In a previous paper (Dehghanpour et al., Phys Rev E 83:065302, 2011a), we showed that relative permeability of mobilized oil, , measured during tertiary gravity drainage, is significantly higher than that of the same oil saturation in other tests where oil is initially a continuous phase. We also showed that tertiary strongly correlates to both water saturation, , water flux (water relative permeability), , and the change in water saturation with time, . To develop a model and understanding of the enhanced oil transport, identifying which of these parameters (, or ) plays the controlling role is necessary, but in the previous experiments these could not be deconvolved. To answer the remaining question, we conduct specific three-phase displacement experiments in which is controlled by applying a fixed water influx, and develops naturally. We obtain by using the saturation data measured in time and space. The results suggest that steady-state water influx, in contrast to transient water displacement, does not enhance . Instead, reducing water influx rate results in excess oil flow. Furthermore, according to our pore scale hydraulic conductivity calculations, viscous coupling and fluid positioning do not sufficiently explain the observed correlation between and . We conclude that tertiary is controlled by the oil mobilization rate, which in turn is linked to the rate of water saturation decrease with time, . Finally, we develop a simple model which relates tertiary to transient two-phase gas/water relative permeability.