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Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method

Published online by Cambridge University Press:  03 June 2015

Heng Li
Heng Li*
Affiliation:
Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China
*
*Corresponding author.Email:[email protected]
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Abstract

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.

Keywords

60H3565C5065M7076S05Conditional simulationprobabilistic collocation methodKarhunen-Loeve expansionpolynomial chaos expansion
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 1010 - 1030
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Ballio, F. and Guadagnini, A.Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour. Res., 40,2004. y04603, doi:10.1029/ 2003WR002876.Google Scholar
[2]Chen, M., Keller, A. A., Zhang, D., Lu, Z., and Zyvoloski, G. A.Stochastic analysis of transient two-phase flow in heterogeneous porous media. Water Resour. Res., 42, 2006. W03425, doi:10.1029/2005WR004257.Google Scholar
[3]Chen, M. and Zhang, D.Stochastic analysis of two phase flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method. Water Resour. Res., 41, 2005. W01006, doi:10.1029/2004WR003412.CrossRefGoogle Scholar
[4]Cushman, J. H.The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. Kluwer Academic, Norwell, MA., 1997.Google Scholar
[5]Dagan, G.Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 1. sponditional simulation and the direct problem. Water Resour Res, 18(4), 1982, 813–33.Google Scholar
[6]Dagan, G.Flow and Transport in Porous Formations. Springer, New York, 1989.CrossRefGoogle Scholar
[7]Gelhar, L. W.Stochastic Subsurface Hydrology. Prentice-Hall, Englewood Cliffs, NJ., 1993.Google Scholar
[8]Ghanem, R. and Spanos, P.Stochastic Finite Element: A Spectral Approach. Springer, New York, 1991.CrossRefGoogle Scholar
[9]Graham, W. D. and McLaughlin, D.Stochastic analysis of nonstationary subsurface solute transport: 1. spnconditional moments. Water Resour. Res., 25(2), 1989, 215–232.Google Scholar
[10]Graham, W. D. and McLaughlin, D.Stochastic analysis of nonstationary subsurface solute transport: 2. sponditional moments. Water Resour. Res., 25(11), 1989, 2331–55.CrossRefGoogle Scholar
[11]Guadagnini, A. and Neuman, S. P.Nonlocal and localized analysis of conditional mean steady state flow in bounded, randomly nonuniform domains: 1. spheory and computational approach. Water Resour. Res., 35(10), 1999, 2999–3018.Google Scholar
[12]Krige, D. G.A statistical analysis of some mine valuation and allied problems on the witwatersrand. Master’s thesis, University of Witwatersrand, 1951.Google Scholar
[13]Li, H. and Zhang, D.Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resour. Res., 43, 2007. W09409, doi:10.1029/2006WR005673.Google Scholar
[14]Li, H. and Zhang, D.Efficient and accurate quantification of uncertainty for multiphase flow with the probabilistic collocation method. SPE Journal, 14(4), 2009, 665–679.Google Scholar
[15]Li, W., Lu, Z., and Zhang, D.Ptochastic analysis of unsaturated flow with probabilistic collo-cation method. Water Resour. Res., 45, 2009. W08425, doi:10.1029/2008WR007530.Google Scholar
[16]Liu, G., Lu, Z., and Zhang, D.Ptochastic uncertainty analysis for solute transport in randomly heterogeneous media using a Karhunen-Loeve based moment equation approach. Water Resour. Res., 43, 2007. W07427, doi:10.1029/2006WR005193.Google Scholar
[17]Lu, Z. and Zhang, D.Conditional simulations of flow in randomly heterogeneous porous media using a KL-based moment-equation approach. Adv. Water Res., 27, 2004,859–874.Google Scholar
[18]Lu, Z. and Zhang, D.Stochastic simulations for flow in nonstationary randomly heterogeneous porous media using a KL-based moment-equation approach. SIAM Multiscale Model. Simul., 6(1), 2007, 228–245.CrossRefGoogle Scholar
[19]Neuman, S. P.Eulerian-Lagrangian theory of transport in spacetime nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximations. Water Resour. Res., 29, 1993, 633–645.CrossRefGoogle Scholar
[20]Neuman, S. P.Stochastic approach to subsurface flow and transport: A view to the future. In Dagan, G. and Neuman, S. P., editors, Subsurface Flow and Transport: A Stochastic Approach, pages 231–241. spambridge University Press, New York, 1997.Google Scholar
[21]Rubin, Y.Prediction of tracer plume migration in disordered porous media by the method of conditional probabilities. Water Resour Res, 27(6), 1991,1291–1308.Google Scholar
[22]Rubin, Y.Spplied Stochastic Hydrogeology. Oxford University Press, New York, 2003.Google Scholar
[23]Shi, L., Yang, J., Zhang, D., and Li, H.Probabilistic collocation method for unconfined flow in heterogeneous media. J. Hydrol., 365, 2009,4–10.Google Scholar
[24]Tatang, M. A., Pan, W., Prinn, R. G., and McRae, G. J.An efficient method for parametric uncertainty analysis of numerical geophysical models. J. Geophy. Res., D18, 1997, 21925–21931.CrossRefGoogle Scholar
[25]Xiu, D. and Karniadakis, G. E.Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Eng., 191, 2002,4927–4948.Google Scholar
[26]Yang, J., Zhang, D., and Lu, Z.Stochastic analysis of saturated-unsaturated flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method. J. Hydrol., 294, 2004,18–38.Google Scholar
[27]Zhang, D.Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded heterogeneous media. Water Resour. Res., 34, 1998,529–538.Google Scholar
[28]Zhang, D.Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media. Water Resour. Res., 35, 1999,1127–1141.CrossRefGoogle Scholar
[29]Zhang, D.Stochastic Methods for Flow in Porous Media: Coping With Uncertainties. Academic Press, San Diego, CA, 2002.Google Scholar
[30]Zhang, D. and Lu, Z.An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Loeve and polynomial expansions. J. Comput. Phys., 194, 2004, 773–794.Google Scholar
[31]Zhang, D. and Neuman, S. P.Eulerian-Lagrangian analysis of transport conditioned on hy-draulic data, 1. spnalytical-numerical approach. Water Resour. Res., 31, 1995,39–51.Google Scholar