In hydraulic fracturing, specially engineered suspensions are pumped at high pressure and rate into the reservoir, causing a propagating fracture to open. When the pressure is released the fracture is supported by the grains of solid proppant that are left behind. Recent trends in the oil industry have included the use of cyclic pumping of a proppant slurry interspersed with clear frac fluid, which is found to increase the subsequent productivity. It is therefore of interest to understand how slugs of proppant pumped in a cyclic fashion can disperse in the pipe on the way and finally in the fracture.
We present a model to describe dispersion of solid particles (proppant) along the fracture in a laminar flow of shear thinning yield stress fluids. We consider two-phase governing equations, assuming that solid and fluid phases can be described as two phases of incompressible continua. We adopt the standard Hele-Shaw type scales and multi-timescale approaches to derive a 1D advection/diffusion model along the streamlines for transport and dispersion of the mean solid particle concentration. We show the effects of lift, drag and centrifugal forces on the dispersion dynamics of the particles along the fracture.