Journel, A.G.: Combining knowledge from diverse sources: an alternative to traditional data independence hypotheses. Math. Geol. 34(5), 573–596 (2002). doi:10.1023/A:1016047012594

    Article  Google Scholar 

  19. Kjønsberg, H., Kolbjørnsen, O.: Markov mesh simulations with data conditioning through indicator kriging. In: Proceedings of the eighth international geostatistics congress, vol. 1, pp 257–266. Kluwer, Dordrecht (2008)

  20. Khodabakhshi, M., Jafarpour, B.: A Bayesian mixture-modeling approach for flow-conditioned multiple-point statistical facies simulation from uncertain training images. Water Resour. Res. 49(1), 328–342 (2013). doi:10.1029/2011WR010787

    Article  Google Scholar 

  21. Kolbjørnsen, O., Stien, M., Kjønsberg, H., Fjellvoll, B., Abrahamsen, P.: Using multiple grids in Markov mesh facies modeling. Math. Geosci. 46(2), 205–225 (2014). doi:10.1007/s11004-013-9499-5

    Article  Google Scholar 

  22. Le Loc’h, G., Galli, A. In: Baafi, E.Y., Schofield, N.A. (eds.) : Truncated plurigaussian method: theoretical and practical points of view, pp 211–222. Kluwer Academic Press , Dordrecht (1997)

  23. Liu, N., Oliver, D.S.: Ensemble Kalman filter for automatic history matching of geologic facies. J. Petrol. Sci. Eng 47(3-4), 147–161 (2005a). doi:10.1016/j.petrol.2005.03.006

    Article  Google Scholar 

  24. Liu, N., Oliver, D.S.: Critical evaluation of the ensemble Kalman filter on history matching of geologic facies. SPE Reserv. Eval. Eng. 8(6), 470–477 (2005b). doi:10.2118/92867-PA

    Article  Google Scholar 

  25. Matheron, G., Beucher, H., de Fouquet, C., Galli, A., Guerillot, D., Ravenne, C.: Conditional simulation of the geometry of fluvio-deltaic reservoirs. In: SPE Annual Technical Conference and Exhibition, 27-30 September, Dallas, pp 123–131 (1987), doi:10.2118/16753-MS

  26. Moreno, D., Aanonsen, S.I.: Stochastic facies modeling using the level set method. In: Petroleum Geostatistics, 10–14 September 2007, Cascais, Portugal, A16, Extended Abstracts Book, EAGE Publications BV, Utrecht (2007)

  27. Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2011). doi:10.1007/s10596-010-9194-2

    Article  Google Scholar 

  28. Oliver, D.S., Chen, Y., Nævdal, G.: Updating Markov chain models using the ensemble Kalman Filter. Comput. Geosci. 15(2), 325–344 (2011). doi:10.1007/s10596-010-9220-4

    Article  Google Scholar 

  29. Riva, M., Guadagnini, A., Fernandez-Garcia, D., Sanchez-Vila, X., Ptak, T.: Relative importance of geostatistical and transport models in describing heavily tailed breakthrough curves at the Lauswiesen site. J. Contam. Hydrol. 101(1–4), 1–13 (2008). doi:10.1016/j.jconhyd.2008.07.004

    Article  Google Scholar 

  30. Sebacher, B., Hanea, R., Heemink, A.: A probabilistic parametrization for geological uncertainty estimation using the ensemble Kalman filter (EnKF). Comput. Geosci. 17 (5), 813–832 (2013). doi:10.1007/s10596-013-9357-z

    Article  Google Scholar 

  31. Stien, M., Kolbjørnsen, O.: Facies modeling using a Markov mesh model specification. Math. Geosci. 43 (6), 611–624 (2011). doi:10.1007/s11004-011-9350-9

    Article  Google Scholar 

  32. Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002). doi:10.1023/A:1014009426274

    Article  Google Scholar 

  33. Tan, X., Tahmasebi, P., Caers, J.: Comparing training-image based algorithms using an analysis of distance. Math. Geosci. 46(2), 149–169 (2014). doi:10.1007/s11004-013-9482-1

    Article  Google Scholar 

  34. Viterbi, A.J.: Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE T. Inform. Theory 13(2), 260–269 (1967). doi:10.1109/TIT.1967.1054010

    Article  Google Scholar 

  35. Zhang, T., Switzer, P., Journel, A.: Filter-based classification of training image patterns for spatial simulation. Math. Geol. 38(1), 63–80 (2006). doi:10.1007/s11004-005-9004-x

    Article  Google Scholar 

  36. Zhang, Y., Oliver, D.S., Chen, Y., Skaug, H.J.: Data assimilation by use of the iterative ensemble smoother for 2D facies models. SPE J. 20(1), 169–185 (2014). doi:10.2118/170248-PA

  37. Winter, C.L., Tartakovsky, D.M., Guadagnini, A.: Moment differential equations for flow in highly heterogeneous porous media. Surv. Geophys. 24(1), 81–106 (2003). doi:10.1023/A:1022277418570

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza L. Da Vinci 32, 20133, Milan, Italy

    M. Panzeri, M. Riva & A. Guadagnini

  2. Eni - S.p.A., Via Emilia 1, 20097, San Donato Milanese, Milan, Italy

    E. L. Della Rossa & L. Dovera

  3. Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ, 85721, USA

    M. Riva & A. Guadagnini

Authors
  1. M. Panzeri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. L. Della Rossa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. Dovera
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. Riva
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Guadagnini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Panzeri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Panzeri, M., Della Rossa, E.L., Dovera, L. et al. Integration of Markov mesh models and data assimilation techniques in complex reservoirs. Comput Geosci 20, 637–653 (2016). https://doi.org/10.1007/s10596-015-9540-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-015-9540-5

Keywords

Search

Navigation