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.
