Non‐Newtonian Backflow in an Elastic Fracture

Abstract

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 p e . We consider a shear-thinning power-law fluid with flow behavior index n. Fracture walls are taken to react with a force proportional to h , with a positive elasticity exponent; for = 1 linear elasticity holds. Constant overload f 0 , 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 p e = 0 and f 0 = 0, the residual aperture and outflow rate scale asymptotically with time t as t −n/(n+ +1) and t −(2n+ +1)/(n+ +1) , respectively, thus generalizing literature results for n = 1 and/or = 1. For nonzero exit pressure and/or overload, the fracture aperture tends asymptotically to a constant value depending on , n, p e , f 0 , 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.

Onset of Darcy–Bénard convection under throughflow of a shear-thinning fluid

“…Other rheological models, more complex than the power law, are available to describe Xanthan Gum mixtures (see, for example, Escudier et al (2001), where a Carreau-Yasuda model is favourably tested). However, the power-law model is suitable to locally describe complex rheologies, with several validations in complex flows geometries -see Longo et al (2013a) and the recent Chiapponi et al (2019). In this regard, we have verified that in the shear rate range of our experiments, a power-law model adequately fits the rheometric data -see figure 1(b) and its caption in the supplementary material.…”

Soft hydraulics: from Newtonian to complex fluid flows through compliant conduits

“…Observe that (61) at steady state (∂p/∂t = 0) reduces to (23) (subject to properly defining the flow rate to match), but not to (37) (due to the linearization of the (h 0 + u r ) 3 term). Similar unsteady problems involving viscous non-Newtonian flows in compliant conduits arise when analyzing the relaxation of an elastic fracture filled with a complex fluid [209,90]. These types of fluid-structure interactions involving complex fluids are common in hydraulic fracturing [210].…”

“…These types of fluid-structure interactions involving complex fluids are common in hydraulic fracturing [210]. In [209,90], as a simplification, the deformation is considered to be uniform in the axial direction, so that the channel (fracture) height is just h(t). Then, from (59), a suitable PDE for the non-Newtonian rheology can be derived.…”

An Experimental Setup to Investigate Non-Newtonian Fluid Flow in Variable Aperture Channels

“…The rheological parameters were measured with a Ubbeholde viscometer, for the Newtonian fluids, and in a parallel-plate rheometer by Anton Paar, at the same temperature of the experiments, for the non-Newtonian mixtures. The rheological parameters of the power-law fluid are listed in Table 1 and were estimated according to the techniques detailed in [18,19,33].…”

“…Experiments with non-Newtonian fluids are even more challenging, also due to the needs of accurate measurements and interpretations of rheological properties of the fluids, with a continuous effort to correctly quantify uncertainties [18]. Recently, Chiapponi et al [19] studied theoretically and experimentally the back-flow of Ostwald-deWaele fluids from a disk-shaped elastic fracture, highlighting the different efficiency of the process in the presence of shear-thinning fluids. In a slightly different context, with gravity currents of HB fluids instead of pressurized flows, the experiments documented in Di Federico et al [20] validated a general analytical solution based on the perturbation of a self-similar solution.…”