Three-Phase Hydraulic Conductances in Angular Capillaries

Abstract

Abstract 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. Introduction Since direct measurement of flow of three immiscible fluids is very difficult, the pore-scale models of three-fluid systems2,5,12,13,17 have blossomed. One of the more important advancements in such models was the approximation of single pore geometries as angular capillaries with square or arbitrary triangular cross-sections. Although real pores are not exactly square or triangular, this approximation allows one to capture the flow of water in the pore corners and the flow of oil and gas in the pore center. As illustrated in Figure 1a, when three fluids are moving in a single angular capillary, the most wetting fluid (water or Fluid 1) resides in the corner and the most nonwetting fluid (gas or Fluid 3) fills the center. The third fluid (oil or Fluid 2) forms an intermediate layer sandwiched between the other two fluids. In some cases of large contact angles and positive spreading coefficients, we may find more than one sandwiched layer, Figure 1b. These intermediate layers are a few micrometers thick and have been observed in micromodel experiments.3,11,16,21 It is drainage through these oil layers that is responsible for the high oil recoveries seen experimentally.4,10,23 Although it was initially thought that only spreading oils could form such layers in angular pores, it has been theoretically predicted and experimentally verified that nonspreading oil can also form intermediate layers in the crevices of the pore space.3,11,25 Therefore, the formation of sandwiched layers is not only related to the positive spreading coefficient, but also depends on the curvatures of the o/w and g/o interfaces, the corner geometry and the contact angles.5 Creeping flow of oil in these intermediate layers is the subject of this paper. In particular, we study the hydraulic conductances of oil flow in stable fluid layers of different sizes and geometries. We provide simple and accurate correlations for these conductances by relating them to the interface geometry, fluid contact angles, and pore geometry. The proposed correlations should be useful in pore network calculations of three-phase relative permeabilities.