MATHICSE Technical Report: A discontinuous Galerkin reduced basis numerical homogenization method for fluid flow in porous media

We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method(DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stokes problems through-out the macroscopic computational domain, the pore geometry is parametrized and a model order reduction algorithm is used to select representative microscopic geometries. Accurate Stokes solutions and related permeabilities are obtained for these representative geometries in an offline stage. In an online stage, the DG-FEM is computed with permeabilities recovered at the desired macroscopic quadrature points from the precomputed Stokes solutions. The multiscale method is shown to be mass conservative at the macro scale and the computational cost for the online stage is similar to the cost of solving a single scale Darcy problem. Numerical experiments for two and three dimensional problems illustrate the efficiency and the performance of the proposed method.