The injection of fluid into a porous medium

The study of fluid flow through soil or rock has many practical applications and is relevant to a number of fields in the natural sciences. This thesis investigates the injection of fluid into a general porous medium, using two different types of mathematical model, each of which is based on different physical assumptions. The central aim is to study the detailed development of wetted plumes in the porous medium, and the way in which they evolve over time.
In our initial work, we consider a model of the injection of some fluid into a completely dry porous medium. In this mathematical model, the wetted plume is allowed to form a sharp boundary between itself and the surrounding dry soil. A linearised solution is presented, and is valid for early times in the development of the plume. After a large time, the plume is expected to eventually form a constant vertical structure, and a steady state asymptotic solution is also presented for the plume’s final shape. These approximate solutions are used to guide the development of a semi-numerical spectral method to solve the fully nonlinear problem over all times; however, we found that the numerical solution is not able to progress past relatively early times. It is suspected that this is due to ill-conditioning relating to a genuine physical effect occurring at the interface between the wetted plume and the dry surrounding medium, through the formation of curvature singularities at the interface. Nevertheless, preliminary results from the numerical spectral method are presented at early times in the plume’s evolution. These results are revisited later in the thesis, and are validated at early times by comparison with a different model of the flow.
In order to overcome the limitations encountered as a result of ill-conditioning caused by curvature singularities at a sharp interface, we instead consider an alternative physical scenario. Here, a fluid solution is injected into a porous medium that is already fully saturated with a different fluid. Boussinesq theory is used to approximate this two-fluid system with an effective single fluid in which the density changes continuously in a narrow interfacial mixing zone between the wetted plume and its surroundings. The injected fluid is denser than the ambient fluid, and so the plume that develops moves downward under the effects of gravity, as time progresses. We first study a slightly simpler geometry, in which the denser injected fluid is introduced through a horizontal line source in the rock, so that the resulting flow is two-dimensional planar flow. A semi-analytical spectral method is developed, using basis functions that are carefully selected to be appropriate for this planar cylindrical geometry, for both the density and the fluid velocity vector. Accurate results are presented, that show in detail the evolution of the plume of the heavier injected fluid.
We go on to consider this fully saturated model further, but now for a situation in which the injected fluid is introduced into the porous medium through a small spherical hole in the rock. Boussinesq theory is again used, and a numerical spectral method is developed for the fully nonlinear problem, using basis functions appropriate for this three-dimensional spherical geometry. The flow is assumed to be axisymmetric, and detailed numerical results are presented for the density and the stream function associated with the flow. It is shown that, at early times, these results are in good agreement with the results of the unsaturated model considered at the beginning of the thesis.
In both the planar geometry and the axisymmetric geometry, asymptotic solutions for the eventual plume shapes are presented, based on the unsaturated model discussed initially. In both cases, these models are shown to agree very closely with the asymptotic plume shapes far from the sources at later times