Two-continua model of suspension flow in a hydraulic fracture

Construction of an asymptotic two-continua model of a suspension flow in a vertical hydraulic fracture with account for a finite particle volume fraction and the particle slip has been demonstrated. Existing models of particle transport in a fracture were formulated within the effective-fluid approximation, where the suspension is assumed to be viscous incompressible fluid with the density and viscosity depending on the particle volume concentration. The dependence of the viscosity on the particle volume concentration is determined by the Scott formula. Particle settling velocity is determined by an empirical formula, which accounts for the decrease in the rate of settling with the increase in the particle volume concentration. It is shown that the difference between the suspension flow model, constructed earlier within the heuristic single-speed approach, and the two-continua model, proposed based on the conservation laws of the mechanics of multiphase media, and it has only a slight effect on the law of fracture propagation, as it follows from the preliminary numerical study of a coupled problem.