Keywords: poroelasticity, homogenization, double porosity, micromechanics, multiscale modelling.

In nature as well as in technical practice one can often find materials with different levels of porosity at different scales. Considering poroelastic [4] description at the smallest scale, the theory of homogenization [3,8] provides a natural way of upscaling this description to higher levels of porosities in a sense that effective poroelastic material coefficients (consistent with the Biot model) at a higher level are obtained by applying homogenization to the lower level. This leads to a suitable hierarchical description of the porous medium, where different porosities associated with different scale levels are taken into account.

In this paper homogenization-based formulae are provided which enable the computation of the poroelasticity coefficients for a given geometry and topology at the micro- and meso-scopic levels. The homogenization at each scale level proceeds in two steps: 1) Find effective (homogenized) coefficients by solving auxiliary problems for several characteristic (or corrector) functions, cf. [7,5]. 2) Compute the homogenized coefficients that can be used for the higher level and, or "global" (homogenized) model of the current level. 3) Solve the homogenized problem with given loading and boundary conditions at the macro-scale (the highest level). As a result of the linearity of the problems, those steps are decoupled in a sense that the computation of the homogenized coefficients for the global level is valid for any point having the corresponding "microstructure".

The two-level upscaling "micro-meso-macro" is presented and the influence of the pore geometry and topology is illustrated. In [6] systems of interconnected pores at different scales were considered; in a steady state there is only one pore fluid pressure. In this paper, a different arrangement of porosities is described which can be mutually separated by a semipermeable interface, however, each porosity can form a separate connected system. In this situation, the homogenized problem results in two different pressures. At the mesoscopic scale we take into account the Darcy flow in the poroelastic matrix, although in the mesoscopic fractures (called channels in our terminology) the fluid is assumed to be static with no pressure gradients.

In the literature, homogenization in poroelasticity is a frequently discussed issue [1], but to our knowledge, the results reported in this paper are novel, namely due to the numerical feedback and computer implementation [2].

1

J.L. Auriault, E. Sanchez-Palencia, "Étude du comportement macroscopique d'un milieu poreux sature deformable", Jour. de Mécanique, 16(4), 575-603, 1977.

2

R. Cimrman, "SfePy: Simple Finite Elements in Python", 2012. http://sfepy.org

3

D. Cioranescu, A. Damlamian, G. Griso, "The periodic unfolding method in homogenization", SIAM Journal on Mathematical Analysis, 40(4), 1585-1620, 2008. doi:10.1137/080713148

4

O. Coussy, "Poromechanics", John Wiley & Sons, 2004.

5

E. Rohan, R. Cimrman, "Multiscale FE simulation of diffusion-deformation processes in homogenized dual-porous media", Math. Comp. Simul., 2011. In Press

6

E. Rohan, S. Naili, R. Cimrman, T. Lemaire, "Hierarchical homogenization of fluid saturated porous solid with multiple porosity scales", Compte Rendus Acad. Sci./Mécanique, 2012. Submitted

7

E. Rohan, S. Naili, R. Cimrman, T. Lemaire, "Multiscale modelling of a fluid saturated medium with double porosity: relevance to the compact bone", Jour. Mech. Phys. Solids, 60, 857-881, 2012. doi:10.1016/j.jmps.2012.01.013

8

E. Sanchez-Palencia, "Non-homogeneous media and vibration theory", Lecture Notes in Physics, 127, Springer, Berlin, 1980.

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