The relation between interfacial tensions and wettability in three-phase systems: Consequences for pore occupancy and relative permeability

In this paper, we consider the relationships between wettability and three-phase pore occupancy in simple systems of arbitrary wettability, defined in terms of the oil - water contact angle, ?_ow of a pore. When O_ow changes, the remaining contact angles ?_gw and O_go also vary, consistent with available limited experimental data and a theoretical constraint equation. This equation was derived originally by Bartell and Osterhof [Ind. Eng. Chem. 19 (11) (1927) 1277] and applied more recently by Zhou and Blunt [J. Contam. Hydrol. 25 (1997)]. The consequences of this theory on several aspects of three-phase flow in mixed-wet and fractionally wet systems are discussed, in particular the issues of fluid-fluid wetting order, capillary displacement invariants, pore occupancy and saturation-dependencies of three-phase relative permeabilities. The model of pore occupancy leads to quite complex saturation-dependencies of the corresponding three-phase relative permeabilities. In case of distributed contact angles in both the water-wet and oil-wet pore clusters, all of the three-phase relative permeabilities depend on two saturations in a complex manner for most saturation combinations. In the general case, where the three-phase relative permeabilities cannot be related simply to the underlying two-phase relative permeabilities, a process-based model based on the pore scale physics is required. Given the great importance of the constraint equation, we recommend further experimental measurements of three-phase interfacial tension-contact angle data to establish the actual relationship between the three contact angles.