A Darcy-Brinkman model of flow in double porous media - Two-level homogenization and computational modelling

The present paper deals with the two-level homogenization of the flow in the rigid double-porous structure described at two characteristic scales: the higher level porosity associated with the mesoscopic structure is constituted by channels in a rigid skeleton which is made of a microporous material. The macroscopic flow model is derived by the asymptotic analysis of the model with respect to two small parameters. The upscaling procedure based on the periodic unfolding is performed in two steps: first the Stokes flow in microchannels interacting with the mesoscopic flow is homogenized, following the idea of Cioranescu et al. (2005). This yields a coupled Darcy-Stokes system of equations governing the mesoscopic flow with simplified Saffman's transmission conditions on the interface between the two porosities. The second step of the homogenization leads to a macroscopic flow model which attains the form of the Darcy-Brinkman model involving the effective parameters given by the microporosity features. The multiscale reconstruction of the flow at the meso- and micro-scale levels is presented. The model has been implemented in our finite element code SfePy which is well-suited for computational homogenization. Numerical illustrations of the proposed model are included.