Non-Newtonian Backflow in an Elastic Fracture

Backflow phenomenon, as a consequence of hydraulic fracturing, is of considerable technical and environmental interest. Here, backflow of a non-Newtonian fluid from a disk-shaped elastic fracture is studied theoretically and experimentally. The fracture is of constant aperture h and the outlet section at constant pressure pe. We consider a shear-thinning power-law fluid with flow behavior index n. Fracture walls are taken to react with a force proportional to hλ, with λ a positive elasticity exponent; for λ=1 linear elasticity holds. Constant overload f0, acting on the fracture, is also embedded in the model. A transient closed-form solution is derived for the (i) fracture aperture, (ii) pressure field, and (iii) outflow rate. The particular case of a Newtonian fluid (n=1) is explicitly provided. For pe=0 and f0=0, the residual aperture and outflow rate scale asymptotically with time t as t−n/(n+λ+1) and t−(2n+λ+1)/(n+λ+1), respectively, thus generalizing literature results for n=1 and/or λ=1. For nonzero exit pressure and/or overload, the fracture aperture tends asymptotically to a constant value depending on λ, n, pe, f0, and other geometrical and physical parameters. The results are provided in dimensionless and dimensional form, including the time to achieve a given percentage of fluid recovery. In addition, an example application (with values of parameters derived from field scale applications) is included to further characterize the influence of fluid rheology. Experimental tests are conducted with Newtonian and shear-thinning fluids and different combinations of parameters to validate the model. Experimental results match well the theoretical predictions, mostly with a slight overestimation.