​In this paper, we extend to three fluid phases a prior finite-element study of hydraulic conductance of two-phase creeping flow in angular capillaries. Previously, we obtained analytic expressions for the hydraulic conductance of water in corner filaments. Here we present the results of a large numerical study with a high-resolution finite element method that solves the three-phase creeping flow approximation of the Navier-Stokes equation. Using the projection-pursuit regression approach, we provide simple analytic expressions for the hydraulic conductance of an intermediate layer of oil sandwiched between water in the corners of the capillary and gas in the center. Our correlations are derived for the oil layers bounded by the concave or convex interfaces that are rigid or allow perfect slip. Therefore, our correlations are applicable to drainage, spontaneous imbibition, and forced imbibition with maximum feasible hysteresis of each contact angle, oil/water and gas/oil. These correlations should be useful in porenetwork calculations of three-phase relative permeabilities of spreading oils. Finally, we compare our results with the existing correlations by Zhou et al., and Hui & Blunt, who assumed thin-film flow with an effective film thickness proportional to the ratio of the average, our correlations are two-four times closer to the numerical results than the corresponding correlations by Zhou et al., and Hui & Blunt.