We study a scalar conservation law u_t+f(x,u)_x=0 with a space-dependent flux function f(\cdot,\cdot). Problems of this type of arise in many applications, in particular in multi-phase flow in porous media, traffic flow, and nonlinear elasticity in heterogeneous materials. We first describe the general framework for stability analysis of the solutions to a finite volume discretization the scalar conservation laws, and review the literature for the case of space-dependent flux function.
Next we introduce a particular multi-phase flow model in porous media, with application to the transport of methane gas in Gas Hydrate Stability zone. The two phases are liquid and hydrate crystal, and they are described with two variables: methane concentration and hydrate saturation which are not independent. In particular, the hydrate phase may appear and disappear, depending on the amount of methane concentration. We consider hydrate formation and dissociation in either equilibrium (EQ) or kinetic (KIN) model. In the equilibrium model (EQ) the two unknowns "live" on a graph represented by a particular nonlinear complementarity constraint. In the kinetic model the variables evolve towards the equilibrium described by the EQ model.
Our results include both rigorous analysis as well as simulations. For EQ model we propose a scheme which combines explicit upwind treatment of transport with an implicit phase equilibrium solver. To conduct numerical analysis of the EQ model we assume that f(x,u) is smooth in x, which is realistic in homogeneous porous media. The main challenge is that the flux function f(x,u), the inverse of a multivalued graph, is not smooth in u, thus we regularize the model. Our first result is the proof of weak stability for the regularized EQ model. We also demonstrate convergence of order O(sqrt(h)) for the original and regularized simple model problems as well as for examples with experimental data, and show that convergence rate is inferior if the smoothness assumptions do not hold. We also derive stability results for the kinetic model (KE) in the product space involving both variables. Here the key is to account for both independent variables. We also show that when the rate of kinetics increases, the solutions to KIN model approach those of the EQ model. This is demonstrated by analysis and by simulations.