Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an H^{1,p} Setting

Abstract

The processes of chemical transport in porous media are extensively studied in the fields of applied mathematics, material science, chemical engineering etc. A porous medium (e.g. concrete, soil, rocks, reservoir etc.) is a multiscale material/medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed by the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods. The transport process in a porous medium is a complex phenomena. In this thesis, the heterogeneities inside a porous medium are assumed to be periodically distributed and diffusion-reaction of a finite number of chemical species are investigated. Two different models are proposed in this work. In model M1, diffusion-reaction of mobile chemical species are considered. The chemical processes are modeled via mass action kinetics and the modeling leads to a system of multi-species diffusion-reaction equations (nonlinear partial differential equations) at the micro scale. For this system of equations, existence of a unique positive global weak solution is proved by the help of a Lyapunov functional and Schaefer's fixed point theorem. The upscaled model of this system is obtained using periodic homogenization which is an averaging method. In model M2, we consider diffusion-advection-reaction of two different types of mobile species (type I and type II). The type II species are supplied via dissolution process due to the presence of immobile species on the surface of the solid parts. The presence of mobile and the immobile species make the model complex and the modeling yields a coupled system of nonlinear partial differential equations. The existence of a unique positive global weak solution of this complex system is shown. Finally, with the help of periodic homogenization, model M2 is upscaled from the micro scale to the macro scale. Numerical simulations are conducted for both models separately. For the purpose of illustration, we restrict ourselves to relatively simple 2-dimensional situations. For models M1 and M2, simulation results at the micro scale and at the macro scale are compared.