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

TL;DR

This paper develops a comprehensive analytical framework for solving three-phase flow problems in porous media with heavy oil, extending previous solutions to broader conditions and verifying stability through numerical experiments.

Contribution

It introduces a classification of Riemann solutions for three-phase flow with heavy oil, extending analytical solutions to wider viscosity regimes and validating stability numerically.

Findings

Classification of Riemann solutions for broader conditions

Extension of analytical solutions beyond previous limits

Verification of solution stability through numerical experiments

Abstract

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.