Computational Geosciences

, Volume 17, Issue 4, pp 671–687

Bayesian updating via bootstrap filtering combined with data-driven polynomial chaos expansions: methodology and application to history matching for carbon dioxide storage in geological formations

Original Paper

Abstract

Model calibration and history matching are important techniques to adapt simulation tools to real-world systems. When prediction uncertainty needs to be quantified, one has to use the respective statistical counterparts, e.g., Bayesian updating of model parameters and data assimilation. For complex and large-scale systems, however, even single forward deterministic simulations may require parallel high-performance computing. This often makes accurate brute-force and nonlinear statistical approaches infeasible. We propose an advanced framework for parameter inference or history matching based on the arbitrary polynomial chaos expansion (aPC) and strict Bayesian principles. Our framework consists of two main steps. In step 1, the original model is projected onto a mathematically optimal response surface via the aPC technique. The resulting response surface can be viewed as a reduced (surrogate) model. It captures the model’s dependence on all parameters relevant for history matching at high-order accuracy. Step 2 consists of matching the reduced model from step 1 to observation data via bootstrap filtering. Bootstrap filtering is a fully nonlinear and Bayesian statistical approach to the inverse problem in history matching. It allows to quantify post-calibration parameter and prediction uncertainty and is more accurate than ensemble Kalman filtering or linearized methods. Through this combination, we obtain a statistical method for history matching that is accurate, yet has a computational speed that is more than sufficient to be developed towards real-time application. We motivate and demonstrate our method on the problem of CO2 storage in geological formations, using a low-parametric homogeneous 3D benchmark problem. In a synthetic case study, we update the parameters of a CO2/brine multiphase model on monitored pressure data during CO2 injection.

Keywords

History matching Arbitrary polynomial chaos Bayesian updating Bootstrap filter CO2 storage Uncertainty quantification 

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References

  1. 1.
    Aanonsen, S.I., Naevdal, G., Reynolds, A.C., Valls, B.: The ensemble Kalman filter in reservoir engineering—a review. SPE J. 14(3), 393–412 (2009)Google Scholar
  2. 2.
    Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)CrossRefGoogle Scholar
  3. 3.
    Bangerth, W., Klie, H., Wheeler, M., Stoffa, P., Sen, M.: On optimization algorithms for the reservoir oil well placement problem. Comput. Geosci. 10(3), 303–319 (2006)CrossRefGoogle Scholar
  4. 4.
    Birkholzer, J.T., Zhou, Q., Tsang, Ch.-F.: Large-scale impact of CO2 storage in deep saline aquifers: a sensitivity study on pressure response in stratified systems. Int. J. of Greenh. Gas Control. 3, 181–194 (2009)CrossRefGoogle Scholar
  5. 5.
    Blatman, G., Sudret, B.: Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Syst. Saf. 95, 1216–1229 (2010)CrossRefGoogle Scholar
  6. 6.
    Chen, J.-S., Wang, L., Hu, H.-Y., Chi, S.-W.: Subdomain radial basis collocation method for heterogeneous media. Int. J. Numer. Meth. Engng. 80, 163–190 (2009)CrossRefGoogle Scholar
  7. 7.
    Class, H., Ebigbo, A., Helmig, R., Dahle, H., Nordbotten, J.N., Celia, M.A., Audigane, P., Darcis, M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S., Jin, M., Krug, S., Labregere, D., Naderi, A., Pawar, R.J., Sbai, A., Sunil, G.T., Trenty, L., Wei, L.: A benchmark study on problems related to CO2 storage in geologic formations. Comput. Geosci. 13, 451–467 (2009)CrossRefGoogle Scholar
  8. 8.
    Cominelli, A., Ferdinandi, F., de Montleau, P.C., Rossi, R.: Using gradients to refine parameterization in field-case history-matching projects. SPE Reserv. Evalu. Eng. 10(3), 233–240 (2007)Google Scholar
  9. 9.
    Cortis, A., Oldenburg, C., Benson, S.M.: The role of optimality in characterizing CO2 seepage from geologic carbon sequestration sites. Int. J. of Greenh. Gas Control. 2, 640–652 (2008)CrossRefGoogle Scholar
  10. 10.
    Crestaux, T., Le Maitre, O., Martinez, J.-M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)CrossRefGoogle Scholar
  11. 11.
    Daoud, A.M.I.: Automatic history matching in Bayesian framework for field-scale applications. Dissertation. Texas A&M University (2004)Google Scholar
  12. 12.
    Ebigbo A., Class H., Helmig R.: CO2 leakage through an abandoned well: problem-oriented benchmarks. Comput. Geosci. 11(2), 103–115 (2007)CrossRefGoogle Scholar
  13. 13.
    Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap (Monographs on Statistics & Applied Probability). Chapman & Hall, London (2010)Google Scholar
  14. 14.
    Evensen, G.: Data Assimilation: the Ensemble Kalman Filter, Berlin (2006)Google Scholar
  15. 15.
    Ewing, R.E., Pilant, M.S., Wade, G.J., A.T. Watson: Estimating parameters in scientific computation. IEEE Comput. Sci. Eng. 1(3), 19–31 (1994)CrossRefGoogle Scholar
  16. 16.
    Fajraoui, N., Ramasomanana, F., Younes, A., Mara, T.A., Ackerer, P., Guadagnini, A.: Use of global sensitivity analysis and polynomial chaos expansion for interpretation of non-reactive transport experiments in laboratory-scale porous media. Water Resour. Res. (2011). doi:10.1029/2010WR009639 Google Scholar
  17. 17.
    Feraille, M., Marrel, A.: Prediction under uncertainty on a mature field. Oil and Gas Science and Technology-Revue dIFP Energies nouvelles 67(2), 193–206 (2012)CrossRefGoogle Scholar
  18. 18.
    Flemisch, B., Fritz, J., Helmig, R., Niessner, J., Wohlmuth, B. In: Ibrahimbegovic, A., Dias, F. (eds.) : ECCO3MAS Thematic Conference on Multi-scale Computational Methods for Solids and Fluids, Cachan, France, 28–30 November 2007Google Scholar
  19. 19.
    Foo, J., Karniadakis, E.G.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229(5), 1536–1557 (2010)CrossRefGoogle Scholar
  20. 20.
    Gao, G., Reynolds, A.C.: An improved implementation of the LBFGS algorithm for automatic history matching. SPE J. 11(1), 5–17 (2006)Google Scholar
  21. 21.
    Gao, G., Reynolds, A.C.: A stochastic optimization algorithm for automatic history matching. SPE J. 12(2), 196–208 (2007)Google Scholar
  22. 22.
    Gavalas, G.R., Shah, P.C., Seinfeld, J.H.: Reservoir history matching by Bayesian estimation. SPE J. 16(6), 337–350 (1976)Google Scholar
  23. 23.
    Ghanem, R., Doostan, A.: On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data. J. Comput. Phys. 217, 63–81 (2006)CrossRefGoogle Scholar
  24. 24.
    Ghanem, R., Spanos, P.: A stochastic Galerkin expansion for nonlinear random vibration analysis. Probab. Eng. Mec. 8, 255–264 (1993)CrossRefGoogle Scholar
  25. 25.
    Ghanem, R., Spanos, P.D.: Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57, 197–202 (1990)CrossRefGoogle Scholar
  26. 26.
    Ghanem, R.G., Spanos, P.D.: Stochastic finite elements: a spectral approach. Springer (1991)Google Scholar
  27. 27.
    Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov chain Monte Carlo in practice. Chapman & Hall, London (1996)Google Scholar
  28. 28.
    Gu, Y., Oliver, D.S.: An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE J 12(4), 438–446 (2007)Google Scholar
  29. 29.
    Hansson, A., Bryngelsson, M.: Expert opinions on carbon dioxide capture and storage: a framing of uncertainties and possibilities. Energy Policy 37, 2273–2282 (2009)CrossRefGoogle Scholar
  30. 30.
    He, J., Sarma, P., Durlofsky, L.J., Chen, W.: Use of reduced-order models for improved data assimilation within an EnKF context. Presented in Reservoir Simulation Symposium, The Woodlands, Texas, USA, SPE 141967 (2011)Google Scholar
  31. 31.
    Hendricks Franssen, H.-J., Kinzelbach, W.: Ensemble Kalman filtering versus sequential self-calibration for inverse modelling of dynamic groundwater flow systems. J. Hydrol. 365(3–4), 261–274 (2009)CrossRefGoogle Scholar
  32. 32.
    IPCC: Special Report on Carbon Dioxide Capture and Storage. Technical Report, Intergovernmental Panel on Climate Change (IPCC), Prepared by Working Group III. Cambridge University Press, Cambridge (2005)Google Scholar
  33. 33.
    Jin, B.: Fast Bayesian approach for parameter estimation. Int. J. Numer. Meth. Engng. 76, 230–252 (2008)CrossRefGoogle Scholar
  34. 34.
    Kitanidis, P.K.: Quasi-linear geostatistical theory for inversing. Water Resour. Res. 31(10), 2411–2419 (1995)CrossRefGoogle Scholar
  35. 35.
    Kopp, A., Class, H., Helmig, H.: Investigations on CO2 storage capacity in saline aquifers—part 1: dimensional analysis of flow processes and reservoir characteristics. Int. J. Greenh. Gas Control. 3, 263–276 (2009)CrossRefGoogle Scholar
  36. 36.
    Helmig, C., Seinfeld, J.H.: Identification of parameters in distributed parameter-systems by regularization. SIAM J. Control Optim. 23(2), 217–241 (1985)CrossRefGoogle Scholar
  37. 37.
    Le Maitre, O., Knio, O.: Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics, New York (2010)Google Scholar
  38. 38.
    Leube, P., Geiges, A., Nowak, W.: Bayesian assessment of the expected data impact on prediction confidence in optimal sampling design. Water Resour. Res. 48, W02501 (2012)Google Scholar
  39. 39.
    Leube, P.C., Nowak, W., Schneider, G.: Temporal moments revisited: why there is no better way for physically-based model reduction in time. Water Resour. Res. (2012). doi:10.1029/2012WR011973 Google Scholar
  40. 40.
    Li, H., Sarma, P., Zhang, D.: A comparative study of the probabilistic collocation and experimental design methods for petroleum reservoir uncertainty quantification. SPE J. 16, 429–439 (2011). SPE–140738–PA–PGoogle Scholar
  41. 41.
    Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43, 44–48 (2007)Google Scholar
  42. 42.
    Li, R., Reynolds, A.C., Oliver, D.S.: Sensitivity coefficients for three-phase flow history matching. J. Can. Pet. Technol. 42(4), 70–77 (2003)Google Scholar
  43. 43.
    Lia, O., Omre, H., Tjelmeland, H., Holden, L., Egeland, T.: Uncertainties in reservoir production forecasts. AAPG Bull. 81(5), 775–802 (1997)Google Scholar
  44. 44.
    Liou, Ch.-L., Lin, Ch.-H.: Applications of the methods of weighted residuals in system science. Int. J. Syst. Sci. 22(9), 1509–1525 (1991)CrossRefGoogle Scholar
  45. 45.
    Liu, N., Oliver, D.S.: Ensemble Kalman filter for automatic history matching of geologic facies. J. Pet. Sci. Eng. 47, 147–161 (2005)CrossRefGoogle Scholar
  46. 46.
    Makhlouf, E.M., Chen, M.L., Wasserman, W.H., Seinfeld, J.H.: A general history matching algorithm for three-phase, three-dimensional petroleum reservoirs. SPE Adv. Technol. 1(2), 83–91 (1993)Google Scholar
  47. 47.
    Maltz, F.H., Hitzl, D.L.: Variance reduction in Monte Carlo computations using multi-dimensional Hermite polynomials. Journal Comput. Phys. 2, 345–376 (1979)CrossRefGoogle Scholar
  48. 48.
    Marzouk, Y., Najm, H., Rahn, L.: Stochastic spectral methods for efficient Bayesian solution of inverse problems. J. Comput. Phys. 224(2), 560–586 (2007)CrossRefGoogle Scholar
  49. 49.
    Morariu, V.I., Srinivasan, B.V., Raykar, V.C., Duraiswami, R., Davis, L.S.: Automatic online tuning for fast gaussian summation. Adv. Neural. Inform. Process. Syst. 21, 1113–1120 (2009)Google Scholar
  50. 50.
    Moritz, H.: Least-squares collocation. Rev. Geophys. Space Phys. 16(3), 421–430 (1978)CrossRefGoogle Scholar
  51. 51.
    Naevdal, G., Hanea, R.G., Oliver, D.S., Valles, B.: Ensemble Kalman filter for model updating—a special issue. Comput. Geosci. 15(2), 223–224 (2011)CrossRefGoogle Scholar
  52. 52.
    Naevdal, G., Johnsen, L.M., Aanonsen, S.I., Vefring, E.H.: Reservoir monitoring and continuous model updating using ensemble Kalman filter. SPE J. 10(1), 66–74 (2005)Google Scholar
  53. 53.
    Nordbotten, J., Celia, M., Bachu, M.: Injection and storage of CO2 in deep saline aquifers: analytical solution for CO2 plume evolution during injection. Transp. Porous Media 58(3), 339–360 (2005)CrossRefGoogle Scholar
  54. 54.
    Nordbotten, J.M., Kavetski, D., Celia, M.A., Bachu, S.: A semi-analytical model estimating leakage associated with CO2 storage in large-scale multi-layered geological systems with multiple leaky wells. Environ. Sci. Technol. 43(3), 743–749 (2009)CrossRefGoogle Scholar
  55. 55.
    Nowak, W.: Best unbiased ensemble linearization and the quasi-linear Kalman ensemble generator. Water Resour. Res. 45, W04431 (2009)CrossRefGoogle Scholar
  56. 56.
    Nowak, W., Rubin, Y., de Barros, F.P.J.: A hypothesis-driven approach to optimal site investigation. Water Resour. Res. (2012). doi:10.1029/2011WR011016 Google Scholar
  57. 57.
    Oladyshkin, S., Class, H., Helmig, R., Nowak, W.: A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations. Adv. Water Resour. 34, 1508–1518 (2011)CrossRefGoogle Scholar
  58. 58.
    Oladyshkin, S., Oladyshkin, H., Helmig, R., Nowak, W.: An integrative approach to robust design and probabilistic risk assessment for CO2 storage in geological formations. Comput. Geosci. 15(3), 565–577 (2011)CrossRefGoogle Scholar
  59. 59.
    Oladyshkin, S., de Barros, F.P.J., Nowak, W.: Global sensitivity analysis: a flexible and efficient framework with an example from stochastic hydrogeology. Adv. Water Resour. 37, 10–22 (2011)CrossRefGoogle Scholar
  60. 60.
    Oladyshkin, S., Nowak, W.: Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliab. Eng. Syst. Saf. 106, 179–190 (2012)CrossRefGoogle Scholar
  61. 61.
    Oladyshkin, S., Nowak, W.: Polynomial Response Surfaces for Probabilistic Risk Assessment and Risk Control via Robust Design. Novel Approaches and Their Applications in Risk. InTech, Manhattan (2012)Google Scholar
  62. 62.
    Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge (2008)Google Scholar
  63. 63.
    Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15, 185–221 (2011)CrossRefGoogle Scholar
  64. 64.
    Oliver, D.S., Reynolds, A.C., Bi, Z., Abacioglu, Y.: Integration of production data into reservoir models. Pet. Geosci. 7, 65–73 (2001)CrossRefGoogle Scholar
  65. 65.
    Slotte, P.A., Smrgrav, E.: Response surface methodology approach for history matching and uncertainty assessment of reservoir simulation models. Europec/EAGE Conference and Exhibition, SPE 113390, Rome, Italy, 9–12 June 2008Google Scholar
  66. 66.
    Wever, U., Paffrath, M.: Adapted polynomial chaos expansion for failure detection. J. Comput. Phys. 226(1), 263–281 (2007)CrossRefGoogle Scholar
  67. 67.
    Pajonk, O., Rosic, B.V., Litvinenko, A., Matthies, H.G.: A deterministic filter for non-Gaussian Bayesian estimation applications to dynamical system estimation with noisy measurements. Physica. D. 241, 775–788 (2012)CrossRefGoogle Scholar
  68. 68.
    Pappenberger, F., Beven, K.J.: Ignorance is bliss: or seven reasons not to use uncertainty analysis. Water Resour. Res. 42(5), 1–8 (2006)CrossRefGoogle Scholar
  69. 69.
    Robert, C.P., Casella, G.: Monte Carlo methods. Springer, New York (2004)CrossRefGoogle Scholar
  70. 70.
    Rodrigues, J.R.P.: Calculating derivatives for automatic history matching. Comput. Geosci. 10, 119–136 (2006)CrossRefGoogle Scholar
  71. 71.
    Saad, G., Ghanem, R.: Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter. Water Resour. Res. 45, W04417 (2009)CrossRefGoogle Scholar
  72. 72.
    Saltelli, A., Ratto, M., Andres, T.: Global Sensitivity Analysis: the Primer. Wiley, New York (2008)Google Scholar
  73. 73.
    Sarma, P., Durlofsky, L.J., Aziz, K.: Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math. Geosci. 40(1), 3–32 (2008)CrossRefGoogle Scholar
  74. 74.
    Sarma, P., Durlofsky, L.J., Aziz, K., Chen, W.H.: Efficient real-time reservoir management using adjoint-based optimal control and model updating. Comput. Geosci. 10(1), 3–36 (2006)CrossRefGoogle Scholar
  75. 75.
    Scheidt, C., Caers, J., Chen, Y., Durlofsky, L.J.: A multi-resolution workflow to generate high-resolution models constrained to dynamic data. Comput. Geosci. 15(3), 545–563 (2011)CrossRefGoogle Scholar
  76. 76.
    Schoeniger, A., Nowak, W., Hendricks, Franssen, H.-J.: Parameter estimation by ensemble Kalman filters with transformed data: Approach and application to hydraulic tomography. Water Resour. Res. 45, W04431 (2012)Google Scholar
  77. 77.
    Smith, A.F.M., Gefland, A.E.: Bayesian statistics without tears: a sampling-resampling perspective. Am. Stat. 46(2), 84–88 (1992)Google Scholar
  78. 78.
    Soize, C., Ghanem, R.: Physical systems with random uncertainties: chaos representations with arbitrary probability measure. SIAM J. Sci. Comput 26(2), 395–410 (2004)CrossRefGoogle Scholar
  79. 79.
    Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)CrossRefGoogle Scholar
  80. 80.
    Sun, N.Z.: Inverse Problems in Groundwater Modeling (Theory and Applications of Transport in Porous Media), New York (1999)Google Scholar
  81. 81.
    Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005)CrossRefGoogle Scholar
  82. 82.
    Villadsen, J., Michelsen, M.L.: Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall, Upper Saddle River (1978)Google Scholar
  83. 83.
    Vrugt, A.J., ter Braak, C.J.F., Diks, C.G.H., Robinson, B.A., Hyman, J.H., Higdon, D.: Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int. J. Nonlinear Sci. Numer. Simul. 10(3), 271–288 (2009)CrossRefGoogle Scholar
  84. 84.
    Walter, L., Binning, P., Oladyshkin, S., Flemisch, B., Class, H.: Int. J. Greenh. Gas. Control. 9(495–506) (2012)Google Scholar
  85. 85.
    Wan, X., Karniadakis, E.G.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006)CrossRefGoogle Scholar
  86. 86.
    Wand, M.P., Jones, M.C.: Kernel smoothing. Monographs on Statistics and Applied Probability 60. Chapman & Hall, Boca Raton (1995)Google Scholar
  87. 87.
    Wang, Y., Li, G., Reynolds, A.C.: Estimation of depths of fluid contacts by history matching using iterative ensemble Kalman smoothers. SPE J. 15(2), 509–529 (2010)Google Scholar
  88. 88.
    Wiener, N.: The homogeneous chaos. Am. J. Math 60, 897–936 (1938)CrossRefGoogle Scholar
  89. 89.
    Wildenborg, A.F.B., Leijnse, A.L., Kreft, E., Nepveu, M.N., Obdam, A.N.M., Orlic, B., Wipfler, E.L., van Kesteren, W., van der Grift, B., Gaus, I., Czernichowski-Lauriol, I., Torfs, P., Wojcik, R.: Risk assessment methodology for CO2 storage: the scenario approach. In: Thomas, D.C., Benson, S.M. (eds.) Carbon Dioxide Capture for Storage in Deep Geologic Formations. Elsevier, London (2005)Google Scholar
  90. 90.
    Williams, M.A., Keating, J.F., Barghouty, M.F.: The stratigraphic method: a structured approach to history-matching complex simulation models. SPE Reserv. Evalu. Eng. 1(2), 169–176 (1998)Google Scholar
  91. 91.
    Witteveen, J.A.S., Bijl, H.: Modeling arbitrary uncertainties using Gram-Schmidt polynomial chaos. 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, AIAA–2006–896 (2006)Google Scholar
  92. 92.
    Witteveen, J.A.S., Sarkar, S., and Bijl, H.: Modeling physical uncertainties in dynamic stall induced fluidstructure interaction of turbine blades using arbitrary polynomial chaos. Comput. Struct. 85, 866–878 (2007)CrossRefGoogle Scholar
  93. 93.
    Xiu, D., Karniadakis, E.G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003)CrossRefGoogle Scholar
  94. 94.
    Zabalza-Mezghani, I., Manceau, E., Feraille, M., Jourdan, A.: Uncertainty management: from geological scenarios to production scheme optimization. J. Pet. Sci. Eng. 44, 11–25 (2004)CrossRefGoogle Scholar
  95. 95.
    Zafari, M., Reynolds, A.C.: Assessing the uncertainty in reservoir description and performance predictions with the ensemble Kalman filter. SPE J. 12(3), 382–391 (2007)Google Scholar
  96. 96.
    Zhang, D., Lu, Z.: An efficient, high-order perturbation approach for flow in random media via Karhunen-Loève and polynomial expansions. J. Comput. Phys. 194, 773–794 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Sergey Oladyshkin
    • 1
  • Holger Class
    • 1
  • Wolfgang Nowak
    • 1
  1. 1.SRC Simulation Technology, Institute for Modelling Hydraulic and Environmental Systems (LH2)University of StuttgartStuttgartGermany

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