An Empirical Model for Three-Phase Relative Permeability

We present an empirical model for three-phase relative permeability that overcomes the limitations of current formulations, such as Stone's methods.1,2 We provide a self-consistent treatment of wettability, changes in hydrocarbon composition, different saturation paths, and the trapping of oil, water and gas. The theoretical development is motivated by a review of recent three-phase experiments. The model is based on saturation-weighted interpolation between the two-phase values.3 To account for the effects of wettability we apply saturation-weighting to all three phases. By writing the relative permeabilities as unique functions of a flowing saturation, the model predicts the behavior for any sequence of saturation changes and accounts for trapping. Layer drainage, which allows oil relative permeabilities to be extrapolated to low saturation, is included for water-wet media. The model ensures smooth changes in relative permeability with changes in hydrocarbon composition and tends to the appropriate limits as the gas and oil become miscible. The model is tested against the data of Oak and co-workers.4,5 We show that it is necessary to include layer drainage and oil trapping to predict three-phase oil relative permeability at low oil saturation accurately.