The talk will start with an exposition on solvers for nonlinear conservation laws. Next we study a nonlinear transport model of methane gas which accounts for the phase equilibrium conditions, and includes formation of hydrate as well as transport of free gas. Methane hydrate is an ice-like substance present in subsea sediments of low temperature and high pressure. Extraction of methane gas from hydrates has been of interest for the energy resources and as environmental hazards. In our problem the challenges are that the flux function is space-dependent and is an inverse of a multivalued graph. We use regularization to approximate the problem to understand the singularities that arise.