A porous medium (concrete, soil, rocks, water reservoir, e.g.)
is a multi‐scale medium
where the heterogeneities present in the medium are characterized by the
micro scale and the
global behaviors of the medium are observed at the macro scale. The
upscaling from the micro
scale to the macro scale can be done via averaging methods.
In this talk, diffusion and reaction of several mobile chemical species
are considered in
the pore space of a heterogeneous porous medium. The reactions amongst the
species are
modelled via mass action kinetics and the modelling leads to a system of
multi‐species diffusion‐
reaction equations (coupled semi‐linear partial differential
equations) at the micro scale where
the highly nonlinear reaction rate terms are present at the right hand
sides of the system of PDEs,
cf. [2]. The existence of a unique positive global weak solution is shown
with the help of a
Lyapunov functional, Schaefer’s fixed point theorem and maximal
Lp‐regularity, cf. [2, 3]. Finally,
with the help of periodic homogenization and two‐scale convergence
we upscale the model from
the micro scale to the macro scale, e.g. [1, 3]. Some numerical
simulations will also be shown in
this talk, however for the purpose of illustration, we restrict ourselves
to some relatively simple dimensional situations.
As an extension to the previous model, we consider the mixture of two
fluids. For such
models, a system of Stokes ;Cahn ;Hilliard equations will be
considered at the micro scale in a
perforated porous medium. We first explain the periodic setting of the
model and the existence
results. At the end homogenization of the model will be shown using some
extension theorems
on Sobolev spaces, two‐scale convergence and periodic unfolding.