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

Zaouter, Tony and Lasseux, Didier and Prat, Marc Gas slip flow in a fracture: local Reynolds equation and upscaled macroscopic model. (2018) Journal of Fluid Mechanics, 837. 413-442. ISSN 0022-1120

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Official URL: https://doi.org/10.1017/jfm.2017.868

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.

Item Type:Article
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:Other partners > Arts et Métiers ParisTech (FRANCE)
Other partners > Bordeaux INP - BINP (FRANCE)
French research institutions > Commissariat à l'Energie Atomique et aux énergies alternatives - CEA (FRANCE)
French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - Toulouse INP (FRANCE)
French research institutions > Institut National de la Recherche Agronomique - INRA (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UT3 (FRANCE)
Other partners > Université de Bordeaux (FRANCE)
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Deposited On:16 Jan 2019 14:10

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