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Abstract. We present a two-scale finite element method for solving Brinkman's and Darcy's equations. These systems of equations model fluid flows in highly porous and porous media, respectively. The method uses a recently proposed discontinuous Galerkin FEM for Stokes' equations by Wang and Ye and the concept of subgrid approximation developed by Arbogast for Darcy's equations. In order to reduce the "resonance error" and to ensure convergence to the global fine solution the algorithm is put in the framework of alternating Schwarz iterations using subdomains around the coarse-grid boundaries. The discussed algorithms are implemented using the Deal.II finite element library and are tested on a number of model problems.

We use a two-scale continuum model to simulate reactive flow and wormhole formation in carbonate rocks under 3-D radial flow conditions. More specifically, we present a new structure-property relationship based on the fractal geometry theory, to describe the evolution of local permeability, pore radius, and specific area with porosity variation. In the numerical calculation, to improve the convergence rate, the heterogeneous medium in question is extended by adding a thin layer of homogeneous porous medium to its inlet. We compare the simulation results with the available experimental observations and find that they are qualitatively consistent with each other. Additionally, sensitivity analysis of the dissolution process with respect to acid injection rate and rock heterogeneity, including heterogeneity magnitude and correlation length, is presented.