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. (English) Zbl 1382.86022

Summary: 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 \(\text{CO}_2\) storage in geological formations, using a low-parametric homogeneous 3D benchmark problem. In a synthetic case study, we update the parameters of a \(\text{CO}_2\)/brine multiphase model on monitored pressure data during \(\text{CO}_2\) injection.


86A60 Geological problems
62F15 Bayesian inference
62F40 Bootstrap, jackknife and other resampling methods
62P12 Applications of statistics to environmental and related topics


KernSmooth; EnKF
Full Text: DOI


[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)
[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) · Zbl 1151.65008
[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) · Zbl 1197.76123
[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)
[5] Blatman, G., Sudret, B.: Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Syst. Saf. 95, 1216–1229 (2010)
[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) · Zbl 1176.74207
[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) · Zbl 1190.86015
[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)
[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)
[10] Crestaux, T., Le Maitre, O., Martinez, J.-M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009)
[11] Daoud, A.M.I.: Automatic history matching in Bayesian framework for field-scale applications. Dissertation. Texas A&M University (2004)
[12] Ebigbo A., Class H., Helmig R.: CO2 leakage through an abandoned well: problem-oriented benchmarks. Comput. Geosci. 11(2), 103–115 (2007) · Zbl 1147.86316
[13] Efron, B., Tibshirani, R.J.: An Introduction to the Bootstrap (Monographs on Statistics & Applied Probability). Chapman & Hall, London (2010)
[14] Evensen, G.: Data Assimilation: the Ensemble Kalman Filter, Berlin (2006) · Zbl 1157.86001
[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) · Zbl 05092116
[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
[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)
[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 2007
[19] Foo, J., Karniadakis, E.G.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229(5), 1536–1557 (2010) · Zbl 1181.65014
[20] Gao, G., Reynolds, A.C.: An improved implementation of the LBFGS algorithm for automatic history matching. SPE J. 11(1), 5–17 (2006)
[21] Gao, G., Reynolds, A.C.: A stochastic optimization algorithm for automatic history matching. SPE J. 12(2), 196–208 (2007)
[22] Gavalas, G.R., Shah, P.C., Seinfeld, J.H.: Reservoir history matching by Bayesian estimation. SPE J. 16(6), 337–350 (1976)
[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) · Zbl 1102.65004
[24] Ghanem, R., Spanos, P.: A stochastic Galerkin expansion for nonlinear random vibration analysis. Probab. Eng. Mec. 8, 255–264 (1993)
[25] Ghanem, R., Spanos, P.D.: Polynomial chaos in stochastic finite elements. J. Appl. Mech. 57, 197–202 (1990) · Zbl 0729.73290
[26] Ghanem, R.G., Spanos, P.D.: Stochastic finite elements: a spectral approach. Springer (1991) · Zbl 0722.73080
[27] Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov chain Monte Carlo in practice. Chapman & Hall, London (1996) · Zbl 0832.00018
[28] Gu, Y., Oliver, D.S.: An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE J 12(4), 438–446 (2007)
[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)
[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)
[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)
[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)
[33] Jin, B.: Fast Bayesian approach for parameter estimation. Int. J. Numer. Meth. Engng. 76, 230–252 (2008) · Zbl 1195.65199
[34] Kitanidis, P.K.: Quasi-linear geostatistical theory for inversing. Water Resour. Res. 31(10), 2411–2419 (1995)
[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)
[36] Helmig, C., Seinfeld, J.H.: Identification of parameters in distributed parameter-systems by regularization. SIAM J. Control Optim. 23(2), 217–241 (1985) · Zbl 0563.93018
[37] Le Maitre, O., Knio, O.: Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics, New York (2010) · Zbl 1193.76003
[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)
[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
[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–P
[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)
[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)
[43] Lia, O., Omre, H., Tjelmeland, H., Holden, L., Egeland, T.: Uncertainties in reservoir production forecasts. AAPG Bull. 81(5), 775–802 (1997)
[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) · Zbl 0742.93032
[45] Liu, N., Oliver, D.S.: Ensemble Kalman filter for automatic history matching of geologic facies. J. Pet. Sci. Eng. 47, 147–161 (2005)
[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)
[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) · Zbl 0437.65005
[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) · Zbl 1120.65306
[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)
[50] Moritz, H.: Least-squares collocation. Rev. Geophys. Space Phys. 16(3), 421–430 (1978)
[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)
[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)
[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)
[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)
[55] Nowak, W.: Best unbiased ensemble linearization and the quasi-linear Kalman ensemble generator. Water Resour. Res. 45, W04431 (2009)
[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
[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)
[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) · Zbl 06116081
[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)
[60] Oladyshkin, S., Nowak, W.: Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliab. Eng. Syst. Saf. 106, 179–190 (2012)
[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)
[62] Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge (2008)
[63] Oliver, D.S., Chen, Y.: Recent progress on reservoir history matching: a review. Comput. Geosci. 15, 185–221 (2011) · Zbl 1209.86001
[64] Oliver, D.S., Reynolds, A.C., Bi, Z., Abacioglu, Y.: Integration of production data into reservoir models. Pet. Geosci. 7, 65–73 (2001)
[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 2008
[66] Wever, U., Paffrath, M.: Adapted polynomial chaos expansion for failure detection. J. Comput. Phys. 226(1), 263–281 (2007) · Zbl 1124.65011
[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) · Zbl 1237.62129
[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)
[69] Robert, C.P., Casella, G.: Monte Carlo methods. Springer, New York (2004) · Zbl 1096.62003
[70] Rodrigues, J.R.P.: Calculating derivatives for automatic history matching. Comput. Geosci. 10, 119–136 (2006) · Zbl 1096.65054
[71] Saad, G., Ghanem, R.: Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter. Water Resour. Res. 45, W04417 (2009)
[72] Saltelli, A., Ratto, M., Andres, T.: Global Sensitivity Analysis: the Primer. Wiley, New York (2008) · Zbl 1161.00304
[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) · Zbl 1144.86004
[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) · Zbl 1161.86303
[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) · Zbl 1254.86029
[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)
[77] Smith, A.F.M., Gefland, A.E.: Bayesian statistics without tears: a sampling-resampling perspective. Am. Stat. 46(2), 84–88 (1992)
[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) · Zbl 1075.60084
[79] Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)
[80] Sun, N.Z.: Inverse Problems in Groundwater Modeling (Theory and Applications of Transport in Porous Media), New York (1999)
[81] Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005) · Zbl 1074.65013
[82] Villadsen, J., Michelsen, M.L.: Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall, Upper Saddle River (1978) · Zbl 0464.34001
[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) · Zbl 06942400
[84] Walter, L., Binning, P., Oladyshkin, S., Flemisch, B., Class, H.: Int. J. Greenh. Gas. Control. 9(495–506) (2012)
[85] Wan, X., Karniadakis, E.G.: Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM J. Sci. Comput. 28(3), 901–928 (2006) · Zbl 1128.65009
[86] Wand, M.P., Jones, M.C.: Kernel smoothing. Monographs on Statistics and Applied Probability 60. Chapman & Hall, Boca Raton (1995)
[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)
[88] Wiener, N.: The homogeneous chaos. Am. J. Math 60, 897–936 (1938) · Zbl 0019.35406
[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)
[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)
[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)
[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)
[93] Xiu, D., Karniadakis, E.G.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187, 137–167 (2003) · Zbl 1047.76111
[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)
[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)
[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) · Zbl 1101.76048
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.