Mathematical and numerical analysis of miscible displacement through porous media

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. <br> <br> 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. <br> <br> 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