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Correspondence between one- and two-equation models for solute transport in two-region heterogeneous porous media

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https://ir.library.oregonstate.edu/concern/articles/4b29b6627

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  • Abstract In this work, we study the transient behavior of homogenized models for solute transport in two-region porous media. We focus on the following three models: (1) a time nonlocal, two-equation model (2eq-nlt). This model does not rely on time constraints and, therefore, is particularly useful in the short-time regime, when the time scale of interest (t) is smaller than the characteristic time (τ₁) for the relaxation of the effective macroscale parameters (i.e., when t ≤ τ₁); (2) a time local, two-equation model (2eq). This model can be adopted when (t) is significantly larger than (τ₁) (i.e., when t ≫ τ₁); and (3) a one-equation, time-asymptotic formulation (1eq∞). This model can be adopted when (t) is significantly larger than the time scale (τ₂) associated with exchange processes between the two regions (i.e., when t ≫ τ₂). In order to obtain insight into this transient behavior, we combine a theoretical approach based on the analysis of spatial moments with numerical and analytical results in several simple cases. The main result of this paper is to show that there is only a weak asymptotic convergence of the solution of (2eq) towards the solution of (1eq∞) in terms of standardized moments but, interestingly, not in terms of centered moments. The physical interpretation of this result is that deviations from the Fickian situation persist in the limit of long times but that the spreading of the solute is eventually dominating these higher order effects.
  • Keywords: homogenization, volume averaging, dispersion, porous media, spatial moments
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  • Davit, Y., Wood, B. D., Debenest, G., & Quintard, M. (2012). Correspondence between one- and two-equation models for solute transport in two-region heterogeneous porous media. Transport in Porous Media, 95(1), 213-238. doi: 10.1007/s11242-012-0040-y
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  • 95
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  • 1
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  • Support from CNRS/GdR 2990 is gratefully acknowledged. The second author (BDW) was supported in part by the Office of Science (BER), U.S. Department of Energy, Grant No. DE-FG02-07ER64417. This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
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