Unsteady Flow of Shear-Thinning Fluids in Porous Media with Pressure-Dependent Properties

In this paper, a model for injection of a power-law shear-thinning fluid in a medium with pressure-dependent properties is developed in a generalized geometry (plane, radial, and spherical). Permeability and porosity are taken to be power functions of the pressure increment with respect to the ambient value. The model mimics the injection of non-Newtonian fluids in fractured systems, in which fractures are already present and are enlarged and eventually extended and opened by the fluid pressure, as typical of fracing technology. Empiric equations are combined with the fundamental mass balance equation. A reduced model is adopted, where the medium permeability resides mainly in the fractures; the fluid and porous medium compressibility coefficients are neglected with respect to the effects induced by pressure variations. At early and intermediate time, the flow interests only the fractures. In these conditions, the problem admits a self-similar solution, derived in closed form for an instantaneous injection (or drop-off) of the fluid, and obtained numerically for a generic monomial law of injection. At late times, the leak-off of the fluid towards the porous matrix is taken into account via a sink term in the mass balance equation. In this case, the original set of governing equations needs to be solved numerically; an approximate self-similar solution valid for a special combination of parameters is developed by rescaling time. An example of application in a radial geometry is provided without and with leak-off. The system behaviour is analysed considering the speed of the pressure front and the variation of the pressure within the domain over time, as influenced by the domain and fluid parameters.