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Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion

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Abstract

In this study, we explore an efficient stochastic approach for uncertainty quantification of unconfined groundwater flow in heterogeneous media, where a sparse polynomial chaos expansion (PCE) surrogate model is constructed with the aid of the feature selection method. The feature selection method is introduced to construct a sparse PCE surrogate model with a reduced number of basis functions, which is accomplished by the least absolute shrinkage and selection operator-modified least angle regression and cross-validation. The training samples are enriched sequentially with the quasi-optimal samples until the results are satisfactory. In this study, we test the performance of the sparse PCE method for unconfined flow with the presence of random hydraulic conductivity and recharge, as well as pumping well. Numerical experiments reveal that, even with large spatial variability and high random dimensionality, the sparse PCE approach is able to accurately estimate the flow statistics with greatly reduced computational efforts compared to Monte Carlo simulations.

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References

  • Babuška, 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)

    Article  Google Scholar 

  • Ballio, F., Guadagnini, A.: Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour Res 40(4), W04603-1 (2004)

    Article  Google Scholar 

  • Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, New York (1972)

    Google Scholar 

  • 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)

    Article  Google Scholar 

  • Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)

    Article  Google Scholar 

  • Chang, H., Zhang, D.: A comparative study of stochastic collocation methods for flow in spatially correlated random fields. Commun. Comput. Phys. 6(3), 509 (2009)

    Google Scholar 

  • Dai, C., Li, H., Zhang, D.: Efficient and accurate global sensitivity analysis for reservoir simulations by use of probabilistic collocation method. SPE J. 19(04), 621–635 (2014)

    Article  Google Scholar 

  • Dash, M., Liu, H.: Feature selection for classification. Intell. Data Anal 1(3), 131–156 (1997)

    Article  Google Scholar 

  • Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)

    Article  Google Scholar 

  • Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)

    Article  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Fajraoui, N., Mara, T.A., Younes, A., Bouhlila, R.: Reactive transport parameter estimation and global sensitivity analysis using sparse polynomial chaos expansion. Water, Air, Soil Pollut. 223(7), 4183–4197 (2012)

    Article  Google Scholar 

  • Fajraoui, N., Marelli, S., Sudret, B.: On optimal experimental designs for sparse polynomial chaos expansions. arXiv preprint arXiv:1703.05312 (2017)

  • Ghanem, R.: Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123–2136 (1998)

    Article  Google Scholar 

  • Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Courier Corporation, New York (2003)

    Google Scholar 

  • Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215–223 (1979)

    Article  Google Scholar 

  • Guyon, I., Elisseeff, A.: An introduction to variable and feature selection. J. Mach. Learn. Res. 3(Mar), 1157–1182 (2003)

    Google Scholar 

  • Hampton, J., Doostan, A.: Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280, 363–386 (2015)

    Article  Google Scholar 

  • Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, New York (2009)

    Book  Google Scholar 

  • Le Maître, O.P., Reagan, M.T., Najm, H.N., Ghanem, R.G., Knio, O.M.: A stochastic projection method for fluid flow: II. Random process. J. Comput. Phys. 181(1), 9–44 (2002)

    Article  Google Scholar 

  • Li, H.: Conditional simulation of flow in random porous media with the probabilistic collocation method. Commun. Comput. Phys. 16(04), 1010–1030 (2014)

    Article  Google Scholar 

  • Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43(9), W09409 (2007)

    Article  Google Scholar 

  • Li, H., Zhang, D.: Stochastic representation and dimension reduction for non-Gaussian random fields: review and reflection. Stoch. Env. Res. Risk Assess. 27(7), 1621–1635 (2013)

    Article  Google Scholar 

  • Li, W., Lu, Z., Zhang, D.: Stochastic analysis of unsaturated flow with probabilistic collocation method. Water Resour Res. 45(8), W08425 (2009)

    Article  Google Scholar 

  • 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(02), 429–439 (2011)

    Article  Google Scholar 

  • Liao, Q., Zhang, D.: Constrained probabilistic collocation method for uncertainty quantification of geophysical models. Comput. Geosci. 19(2), 311–326 (2015)

    Article  Google Scholar 

  • Meng, J., Li, H.: An efficient stochastic approach for flow in porous media via sparse polynomial chaos expansion constructed by feature selection. Adv. Water Resour. 105, 13–28 (2017)

    Article  Google Scholar 

  • Nezhad, M.M., Javadi, A., Abbasi, F.: Stochastic finite element modelling of water flow in variably saturated heterogeneous soils. Int. J. Numer. Anal. Meth. Geomech. 35(12), 1389–1408 (2011)

    Article  Google Scholar 

  • Oladyshkin, S., de Barros, F., Nowak, W.: Global sensitivity analysis: a flexible and efficient framework with an example from stochastic hydrogeology. Adv. Water Resour. 37, 10–22 (2012)

    Article  Google Scholar 

  • Polmann, D.J., McLaughlin, D., Luis, S., Gelhar, L.W., Ababou, R.: Stochastic modeling of large-scale flow in heterogeneous unsaturated soils. Water Resour. Res. 27(7), 1447–1458 (1991)

    Article  Google Scholar 

  • Saeys, Y., Inza, I., Larrañaga, P.: A review of feature selection techniques in bioinformatics. Bioinformatics 23(19), 2507–2517 (2007)

    Article  Google Scholar 

  • Seshadri, P., Narayan, A., Mahadevan, S.: Optimal quadrature subsampling for least squares polynomial approximations. ArXiv e-prints (2016)

  • Shi, L., Yang, J., Zhang, D., Li, H.: Probabilistic collocation method for unconfined flow in heterogeneous media. J. Hydrol. 365(1), 4–10 (2009)

    Article  Google Scholar 

  • Shin, Y., Xiu, D.: Nonadaptive quasi-optimal points selection for least squares linear regression. SIAM J. Sci. Comput. 38(1), A385–A411 (2016)

    Article  Google Scholar 

  • Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58, 267–288 (1996)

    Google Scholar 

  • Webster, M.D., Tatang, M.A., McRae, G.J.: Application of the probabilistic collocation method for an uncertainty analysis of a simple ocean model. Massachusetts Institute of Technology Technical report, MIT Joint Program on the Science and Policy of Global Change Reports Series No. 4. (1996)

  • Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  Google Scholar 

  • Xiu, D., Karniadakis, G.E.: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Eng. 191(43), 4927–4948 (2002)

    Article  Google Scholar 

  • Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  Google Scholar 

  • Yan, L., Guo, L., Xiu, D.: Stochasic collocation algorithms using L1-minimization. Int. J. Uncertain. Quantif. 2(3), 279–293 (2012)

    Article  Google Scholar 

  • Zhang, D.: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, Cambridge (2001)

    Google Scholar 

  • 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)

    Article  Google Scholar 

Download references

Acknowledgements

This work is partially funded by the National Science and Technology Major Project of China ( Grant nos. 2017ZX05039-005 and 2016ZX05014-004-006).

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Authors and Affiliations

  1. Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing, 100871, China

    Jin Meng & Heng Li

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  1. Jin Meng
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  2. Heng Li
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Correspondence to Heng Li.

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Meng, J., Li, H. Efficient Uncertainty Quantification for Unconfined Flow in Heterogeneous Media with the Sparse Polynomial Chaos Expansion. Transp Porous Med 126, 23–38 (2019). https://doi.org/10.1007/s11242-017-0974-1

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  • DOI: https://doi.org/10.1007/s11242-017-0974-1

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