Some Moving Boundary Problems in Underground Energy Processes

Abstract

In this thesis, we study four moving boundary problems motivated by underground energy processes such as enhancing oil recovery and geological CO2 sequestration. The major focus is the time evolution of the fluid-fluid interface under the effect of confinement, heterogeneity, drainage, and unfavorable viscosity ratio.

Specifically, in Chapter 2, theoretical and numerical investigations are combined to study the effect of confinement when a fluid is injected into a porous medium saturated with another immiscible fluid. In the early time period, the effect of confinement is negligible, whereas the confinement effect becomes significant in the late time period. The flow behaviour is summarized in a regime diagram with five distinct dynamical regimes that characterize the shape of the interface: a nonlinear diffusion regime, a transition regime, a traveling wave regime, an equal viscosity regime, and a rarefaction wave regime.

The thesis continues in Chapter 3 with a theoretical study based on a phase-plane analysis of the effect of horizontal heterogeneity, which eventually reveals the existence of a second-kind self-similar solution when a viscous gravity current is propagating toward the origin. Scaling arguments alone do not work in this case, because there exists a natural length scale, i.e., the distance between the origin and the initial location of the fluid, and hence a natural time scale, i.e., the time for the front to reach the origin. Experimental and numerical results are also provided to support the theoretical predictions.

Our study on drainage, described in Chapter 4, provides a self-similar solution to describe the buoyancy-driven fluid drainage process. Previous reports mainly focus on the propagation problem when the information on the global mass is given. In contrast, in a fluid drainage process the time evolution of the global mass is obtained as the solution of the problem.

The thesis closes in Chapter 5, with theoretical and experimental investigations on control and stabilization of the immiscible viscous fingering in a radial Hele-Shaw cell. Classic experiments exhibit that small perturbations grow into viscous fingers, which extend and eventually split into more fingers. In contrast, we provide a series of time-dependent strategies to manipulate the physical properties such as the injection rate and the gap thickness, which either stabilize the fluid displacement process or maintain a series of non-splitting viscous fingers.