Geometric Measure Theory and Regularity of Free Boundary Problems in Models of Oil Reservoir Simulation

This dissertation applies geometric measure theory to the analysis of free boundary regularity in mathematical models of fluid flow in porous media arising in oil reservoir simulation, contributing original regularity theory for a class of problems directly relevant to hydrocarbon extraction in the Niger Delta and related Nigerian petroleum-producing basins. Free boundary problems, in which the domain of definition of the solution is itself an unknown of the problem, arise naturally in reservoir simulation through the moving oil-water contact and the advancing displacement front during secondary recovery operations, and their regularity properties determine the mathematical validity and numerical stability of reservoir simulation models. The dissertation proves novel Lipschitz and C1,alpha regularity results for the free boundary in a two-phase Darcy flow problem with variable permeability coefficients, using the geometric measure theory framework of monotonicity formulas and blow-up analysis following the approach of Alt and Caffarelli for the one-phase problem and extending it to the two-phase degenerate case. Sharp estimates on the Hausdorff dimension of the singular set of the free boundary are derived using a second variation argument adapted to the degenerate elliptic operator structure. The regularity results are applied to derive a priori error estimates for finite element discretisations of the free boundary problem, establishing convergence rates that account for the regularity of the free boundary. Computational validation against benchmark two-phase flow problems in petroleum reservoir simulation literature confirms the theoretical predictions. Keywords: geometric measure theory, free boundary regularity, porous media, oil reservoir simulation, Darcy flow