Advertisement

Computational Geosciences

, Volume 19, Issue 5, pp 979–998 | Cite as

Robust optimization of subsurface flow using polynomial chaos and response surface surrogates

  • Masoud Babaei
  • Ali Alkhatib
  • Indranil Pan
ORIGINAL PAPER

Abstract

This study employs an inclusive framework for surrogate model-based optimization in the presence of parametric and spatial uncertainties. The framework is applied to optimize water injection rate for optimal hydrocarbon recovery from a synthetic subsurface model with uncertainty in the geological and fluid relative permeability properties. In one model of parametric uncertainty, geological properties such as the channel’s absolute permeability and the fault transmissibility multiplier and the fluid relative permeability parameters such as the residual oil saturation to water and the water relative permeability at residual oil are assumed to be non-informative. In another model, the channels positions are assumed uncertain and various realizations of the channelized permeability are parameterized and the spatial uncertainty is accounted for in the optimization. The uncertainty is quantified in each evaluation of the objective function via polynomial chaos expansions. The coefficients of polynomial chaos expansion are solved by probabilistic collocation method. The objective function is assigned with a risk-averse net present value computed from a distribution of values obtained from the probabilistic proxies. The proxies are updated for each round of objective function evaluation. Monte-Carlo simulations are also conducted to verify accuracy and to demonstrate the computational efficiency of the probabilistic collocation approach. The optimization is conducted in various random input cases (depending on the number of uncertain parameters) and for each case net present value is successfully maximized and optimal solutions of the water injection rates are determined.

Keywords

Robust optimization Polynomial chaos expansion Probabilistic collocation method Uncertainty quantification Subsurface multiphase flow 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aanonsen, S.I., Eide, A.L., Holden, L., Aasen, J.O.: Optimizing reservoir performance under uncertainty with application to well location. In: the SPE Annual Technical Conference and Exhibition held in Dallas, U.S.A., 22-25 October, doi: 10.2118/30710-MS (1995)
  2. 2.
    Aitokhuehi, I., Durlofsky, L.J.: Optimizing the performance of smart wells in complex reservoirs using continuously updated geological models. J. Pet. Sci. Eng. 48(3), 254–264 (2005)CrossRefGoogle Scholar
  3. 3.
    Alkhatib, A., King, P.R.: An approximate dynamic programming approach to decision making in the presence of uncertainty for surfactant-polymer flooding. Comput. Geosci. 18(2), 243–263 (2014a)CrossRefGoogle Scholar
  4. 4.
    Alkhatib, A., King, P.R.: Robust quantification of parametric uncertainty for surfactant–polymer flooding. Comput. Geosci. 18(1), 77–101 (2014b)CrossRefGoogle Scholar
  5. 5.
    Artus, V., Durlofsky, L.J., Onwunalu, J., Aziz, K.: Optimization of nonconventional wells under uncertainty using statistical proxies. Comput. Geosci. 10(4), 389–404 (2006)CrossRefGoogle Scholar
  6. 6.
    Ashraf, M., Oladyshkin, S., Nowak, W.: Geological storage of CO 2 : application, feasibility and efficiency of global sensitivity analysis and risk assessment using the arbitrary polynomial chaos. Int. J. Greenh. Gas Control 19, 704–719 (2013). doi: 10.1016/j.ijggc.2013.03.023 CrossRefGoogle Scholar
  7. 7.
    Ben-Tal, A., Nemirovski, A.: Robust optimization–methodology and applications. Math. Prog. 92(3), 453–480 (2002)CrossRefGoogle Scholar
  8. 8.
    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)CrossRefGoogle Scholar
  9. 9.
    Beyer, H.G., Sendhoff, B.: Robust optimization—a comprehensive survey. Comput. Methods Appl. Mech. Eng. 196(33), 3190–3218 (2007)CrossRefGoogle Scholar
  10. 10.
    Blatman, G., Sudret, B.: Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Syst. Saf. 95(11), 1216–1229 (2010)CrossRefGoogle Scholar
  11. 11.
    Burton, M., Kumar, N., Bryant, S.L.: CO 2 injectivity into brine aquifers: why relative permeability matters as much as absolute permeability. Energy Proc. 1(1), 3091–3098 (2009)CrossRefGoogle Scholar
  12. 12.
    Busby, D., Farmer, C.L., Iske, A.: Hierarchical nonlinear approximation for experimental design and statistical data fitting. SIAM J. Sci. Comput. 29(1), 49–69 (2007)CrossRefGoogle Scholar
  13. 13.
  14. 14.
    Chen, Y., Oliver, D.: Ensemble-based closed-loop optimization applied to Brugge field. SPE Reserv. Eval. Eng. 13(1), 56–71 (2010)CrossRefGoogle Scholar
  15. 15.
    Cinnella, P., Hercus, S.: Robust optimization of dense gas flows under uncertain operating conditions. Comput. Fluids 39(10), 1893–1908 (2010)CrossRefGoogle Scholar
  16. 16.
    Da Cruz, P.S., Horne, R.N., Deutsch, C.V.: The quality map: a tool for reservoir uncertainty quantification and decision making. SPE Reserv. Eval. Eng. 7(01), 6–14 (2004)CrossRefGoogle Scholar
  17. 17.
    Cushman, J.H.: The physics of fluids in hierarchical porous media: Angstroms to miles. Kluwer Academic Publishers Dordrecht, The Netherlands (1997)CrossRefGoogle Scholar
  18. 18.
    Dagan, G.: Flow and transport in porous formations. Springer-Verlag GmbH & Co. KG (1989)Google Scholar
  19. 19.
    Dagan, G., Neuman S.P. Cambridge University Press, Subsurface flow and transport, A stochastic approach (2005)Google Scholar
  20. 20.
    Deutsch, C.V.: Geostatistical reservoir modeling. Oxford University Press (2002)Google Scholar
  21. 21.
    Dodson, M., Parks, G.T.: Robust aerodynamic design optimization using polynomial chaos. J. Aircr. 46 (2), 635–646 (2009)CrossRefGoogle Scholar
  22. 22.
    Dwight, R.P., Han, Z.H.: Efficient uncertainty quantification using gradient-enhanced kriging. AIAA Paper 2276 (2009)Google Scholar
  23. 23.
    Eldred, M.S.: Design under uncertainty employing stochastic expansion methods. Int. J. Uncertain. Quantif. 1(2) (2011)Google Scholar
  24. 24.
    Elsheikh, A.H., Hoteit, I., Wheeler, M.F.: Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates. Comput. Methods Appl. Mech. Eng. 269, 515–537 (2014)CrossRefGoogle Scholar
  25. 25.
    Van Essen, G., Zandvliet, M., Van den Hof, P., Bosgra, O., Jansen, J.D.: Robust waterflooding optimization of multiple geological scenarios. SPE J. 14(1), 202–210 (2009)CrossRefGoogle Scholar
  26. 26.
    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 nonreactive transport experiments in laboratory-scale porous media. Water Resour. Res. 47(2) (2011)Google Scholar
  27. 27.
    Feinberg J: Probabilistic collocation method module POLYCHAOS., https://bitbucket.org/jonathf/polychaos/src(2012)
  28. 28.
    Field, R., Grigoriu, M.: Convergence properties of polynomial chaos approximations for L 2 random variables. Public Report, Sandia National Laboratories, Albuquerque (2007)CrossRefGoogle Scholar
  29. 29.
    Foo, J., Karniadakis, G.E.: Multi-element probabilistic collocation method in high dimensions. J. Comput. Phys. 229(5), 1536–1557 (2010)CrossRefGoogle Scholar
  30. 30.
    Gautschi, W.: Algorithm 726: ORTHPOL—A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Softw. (TOMS) 20(1), 21–62 (1994)CrossRefGoogle Scholar
  31. 31.
    Gelhar, L.W.: Stochastic subsurface hydrology from theory to applications. Water Resources Research 22 (9S), 135S–145S (1986)CrossRefGoogle Scholar
  32. 32.
    Ghanem, R., Spanos, P.: A stochastic Galerkin expansion for nonlinear random vibration analysis. Probabilistic Eng. Mech. 8(3), 255–264 (1993)CrossRefGoogle Scholar
  33. 33.
    Glaz, B., Goel, T., Liu, L., Friedmann, P.P., Haftka, R.T.: Multiple-surrogate approach to helicopter rotor blade vibration reduction. AIAA J. 47(1), 271–282 (2009)CrossRefGoogle Scholar
  34. 34.
    Golub, G.H., Welsch, J.H.: Calculation of gauss quadrature rules. Math. Comput. 23(106), 221–230 (1969)CrossRefGoogle Scholar
  35. 35.
    Gorissen, D., Couckuyt, I., Laermans, E., Dhaene, T.: Multiobjective global surrogate modeling, dealing with the 5-percent problem. Eng. Comput. 26(1), 81–98 (2010)CrossRefGoogle Scholar
  36. 36.
    Güyagüler, B.: Optimization of well placement and assessment of uncertainty. PhD thesis, Stanford university (2002)Google Scholar
  37. 37.
    Güyagüler, B., Horne, R.N.: Uncertainty assessment of well-placement optimization. SPE Reserv. Eval. Eng. 7(1), 24–32 (2004)CrossRefGoogle Scholar
  38. 38.
    Huang, S., Quek, S., Phoon, K.: Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes. Int. J. Numer. Methods Eng. 52(9), 1029–1043 (2001)CrossRefGoogle Scholar
  39. 39.
    Isukapalli, S., Roy, A., Georgopoulos, P.: Stochastic response surface methods SRSMs for uncertainty propagation: Application to environmental and biological systems. Risk Anal. 18(3), 351–363 (1998)CrossRefGoogle Scholar
  40. 40.
    Jafarpour, B., McLaughlin, D.B.: History matching with an ensemble Kalman filter and discrete cosine parameterization. Comput. Geosci. 12(2), 227–244 (2008)CrossRefGoogle Scholar
  41. 41.
    Kalla, S., White, C.D.: Efficient design of reservoir simulation studies for development and optimization. SPE Reserv. Eval. Eng. 10(06), 629–637 (2007)CrossRefGoogle Scholar
  42. 42.
    Keese, A., Matthies, H.G.: Sparse quadrature as an alternative to Monte Carlo for stochastic finite element techniques. Proc. Appl. Math. Mech. 3(1), 493–494 (2003)CrossRefGoogle Scholar
  43. 43.
    Khu, S.T., Werner, M.G.: Reduction of Monte-Carlo simulation runs for uncertainty estimation in hydrological modelling. Hydrol. Earth Syst. Sci. Discuss. 7(5), 680–692 (2003)CrossRefGoogle Scholar
  44. 44.
    Kim, N.H., Wang, H., Queipo, N.V.: Efficient shape optimization under uncertainty using polynomial chaos expansions and local sensitivities. AIAA J. 44(5), 1112–1116 (2006)CrossRefGoogle Scholar
  45. 45.
    Krevor, S., Pini, R., Zuo, L., Benson, S.M.: Relative permeability and trapping of CO 2 and water in sandstone rocks at reservoir conditions. Water Resour. Res. 48(2) (2012)Google Scholar
  46. 46.
    Kruisselbrink, J., Emmerich, M., Bäck, T.: An archive maintenance scheme for finding robust solutions. In: Parallel Problem Solving from Nature, PPSN XI, pp 214–223. Springer (2010a)Google Scholar
  47. 47.
    Kruisselbrink, J., Emmerich, M., Deutz, A., Bäck, T.: Exploiting overlap when searching for robust optima. In: Parallel Problem Solving from Nature, PPSN XI, pp 63–72. Springer (2010b)Google Scholar
  48. 48.
    Laloy, E., Rogiers, B., Vrugt, J.A., Mallants, D., Jacques, D.: Efficient posterior exploration of a high-dimensional groundwater model from two-stage Markov chain Monte Carlo simulation and polynomial chaos expansion. Water Resour. Res. 49(5), 2664–2682 (2013)CrossRefGoogle Scholar
  49. 49.
    Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resour. Res. 43(9) (2007)Google Scholar
  50. 50.
    Li, H., Zhang, D.: Efficient and accurate quantification of uncertainty for multiphase flow with the probabilistic collocation method. SPE J. 14(4), 665–679 (2009)CrossRefGoogle Scholar
  51. 51.
    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(2), 429–439 (2011)CrossRefGoogle Scholar
  52. 52.
    Lie, K.A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source MATLAB implementation of consistent discretizations on complex grids. Comput. Geosci. 16(2), 297–322 (2012)CrossRefGoogle Scholar
  53. 53.
    Lin, G., Tartakovsky, A.M.: An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media. Adv. Water Resour. 32(5), 712–722 (2009)CrossRefGoogle Scholar
  54. 54.
    Loeven, G., Bijl, H.: Probabilistic collocation used in a two-step approach for efficient uncertainty quantification in computational fluid dynamics. Comput. Model. Eng. Sci. 36(3), 193–212 (2008)Google Scholar
  55. 55.
    Loeven, G., Witteveen, J., Bijl, H.: Probabilistic collocation: an efficient non-intrusive approach for arbitrarily distributed parametric uncertainties. In: Proceedings of the 45th AIAA Aerospace Sciences Meeting, vol. 6, pp 3845–3858 (2007)Google Scholar
  56. 56.
    Lophaven, S.N., Nielsen, H.B., Søndergaard, J.: DACE-a Matlab Kriging toolbox, version 2.0. Technical Report (2002)Google Scholar
  57. 57.
    Mandur, J., Budman, H.: Robust optimization of chemical processes using Bayesian description of parametric uncertainty. J. Process Control 24(2), 422–430 (2013)CrossRefGoogle Scholar
  58. 58.
    Manzocchi, T., Walsh, J., Nell, P., Yielding, G.: Fault transmissibility multipliers for flow simulation models. Pet. Geosci. 5(1), 53–63 (1999)CrossRefGoogle Scholar
  59. 59.
    Mathelin, L., Hussaini, M.Y.: A stochastic collocation algorithm for uncertainty analysis, Technical report. Florida State University (2003)Google Scholar
  60. 60.
    Matheron, G.: Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature, Masson Paris (1965)Google Scholar
  61. 61.
    Mathias, S.A., Gluyas, J.G., Martínez, G., De Miguel, G.J., Bryant, S.L., Wilson, D.: On relative permeability data uncertainty and CO 2 injectivity estimation for brine aquifers. Int. J. Greenh. Gas Control 12, 200–212 (2013)CrossRefGoogle Scholar
  62. 62.
    Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(12), 1295–1331 (2005)CrossRefGoogle Scholar
  63. 63.
    Mohaghegh, S.D., Modavi, A., Hafez, H., Haajizadeh, M.: Development of surrogate reservoir model SRM for fast track analysis of a complex reservoir. International Journal of Oil. Gas Coal Technol. 2(1), 2–23 (2009)CrossRefGoogle Scholar
  64. 64.
    Molina-Cristobal, A., Parks, G., Clarkson, P.: Finding robust solutions to multi-objective optimisation problems using polynomial chaos. In: Proceedings of the 6th ASMO UK/ISSMO Conference on Engineering Design Optimization. Citeseer (2006)Google Scholar
  65. 65.
    Mondal, A., Efendiev, Y., Mallick, B., Datta-Gupta, A.: Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain monte carlo methods. Adv. Water Res. 33(3), 241–256 (2010)CrossRefGoogle Scholar
  66. 66.
    Müller, J.: Surrogate model optimization toolbox. Technical report. Tampere University of Technology (2012)Google Scholar
  67. 67.
    Müller, J., Piché, R.: Mixture surrogate models based on Dempster-Shafer theory for global optimization problems. J. Global Optim. 51(1), 79–104 (2011)CrossRefGoogle Scholar
  68. 68.
    Müller, J., Shoemaker, C.A.: Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems, pp 1–22. Journal of Global Optimization (2014)Google Scholar
  69. 69.
    Nagy, Z., Braatz, R.: Distributional uncertainty analysis using power series and polynomial chaos expansions. J. Process Control 17(3), 229–240 (2007)CrossRefGoogle Scholar
  70. 70.
    Okano, H., Pickup, G., Christie, M., Subbey, S., Sambridge, M., Monfared, H.: Quantification of uncertainty in relative permeability for coarse-scale reservoir simulation. In: The SPE Europec/EAGE Annual Con in Madrid, pp 13–16. Society of Petroleum Engineers, Spain (2005)Google Scholar
  71. 71.
    Oladyshkin, S., Nowak, W.: Polynomial response surfaces for probabilistic risk assessment and risk control via robust design. doi: 10.5772/38170 (2012)
  72. 72.
    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(11), 1508–1518 (2011)CrossRefGoogle Scholar
  73. 73.
    Onorato, G., Loeven, G., Ghorbaniasl, G., Bijl, H., Lacor, C.: Comparison of intrusive and non-intrusive polynomial chaos methods for CFD applications in aeronautics. In: Proceedings of the 5th European conference on computational fluid dynamics. ECCOMAS CFD, Lisbon, Portugal (2010)Google Scholar
  74. 74.
    Onwunalu, J.E., Durlofsky, L.J.: Application of a particle swarm optimization algorithm for determining optimum well location and type. Comput. Geosci. 14(1), 183–198 (2010)CrossRefGoogle Scholar
  75. 75.
    Pan, Y., Horne, R.N.: Improved methods for multivariate optimization of field development scheduling and well placement design. In: The 1998 SPE Annual Technical Conference and Exhibition, pp 27–30. Society of Petroleum Engineers, Held in New Orleans, Louisiana (1998)Google Scholar
  76. 76.
    Petvipusit, K.R., Elsheikh, A.H., King, P.R., Blunt, M.J.: Robust optimisation using spectral high dimensional model representation-an application to CO2 sequestration strategy. In: ECMOR XIV-14th European conference on the mathematics of oil recovery (2014a)Google Scholar
  77. 77.
    Petvipusit, K.R., Elsheikh, A.H., LaForce, T.C., King, P.R., Blunt, M.J.: Robust optimisation of CO2 sequestration strategies under geological uncertainty using adaptive sparse grid surrogates. Comput. Geosci. 18(5), 763–778 (2014b)CrossRefGoogle Scholar
  78. 78.
    Petvipusit, K.R., Elsheikh, A.H., King, P.R., Blunt, M.J.: An efficient optimisation technique using adaptive spectral high-dimensional model representation: Application to CO2 sequestration strategies. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2015)Google Scholar
  79. 79.
    Queipo, N.V., Haftka, R.T., Shyy, W., Goel, T., Vaidyanathan, R., Kevin Tucker, P.: Surrogate-based analysis and optimization. Progress Aerosp. Sci. 41(1), 1–28 (2005)CrossRefGoogle Scholar
  80. 80.
    Rashid, K., Bailey, W.J., Couet, B., Wilkinson, D.: An efficient procedure for expensive reservoir-simulation optimization under uncertainty. SPE Econ. Manag. 5(4), 21–33 (2013)CrossRefGoogle Scholar
  81. 81.
    Razavi, S., Tolson, B.A., Burn, D.H.: Review of surrogate modeling in water resources. Water Resour. Res. 48(7) (2012)Google Scholar
  82. 82.
    Reagana, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M.: Uncertainty quantification in reacting-flow simulations through non-intrusive spectral projection. Combust. Flame 132(3), 545–555 (2003)CrossRefGoogle Scholar
  83. 83.
    Remy N: S-gems: the Stanford geostatistical modeling software: a tool for new algorithms development. In: Geostatistics Banff 2004, pp 865–871. Springer (2005)Google Scholar
  84. 84.
    Rohmer, J., Bouc, O.: A response surface methodology to address uncertainties in cap rock failure assessment for CO 2 geological storage in deep aquifers. Int. J. Greenh. Gas Control 4(2), 198–208 (2010)CrossRefGoogle Scholar
  85. 85.
    Rubin, Y.: Applied stochastic hydrogeology. Oxford University Press (2003)Google Scholar
  86. 86.
    Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–423 (1989)CrossRefGoogle Scholar
  87. 87.
    Sahinidis, N.V.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28(6), 971–983 (2004)CrossRefGoogle Scholar
  88. 88.
    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
  89. 89.
    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
  90. 90.
    Schölkopf, B., Smola, A., Müller, K.R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10(5), 1299–1319 (1998)CrossRefGoogle Scholar
  91. 91.
    Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002)CrossRefGoogle Scholar
  92. 92.
    Subbey, S., Monfared, H., Christie, M., Sambridge, M.: Quantifying uncertainty in flow functions derived from scal data. Trans. Porous Media 65(2), 265–286 (2006)CrossRefGoogle Scholar
  93. 93.
    Sun, A.Y., Zeidouni, M., Nicot, J.P., Lu, Z., Zhang, D.: Assessing leakage detectability at geologic CO 2 sequestration sites using the probabilistic collocation method. Adv. Water Res. 56, 49–60 (2013)CrossRefGoogle Scholar
  94. 94.
    Tatang, M.A.: Direct incorporation of uncertainty in chemical and environmental engineering systems. PhD thesis, Massachusetts Institute of Technology (1995)Google Scholar
  95. 95.
    Tatang, M.A., Pan, W., Prinn, R.G., McRae, G.J.: An efficient method for parametric uncertainty analysis of numerical geophysical models. J. Geophys. Res. 102(D18), 21,925–21,932 (1997)CrossRefGoogle Scholar
  96. 96.
    Tsutsui, S., Ghosh, A.: Genetic algorithms with a robust solution searching scheme. Evolutionary Computation. IEEE Trans. Evol. Comput. 1(3), 201–208 (1997)CrossRefGoogle Scholar
  97. 97.
    Viana, F.A., Gogu, C., Haftka, R.T.: Making the most out of surrogate models: tricks of the trade. In: ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp 587?-598. American Society of Mechanical Engineers (2010)Google Scholar
  98. 98.
    Vincent, G., Corre, B., Thore, P.: Managing structural uncertainty in a mature field for optimal well placement. SPE Reserv. Eval. Eng 2(04), 377–384 (1999)CrossRefGoogle Scholar
  99. 99.
    Vo, H.X., Durlofsky, L.J.: A new differentiable parameterization based on Principal Component Analysis for the low-dimensional representation of complex geological models. Mathematical Geosciences (2014)Google Scholar
  100. 100.
    Wang, H., Echeverría-Ciaurri, D., Durlofsky, L.J., Cominelli, A.: Optimal well placement under uncertainty using a retrospective optimization framework. SPE J. 17(1), 112–121 (2012)CrossRefGoogle Scholar
  101. 101.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)CrossRefGoogle Scholar
  102. 102.
    Xiong, F., Xue, B., Yan, Z., Yang, S.: Polynomial chaos expansion based robust design optimization. In: International Conference on Quality, Reliability, Risk, Maintenance, and Safety Engineering (ICQR2MSE), pp 868–873. IEEE (2011)Google Scholar
  103. 103.
    Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)CrossRefGoogle Scholar
  104. 104.
    Yeh, W.W.G.: Reservoir management and operations models: a state-of-the-art review. Water Resour. Res. 21(12), 1797–1818 (1985)CrossRefGoogle Scholar
  105. 105.
    Yeten, B., Durlofsky, L.J., Aziz, K.: Optimization of nonconventional well type, location, and trajectory. SPE J. 8(3), 200–210 (2003)CrossRefGoogle Scholar
  106. 106.
    Zein, S.: A polynomial chaos expansion trust region method for robust optimization. Commun. Comput. Phys. 14(2), 412–424 (2013)Google Scholar
  107. 107.
    Zhang, D.: Stochastic methods for flow in porous media: coping with uncertainties. Academic Press (2001)Google Scholar
  108. 108.
    Zhang, D., Lu, Z.: An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loeve and polynomial expansions. J. Comput. Phys. 194(2), 773–794 (2004)CrossRefGoogle Scholar
  109. 109.
    Zhang, J., Chowdhury, S., Messac, A.: An adaptive hybrid surrogate model. Struct. Multidiscip. Optim. 46(2), 223–238 (2012)CrossRefGoogle Scholar
  110. 110.
    Zhang, Y., Sahinidis, N.V.: Uncertainty quantification in CO 2 sequestration using surrogate models from polynomial chaos expansion. Ind. Eng. Chem. Res. 52(9), 3121–3132 (2012)CrossRefGoogle Scholar
  111. 111.
    Zhou, Z., Ong, Y.S., Lim, M.H., Lee, B.S.: Memetic algorithm using multi-surrogates for computationally expensive optimization problems. Soft Comput. 11(10), 957–971 (2007)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Chemical Engineering and Analytical ScienceUniversity of ManchesterManchesterUnited Kingdom
  2. 2.Saudi AramcoDhahranSaudi Arabia
  3. 3.Imperial College LondonLondonUnited Kingdom

Personalised recommendations