Consistency of three-phase capillary entry pressures and pore phase occupancies

Using the general equation for the true three-phase capillary entry pressures [van Dijke MIJ, Sorbie KS. Three-phase capillary entry conditions in pores of non-circular cross-section. J Colloid Interf Sci 2003;260:385–97, van Dijke MIJ, Lago M, Sorbie KS, Araujo M. Free energy balance for three fluid phases in a capillary of arbitrarily shaped cross-section: capillary entry pressures and layers of the intermediate-wetting phase. J Colloid Interf Sci 2004;277:184–201], we investigate for a single pore of uniform but arbitrary wettability the consistency of the relation between the pressure combinations and the pore phase occupancies in a three-phase system. The solutions of this equation corresponding to all possible displacements arising in three-phase flow are plotted in a single graph for two of the radii of curvature, which are equivalent to the phase pressure combinations. Consequently, this graph presents a delineation of the radius (pressure) space corresponding to different pore phase occupancies and we show that the underlying pressure–occupancy relation is unique using a free energy argument. The relation is presented for two very different parameter combinations. We explore how it can be extended when additional restrictions apply, for example those arising from lack of phase availability in an interconnected network. Using a capillary bundle of pores with triangular cross-section, we show the consequences for three-phase displacement processes if capillary entry pressures are and are not chosen consistently, the latter being common practice in all current three-phase pore network models. Displacements in both a water-wet and in a mixed-wet system are studied. The main conclusions from this work are that for a single pore different parameter combinations may lead to very different, but unique, pressure–occupancy relations. The saturation paths resulting from three-phase flow calculations in a capillary bundle of triangular pores are often significantly different using the true three-phase conditions as opposed to using inconsistent (two-phase) conditions, in particular in a mixed-wet system. Such large differences may also occur in interconnected pore network models, but only when continuity of the phases is high. The results indicate that, when the true conditions are used, there exists also a unique relation between the pressure combinations and the (bulk) occupancies in a bundle of pores with non-circular cross-section, similar to the relation for a bundle of cylindrical pores.