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Upscaling diffusion–reaction in porous media

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Abstract

In this work, we present the outlines of the periodic homogenization of the diffusion equation with chemical reaction at the interface, for different orders of magnitude of the Damköhler number. For large values of the Damköhler number, a non-classical homogenized model is obtained, where the homogenized diffusion tensor is strongly coupled with the chemical reaction rate. This homogenized model is particularly well adapted to describe, at the macroscopic level, diffusion with strong chemical reactions at the pore interfaces. The aim of this article is to highlight the transition between different regimes of diffusion–reaction according to the order of magnitude of the Damköhler number. In the last part of this work, we first consider a simple analytical example between two parallel plates, to understand the transition between the different possible regimes of diffusion–reaction. Finally, numerical simulations are performed on more complex two-dimensional elementary cells.

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Notes

  1. The variables with \((*)\) are dimensional variables.

  2. Let us quote that \(\left\langle c\right\rangle _{\varOmega _f}=c\).

  3. Where \(\left\langle c \right\rangle _{\varOmega _{f}}=\left\langle c \right\rangle _{\varOmega _{f}}\left( {\mathbf {x}},t\right) \).

  4. The subscript r is used for distinguishing the variable \(\mathbf {\chi }_{r}\) from the variable \(\mathbf {\chi }\) solution of problem (15), (16).

  5. We adopt the following simplification of notation: \(Da_l \equiv Da \) in Sect. 4

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Acknowledgements

The authors wish to acknowledge the financial support by the GdRI Multi-physics and Multi-scale Couplings in Geo-environmental Mechanics (GdRI GeoMech).

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Authors and Affiliations

  1. LGCGM EA 3913, INSA Rennes, 20 Avenue des Buttes de Coesmes, 35700, Rennes, France

    M. K. Bourbatache

  2. LaSIE - UMR 7356, University of La Rochelle, Avenue Michel Crépeau, 17042, La Rochelle Cedex 1, France

    M. K. Bourbatache & O. Millet

  3. LEMTA - UMR 7563, University of Lorraine, 54000, Nancy, France

    C. Moyne

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  1. M. K. Bourbatache
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Correspondence to M. K. Bourbatache.

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Bourbatache, M.K., Millet, O. & Moyne, C. Upscaling diffusion–reaction in porous media. Acta Mech 231, 2011–2031 (2020). https://doi.org/10.1007/s00707-020-02631-9

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