Reduced‐Order Models Unravel the Joint Impact of Aperture Heterogeneity and Shear‐Thinning Rheology on Fracture‐Scale Flow Metrics

Subsurface industrial operations often make use of complex engineered fluids. In fractured media, the hydraulic behavior of a geological fracture is affected by the shear-thinning (ST) rheology of such fluids, depending on the imposed pressure gradient. Besides, owing to the stochastic nature of heterogeneities in the fracture's local apertures, obtaining the generic behavior of a fracture under ST flow requires large statistics of fractures whose aperture fields have the same statistical properties, namely their mean value, standard deviation, correlation length Lc, and Hurst exponent (which controls the scale invariances at scales smaller than Lc). The first such Monte Carlo study was recently proposed by Lenci, Putti, et al. (2022) using a new lubrication-based model relying on a non-linear Reynolds equation. This model describes the flow of an Ellis fluid, with both a low shear rate quasi-Newtonian viscosity plateau and a large shear rate power law ST behavior, as measured for engineered fluids such as biopolymer solutions (e.g., xanthan gum). Here we aim to obtain the dependence of the fracture transmissivity and its dispersion over the statistics, as well as of the flow correlation length, on fracture closure, geometry correlation length, and applied pressure gradient, over a vast volume of the parameter space, and in a simple mathematical form. Employing reduced-order models based on the polynomial chaos expansion theory