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On non-Fickian hyperbolic diffusion

Auriault J. L., Lewandowska J., Royer P.

Studia Geotechnica et Mechanica

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2008

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Vol. 30, nr 1-2

139-146

EN

Fick's law expresses the proportionality of solute flux with respect to concentration gradient. Similar relations are given by Darcy's law for the fluid flow in porous media, Ohm's law for the electric flux and Fourier's law for heat transfers. When introduced in the corresponding balance equations, these flux laws yield diffusion equations of parabolic character. Different attempts have been made to obtain hyperbolic equations so as to point out propagative phenomena. This was done by adding a time derivative flux term to the flow law. In this paper, we focus on solute transport. Two possible non-Fickian diffusion cases are addressed. We firstly investigate diffusion in fluids by a mechanistic approach. Secondly, we study the macroscopic diffusion law in composite materials with large contrast of diffusion coefficient. We show that the diffusion law obtained yields hyperbolicity for drastically short characteristic times or non-propagative waves.

2

Contaminant transport in fractured porous media

Royer P., Auriault J. L., Lewandowska J., Serres C.

Studia Geotechnica et Mechanica

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2003

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Vol. 25, nr 1-2

171--180

EN

This work is aimed at deriving mathematical models that describe pollutant migration through fractured porous media. A homogenisation method is used, i.e. macroscopic models are rigorously deduced from the physical description which is valid within a Representative Elementary Volume (REV). The fundamental assumption behind homogenisation is the separation of scales which is expressed by: l/L = E<<1. In the present work, l denotes the characteristic size of the REV, i.e. at the fracture's scale, and L is the characteristic macroscopic size. The approach introduced by Auriault [1] is used. This methodology is based on the definition and estimation of dimensionless numbers arising from the description at the REV's scale. It is shown that the macroscopic behaviour strongly depends upon the local transport regime characterised by the Peclet number in the fractures. Four distinct macroscopic models for solute transport in fractured porous media are derived.

3

Homogenization of compressible fluid flow in porous media with interfacial flow barrier

Royer P., Auriault J.

Archives of Mechanics

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1999

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Vol. 51, nr 3-4

469-486

EN

This work is concerned with modelling compressible fluid flow in a composite porous medium with interfacial flow barrier. The macroscopic behaviour and the effective permeability are obtained by homogenization, i.e. by upscaling the description at the heterogeneity scale. Five distinct macroscopic models are derived that relate to five relative orders of magnitude of the interfacial conductance with respect to the permeabilities of the constituents.