A Domain Decomposition Approach to Finite-Epsilon Homogenization of Scalar Transport in Porous Media

Abstract

Modeling scalar transport by advection and diffusion in multiscale porous structures is a challenging problem, particularly in the preasymptotic regimes when non-Fickian effects are prominent. Mathematically, one of the main difficulties is to obtain macroscale models from the homogenization of conservation equations at microscale when epsilon, the ratio of characteristic lengthscales between the micro- and macroscale, is not extremely small compared to unity. Here, we propose the basis of a mathematical framework to do so. The focal idea is to decompose the spatial domain at pore-scale into a set of N subdomains to capture characteristic times associated with exchanges between these subdomains. At macroscale, the corresponding representation consists of a system of N coupled partial differential equations describing the transport of the spatially averaged scalar variable within each subdomain. Besides constructing the framework, we also compare numerically the results of our models to a complete resolution of the problem at the pore-scale, which shows great promises for capturing preasymptotic regimes, non-Fickian transport, and going toward finite-epsilon homogenization.