Abstract
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.
Highlights
Non-Newtonian fluid flow in fractured media is of interest for many environmentally related applications, such as hydraulic fracturing, drilling operations, enhanced oil recovery, and subsurface contamination and remediation
Flow modeling at the single-fracture scale leads to the determination of the flow rate under a given pressure gradient as a function of the parameters that describe the variability of the aperture field or of the confining walls
The power-law model presents a linear trend in log-log coordinates, while two different behaviors can be observed in the Prandtl–Eyring model: for low-pressure gradients, the slope of the red solid line is minor compared with the one of the blue solid line but higher than the Newtonian case; on the other hand, when the average fracture shear rates of the two fluids are sufficiently different, the Prandtl–Eyring model rapidly increases its steepness
Summary
Non-Newtonian fluid flow in fractured media is of interest for many environmentally related applications, such as hydraulic fracturing, drilling operations, enhanced oil recovery, and subsurface contamination and remediation. The basic building block in fractured media modeling is a thorough understanding of flow and transport in a single fracture [1]. As a result of the heterogeneity of these surfaces, the fracture aperture is spatially variable To model this variability, two basic approaches have been adopted. The second envisages the aperture variability as the outcome of the joint variation in self-affine surfaces, correlated at all scales [3]. In both cases, flow modeling at the single-fracture scale leads to the determination of the flow rate under a given pressure gradient as a function of the parameters that describe the variability of the aperture field or of the confining walls. A hydraulic aperture can be derived from the flow rate [4] as the aperture of a smooth-walled conduit that would produce the same flow rate under a given pressure gradient as the real rough-walled fracture
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