Homogenization of stochastic partial differential equations in perforated porous media

Abstract

In this thesis, we study the homogenization of a stochastic model of groundwater pollution in periodic porous media and the homogenization of a stochastic model of a single-phase uid ow in partially ssured media. In the rst study, we investigated the ow of a uid carrying reacting substances through a porous medium. We modeled this ow using a coupled system of equations; the velocity of the uid is modeled using steady Stokes equations, the concentration of the solute while being moved by the uid under the action of random forces is modeled by a stochastic convection-di usion equation driven by a Wiener type random force and the concentration of the solute on the surface of the pore skeleton is modeled using reaction-di usion equations. The homogenization process was carried out using the multiple scale expansion, Tartar's method of oscillating test functions and stochastic calculus together with deep probability compactness results due to Prokhorov and Skorokhod. This part of the thesis is the rst in the scienti c literature dealing with the important problem of groundwater pollution using stochastic partial di erential equations. Our results in this regard are original. Also as a by-product of our work, we establish the rst homogenization result for stochastic convection-di usion equation The second study is devoted to a single-phase ow under the in uence of external random forces through partially ssured media arising in reservoir engineering (oil and gas industries). We undertake to model this ow using a system of nonlinear stochastic di usion equations with monotone operators in the pore system and the ssure system; on the interface of the pores and ssures, we prescribe transmission boundary conditions. We carried out the homogenization process using the two-scale convergence method, Prokhorov- Skorokhod compactness process and Minty's monotonicity method. While some works have been undertaken in the deterministic case and in the case of nonlinear di usion equations with randomly oscillating coe cients, our work is novel in the sense that it uses the more advanced tool of stochastic partial di erential equations driven by random forces to investigate the in uence of random uctuations on the ow. To the best of our knowledge, our work also initiates the study of stochastic evolution transmission problems by means of homogenization.