%0 Thesis
%A Talbot, Kyle
%D 2017
%T Mathematical and numerical analysis of miscible displacement through porous media
%U https://bridges.monash.edu/articles/thesis/Mathematical_and_numerical_analysis_of_miscible_displacement_through_porous_media/4814125
%R 10.4225/03/58e2cfb78b194
%2 https://bridges.monash.edu/ndownloader/files/7977463
%K Partial differential equations
%K Porous media
%K Elliptic and parabolic problems
%X This thesis documents
some recent advances in the mathematical and numerical analysis of a model
describing the single-phase, miscible displacement through a porous medium of
one incompressible fluid by another. The model is an initial-boundary value
problem for a nonlinearly-coupled elliptic-parabolic system for the pressure of
the fluid mixture, and the concentration of one of the components in the
mixture.
The thesis proves three main results for this model.
Following standard approximation techniques, we prove the existence of weak
solutions to the model with measure source terms and a diffusion-dispersion
tensor that grows linearly with the Darcy velocity. Retaining the measure
source terms, we extend this result to establish existence as the molecular
diffusion in the diffusion-dispersion tensor vanishes. The third result is a
unified convergence analysis of numerical schemes for the model, conducted
using the framework of the Gradient Discretisation Method. Without assuming
regularity or uniqueness of the solution to the model, we show that approximate
concentrations converge uniformly-in-time.
The last result of the thesis studies the behaviour of
solutions to a class of doubly nonlinear degenerate parabolic problems when the
data are perturbed. The class includes the Richards equation, the Stefan
problem and the parabolic Leray--Lions equation. We show that, up to a subsequence,
weak solutions of the perturbed problem converge uniformly-in-time to weak
solutions of the original problem as the perturbed data approach the original
data.
%I Monash University