Displacement of three-phase flow for Heavy Oil: Riemann Solutions

This work presents the Riemann solution for three-phase flow in porous media under the condition that oil viscosity exceeds that of water and gas. We classify all Riemann solution problems for scenarios where the left states $L$ lie along the edge $G$-$W$, and the right states $R$ span nearly the entire saturation triangle, excluding small regions near the boundaries $G$-$O$ and $W$-$O$. We use the wave curve method to determine the Riemann solution for initial and injection data within the above-mentioned class. This study extends previous analytical solutions, which were limited to right states near the corner $O$ or within the quadrilateral $O$-$E$-$\mathcal{U}$-$D$. Notably, this classification remains valid for all viscosity variations satisfying the inequalities \eqref{eq:classical}, corresponding to viscosity regimes where the umbilic point is close to the vertex $O$. We verify the $L^1_{loc}$-stability of the Riemann solution with respect to variations in the data. While we do not establish the uniqueness of the Riemann solution, extensive numerical experiments confirm its validity. Our findings provide a comprehensive framework for understanding three-phase flow dynamics in porous media under a wide range of conditions.