Analytical expressions for three-phase generalized relative permeabilities in water- and oil-wet capillary tubes

We analyze three-phase flow of immiscible fluids taking place within an elementary capillary tube with circular cross-section under water- and oil-wet conditions. We account explicitly for momentum transfer between the moving phases, which leads to the phenomenon of viscous coupling, by imposing continuity of velocity and shear stress at fluid-fluid interfaces. The macroscopic flow model which describes the system at the Darcy scale includes three-phase effective relative permeabilities, Kij,r, accounting for the flux of the ith phase due to the presence of the jth phase. These effective parameters strongly depend on phase saturations, fluid viscosities, and wettability of the solid matrix. In the considered flow setting, Kij,r reduce to a set of nine scalar quantities, Kij,r. Our results show that Kij,r of the wetting phase is a function only of the fluid phase own saturation. Otherwise, Kij,r of the non-wetting phase depends on the saturation of all fluids in the system and on oil and water viscosities. Viscous coupling effects (encapsulated in Kij,r with i ≠ j) can be significantly relevant in both water- and oil-wet systems. Wettability conditions influence oil flow at a rate that increases linearly with viscosity ratio between oil and water phases.