Abstract
We leverage on information theory to assess the fidelity of approximated numerical stochastic groundwater flow simulations. We consider flow in saturated heterogeneous porous media, where the Karhunen–Loève (KL) expansion is used to express the hydraulic conductivity as a spatially correlated random field. We quantify the impact of the KL expansion truncation on the uncertainty associated with punctual values of hydraulic conductivity and flow velocity. In particular, we compare the statistical dependence between variables by considering (a) linear correlation metrics (Pearson coefficient of correlation) and (b) metrics capable of accounting for nonlinear dependence (coefficient of uncertainty based on mutual information). We test the selected metrics by analyzing the relationship between hydraulic conductivity fields generated via Monte Carlo sampling with different levels of truncation of the KL expansion and the corresponding fluid velocity fields, obtained through the numerical solution of Darcy’s flow. Our analysis shows that employing linear correlation metrics leads to a general overestimation of the correlation level and information theory based indicators are valuable tools to assess the impact of the KL truncation on the output velocity values. We then analyze the impact of the number of retained modes on the spatial organization of the velocity field. Results indicates that (i) as the number of modes decrease the spatial correlations of the velocity field increases; (ii) linear indicators of spatial correlation are again larger than their nonlinear counterparts.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bianchi, M., Pedretti, D.: Geological entropy and solute transport in heterogeneous porous media. Water Resour. Res. 53(6), 4691–4708 (2017). https://doi.org/10.1002/2016WR020195
Bianchi, M., Pedretti, D.: An entrogram-based approach to describe spatial heterogeneity with applications to solute transport in porous media. Water Resour. Res. 54(7), 4432–4448 (2018). https://doi.org/10.1029/2018WR022827
Butera, I., Vallivero, L., Ridolfi, L.: Mutual information analysis to approach nonlinearity in groundwater stochastic fields. Stoch. Environ. Res. Risk Assess. 32(10), 2933–2942 (2018). https://doi.org/10.1007/s00477-018-1591-4
Comolli, A., Hakoun, V., Dentz, M.: Mechanisms, upscaling, and prediction of anomalous dispersion in heterogeneous porous media. Water Resour. Res. 55(10), 8197–8222 (2019). https://doi.org/10.1029/2019WR024919
Cover, T.M., Thomas, J.A.: Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, Hoboken (2006)
Dagan, G.: Flow and Transport in Porous Formations. Springer, Berlin (1989)
de Barros, F.P.J., Rubin, Y.: A risk-driven approach for subsurface site characterization. Water Resour. Res. (2008). https://doi.org/10.1029/2007WR006081
Dell’Oca, A., Porta, G., Guadagnini, A., Riva, M.: Space-time mesh adaptation for solute transport in randomly heterogeneous porous media. J. Contam. Hydrol. 212, 28–40 (2018). https://doi.org/10.1016/j.jconhyd.2017.07.001
Esfandiar, B., Porta, G., Perotto, S., Guadagnini, A.: Anisotropic mesh and time step adaptivity for solute transport modeling in porous media. In: Perotto, S., Formaggia, L. (eds.) New Challenges in Grid Generation and Adaptivity for Scientific Computing. SEMA SIMAI Springer Series, pp. 1–25. Springer, Milan (2014)
Fiori, A., Jankovic, I.: On preferential flow, channeling and connectivity in heterogeneous porous formations. Math. Geosci. 44(2), 133–145 (2012). https://doi.org/10.1007/s11004-011-9365-2
Gelhar, L.: Stochastic Subsurface Hydrology. Prentice-Hall, Upper Saddle River (1993)
Goodwell, A.E., Kumar, P.: Temporal information partitioning: characterizing synergy, uniqueness, and redundancy in interacting environmental variables. Water Resour. Res. 53(7), 5920–5942 (2017). https://doi.org/10.1002/2016WR020216
Hakoun, V., Comolli, A., Dentz, M.: Upscaling and prediction of Lagrangian velocity dynamics in heterogeneous porous media. Water Resour. Res. (2019). https://doi.org/10.1029/2018WR023810
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3–4), 251–265 (2012)
Kaiser, A., Schreiber, T.: Information transfer in continuous processes. Physica D 166(1), 43–62 (2002). https://doi.org/10.1016/S0167-2789(02)00432-3
Katsoulakis, M.A., Plechaĉ, P.: Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems. J. Chem. Phys. 139(7), 074115 (2013). https://doi.org/10.1063/1.4818534
Kikuchi, C.P., Ferré, T.P.A., Vrugt, J.A.: On the optimal design of experiments for conceptual and predictive discrimination of hydrologic system models. Water Resour. Res. 51(6), 4454–4481 (2015). https://doi.org/10.1002/2014WR016795
Le Borgne, T., de Dreuzy, J.R., Davy, P., Bour, O.: Characterization of the velocity field organization in heterogeneous media by conditional correlation. Water Resour. Res. (2007). https://doi.org/10.1029/2006WR004875
Lin, G., Tartakovsky, A.: 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). https://doi.org/10.1016/j.advwatres.2008.09.003
Liu, Z., Liu, Z., Peng, Y.: Dimension reduction of Karhunen–Loeve expansion for simulation of stochastic processes. J. Sound Vib. 408, 168–189 (2017). https://doi.org/10.1016/j.jsv.2017.07.016
Loève, M.: Probability Theory. Graduate Texts in Mathematics. Springer, New York (1978)
Marzouk, Y.M., Najm, H.N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems. J. Comput. Phys. 228(6), 1862–1902 (2009). https://doi.org/10.1016/j.jcp.2008.11.024
Moslehi, M., de Barros, F.: Uncertainty quantification of environmental performance metrics in heterogeneous aquifers with long-range correlations. J. Contam. Hydrol. 196, 21–29 (2017). https://doi.org/10.1016/j.jconhyd.2016.12.002
Nearing, G.S., Ruddell, B.L., Clark, M.P., Nijssen, B., Peters-Lidard, C.: Benchmarking and process diagnostics of land models. J. Hydrometeorol. 19(11), 1835–1852 (2018). https://doi.org/10.1175/JHM-D-17-0209.1
Neuman, S.P., Guadagnini, A., Riva, M., Siena, M.: Recent advances in statistical and scaling analysis of earth and environmental variables. In: Mishra, P.K., Kuhlman, K.L. (eds.) Advances in Hydrogeology, pp. 1–25. Springer, New York (2013). https://doi.org/10.1007/978-1-4614-6479-21
Oberst, S., Niven, R.K., Lester, D.R., Ord, A., Hobbs, B., Hoffmann, N.: Detection of unstable periodic orbits in mineralising geological systems. Chaos Interdiscip. J. Nonlinear Sci. 28(8), 085711 (2018). https://doi.org/10.1063/1.5024134
Renard, P., Allard, D.: Connectivity metrics for subsurface flow and transport. Adv. Water Resour. 51, 168–196 (2013). https://doi.org/10.1016/j.advwatres.2011.12.001
Ruddell, B.L., Kumar, P.: Ecohydrologic process networks: 1. Identification. Water Resour. Res. (2009). https://doi.org/10.1029/2008WR007279
Salandin, P., Fiorotto, V.: Solute particles in highly heterogeneous aquifers. Water Resour. Res. 34(5), 949–961 (1998)
Shannon, C.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948). https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
Theil, H.: Statistical Decomposition Analysis. North-Holland Publishing Co, Amsterdam (1972). https://doi.org/10.1017/S0770451800006916
Wright, E., Sund, N., Richter, D., Porta, G., Bolster, D.: Upscaling mixing in highly heterogeneous porous media via a spatial Markov model. Water 11(1), 53 (2018). https://doi.org/10.3390/w11010053
Zahm, O., Constantine, P., Prieur, C., Marzouk, Y.: Gradient-based dimension reduction of multivariate vector-valued functions. (2018). arXiv:1801.07922
Zhang, D., Lu, Z.: An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loève and polynomial expansions. J. Comput. Phys. 194(2), 773–794 (2004). https://doi.org/10.1016/j.jcp.2003.09.015
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dell’Oca, A., Porta, G.M. Characterization of flow through random media via Karhunen–Loève expansion: an information theory perspective. Int J Geomath 11, 18 (2020). https://doi.org/10.1007/s13137-020-00155-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13137-020-00155-x