During oil production several processes are used for extracting oil from underground reservoirs at different stages of the production process. After exploiting so called primary recovery, that depends on the "natural" depletion of the reservoir, other techniques are applied in subsequent stages. In tertiary recovery, foam can be injected and used as the displacing fluid. Foams have the capacity to provide a better percentage of recovery compared to other fluids because foams lower gas mobility, permitting a more uniform and efficient sweep of oil in the formation.However foams are complex fluids and the study of their flow within porous systems, like oil reservoirs, is challenging. Therefore the aim of this work is to study the propagation of a foam front within reservoirs in the context of improved oil recovery. The perspective that is adopted here is to use a simple model for foam rheology known as pressure-driven growth, to describe the foam displacement process using numerical simulations and (in some cases) solving the system analytically. The pressure-driven growth model is a limiting case of the viscous froth model, where terms for surface tension and curvature are removed. Taking this particular limiting case has consequences for the numerical solution of the system as the governing equations become far less stable both physically and numerically.An injection strategy called surfactant alternating gas is described by pressure-driven growth, where all resistance to motion in the advance of the foam is assumed to be focused on a region of wet small bubbles (the foam front) forming the interface of the water and gas phases. This front can be considered to be a one-dimensional curve. We then follow the propagation of the foam front over time, obtaining the front location and its shape.For the case of a homogeneous reservoir with constant driving pressure, the foam front is expected to have a convex shape. However, the numerical solution of pressure-driven growth can admit the formation of concavities in the front shape. These prove to be difficult to handle numerically since they focus down into sharp concave corners. As a consequence, robust numerical schemes are needed, and such schemes can be derived informed by the analysis of asymptotic solutions for the process. In order to deal with concavities, a modification is applied to the velocity of concave corners, which is used to recover the expected convex shape for the entire foam front.Other cases of interest arise in the scope of this study, where the development of concavities are expected due to the nature of the processes themselves, rather than being a mere numerical artifact. These are the case when there is surfactant slumping (i.e. downward migration of surfactant), the case when driving pressure is increased part-way through the process, and the case when the reservoir itself is heterogeneous. The pressure-driven growth model can be used in all these cases with the appropriate modifications to front velocities that each case requires within concavities, and spurious behaviour that would otherwise affect numerical results is thereby prevented.