Mathematical Complexities in Porous Media Flow

Abstract

Multiphase flow through porous media plays an important role in many practical applications, from groundwater modelling, oil and gas recovery to CO2 sequestration. In the current work, we address two challenges related to accurate modelling and simulation of such processes. The first is to incorporate non-equilibrium effects such as hysteresis and dynamic capillarity in the models. Experiments have shown that under certain circumstances, phenomena like saturation overshoot and finger formation occur, that cannot be explained by the standard (equilibrium) models. Hence, an extension of these models needs to be considered. The second is to develop fast, stable and preferably simple numerical techniques that solve the highly nonlinear and possibly degenerate equations governing flow in the extremely heterogeneous porous domains of the real world. Accordingly, this work is divided into two parts: (PART I: Non-equilibrium effects) First, a new model is proposed for hysteresis in capillary pressure, which extends and improves the play-type hysteresis model. It is shown that this model is physically consistent and approximates experimentally obtained hysteresis curves. It is then used to solve the problem of horizontal redistribution of water and air, demonstrating that ‘unconventional’ flow, predicted earlier in literature, does indeed occur in certain cases. To follow-up, we show that the model is mathematically well-posed. Next, gravity-driven infiltration of water into relatively dry soil is considered when the wetting front has the form of a downward propagating travelling wave, i.e., the wetting front moves at a constant speed and shape. We consider various cases with increasing complexity in a number of chapters. In the first, we study the behaviour of fronts when either hysteresis or dynamic capillarity is included. In the second chapter, both effects are included simultaneously and both the play-type and the extended play-type models, mentioned in the previous paragraph, are considered. The existence of travelling waves is proved and criteria for the occurrence of overshoots and the system to reach full saturation are made precise. The techniques developed are further used in the third chapter to describe viscous fingering and to derive the propagation speed of the fingers. Finally, fronts are analysed for the two-phase case in a very general setting where the relative permeabilities, as well as the capillary pressure, are hysteretic and dynamic capillary effect is included. Existence of all possible travelling wave solutions is shown and a number of qualitative properties are established. The travelling wave solutions are then used to derive admissibility conditions for shocks in the hyperbolic limit. The entropy solutions derived in this way are much broader compared to the standard entropy solutions of the Buckley-Leverett equation since they can be non-monotone and have multiple shocks. These results are used to explain experimental observations such as non-monotone saturation profiles and stable saturation plateaus, which were previously not well-understood. (PART II: Numerical methods) A linear domain decomposition scheme is proposed for heterogeneous and in particular, layered porous media. Apart from being parallelizable, it is unconditionally convergent for a mild restriction on the time step. Moreover, it is, in general, more stable and better conditioned than standard monolithic schemes such as the Newton or the Picard scheme, while being comparable in speed. The issue of nonlinearity is handled in the following chapter where a linear iterative scheme is proposed for solving the nonlinear diffusion equations that arise in porous flow problems. Being a modified version of the L-scheme, it converges linearly for a mild restriction on the time step, having convergence rate proportional to an exponent of the time step size. The convergence is also guaranteed for degenerate cases. This makes it faster than both the L-scheme and the Picard scheme and more stable than the Newton and the Picard scheme. Numerical results are provided that support the analytical findings. Finally, a mixed finite element method is proposed for the two-phase flow model with dynamic capillarity effect. Error estimates are derived showing that the scheme is first order in both space and time. The numerical results support our conclusion.