Dynamic capillarity in porous media : mathematical analysis

In this thesis we investigate pseudo-parabolic equations modelling the two-phase flow in porous media, where dynamic effects in the difference of the phase pressures are included. Specifically, the time derivative of saturation is taken into account. The first chapter investigates the travelling wave (TW) solutions, the existence anduniqueness of smooth TW solutions are proved by ordinary differential equation techniques. The existence depends on the parameters involved. There is a threshold value for one of the parameters, the damping coefficient. When the damping coefficient is beyond the threshold value, smooth TW solutions do not exist any more. Instead, non-smooth (sharp) TW solutions are introduced and their existence is shown. The theoretical results for both smooth and non-smooth TW solutions are confirmed by numerical computations. In the next chapter, the analysis is extended to weak solutions. In a simplified case where the mixed (time and space) order-three derivative term is linear, the existence and uniqueness of a weak solution are obtained. Next, the complex model involving nonlinear and possibly degenerate capillary induced diffusion function is considered. Then the existence is obtained by employing regularization techniques, compensated compactness and equi-integrability arguments. Finally, inspired by the nature of the capillary pressure, different formulations of the equation are introduced and their equivalence is proved. The corresponding numerical scheme is implemented and numerical results are given, which agree well with the theoretical results.