Backflow of non-Newtonian fluids in smooth fractures: theoretical and experimental investigations

Backflow of a non-Newtonian fluid in a plane smooth fracture is modeled both theoretically and experimentally to simulate one important phase of hydrofracturing operations. An Ellis or power-law constitutive equation is considered for flow in a smooth fracture closing under the action of relaxating walls contrasted by the internal fluid pressure. A generally nonlinear relationship between pressure and aperture is considered, to take into account the progressive softening or stiffening of the boundary associated with the properties of the surrounding rock. Coupling this relationship with mass balance and 1-D flow equations results in an integro-differential problem admitting a numerical solution, reducing to a closed-form implicit solution for the special case of power-law fluid. The behaviour of pressure, aperture, and drainage time is discussed as function of time, space and problem parameters. A comparison is conducted between the drainage time in the plane and radial geometry. The approach is further generalized by introducing leak-off. To validate the theoretical results, 14 experiments are conducted with ad-hoc replicas of radial or rectangular fractures. The latter have an aspect ratio 2.5-2.7, with a maximum height around 2 mm; the elastic reaction of the walls is due to o-rings, sealing the fracture without adding friction disturbances. Fluids with different rheology, both Newtonian and power-law shear-thinning, are associated with different boundary conditions of external pressure and overload. The match between theory and experiments is fairly good, with discrepancies of a few percent.