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Transport in Porous Media

, Volume 126, Issue 1, pp 23–38 | Cite as

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

  • Jin Meng
  • Heng Li
Article

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.

Keywords

Uncertainty quantification Unconfined flow Sparse PCE Feature selection Quasi-optimal sampling 

Notes

Acknowledgements

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

References

  1. 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)Google Scholar
  2. Ballio, F., Guadagnini, A.: Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour Res 40(4), W04603-1 (2004)Google Scholar
  3. Bear, J.: Dynamics of Fluids in Porous Media. Courier Corporation, New York (1972)Google Scholar
  4. 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)Google Scholar
  5. Blatman, G., Sudret, B.: Adaptive sparse polynomial chaos expansion based on least angle regression. J. Comput. Phys. 230(6), 2345–2367 (2011)Google Scholar
  6. 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
  7. 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)Google Scholar
  8. Dash, M., Liu, H.: Feature selection for classification. Intell. Data Anal 1(3), 131–156 (1997)Google Scholar
  9. Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)Google Scholar
  10. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)Google Scholar
  11. 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)Google Scholar
  12. 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)Google Scholar
  13. Fajraoui, N., Marelli, S., Sudret, B.: On optimal experimental designs for sparse polynomial chaos expansions. arXiv preprint arXiv:1703.05312 (2017)
  14. Ghanem, R.: Scales of fluctuation and the propagation of uncertainty in random porous media. Water Resour. Res. 34(9), 2123–2136 (1998)Google Scholar
  15. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Courier Corporation, New York (2003)Google Scholar
  16. 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)Google Scholar
  17. Guyon, I., Elisseeff, A.: An introduction to variable and feature selection. J. Mach. Learn. Res. 3(Mar), 1157–1182 (2003)Google Scholar
  18. Hampton, J., Doostan, A.: Compressive sampling of polynomial chaos expansions: convergence analysis and sampling strategies. J. Comput. Phys. 280, 363–386 (2015)Google Scholar
  19. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning, 2nd edn. Springer, New York (2009)Google Scholar
  20. 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)Google Scholar
  21. Li, H.: Conditional simulation of flow in random porous media with the probabilistic collocation method. Commun. Comput. Phys. 16(04), 1010–1030 (2014)Google Scholar
  22. Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43(9), W09409 (2007)Google Scholar
  23. 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)Google Scholar
  24. Li, W., Lu, Z., Zhang, D.: Stochastic analysis of unsaturated flow with probabilistic collocation method. Water Resour Res. 45(8), W08425 (2009)Google Scholar
  25. 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)Google Scholar
  26. Liao, Q., Zhang, D.: Constrained probabilistic collocation method for uncertainty quantification of geophysical models. Comput. Geosci. 19(2), 311–326 (2015)Google Scholar
  27. 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)Google Scholar
  28. 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)Google Scholar
  29. 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)Google Scholar
  30. 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)Google Scholar
  31. Saeys, Y., Inza, I., Larrañaga, P.: A review of feature selection techniques in bioinformatics. Bioinformatics 23(19), 2507–2517 (2007)Google Scholar
  32. Seshadri, P., Narayan, A., Mahadevan, S.: Optimal quadrature subsampling for least squares polynomial approximations. ArXiv e-prints (2016)Google Scholar
  33. Shi, L., Yang, J., Zhang, D., Li, H.: Probabilistic collocation method for unconfined flow in heterogeneous media. J. Hydrol. 365(1), 4–10 (2009)Google Scholar
  34. Shin, Y., Xiu, D.: Nonadaptive quasi-optimal points selection for least squares linear regression. SIAM J. Sci. Comput. 38(1), A385–A411 (2016)Google Scholar
  35. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.) 58, 267–288 (1996)Google Scholar
  36. 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)Google Scholar
  37. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)Google Scholar
  38. 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)Google Scholar
  39. Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)Google Scholar
  40. Yan, L., Guo, L., Xiu, D.: Stochasic collocation algorithms using L1-minimization. Int. J. Uncertain. Quantif. 2(3), 279–293 (2012)Google Scholar
  41. Zhang, D.: Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, Cambridge (2001)Google Scholar
  42. 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)Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Energy and Resources Engineering, College of EngineeringPeking UniversityBeijingChina

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