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

Abstract : The slightly compressible flow of a gas in the slip regime within a rough fracture featuring a heterogeneous aperture field is analyzed 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 that 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, that 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 sub-problems. 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.
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Submitted on : Wednesday, September 25, 2019 - 11:34:21 AM
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Tony Zaouter, Didier Lasseux, Marc Prat. Gas slip flow in a fracture: local Reynolds equation and upscaled macroscopic model. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2018, 837, pp.413-442. ⟨10.1017/jfm.2017.868⟩. ⟨hal-02366909v2⟩

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