Approximate solutions of the Riemann problem for a two-phase flow of immiscible liquids based on the Buckley–Leverett model

The article proposes an approximate method based on the "vanishing viscosity" method, which ensures the smoothness of the solution without taking into account the capillary pressure. We will consider the vanishing viscosity solution to the Riemann problem and to the boundary Riemann problem. It is not a weak solution, unless the system is conservative. One can prove that it is a viscosity solution actually meaning the extension of the semigroup of the vanishing viscosity solution to piecewise constant initial and boundary data. It is known that without taking into account the capillary pressure, the Buckley–Leverett model is the main one. Typically, from a computational point of view, approximate models are required for time slicing when creating computational algorithms. Analysis of the flow of a mixture of two immiscible liquids, the viscosity of which depends on pressure, leads to a further extension of the classical Buckley–Leverett model. Some two-phase flow models based on the expansion of Darcy’s law include the effect of capillary pressure. This is motivated by the fact that some fluids, e.g., crude oil, have a pressure-dependent viscosity and are noticeably sensitive to pressure fluctuations. Results confirm the insignificant influence of cross-coupling terms compared to the classical Darcy approach.