Measurement and modeling of three-phase relative permeability as a function of saturation path
Three-phase flow occurs in many petroleum recovery processes, especially during tertiary recovery. One of the most important parameters to accurately model these complex processes at large scale is relative permeability to each of the flowing fluids. However, relative permeability in three-phase systems becomes extremely complicated due to different flow mechanisms involved with three-phase flow as well as dependence of relative permeability on saturation path (water and oil saturations at each time in three-phase saturation space). During the past several decades, many studies reported experimental measurements of relative permeabilities in three-phase systems under different rock and fluid properties. Based on these measurements, several empirical models have been developed to predict relative permeabilities in three-phase space. However, the performance of these models have been frequently reported to be poor in reproducing experimental data. This indicates that despite the great effort during the past decades, there are still some aspects of three-phase space that have not been completely understood. Therefore, in this study we attempt to obtain a better physical understanding of three-phase relative permeability through extending relative permeability measurements over the entire three-phase phase space with different fluids and porous media. To do so, we use a gravity drainage method to measure three-phase relative permeabilities. However, previous applications of this method suffered from three main issues: a) relative permeabilities at medium to high saturations were not accessible due to quick saturation changes at early times of experiments, b) saturation path of experiments was out of control, and c) it was only applicable to unconsolidated sandpacks. In this study, we develop new procedures and methods to overcome the mentioned shortcomings of this method, by combining the method with steady-state method and applying a small gas pressure gradient. These improvements to the method allows us to measure relative permeabilities along different saturation paths, in both consolidated and unconsolidated porous media while extending the relative permeability measurements to higher saturation regions. In particular, we measure three-phase relative permeabilities along several saturation paths during unsteady-state gravity drainage experiments in two consolidated and unconsolidated media, with oils varying in composition, viscosity, and spreading coefficient. The method involves measurement of in-situ saturations along the porous media at different times during the experiments. These saturation profiles are then used to directly obtain three-phase relative permeabilities. We find that in our water-wet media, oil relative permeability varies significantly depending on the saturation path, while water relative permeability remains unchanged. In addition, oil relative permeabilities along different saturation paths extrapolate to different residual oil saturations. Interestingly, we find that once the measured relative permeabilities along different saturation paths are plotted as a function of mobile oil saturation, So-Sor, all of the differences between relative permeabilities vanish and they all form a single relative permeability curve. We compare our data with the relative permeability data published over the past several decades. We find that literature data are in complete agreement with our data. In addition, literature data for water-wet media suggest that for each media, experiments which end up at a different residual saturation result in a different relative permeability curve. This is while the experiments which reach to the same residual oil saturation result in the same oil relative permeability, regardless of their saturation paths. We examine the importance of residual oil saturation using three most commonly used relative permeability models against our data. These models consist of Corey, saturation weighted interpolation (SWI), and Stone. The idea is simply to change residual oil saturation while keeping all the other parameters constant, to fit experimental data along different saturation paths. We find that, by changing residual saturations, Corey and SWI models fit the data well, while the Stone model fails at low saturations. Overall, we find that the key to modeling relative permeability of water-wet media in three-phase space is residual oil saturation. Our data suggests that three-phase oil relative permeability is only a function of mobile oil saturation, and residual oil saturation is a non-linear function of water saturation.