**Keywords:**Flows in porous media | Riemann problem | capillary pressure | viscous profiles; conservation laws

We study the physical diffusive effect caused by capillary pressure between phases in three-phase flow in porous media, disregarding gravitational effects. The problem is modeled by a system of two nonlinear conservation laws. We solve a class of Riemann problems for this model where one of the viscosities is higher than the other two. To this end, we first developed a methodology using artificial diffusion and identified the transitional surfaces and associated shocks, which appear as a result of loss of strict hyperbolicity at an isolated point in the space of saturations. We identify the surfaces that characterize solutions which require transitional shocks as part of the solution to Riemann problems. We use the wave curve method to determine the solutions for arbitrary Riemann data (i.e., left and right states), except for a small set of right states that utilize transitional rarefactions for the corresponding solutions. This methodology combines theoretical analysis with numerical experiments to furnish scientific evidence for the existence (and uniqueness) of solutions with continuity under variation of data. Finally, we present the transitional surface for the general case where diffusion arises from capillary effects.