Numerical upscaling of the free boundary dam problem in multiscale high-contrast media

Abstract

In this paper, we address the numerical homogenization approximation of a free-boundary dam problem posed in a heterogeneous media. More precisely, we propose a Generalized Multiscale Finite Element Method (GMsFEM) for the heterogeneous dam problem. The motivation of using the GMsFEM approach comes from the multiscale nature of the porous media due to its high-contrast permeability. Thus, although we can classically formulate the free-boundary dam problem as in the homogeneous case, a very high resolution will be needed by a standard finite element approximation in order to obtain realistic results that recover the multiscale nature. First, we introduce a fictitious time variable which motivates a suitable time discretization that can be understood as a fixed point iteration to the steady state solution, and we use a duality method to deal with the involved multivalued nonlinear terms. Next, we compute efficient approximations of the pressure and the saturation by using the GMsFEM method and we can identify the free boundary. More precisely, the GMsFEM method provides numerical results that capture the behavior of the solution due to the variations of the coefficient at the fine-resolution, by just solving linear systems with the size proportional to the number of coarse blocks of a coarse-grid (that does not need to be adapted to the variations of the coefficient). Finally, we present illustrative numerical results to validate the proposed methodology.