Comptes Rendus
no. 8
Wave propagation in a strongly heterogeneous elastic porous medium: Homogenization of Biot medium with double porosities
[Propagation d'ondes dans un milieu élastique poreux fortement hétérogène : homogénéisation d'un milieu de Biot avec porosités doubles]
Eduard Rohan1 ; Salah Naili2  ; Vu-Hieu Nguyen2
1 European Centre of Excellence, NTIS – New Technologies for Information Society Faculty of Applied Sciences, University of West Bohemia, Univerzitnií 22, 30614 Pilsen, Czech Republic
2 Université Paris-Est, Laboratoire “Modélisation et simulation multi-échelle”, MSME UMR 8208 CNRS, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France
Comptes Rendus. Mécanique, Volume 344 (2016) no. 8, pp. 569-581.

Nous étudions la propagation des ondes dans un milieu poreux élastique dont le réseau poreux est interconnecté et saturé par un fluide newtonien compressible. On suppose que la taille caractéristique des micropores est très hétérogène dans le réseau poreux. Le milieu est caractérisé par un contraste élevé de perméabilités et d'élasticités anisotropes et par un contraste modéré pour le coefficient de couplage de Biot. À l'échelle mésoscopique, le mouvement du fluide est régi par le modèle d'écoulement de Darcy étendu avec des termes d'inertie et par l'équation de conservation de la masse. Le matrice poreuse est décrite en utilisant le modèle de Biot. La méthode d'homogénéisation via l'analyse asymptotique à double échelle est utilisée pour obtenir un modèle macroscopique pour des contrastes élevés de perméabilité et d'élasticité anisotrope, mais avec des contrastes modérés du coefficient de couplage de Biot, lesquels ont été mis à l'échelle par rapport à la taille des hétérogénéités. À partir de l'estimation des longueurs d'onde dans les deux réseaux, il est montré que, non seulement la description macroscopique dépend du contraste de perméabilité statique entre les pores et micropores et de la fréquence, mais aussi que la double porosité est responsable des effets de mémoire via la loi de comportement poroviscoélastique.

We study wave propagation in an elastic porous medium saturated with a compressible Newtonian fluid. The porous network is interconnected whereby the pores are characterized by two very different characteristic sizes. At the mesoscopic scale, the medium is described using the Biot model, characterized by a high contrast in the hydraulic permeability and anisotropic elasticity, whereas the contrast in the Biot coupling coefficient is only moderate. Fluid motion is governed by the Darcy flow model extended by inertia terms and by the mass conservation equation. The homogenization method based on the asymptotic analysis is used to obtain a macroscopic model. To respect the high contrast in the material properties, they are scaled by the small parameter, which is involved in the asymptotic analysis and characterized by the size of the heterogeneities. Using the estimates of wavelengths in the double-porosity networks, it is shown that the macroscopic descriptions depend on the contrast in the static permeability associated with pores and micropores and on the frequency. Moreover, the microflow in the double porosity is responsible for fading memory effects via the macroscopic poroviscoelastic constitutive law.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2016.05.001
Keywords: Poroelasticity, Asymptotic homogenization, Double-porosity, Wave propagation, Effective properties
Mot clés : Poroélasticité, Homogénéisation asymptotique, Double porosité, Propagation d'ondes, Propriétés effectives
Eduard Rohan 1 ; Salah Naili 2 ; Vu-Hieu Nguyen 2

1 European Centre of Excellence, NTIS – New Technologies for Information Society Faculty of Applied Sciences, University of West Bohemia, Univerzitnií 22, 30614 Pilsen, Czech Republic
2 Université Paris-Est, Laboratoire “Modélisation et simulation multi-échelle”, MSME UMR 8208 CNRS, 61, avenue du Général-de-Gaulle, 94010 Créteil cedex, France 
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Eduard Rohan; Salah Naili; Vu-Hieu Nguyen. Wave propagation in a strongly heterogeneous elastic porous medium: Homogenization of Biot medium with double porosities. Comptes Rendus. Mécanique, Volume 344 (2016) no. 8, pp. 569-581. doi : 10.1016/j.crme.2016.05.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.05.001/

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