In this thesis we develop mathematical treatment for two important applications: (i) evolution of methane in coalbeds and the associated phenomena of adsorption, and (ii) evolution of methane in methane hydrates. We use simplified models for (i) and (ii) since we are more interested in qualitative properties of the solutions rather than direct applications to engineering.
For methane hydrates we focus on a scalar problem with diffusion only, and we discuss it as a nonlinear parabolic problem in a single variable with monotone operators. We show how the problem can be cast in the framework of a free boundary problem. The particular nonlinearity that we deal with comes from a constraint on one of the variables. For the simplified model of methane hydrates, we establish well-posedness of the problem in an abstract weak setting. We also perform simulations with a novel approach based on semismooth Newton methods. We demonstrate convergence rates of the numerical approximation which are similar to those for Stefan free boundary value problem.
On the other hand, for adsorption problems, we focus on their structure as systems of conservation laws, with equilibrium and non-equilibrium type nonlinearities, where the latter are associated with microscale diffusion. We also work with an unusual type isotherm called Ideal Asorbate Solution, which is defined implicitly. For IAS adsorption system, we show sufficient conditions that render the system hyperbolic. We also construct numerical approximations for equilibrium and nonequilibrium models.