Coupling surface flow with porous media flow

This thesis proposes a model for the interaction between ground flow and surface flow using a coupled system of the Navier-Stokes and Darcy equations. The coupling of surface flow with porous media flow has important applications in science and engineering. This work is motivated by applications to geo-sciences.This work couples the two flows using interface conditions that incorporate the continuity of the normal component, the balance of forces and the Beaver-Joseph-Saffman Law. The balance of forces condition can be written with or without inertial forces from the free fluid region. This thesis provides both theoretical and numerical analysis of the effect of the inertial forces on the model. Flow in porous media is often simulated over large domains in which the actual permeability is heterogeneous with discontinuities across the domain. The discontinuous Galerkin method is well suited to handle this problem. On the other hand, the continuous finite element is adequate for the free flow problems considered in this work. As a result this thesis proposes coupling the continuous finite element method in the free flow region with the discontinuous Galerkin method in the porous medium.Existence and uniqueness results of a weak solution and numerical scheme are proved. This work also provides derivations of optimal a priori error estimates for the numerical scheme. A two-grid approach to solving the coupled problem is analyzed. This method will decouple the problem naturally into two problems, one in the free flow domain and other in the porous medium. In applications for this model, it is often the case that the areas of interest (faults, kinks) in the porous medium are small compared to the rest of the domain. In view of this fact, the rest of the thesis is dedicated to a coupling of the Discontinuous Galerkin method in the problem areas with a cheaper method on the rest of the domain. The finite volume method will be coupled with the Discontinuous Galerkin method on parts of the domain on which the permeability field varies gradually to decrease the problem sizes and thus make the scheme more efficient.