Stokes-Brinkman-Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

摘要

An analytical study is devised for the problem of bimodal porous flow across a periodic array of permeable cylindrical inclusions. Such a configuration is particularly relevant for porous media systems of dual granulometry, an idealization often taken, e.g. in the modelling of membranes and fibrous applications. The double-porosity system is governed by the Stokes-Brinkman-Darcy equations, the most general description in this class of flow problems characterized by the permeabilities of the surrounding matrix and inclusions, their porosities and the relative volume fraction. We solve this problem with the Kuwabara cell model and lubrication approach, providing analytical solutions for the system effective permeability in closed analytical form. The ensemble of results demonstrates the self-consistency of the bimodal solutions in eight possible limit configurations and supports the validity of the Beavers-Joseph interface stress jump condition for transmission from the open Stokes flow to low-permeable Darcy region. At the same time, these solutions bring further insight on the relative significance of the governing parameters on the effective permeability, with a focus on the role of the effective viscosity (porosity) distribution. Furthermore, although the cell model is restricted to relatively small volume fractions in open flow, its validity extends in less-permeable background flow inside Brinkman/Brinkman description. In turn, the lubrication approximation remains more adequate in the opposite limit of the dense impermeable inclusions. These conclusions are drawn from comparisons with the numerical solutions obtained with the developed lattice Boltzmann model and the standard finite element method. The two methods principally differ in the treatment of the interface conditions: implicit and explicit, respectively. The purpose of this task is therefore twofold. While the numerical schemes help quantifying the validity limits of the theoretical approach, the analytical solutions offer a non-trivial benchmark for numerical schemes in highly heterogeneous soil.