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, K i j , r , accounting for the flux of the i th phase due to the presence of the j th phase. These effective parameters strongly depend on phase saturations, fluid viscosities, and wettability of the solid matrix. In the considered flow setting, K i j , r reduce to a set of nine scalar quantities, K i j , r . Our results show that K i j , r of the wetting phase is a function only of the fluid phase own saturation. Otherwise, K i j , 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 K i j , 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.