Non-Newtonian backflow in an elastic fracture

Backflow phenomenon, as a consequence of hydraulic fracturing, is of considerable tech-nical and environmental interest. Here, backflow of a non-Newtonian fluid from a disc-shaped elastic fracture is studied theoretically and experimentally. The fracture is of con-stant aperturehand the outlet section at constant pressurepe. We consider a shear-thinningpower-law fluid with flow behavior indexn. Fracture walls are taken to react with a forceproportional tohλ, withλa positive elasticity exponent; forλ=1linear elasticity holds.Constant overloadf0, 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=0and f0=0, the residual aperture and outflow rate scale asymptotically with time tast−n/(n+λ+1) and t−(2n+λ+1)/(n+λ+1) respectively, thus generalizing literature results for n=1and/orλ=1. For non-zero exit pressure and/or overload, the fracture aperture tends asymptotically to a constant value depending onλ,n,pe,f0, and other geometrical and physical parameters. 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.