Non-Newtonian fluid flow in fractured media is relevant for hydraulic fracturing operations aimed at the exploitation of oil, gas and thermal reservoirs. Rheologically complex fluids also interact with pre-existing rock fractures during drilling operations, environmental remediation, enhanced oil recovery, and natural phenomena such as lava and sand intrusions. Even Newtonian flow in single natural or artificial fractures is known to be typically tortuous and preferential, as the fracture aperture is characterized by a strong degree of variability, which in turn is modeled either as a 2-D random field or as the space between self-affine fracture walls. Specific challenges and compound effects arise from the interaction between the nonlinearity of the flow and the inherent multiscale heterogeneity characterizing the fractured medium. To capture such interactions and provide a benchmark for numerical models, we adopt a simplified geometric model to describe the aperture variability, consisting of adjacent one-dimensional channels with constant aperture, each drawn from a stochastic aperture distribution of given mean and variance. The flow rate in such a fracture subject to an external pressure gradient is then derived under the lubrication approximation for the two limiting cases of a pressure gradient which is perpendicular/parallel to aperture variation; these parallel/serial arrangements (PA/SA) provide an upper/lower bound to the fracture conductance.

The fluid rheology is described via a Prandtl-Eyring rheological model. Novel closed-form results for the flowrate in the PA/SA cases are then derived and discussed; different distributions and combinations of the parameters describing the fluid rheology and the variability of the aperture field are considered. Results are compared with those valid for a power-law fluid.