# Shear-Thinning Fluid Flow in Variable-Aperture Channels

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Dipartimento di Ingegneria Civile, Ambientale e dei Materiali (DICAM), Università di Bologna Alma Mater Studiorum, 40136 Bologna, Italy

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Dipartimento di Ingegneria e Architettura (DIA), Università di Parma, Parco Area delle Scienze, 181/A, 43124 Parma, Italy

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Author to whom correspondence should be addressed.

Received: 2 March 2020 / Revised: 7 April 2020 / Accepted: 13 April 2020 / Published: 17 April 2020

(This article belongs to the Special Issue Urban Water Management: A Pragmatic Approach)

Non-Newtonian fluid flow in a single fracture is a 3-D nonlinear phenomenon that is often averaged across the fracture aperture and described as 2-D. To capture the key interactions between fluid rheology and spatial heterogeneity, we adopt a simplified geometric model to describe the aperture variability, consisting of adjacent one-dimensional channels with constant aperture, each drawn from an assigned aperture distribution. The flow rate is then derived under the lubrication approximation for the two limiting cases of an external pressure gradient that is parallel/perpendicular to the channels; these two arrangements provide upper and lower bounds to the fracture conductance. The fluid rheology is described by the Prandtl–Eyring shear-thinning model. Novel closed-form results for the flow rate and hydraulic aperture are derived and discussed; different combinations of the parameters that describe the fluid rheology and the variability of the aperture field are considered. The flow rate values are very sensitive to the applied pressure gradient and to the shape of the distribution; in particular, more skewed distribution entails larger values of a dimensionless flow rate. Results for practical applications are compared with those valid for a power-law fluid and show the effects on the fracture flow rate of a shear stress plateau.