Dispersion of solids in fracturing flows of yield stress fluids

Recent trends in the hydraulic fracturing have included the use of cyclic pumping of solid particles (proppant) interspersed with yield stress fracturing fluid, which is found to increase the subsequent productivity. A mathematical model is derived to estimate the dispersion of proppant slugs pumped in a cyclic fashion within the fracture. The flows here are challenging in that the combination of non-Newtonian rheology and particle migration results in sharp variations in shear rate and effective viscosity. Therefore, the suspensions vary from Stokesian behaviour to inertial behaviour across the width of the fracture. In this paper the Suspension Balance Model of Nott and Brady (J. Fluid Mech 275 (1994)) is extended to include non-Newtonian and inertial effects. This results in a set of scaled equations of motion for the mixture of noncolloidal particles suspended in a yield stress fluid plus a transport equation for the solid volume fraction. Asymptotic approach is used to derive the leading order governing equations in the limit of long and thin fracture. It is shown that an one dimensional nonlinear advection equation governs the streamwise dispersion of solid volume fraction to the leading order. The flow profiles and solid dispersion are computed for a range of dimensionless flow parameters. The results show that the solid dispersion strongly depends on inertial and non-Newtonian effects. In general, use of yield stress background fluid will limit the particle phase dispersion within the fracture. For a range of yield stress; however, the slurry convection and consequently fracture closure may occur due to particles settling in the direction of gravity and particles shear-induced migration toward the regions of minimum shear rate (i.e., central pseudo-plug region).