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Gas slip flow in a fracture: local Reynolds equation and upscaled macroscopic model

Published online by Cambridge University Press:  21 December 2017

Tony Zaouter ,
Didier Lasseux Open the ORCID record for Didier Lasseux [Opens in a new window]  and
Marc Prat
Tony Zaouter
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, 2, Allée du Prof. Camille Soula, 31400 Toulouse, France CEA, DEN, SEAD, Laboratoire d’Étanchéité, 30207 Bagnols-sur-Cèze, France
Didier Lasseux*
Affiliation:
CNRS, I2M, UMR 5295 – Esplanade des Arts et Métiers, 33405 Talence, CEDEX, France
Marc Prat
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, 2, Allée du Prof. Camille Soula, 31400 Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Low-Reynolds-number flows: Lubrication theory Low-Reynolds-number flows: Porous media Rarefied Gas Flow: Rarefied Gas Flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 413 - 442
Copyright
© 2017 Cambridge University Press 

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