Two-phase flow of a shear thinning fluid in reservoir rock

Abstract

The flow of a purely viscous rheologically complex fluid as the aqueous phase in two-phase flow in a porous medium is developed. The relationship relative permeability vs. saturation is related to the pore-size distribution of the medium using a “bundle of capillaries”, generalized to allow for flow paths of various diameters and lengths. The pore size distribution is determined by a capillary pressure vs. saturation curve. A weighted shear rate and saturation-dependent viscosity function is used to calculate the aqueous phase flow rate from the multiphase Darcy equation using conventional water relative permeability values. A similar approach can be applied to the hydrocarbon phase if one suspects the displaced phase is rheologically complex. The viscosity function is represented by the empirical power law model. However, the viscosity function may also be represented by a variation of the generalized scale-up method for single-phase non-Newtonian flow in porous media. The analysis predicts interactions between advance rate, rheology, shear rate, reservoir rock morphology, and the displacing phase saturation. Some predicted interactions are absent in the case of a Newtonian/Newtonian fluid pair. The analysis is extended to a consideration of the mobility ratio referenced with respect to the displacement front between the aqueous and hydrocarbon phases. The predicted relationship between the apparent viscosity of the aqueous phase and the saturation level suggests sharp changes in polymer phase mobility can occur over rather small intervals of saturation. These results are used to develop a modified fractional flow expression for non-Newtonian two-phase flow. Fractional flow for a non-Newtonian/Newtonian fluid pair is illustrated for a shear thinning aqueous phase using the properties of a xanthan gum solution.