| We study systems of two conservation laws arising in the modeling of one-dimensional immiscible three-phase flow in a porous medium. Using relative permeability functions of Corey-Pope type, and taking capillary pressure differences into account, nonclassical (transitional) shock waves are found.; We determine the transitional region, which defines the set of transitional shock waves, by constructing its boundary. Our numerical and analytical results describe connecting orbits for the shock waves along the transitional boundary. There are four types of such connecting orbits: saddle to repeller-saddle, saddle-attractor to saddle, distinguished repeller-saddle to saddle, and saddle to distinguished saddle-attractor. We show that near the umbilic point, if we approximate the flux function by a homogeneous quadratic function and assume the diffusion matrix to be symmetric positive definite, then connecting orbits are only of distinguished type. Away from the umbilic point, our numerical result shows that another type of connecting orbit arises, namely, saddle-attractor to saddle, in addition to connecting orbits of distinguished type. |
