The slightly compressible flow of a gas in the slip regime within a rough fracture featuring a heterogeneous aperture field is analysed in depth in this work. Starting from the governing Navier–Stokes, continuity and gas state law equations together with a first-order slip boundary condition at the impermeable walls of the fracture, the two-dimensional slip-corrected Reynolds model is first derived, which is shown to be second-order-accurate in the local slope of the roughness asperities while being first-order-accurate in the Knudsen number. Focusing the interest on the flow-rate to pressure-gradient relationship over a representative element of the fracture, an upscaling procedure is applied to the local Reynolds equation using the method of volume averaging, providing a macroscopic model for which the momentum conservation equation has a Reynolds-like form. The effective macroscopic transmissivity tensor, which is characteristic of the representative element, is shown to be given by a closure problem that is non-intrinsic to the geometrical structure of the fracture only due to the slip effect. An expansion to the first order in the Knudsen number is carried out on the closure, yielding a decomposition of the effective transmissivity tensor into its purely viscous part and its slip correction, both being given by the solution of intrinsic closure subproblems. Numerical validations of the solution to the closure problem are performed with analytical predictions for simple fracture geometries. Comparison between the macroscopic transmissivity tensor, obtained from the solution of the closure problem, and its first-order approximation is illustrated on a randomly rough correlated Gaussian fracture.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.