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I present a
model to describe transport in fractured media. In the small fractures
we assume a Stokes flow and the chemistry is modelled by a convection
diffusion equation, while a diffusion equation in the large (porous)
parts of the domain is considered. The particularity of the model is in
the boundary conditions at the common interface, where a nonlinear
transmission condition is imposed. We consider a simple geometry, where
a large rectangular porous domain is coupled to a rectangular fracture
with small thickness.
After making the model dimensionless, we use asymptotic expansion
methods to formally derive upscaled models. This is done by averaging
in the transversal direction of the fracture.
Next, using Rothe's method, we prove existence of solutions to our
model, assuming existence of the time discretized solutions. Also,
positivity and boundedness in L∞ are proven. A domain decomposition
method is used to prove existence of the time discretized solutions
when the transmission condition is linear. For the nonlinear one
dimensional case it is shown that the domain decomposition method
converges.
In each step of the domain decomposition iteration, a nonlinear problem
has to be solved, for which we propose a simple fixed-point iterative
procedure. Also, we prove existence of solutions to the linear elliptic
transmission problem, by using variational arguments.
Next, it is rigorously proven that as the ratio between the thickness
and the length of the fracture vanishes, we obtain the model derived by
formal asymptotics.
The obtained numerical schemes are implemented and numerical
computations are presented to compare the outcome of the effective
model with (a transversally averaged version of) the full model.
Finally, I formulate currently open problems, inspired by the research.
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