Analytical expressions for upscaled relative permeabilities in three-phase flow

We present analytical solutions for the relative permeabilities governing a Darcy scale description of threephase flow of immiscible fluids. We consider flow taking place within a capillary tube with circular crosssection for two settings corresponding to (a) a water wet and (b) an oil wet configuration. Momentum transfer between the moving phases, which leads to the phenomenon of viscous coupling, is explicitly accounted by imposing continuity of velocity and shear stress at the fluid-fluid interfaces. The macroscopic model describing the system at the Darcy scale includes three-phase effective relative permeabilities, Kij, r, accounting for the flow rate of the ith-phase due to the presence of the jth-phase. These effective coefficients are function of phases saturation, fluids viscosity and wettability of the medium. Our results show that (i) the relative permeability Kii, r of the wetting phase is only a function of its own saturation while Kii, r of the non-wetting phases depend on the saturations of all the fluids; (ii) viscous coupling effects (elucidated by Kij, r with i ≠, j) can be relevant in water wet and oil wet systems.