Skip to main content
SpringerLink
Log in

Ensemble-based multi-scale history-matching using second-generation wavelet transform

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

Ensemble-based optimization methods are of- ten efficiently applied to history-matching problems. Although satisfactory matches can be obtained, the updated realizations, affected by spurious correlations, generally fail to preserve prior information when using a small ensemble, even when localization is applied. In this work, we propose a multi-scale approach based on grid-adaptive second-generation wavelets. These wavelets can be applied on irregular reservoir grids of any dimensions containing dead or flat cells. The proposed method starts by modifying a few low frequency parameters (coarse scales) and then progressively allows more important updates on a limited number of sensitive parameters of higher resolution (fine scales). The Levenberg-Marquardt ensemble randomized maximum likelihood (LM-enRML) is used as optimization method with a new space-frequency distance-based localization of the Kalman gain, specifically designed for the multi-scale scheme. The algorithm is evaluated on two test cases. The first test is a 2D synthetic case in which several inversions are run using independent ensembles. The second test is the Brugge benchmark case with 10 years of history. The efficiency and quality of results of the multi-scale approach are compared with the grid-block-based LM-enRML with distance-based localization. We observe that the final realizations better preserve the spatial contrasts of the prior models and are less noisy than the realizations updated using a standard grid-block method, while matching the production data equally well.

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. Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53(4), 343–367 (2003)

    Article  Google Scholar 

  2. 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)

    Article  Google Scholar 

  3. Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–221 (2011)

    Article  Google Scholar 

  4. Chen, Y., Oliver, D.S.: Multiscale parameterization with adaptive regularization for improved assimilation of nonlocal observation. Water Resour. Res. 48(4) (2012)

  5. Bhark, E., Jafarpour, B., Datta-Gupta, A.: An adaptively scaled frequency-domain parameterization for history matching. J. Pet. Sci. Eng. 75(3–4), 289–303 (2011)

    Article  Google Scholar 

  6. Sahni, I., Horne, R.: Multiresolution wavelet analysis for improved reservoir description. SPE Reserv. Eval. Eng. 8(1), 53–69 (2005)

    Article  Google Scholar 

  7. Soubaras, R., Dowle, R.: Variable-depth streamer—a broadband marine solution. first break 28(12) (2010)

  8. Gu, Y., Oliver, D.S.: An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE J. 12(4), 438–446 (2007)

    Article  Google Scholar 

  9. Chen, Y., Oliver, D.S.: Levenberg-marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Comput. Geosci. 17(4), 689–703 (2013)

    Article  Google Scholar 

  10. Ben Ameur, H., Chavent, G., Jaffré, J.: Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic transmissivities. Inverse Probl. 18(3), 775 (2002)

    Article  Google Scholar 

  11. Grimstad, A.A., Mannseth, T., Nævdal, G., Urkedal, H.A.: Adaptive multiscale permeability estimation. Comput. Geosci. 7(1), 1–25 (2003)

    Article  Google Scholar 

  12. Ben Ameur, H., Clément, F., Weis, P., Chavent, G.: The multidimensional refinement indicators algorithm for optimal parameterization. J. Inverse Ill-Posed Probl. 16(2), 107–126 (2008)

    Google Scholar 

  13. Lu, P., Horne, R.: A multiresolution approach to reservoir parameter estimation using wavelet analysis. In: SPE annual technical conference and exhibition (2000)

  14. Bhark, E., Rey, A., Datta-Gupta, A., Jafarpour, B.: A multiscale workflow for history matching in structured and unstructured grid geometries. SPE J. 17(3), 828–848 (2012)

    Article  Google Scholar 

  15. Bhark, E., Datta-Gupta, A., Jafarpour, B.: Subsurface flow model calibration with a spectral-domain parameterization adaptive to grid connectivity and prior model information. Math. Geosci. 44(6), 673–710 (2012)

    Article  Google Scholar 

  16. Liu, J.: A multiresolution method for distributed parameter estimation. Sci. Comput. 14(2), 389–405 (1993)

    Google Scholar 

  17. Mannseth, T.: Permeability identification from pressure observations: some foundations for multiscale regularization. Multiscale Model. Simul. 5(1), 21–44 (2006)

    Article  Google Scholar 

  18. Chavent, G.: Nonlinear least squares for inverse problems: theoretical foundations and step-by-step guide for applications. Springer Verlag (2009)

  19. Lien, M., Brouwer, D., Mannseth, T., Jansen, J.-D.: Multiscale regularization of flooding optimization for smart field management. SPE J. 13(2), 195–204 (2008)

    Article  Google Scholar 

  20. Gardet, C., Ravalec, M., Gloaguen, E.: Multiscale parameterization of petrophysical properties for efficient history-matching. Math. Geosci., 1–22 (2013)

  21. Reynolds, A.C., He, N., Chu, L., Oliver, D.S.: Reparameterization techniques for generating reservoir descriptions conditioned to variograms and well-test pressure data. SPE J. 1(4), 413–426 (1996)

    Article  Google Scholar 

  22. Efendiev, Y., Hou, T., Luo, W.: Preconditioning Markov chain Monte Carlo simulations using coarse-scale models. SIAM J. Sci. Comput. 28(2), 776–803 (2006)

    Article  Google Scholar 

  23. Jafarpour, B., McLaughlin, D.: Reservoir characterization with the discrete cosine transform. SPE J. 14(1), 182–201 (2009)

    Article  Google Scholar 

  24. Li, L., Jafarpour, B.: A sparse Bayesian framework for conditioning uncertain geologic models to nonlinear flow measurements. Adv. Water Resour. 33(9), 1024–1042 (2010)

    Article  Google Scholar 

  25. Bhark, E.W., Jafarpour, B., Datta-Gupta, A.: A generalized grid connectivity-based parameterization for subsurface flow model calibration. Water Resour. Res. 47(6) (2011)

  26. Feng, T., Mannseth, T.: Improvements on a predictor-corrector strategy for parameter estimation with several data types. Inverse Probl. 25(10), 105012–105032 (2009)

    Article  Google Scholar 

  27. Hamill, T.M., Whitaker, J.S., Snyder, C.: Distance-depen- dent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Weather Rev. 129(11), 2776–2790 (2001)

    Article  Google Scholar 

  28. Houtekamer, P.L., Mitchell, H.L.: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weather Rev. 129(1), 123–137 (2001)

    Article  Google Scholar 

  29. Chen, Y., Oliver, D.S.: Cross-covariances and localization for EnKF in multiphase flow data assimilation. Comput. Geosci. 14, 579–601 (2010)

    Article  Google Scholar 

  30. Chen, Y., Oliver, D.S.: Ensemble-based closed-loop optimization applied to Brugge field. SPE Reserv. Eval. Eng. 13(1), 56–71 (2010)

    Article  Google Scholar 

  31. Furrer, R., Bengtsson, T.: Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. J. Multivar. Anal. 98(2), 227–255 (2007)

    Article  Google Scholar 

  32. Anderson, J.L.: Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Phys. D: Nonlinear Phenom. 230(1–2), 99–111 (2007)

    Article  Google Scholar 

  33. Zhang, Y., Oliver, D.S.: Improving the ensemble estimate of the Kalman gain by bootstrap sampling. Math. Geosci. 42(3), 327–345 (2010)

    Article  Google Scholar 

  34. Grossmann, A., Morlet, J.: Decomposition of hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)

    Article  Google Scholar 

  35. Mallat, S.G.: A theory for multiresolution signal decomposition: the wavelet representation. Pattern Anal. Mach. Intell., IEEE Trans. 11(7), 674–693 (1989)

    Article  Google Scholar 

  36. Burrus, C.S., Gopinath, R.A., Guo, H., Odegard, J.E., Selesnick, I.W.: Introduction to wavelets and wavelet transforms: a primer, vol. 23. Prentice hall Upper Saddle River (1998)

  37. Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29(2), 511–546 (1998)

    Article  Google Scholar 

  38. Sweldens, W., Schröder, P.: Building your own wavelets at home. In: Klees, R., Haagmans, R. (eds.) Wavelets in the Geosciences of Lecture Notes in Earth Sciences, vol. 90, pp. 72–107. Springer, Berlin, Heidelberg (2000)

    Book  Google Scholar 

  39. Chen, Y., Oliver, D.S.: Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math. Geosci. 44(1), 1–26 (2012)

    Article  Google Scholar 

  40. Liu, N., Oliver, D.S.: Evaluation of Monte Carlo methods for assessing uncertainty. SPE J. 8(2), 188–195 (2003)

    Article  Google Scholar 

  41. 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), 506–515 (2006)

    Article  Google Scholar 

  42. Golub, G.H., Van Loan, C.F.: Matrix Computations, 2nd edn. Johns Hopkins University Press, Baltimore (1989)

    Google Scholar 

  43. Li, R., Reynolds, A.C., Oliver, D.S.: History matching of three-phase flow production data. SPE J. 8(4), 328–340 (2003)

    Article  Google Scholar 

  44. Iglesias, M.A., Dawson, C.: The regularizing Levenberg-Marquardt scheme for history matching of petroleum reservoirs. Comput. Geosci. 17(6), 1033–1053 (2013)

    Article  Google Scholar 

  45. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press (2008)

  46. Donoho, D.: Compressed sensing. Inf. Theory, IEEE Trans. 52, 1289–1306 (2006)

    Article  Google Scholar 

  47. Peters, L., Arts, R., Brouwer, G., Geel, C., Cullick, S., Lorentzen, R., Chen, Y., Dunlop, N., Vossepoel, F., Xu, R., et al.: Results of the Brugge benchmark study for flooding optimization and history matching. SPE SPE Reserv. Eval. Eng. 13(3), 391–405 (2010)

    Article  Google Scholar 

  48. Grimstad, A.-A., Mannseth, T.: Nonlinearity, scale, and sensitivity for parameter estimation problems. SIAM J. Sci. Comput. 21(6), 2096–2113 (2000)

    Article  Google Scholar 

  49. Fernandez, G., Periaswamy, S., Sweldens, W.: LIFTPACK: A software package for wavelet transforms using lifting. Proc. SPIE 2825, 396 (1996)

    Article  Google Scholar 

  50. Girardi, M., Sweldens, W.: A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces. J. Fourier Anal. Appl. 3(4), 457–474 (1997)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. GeoRessources (UMR 7359 Université de Lorraine-ENSG, CNRS, CREGU), 2 Rue du Doyen Marcel Roubault, 54518, Vandœuvre-les-Nancy, France

    Théophile Gentilhomme & Guillaume Caumon

  2. CGG - GeoConsulting, 27 Avenue Carnot, 91300, Massy, France

    Théophile Gentilhomme & Rémi Moyen

  3. Uni CIPR, Uni Research, P.O. Box 7800, 5020, Bergen, Norway

    Dean S. Oliver & Trond Mannseth

  4. Department of Mathematics, University of Bergen, P.O. Box 7800, 5020, Bergen, Norway

    Trond Mannseth

  5. CGG - GeoConsulting, Crompton Way, Crawley, RH10 9QN, UK

    Philippe Doyen

Authors
  1. Théophile Gentilhomme
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dean S. Oliver
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Trond Mannseth
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Guillaume Caumon
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Rémi Moyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Philippe Doyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Théophile Gentilhomme.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gentilhomme, T., Oliver, D.S., Mannseth, T. et al. Ensemble-based multi-scale history-matching using second-generation wavelet transform. Comput Geosci 19, 999–1025 (2015). https://doi.org/10.1007/s10596-015-9517-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-015-9517-4

Keywords

Search

Navigation