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Computational Geosciences

, Volume 23, Issue 5, pp 981–996 | Cite as

Bayesian model calibration and optimization of surfactant-polymer flooding

  • Pratik Naik
  • Piyush Pandita
  • Soroush Aramideh
  • Ilias Bilionis
  • Arezoo M. ArdekaniEmail author
Original Paper
  • 65 Downloads

Abstract

The physical models governing surfactant-polymer (SP) flooding process are subject to parametric uncertainties, accurate quantification of which is crucial for improved decision making. Moreover, history matching of SP flooding is an ill-posed problem, typically characterized by a multimodal posterior distribution of these model parameters. This paper presents a systematic approach for Bayesian history matching and uncertainty quantification in the model calibration stage of SP flooding using coreflood experimental data. The approach is as follows. First, we construct a surrogate of the computationally expensive physics-based model using a polynomial chaos expansion (PCE-proxy). Second, we formulate a Bayesian calibration problem for inferring the model parameters from a single coreflood experiment that measures pressure drop and oilcut profiles. Third, we solve the Bayesian calibration problem by sampling directly from the posterior using Markov chain Monte Carlo (MCMC). We validate the calibrated parameters by successfully predicting the result of two other coreflood experiments. Then, we extend this framework to stochastic multiobjective optimization of injection slug size design under uncertainties in model parameters (captured by the posterior of Bayesian calibration problem) and oil price (modeled as a geometric random walk with constant drift and volatility). To identify the Pareto frontier of the stochastic multiobjective optimization problem, we employ a variant of Bayesian global optimization (BGO), a class of algorithms capable of optimizing black-box, gradient-free, computationally expensive functions. In particular, we use the extended expected improvement over the dominated hypervolume to sequentially select simulations that seek to reveal the Pareto frontier. An addendum of the implemented BGO is that it quantifies the epistemic uncertainty about the Pareto frontier as induced by the limited number of simulations used to construct it.

Keywords

Surfactant-polymer flooding model Bayesian history matching Polynomial chaos expansion Markov chain Monte Carlo Model calibration Bayesian global optimization Uncertainty quantification 

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Notes

Acknowledgments

The authors would like to thank Bryan Clayton for his support and useful discussion.

Funding information

This research is partially supported by a grant from the Pioneer Oil Company.

References

  1. 1.
    Aanonsen, S., Naevdal, G., Oliver, D., Reynolds, A., Valles, B.: The ensemble Kalman filter in reservoir engineering —- a review. SPE J, 14(3) (2009)Google Scholar
  2. 2.
    Abidin, A., Puspasari, T., Nugroho, W.: Polymers for enhanced oil recovery technology. Procedia Chem. 4, 11–16 (2012).  https://doi.org/10.1016/j.proche.2012.06.002 Google Scholar
  3. 3.
    Ahmed, H., Awotunde, A.A., Sultan, A.S., Al-Yousef, H.Y., et al.: Stochastic optimization approach to surfactant-polymer flooding. In: SPE/PAPG Pakistan Section Annual Technical Conference and Exhibition. Society of Petroleum Engineers (2017)Google Scholar
  4. 4.
    Al-Sofi, A.M., Blunt, M.J.: The design and optimization of polymer flooding under uncertainty. SPE Enhanced Oil Recovery Conference.  https://doi.org/10.2118/145110-ms (2011)
  5. 5.
    Alfi, M., Hosseini, S.A.: Integration of reservoir simulation, history matching, and 4d seismic for co2-eor and storage at Cranfield, Mississippi, USA. Fuel 175, 116–128 (2016).  https://doi.org/10.1016/j.fuel.2016.02.032 Google Scholar
  6. 6.
    Alkhatib, A., King, P.: An approximate dynamic programming approachto decision making in the presence of uncertainty for surfactant-polymer flooding. Comput. Geosci. 18(2), 243–263 (2014).  https://doi.org/10.1007/s10596-014-9406-2 Google Scholar
  7. 7.
    Alsofi, A.M., Liu, J.S., Han, M., Aramco, S.: Numerical simulation of surfactant–polymer coreflooding experiments for carbonates. J. Pet. Sci. Eng. 111, 184–196 (2013)Google Scholar
  8. 8.
    Anderson, G.A.: Simulation of chemical flood enhanced oil recovery processes including the effects of reservoir wettability. Ph.D. thesis University of Texas at Austin (2006)Google Scholar
  9. 9.
    Andonyadis, P.: Decision support for enhanced oil recovery projects. Ph.D thesis (2010)Google Scholar
  10. 10.
    Aramideh, S., Borgohain, R., Naik, P.K., Johnston, C.T., Vlachos, P.P., Ardekani, A.M.: Multi-objective history matching of surfactant-polymer flooding. Fuel 228, 418–428 (2018).  https://doi.org/10.1016/j.fuel.2018.04.069. https://www.sciencedirect.com/science/article/pii/S0016236118307014 Google Scholar
  11. 11.
    Aramideh, S., Vlachos, P.P., Ardekani, A.M.: Unstable displacement of non-aqueous phase liquids with surfactant and polymer. Transp. Porous Media, 1–20 (2018)Google Scholar
  12. 12.
    Bailey, R., Baù, D.: Ensemble smoother assimilation of hydraulic head and return flow data to estimate hydraulic conductivity distribution, Water Resour. Res., 46(12) (2010)Google Scholar
  13. 13.
    Bazargan, H., Christie, M.: Bayesian model selection for complex geological structures using polynomial chaos proxy. Comput. Geosci. 21(3), 533–551 (2017).  https://doi.org/10.1007/s10596-017-9629-0 Google Scholar
  14. 14.
    Bazargan, H., Christie, M., Elsheikh, A.H., Ahmadi, M.: Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion. Adv. Water Resour. 86, 385–399 (2015).  https://doi.org/10.1016/j.advwatres.2015.09.009 Google Scholar
  15. 15.
    Bilionis, I., Zabaras, N.: Multi-output local gaussian process regression: Applications to uncertainty quantification. J. Comput. Phys. 231(17), 5718–5746 (2012)Google Scholar
  16. 16.
    Bilionis, I., Zabaras, N.: Solution of inverse problems with limited forward solver evaluations: A Bayesian perspective. Inverse Probl. 30(1), 015004 (2013)Google Scholar
  17. 17.
    Bilionis, I., Zabaras, N., Konomi, B.A., Lin, G.: Multi-output separable gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification. J. Comput. Phys. 241, 212–239 (2013)Google Scholar
  18. 18.
    Blatman, G., Sudret, B.: An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probab. Eng. Mech. 25(2), 183–197 (2010).  https://doi.org/10.1016/j.probengmech.2009.10.003 Google Scholar
  19. 19.
    Bonet-Cunha, L., Oliver, D., Redner, R., Reynolds, A.: A hybrid markov chain monte carlo method for generating permeability fields conditioned to multiwell pressure data and prior information. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/36566-ms (1996)
  20. 20.
    Box, G.E., Tiao, G.C.: Bayesian Inference in Statistical Analysis, vol. 40. Wiley (2011)Google Scholar
  21. 21.
    Brown, C., Smith, P.: The evaluation of uncertainty in surfactant eor performance prediction. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/13237-ms (1984)
  22. 22.
    Caers, J.: Efficient gradual deformation using a streamline-based proxy method. J. Pet. Sci. Eng. 39(1-2), 57–83 (2003).  https://doi.org/10.1016/s0920-4105(03)00040-8 Google Scholar
  23. 23.
    Caers, J.: Comparing the gradual deformation with the probability perturbation method for solving inverse problems. Mathematical Geology.  https://doi.org/10.1007/s11004-007-9119-3 (2007)
  24. 24.
    Chang, H., Liao, Q., Zhang, D.: Surrogate model based iterative ensemble smoother for subsurface flow data assimilation. Adv. Water Resour. 100, 96–108 (2017)Google Scholar
  25. 25.
    Chatzis, I., Morrow, N.R.: Correlation of capillary number relationships for sandstone. Soc. Petroleum Eng. J. 24(05), 555–562 (1984).  https://doi.org/10.2118/10114-pa Google Scholar
  26. 26.
    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)Google Scholar
  27. 27.
    Class, H., Mahl, L., Ahmed, W., Norden, B., Khn, M., Kempka, T.: Matching pressure measurements and observed co2 arrival times with static and dynamic modelling at the ketzin storage site. Energy Procedia 76, 623–632 (2015).  https://doi.org/10.1016/j.egypro.2015.07.883 Google Scholar
  28. 28.
    Cui, H., Kelkar, M.G.: Automatic history matching of naturally fractured reservoirs and a case study. SPE Western Regional Meeting.  https://doi.org/10.2118/94037-ms (2005)
  29. 29.
    Dachanuwattana, S., Jin, J., Zuloaga-Molero, P., Li, X., Xu, Y., Sepehrnoori, K., Yu, W., Miao, J.: Application of proxy-based mcmc and edfm to history match a vaca muerta shale oil well. Fuel 220, 490–502 (2018).  https://doi.org/10.1016/j.fuel.2018.02.018 Google Scholar
  30. 30.
    Dachanuwattana, S., Yu, W., Zuloaga-Molero, P., Sepehrnoori, K.: Application of assisted-history-matching workflow using proxy-based mcmc on a shale oil field case. J. Pet. Sci. Eng. 167, 316–328 (2018).  https://doi.org/10.1016/j.petrol.2018.04.029 Google Scholar
  31. 31.
    Delshad, M., Pope, G.: Comparison of the three-phase oil relative permeability models Transport in Porous Media 4(1) (1989)Google Scholar
  32. 32.
    Douarche, F., Da Veiga, S., Feraille, M., Enchéry, G., Touzani, S., Barsalou, R.: Sensitivity analysis and optimization of surfactant-polymer flooding under uncertainties. Oil & Gas Science and Technology–Revue d’IFP Energies nouvelles 69(4), 603–617 (2014)Google Scholar
  33. 33.
    Eaton, M.L., Sudderth, W.D.: Invariance of posterior distributions under reparametrization. Sankhya A 72(1), 101–118 (2010)Google Scholar
  34. 34.
    Elsheikh, A.H., Jackson, M.D., Laforce, T.C.: Bayesian reservoir history matching considering model and parameter uncertainties. Math. Geosci. 44(5), 515–543 (2012).  https://doi.org/10.1007/s11004-012-9397-2 Google Scholar
  35. 35.
    Emerick, A.A., Reynolds, A.C.: Combining the ensemble Kalman filter with Markov chain monte carlo for improved history matching and uncertainty characterization. SPE Reservoir Simulation Symposium.  https://doi.org/10.2118/141336-ms (2011)
  36. 36.
    Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013)Google Scholar
  37. 37.
    Emmerich, M., Deutz, A.H., Klinkenberg, J.W.: Hypervolume-based expected improvement: Monotonicity properties and exact computation. In: 2011 IEEE Congress on Evolutionary Computation (CEC), pp 2147–2154. IEEE (2011)Google Scholar
  38. 38.
    Emmerich, M., Giannakoglou, K.C., Naujoks, B.: Single-and multiobjective evolutionary optimization assisted by gaussian random field metamodels. IEEE Trans Evolut Comput 10(4), 421–439 (2006)Google Scholar
  39. 39.
    Evensen, G., Hove, J., Meisingset, H., Reiso, E., Seim, K.S., Espelid: Using the enkf for assisted history matching of a north sea reservoir model. SPE Reservoir Simulation Symposium.  https://doi.org/10.2118/106184-ms (2007)
  40. 40.
    Fajraoui, N., Marelli, S., Sudret, B.: Sequential design of experiment for sparse polynomial chaos expansions. SIAM/ASA J. Uncert. Quantif. 5(1), 1061–1085 (2017).  https://doi.org/10.1137/16m1103488 Google Scholar
  41. 41.
    Frazier, P., Powell, W., Dayanik, S.: The knowledge-gradient policy for correlated normal beliefs. Informs J. Comput. 21(4), 599–613 (2009).  https://doi.org/10.1287/ijoc.1080.0314 Google Scholar
  42. 42.
    Frazier, P.I., Powell, W.B., Dayanik, S.: A knowledge-gradient policy for sequential information collection. SIAM J. Control Optim. 47(5), 2410–2439 (2008).  https://doi.org/10.1137/070693424 Google Scholar
  43. 43.
    Fu, Y., Ding, J., Wang, H., Wang, J.: Two-objective stochastic flow-shop scheduling with deteriorating and learning effect in industry 4.0-based manufacturing system. Appl. Soft Comput. 68, 847–855 (2018)Google Scholar
  44. 44.
    Geir, N., Johnsen, L.M., Aanonsen, S.I., Vefring, E.H., et al.: Reservoir monitoring and continuous model updating using ensemble Kalman filter (2003)Google Scholar
  45. 45.
    Geweke, J., et al.: Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, vol. 196. Federal Reserve Bank of Minneapolis, Research Department Minneapolis, MN, USA (1991)Google Scholar
  46. 46.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer (1991)Google Scholar
  47. 47.
    Gu, Y., Oliver, D.S.: History matching of the punq-s3 reservoir model using the ensemble kalman filter. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/89942-ms (2004)
  48. 48.
    Hastings, W.K.: Monte carlo sampling methods using markov chains and their applications (1970)Google Scholar
  49. 49.
    Heidari, L., Gervais, V., Ravalec, M.L., Wackernagel, H.: History matching of petroleum reservoir models by the ensemble Kalman filter and parameterization methods. Comput. Geosci. 55, 84–95 (2013).  https://doi.org/10.1016/j.cageo.2012.06.006 Google Scholar
  50. 50.
    Hirasaki, G.J., Miller, C.A., Puerto, M.: Recent advances in surfactant eor. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/115386-ms (2008)
  51. 51.
    Hu, L.: Gradual deformation and iterative calibration of gaussian-related stochastic models. Math, Geol., 87—-108 (2000)Google Scholar
  52. 52.
    Huang, D., Allen, T.T., Notz, W.I., Zeng, N.: Global optimization of stochastic black-box systems via sequential Kriging meta-models. J. Global Optim. 34(3), 441–466 (2006)Google Scholar
  53. 53.
    Jahangiri, H.R., Zhang, D., et al.: Optimization of the net present value of carbon dioxide sequestration and enhanced oil recovery. In: Offshore Technology Conference. Offshore Technology Conference (2011)Google Scholar
  54. 54.
    Johnson, R.L., Greenstreet, C.W.: Managing uncertainty related to hydraulic fracturing modeling in complex stress environments with pressure-dependent leakoff. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/84492-ms (2003)
  55. 55.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)Google Scholar
  56. 56.
    Kamal, M.S., Hussein, I.A., Sultan, A.S.: Review on surfactant flooding: Phase behavior, retention, ift, and field applications. Energy Fuels 31(8), 7701–7720 (2017).  https://doi.org/10.1021/acs.energyfuels.7b00353 Google Scholar
  57. 57.
    Khaninezhad, M.M., Jafarpour, B.: Sparse randomized maximum likelihood (sprml) for subsurface flow model calibration and uncertainty quantification. Adv. Water Resour. 69, 23–37 (2014)Google Scholar
  58. 58.
    Lake, L.W.: Fundamentals of enhanced oil recovery. Society of Petroleum Engineers (2014)Google Scholar
  59. 59.
    Le Van, S., Chon, B.H.: Chemical flooding in heavy-oil reservoirs: From technical investigation to optimization using response surface methodology. Energies 9(9), 711 (2016)Google Scholar
  60. 60.
    Leray, S., Douarche, F., Tabary, R., Peysson, Y., Moreau, P., Preux, C.: Multi-objective assisted inversion of chemical eor corefloods for improving the predictive capacity of numerical models. J. Pet. Sci. Eng. 146, 1101–1115 (2016)Google Scholar
  61. 61.
    Li, G., Reynolds, A.C.: Uncertainty quantification of reservoir performance predictions using a stochastic optimization algorithm. Comput. Geosci. 15(3), 451–462 (2011)Google Scholar
  62. 62.
    Li, Q., Xing, H., Liu, J., Liu, X.: A review on hydraulic fracturing of unconventional reservoir. Petroleum 1(1), 8–15 (2015).  https://doi.org/10.1016/j.petlm.2015.03.008 Google Scholar
  63. 63.
    Luo, X., Stordal, A.S., Lorentzen, R.J., Nævdal, G.: Iterative ensemble smoother as an approximate solution to a regularized minimum-average-cost problem: theory and applications. SPE J. 20(05), 962–982 (2015).  https://doi.org/10.2118/176023-PA Google Scholar
  64. 64.
    Pope, M., Delshad G.K.S.: Utchem version 9.82 technical documentation. Center for Petroleum and Geosystems Engineering (2000)Google Scholar
  65. 65.
    Ma, X., Al-Harbi, M., Datta-Gupta, A., Efendiev, Y.: An efficient two-stage sampling method for uncertainty quantification in history matching geological models. SPE J. 13(01), 77–87 (2008).  https://doi.org/10.2118/102476-pa Google Scholar
  66. 66.
    Ma, X., Datta-Gupta, A., Efendiev, Y.: A multistage mcmc method with nonparametric error model for efficient uncertainty quantification in history matching. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/115911-ms (2008)
  67. 67.
    Marelli, S., Sudret, B.: UQLab user manual – polynomial chaos expansions. Report UQLab-V1.0-104, http://www.uqlab.com/download (2017)
  68. 68.
    Maschio, C., Schiozer, D.J.: Bayesian history matching using artificial neural network and Markov chain Monte Carlo. J. Pet. Sci. Eng. 123, 62–71 (2014).  https://doi.org/10.1016/j.petrol.2014.05.016 Google Scholar
  69. 69.
    Maschio, C., Schiozer, D.J.: A new methodology for Bayesian history matching using parallel interacting Markov chain monte carlo. Inverse Problems Sci. Eng. 26(4), 498–529 (2017).  https://doi.org/10.1080/17415977.2017.1322078 Google Scholar
  70. 70.
    McKay, M.D., Beckman, R.J., Conover, W.J.: Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2), 239–245 (1979)Google Scholar
  71. 71.
    Mohamed, L., Calderhead, B., Filippone, M., Christie, M., Girolami, M.: Population mcmc methods for history matching and uncertainty quantification. Comput. Geosci. 16(2), 423–436 (2011).  https://doi.org/10.1007/s10596-011-9232-8 Google Scholar
  72. 72.
    Mondal, A., Mallick, B., Efendiev, Y., Datta-Gupta, A.: Bayesian uncertainty quantification for subsurface inversion using a multiscale hierarchical model. Technometrics 56(3), 381–392 (2014)Google Scholar
  73. 73.
    Naik, P., Aramideh, S., Ardekani, A.M.: History matching of surfactant-polymer flooding using polynomial chaos expansion. Journal of Petroleum Science and Engineering (2018)Google Scholar
  74. 74.
    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)Google Scholar
  75. 75.
    Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press (2008)Google Scholar
  76. 76.
    Pandita, P., Bilionis, I., Panchal, J., Gautham, B., Joshi, A., Zagade, P.: Stochastic multiobjective optimization on a budget: Application to multipass wire drawing with quantified uncertainties. Int. J. Uncertain. Quantif., 8(3) (2018)Google Scholar
  77. 77.
    Patil, A., Huard, D., Fonnesbeck, C.J.: Pymc: Bayesian stochastic modelling in python. J. Statist. Softw. 35(4), 1 (2010)Google Scholar
  78. 78.
    Paul, B.K., Moulik, S.P.: The viscosity behaviors of micro emulsions: An overview (2000)Google Scholar
  79. 79.
    Pope, G., Wang, B., Tsaur, K.: A sensitivity study of micellar/polymer flooding. Soc. Pet. Eng. J. 19 (06), 357–368 (1979).  https://doi.org/10.2118/7079-pa Google Scholar
  80. 80.
    Pyrcz, M.J., White, C.D.: Uncertainty in reservoir modeling. Interpretation, 3(2).  https://doi.org/10.1190/int-2014-0126.1 (2015)
  81. 81.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge (2006). Tableofcontentsonly http://www.loc.gov/catdir/toc/fy0614/2005053433.html Google Scholar
  82. 82.
    Riazi, S.H., Zargar, G., Baharimoghadam, M., Moslemi, B., Darani, E.S.: Fractured reservoir history matching improved based on artificial intelligent. Petroleum 2(4), 344–360 (2016)Google Scholar
  83. 83.
    Roggero, F., Hu, L.: Gradual deformation of continuous geostatistical models for history matching. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/49004-ms (1998)
  84. 84.
    Schulze-Makuch, D.: Longitudinal dispersivity data and implications for scaling behavior. Ground Water 43 (3), 443–456 (2005).  https://doi.org/10.1111/j.1745-6584.2005.0051.x Google Scholar
  85. 85.
    Shah, D.O., Schechter, R.S.: Improved Oil Recovery by Surfactant and Polymer Flooding. Academic Press (1977)Google Scholar
  86. 86.
    Sheng, J.: Modern Chemical Enhanced Oil Recovery: Theory and Practice. Gulf Professional Publishing (2010)Google Scholar
  87. 87.
    Sheng, J.J.: Surfactant-polymer flooding. Modern Chemical Enhanced Oil Recovery, 371–387.  https://doi.org/10.1016/b978-1-85617-745-0.00009-7 (2011)
  88. 88.
    Sheng, J.J.: Status of surfactant eor technology. Petroleum 1(2), 97–105 (2015).  https://doi.org/10.1016/j.petlm.2015.07.003 Google Scholar
  89. 89.
    Solairaj, S., Britton, C., Kim, D.H., Weerasooriya, U., Pope, G.A.: Measurement and analysis of surfactant retention. SPE Improved Oil Recovery Symposium.  https://doi.org/10.2118/154247-ms (2012)
  90. 90.
    Stein, M.: Large sample properties of simulations using latin hypercube sampling. Technometrics 29(2), 143–151 (1987)Google Scholar
  91. 91.
    Suniga, P.T., Fortenberry, R., Delshad, M.: Observations of microemulsion viscosity for surfactant EOR processes. SPE Improved Oil Recovery Conference.  https://doi.org/10.2118/179669-ms (2016)
  92. 92.
    Supee, A., Idris, A.K.: Effects of surfactant-polymer formulation and salinities variation towards oil recovery. Arab. J. Sci. Eng. 39(5), 4251–4260 (2014).  https://doi.org/10.1007/s13369-014-1025-7 Google Scholar
  93. 93.
    Tagavifar, M., Herath, S., Weerasooriya, U.P., Sepehrnoori, K., Pope, G.: Measurement of microemulsion viscosity and its implications for chemical eor. SPE Improved Oil Recovery Conference.  https://doi.org/10.2118/179672-ms (2016)
  94. 94.
    Tavakoli, R., Reynolds, A.C.: Monte carlo simulation of permeability fields and reservoir performance predictions with svd parameterization in rml compared with enkf. Comput. Geosci. 15(1), 99–116 (2010).  https://doi.org/10.1007/s10596-010-9200-8 Google Scholar
  95. 95.
    Tsay, R.S.: Analysis of Financial Time Series, vol. 543. Wiley (2005)Google Scholar
  96. 96.
    Walker, D., Britton, C., Kim, D.H., Dufour, S., Weerasooriya, U., Pope, G.A.: The impact of microemulsion viscosity on oil recovery. SPE Improved Oil Recovery Symposium.  https://doi.org/10.2118/154275-ms(2012)
  97. 97.
    Wantawin, M., Yu, W., Sepehrnoori, K.: An iterative work flow for history matching by use of design of experiment, response-surface methodology, and Markov chain Monte Carlo algorithm applied to tight oil reservoirs. SPE Reserv. Eval. Eng. 20(03), 613–626 (2017).  https://doi.org/10.2118/185181-pa Google Scholar
  98. 98.
    Weiss, W., Baldwin, R.: Planning and implementing a large-scale polymer flood. J. Petrol. Tech. 37(04), 720–730 (1985).  https://doi.org/10.2118/12637-pa Google Scholar
  99. 99.
    Xiu, D., Karniadakis, G.E.: The wiener-askey polynomial chaos for stochastic differential equations.  https://doi.org/10.21236/ada460654 (2003)
  100. 100.
    Xu, T., Gómez-Hernández, J.J., Zhou, H., Li, L.: The power of transient piezometric head data in inverse modeling: An application of the localized normal-score enkf with covariance inflation in a heterogenous bimodal hydraulic conductivity field. Adv. Water Resour. 54, 100–118 (2013)Google Scholar
  101. 101.
    Yustres, Á., Asensio, L., Alonso, J., Navarro, V.: A review of Markov chain Monte Carlo and information theory tools for inverse problems in subsurface flow. Comput. Geosci. 16(1), 1–20 (2012)Google Scholar
  102. 102.
    Zeng, L., Zhang, D.: A stochastic collocation based Kalman filter for data assimilation. Comput. Geosci. 14(4), 721–744 (2010)Google Scholar
  103. 103.
    Zhang, D., Lu, Z., Chen, Y., et al.: Dynamic reservoir data assimilation with an efficient, dimension-reduced Kalman filter. Spe J. 12(01), 108–117 (2007)Google Scholar
  104. 104.
    Zhang, F., Skjervheim, J.A., Reynolds, A., Oliver, D.: Automatic history matching in a Bayesian framework, example applications. SPE Annual Technical Conference and Exhibition.  https://doi.org/10.2118/84461-ms (2003)
  105. 105.
    Zhang, J., Delshad, M., Sepehrnoori, K., Pope, G.A.: An efficient reservoir-simulation approach to design and optimize improved oil-recovery-processes with distributed computing. SPE Latin American and Caribbean Petroleum Engineering Conference.  https://doi.org/10.2118/94733-ms (2005)
  106. 106.
    Zheng, Z.: History matching and optimization using stochastic methods: Applications to chemical flooding. PhD Thesis. http://hdl.handle.net/1969.1/153874 (2014)
  107. 107.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Da Fonseca, V.G.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)Google Scholar

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Authors and Affiliations

  • Pratik Naik
    • 1
  • Piyush Pandita
    • 1
  • Soroush Aramideh
    • 1
  • Ilias Bilionis
    • 1
  • Arezoo M. Ardekani
    • 1
    Email author
  1. 1.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA

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