Analytical solution to the Riemann problem of three-phase flow in porous media

In this paper we study one-dimensional three-phase flow through porous media of immiscible, incompressible fluids. The model uses the common multiphase flow extension of Darcy's equation, and does not include gravity and capillarity effects. Under these conditions, the mathematical problem reduces to a 2 × 2 system of conservation laws whose essential features are: (1) the system is strictly hyperbolic; (2) both characteristic fields are nongenuinely nonlinear, with single, connected inflection loci. These properties, which are natural extensions of the two-phase flow model, ensure that the solution is physically sensible. We present the complete analytical solution to the Riemann problem (constant initial and injected states) in detail, and describe the characteristic waves that may arise, concluding that only nine combinations of rarefactions, shocks and rarefaction-shocks are possible. We demonstrate that assuming the saturation paths of the solution are straight lines may result in inaccurate predictions for some realistic systems. Efficient algorithms for computing the exact solution are also given, making the analytical developments presented here readily applicable to interpretation of lab displacement experiments, and implementation of streamline simulators.