Mathematical and numerical methods for vertical equilibrium in porous media

We consider the motion of two-phases flow in a porous medium under the condition of vertical equilibrium. The mathematical model is deduced and boundary and initial condition are prescribed. The flux function has a bell shape, that is, it changes concavity twice. We solve analytically the corresponding Riemann problem with boundary conditions for short time. Then we design numerical schemes of Godunov type for the mathematical model with and without viscosity. Using these schemes we compute numerically the solutions for long time. These numerical solutions agree very sharply with the analytical ones for short time, what validates our schemes. Furthermore, our work is motivated by a laboratory experiment that presents hysteresis effects. We also present analytical and numerical solutions of our model with hysteresis.