Buckley--Leverett Equation with Viscosities and Relative Permeabilities Dependending on Pressure

The aim of this note is to analyze the flow of a mixture of two immiscible fluids whose viscosity depends on pressure, generalizing the classical Buckley–Leverett model. Such extension is mainly motivated by the overwhelming evidence that many fluids have a viscosity that depends on pressure, a typical example being crude oils, where viscosity is extremely sensitive to pressure variations. We model the flow of two immiscible fluids through a channel neglecting nonlinear effects and assuming that the fluids are separated by a smooth interface. The solid matrix is treated as an impervious rigid body and a classical no-slip condition is imposed on its boundary. We end up with a system of equations for the saturation and for the pressure which reduces to the classical Buckley–Leverett equation when both viscosities do not depend on pressure. In doing so, we find that the relative permeabilities depend on the fluid pressure, and when the solid matrix is treated as rigid. Moreover, we study analytically the steady state problem for generic pressure-dependent viscosities and for the general case we provide some numerical simulations.