Dynamic bridging of proppant particles in a hydraulic fracture

Abstract

This work is focused on the development of a dynamic criterion for the arching and bridging of spherical particles in a 3D suspension flow through a channel with plane walls. Elasticity of the particles and the channel walls are taken into account, as well as friction between the particles and with the walls. The carrier fluid is viscous, incompressible and Newtonian. Bridging occurs under the balance of the hydrodynamic force exerted from the fluid on the particles and the friction forces exerted from the walls and in between the particles. The 3D motion of particles in fluid is analyzed by means of direct numerical simulation. Various geometrical configurations are considered: three and four particles across the slot, and loose and close packing. Stability of the bridge is studied. The particles behave as a system with jumps (or a system with non-adjacent equilibrium positions), where the transition to the new equilibrium position occurs due to the accumulated elastic energy in the system (when the central particle is pushed through between the side ones). In this sense, the behavior of the grains is similar to the von Mizes truss. The bridging criterion is formulated as a domain on the plane in terms of the two nondimensional parameters: the particle size to channel width ratio and the flow velocity. For each particle-to-width size ratio there is a range of velocities, in which bridging occurs. The dynamic bridging criterion is different from the earlier purely kinematic criteria (e.g., w/d=2.5), which were formulated in terms of the particle-to-channel width ratio only. The bridging criterion is implemented into the 2D width-averaged lubrication model of suspension flow through a fracture, and illustrative simulations are conducted. The proposed model can be further used in the development of proppant transport models for fracture growth simulators, which are used in the oilfield industry for the design of hydraulic fracturing operations.