We use a three-dimensional mixed-wet random network model representing Berea sandstone to compute displacement paths and relative permeabilities for water alternating gas (WAG) flooding. First we reproduce cycles of water and gas injection observed in previously published experimental studies. We predict the measured oil, water and gas relative permeabilities accurately. We discuss the hysteresis trends in the water and gas relative permeabilities and compare the behavior of water-wet and oil-wet media. We interpret the results in terms of pore-scale displacements. In water-wet media the water relative permeability is lower in the presence of gas during waterflooding due to an increase in oil/water capillary pressure that causes a decrease in wetting layer conductance. The gas relative permeability is higher for displacement cycles after first gas injection at high gas saturation due to cooperative pore filling, but lower at low saturation due to trapping. In oil-wet media, the water relative permeability remains low until water-filled elements span the system at which point the relative permeability increases rapidly. The gas relative permeability is lower in the presence of water than oil because it is no longer the most non-wetting phase.
We show how to use network modeling to develop a physically-based empirical model for three-phase relative permeability. We demonstrate that the relative permeabilities are approximately independent of saturation path when plotted as a function of flowing saturation. The flowing saturation is the saturation minus the amount that is trapped. The amount of oil and gas that is trapped shows a surprising trend with wettability - weakly water-wet media show more trapping of oil and gas than a water-wet system due to the complex competition of different three-phase displacement processes. Further work is needed to explore the full range of behavior as a function of wettability and displacement path.