Relaxation-induced flow in a smooth fracture for Ellis rheology

Hydraulic fracturing is a process aimed at improving the productivity of oil, gas or geothermal reservoirs. During hydrofracturing, backflow follows injection and represents the second phase of the process, when part of the fracturing fluid returns from fractures to well, and from well to surface. A conceptual model is presented to grasp the essential features of the phenomenon, conceiving the draining subsurface domain as a planar and rigid fracture. Backflow against an outlet pressure in the injection well is induced by the relaxation of the fracture wall, exerting a force on the fluid proportional to with the time-variable aperture and a non-negative exponent; an overload on the fracture may contribute to slowing or accelerating the closure process. The fluid rheology is described by the three-parameter Ellis constitutive equation, well representing the shear-thinning rheology typical of hydrofracturing fluids and coupling Newtonian and power-law behaviour. The interplay between these tendencies is modulated by a dimensionless number encapsulating most problem parameters; the range of variation of is discussed and found to vary around unity. The time-variable aperture and discharge rate, the space-time variable pressure field, and the time to drain a specified fraction of the fracture volume are derived as functions of geometry (length and initial aperture), wall elastic parameters, fluid properties, outlet pressure and overload . The late-time behaviour of the system is practically independent from rheology as the Newtonian nature of the fluid prevails at low shear stress. In particular, aperture and discharge scale asymptotically with time as and for ; else, the aperture tends to a constant, residual value proportional to . A case study with equally spaced fractures adopting realistic geometric, mechanical and rheological parameters is examined: two fluids normally used in fracking technology show completely different behaviours, with backflow dynamics and drainage times initially not dissimilar, later varying by orders of magnitude.