Multiscale Method For Unsaturated Flow In Porous Medium

In this thesis, we mainly study the problem which is usually used to describe the motion of flow in unsaturated porous media. This paper is divided into four parts.In Chapter1, we will introduce the physical background of the motion of flow in unsaturated porous media and the research status of this problem. A summary of this thesis is given in the end.Richards’equation is often used to describe the flow in unsaturated porous media. It is a nonlinear convection-diffusion parabolic equation. In Chapter2, combing the heterogenous multiscale method (HMM) with interior penalty discontinuous Galerkin (IPDG) finite element method, we set up a completely discrete IPDG-HMM to solve the problem. The macro element flux and inter edge flux should be constructed by solv-ing the two different micro cell problems. Here’fully discrete’means that we consider not only the discretization both in time and space, the effect of numerical quadrature over each macro-scale element and interface for IPDG, but also the numerical approx-imation for the local cell problems at fine scale. Then, we prove the error estimate between the solutions of IPDG-HMM and the solution of homogenized problem un-der the assumption that the coefficients are periodic. Some numerical examples for the Gardener model and Van Genuchten-Mualen model are presented respectively to demonstrate the accuracy and efficiency of our method.In Chapter3, by assuming that the coefficients are periodic, we will develop the homogenized theory of Richards’equation of van Genuchten-Mualem model, which is a nonlinear degenerate parabolic differential equation. The homogenization theory of some degenerated parabolic problem has been studied in recently years, but the degenerated condition is not fit for the Richards’equation of van Genuchten-Mualem model. Applying the Kirchhoff transformation to the equation firstly, we obtain a simpler equivalent equation with a linear oscillated diffusion term. Then under the real assumptions for van Genuchten-Mualem model, we obtain the homogenized equation by using the two-scale convergence theory and compactness theory. Some results on the first order corrector are also presented.In Chapter4, under the regularization step, a completely discrete HMM-FEM is developed for solving the degenerated Richards’equation of van Genuchten-Mualem model. Here, we also take the effect of numerical quadrature over each macro-scale element into account. Error estimates between the numerical solution and the solution of homogenized problem are also derived under the assumption that the permeability is periodic. At last, we also present some numerical examples to test our method.