An Analytical Equation to Predict Oil-Gas-Water Three-Phase Relative Permeability Curves in Fractures

As fractures are the major flow channels for multiphase flow in naturally and hydraulically fractured reservoirs, the accurate prediction of multiphase flow in fractures is highly important. The oil-gas-water three-phase relative permeability relations in fractures define the hydrodynamics of multiphase fluids flow and are necessary for modeling of multi-phase flow in fractured reservoirs. In this work, a novel flow model based on the concept of shell momentum balance, Newton's law of viscosity, and the cubic law, is derived to determine analytic functions for the three-phase relative permeability curves versus phase saturation and viscosity in a single fracture. The results show that the equations describing three-phase relative permeability curves in a fracture are function of saturations and viscosities. Water phase relative permeability depends on water saturation, gas phase relative permeability depends on gas saturation when μg is much lower than μo and μw. However, oil phase relative permeability is function of all-phase saturations. The isoperms of water phase and gas phase are straight lines. However, oil phase isoperms are functions of all phase saturations and have significant curvature. The curvatures of oil phase isoperms increase with the increase of μo. Gas saturation decreases oil phase relative permeability with a given oil saturation, while the viscosity ratio increases it. Introduction Multiphase flow in naturally fractured reservoirs and hydraulically fractured reservoirs, which holds major part of the world's remaining hydrocarbon reservoirs, is strongly influenced by fractures in the geological formations (Lei et al., 2014). Fractures are the major flow channels for fluid flowing in fractured reservoirs. So the accurate prediction of multiphase flow in fractures is highly important. The three-phase relative permeability relations for fractures define the hydrodynamics of fluid flow and are necessary for modeling of multiphase flow in reservoirs. The study of three-phase relative permeability was reported as early as 1941 by Leverett and Lewis (1941). They conducted steady-state three-phase relative permeability measurements in a tightly packed sand core. Corey et al. (1956) reported results of three-phase relative permeability measurements in Berea sandstone and proposed a model for prediction of three-phase relative permeability with assuming that the oil relative permeability depends on two saturations due to the dependence of residual oil saturation on two saturations. They suggested that the water phase isoperms and gas phase isoperms were straight lines. Sarem (1966) modified three-phase relative permeability measuring techniques by using unsteady-state technique. Donaldson and Dean (1965) used Sarem’s technique to measure three-phase relative permeability in Berea sandstone. Saraf and Fatt (1967) developed a new technique using NMR for in-situ saturation measurements in three-phase flow system. Stone (1970) proposed the StoneI model for prediction of three-phase relative permeability. Stone (1973) proposed the StoneII model using four two-phase flow relative permeability curves (two for oil-water and two for oil-gas) for predicting three-phase relative permeability. Dietich and Bonder (1976) proposed a model accounted for reduction in oil relative permeability due to the presence of a third phase. Spronsen (1982) measured a three-phase system in Berea sandstone using the centrifuge method. Saraf, et al. (1982), Grader and O’Meara (1988), and Maini, et al. (1990) measured the three-phase relative permeability using steady-state and unsteady-state methods. In addition, there have been more models (Maini, et al. 1989; Hustad and Hansen, 1995; Oosrom and Lenhard, 1998; Balbinski, et al., 1999) proposed for prediction of three-phase relative permeability. Oak, et al. (1990) and Oak (1990) conducted a three-phase relative permeability measurement on water-wet fired Berea sandstone core and presented about 1,800 data collected