Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T11:26:05.716Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Agrawal, A. & Prabhu, S. V. 2008 Survey on measurement of tangential momentum accommodation coefficient. J. Vac. Sci. Technol. A 26 (4), 634–645.10.1116/1.2943641Google Scholar
Arkilic, E. B., Breuer, K. S. & Schmidt, M. A. 2001 Mass flow and tangential momentum accommodation in silicon micromachined channels. J. Fluid Mech. 437, 29–43.10.1017/S0022112001004128Google Scholar
Arkilic, E. B., Schmidt, M. A. & Breuer, K. S. 1997 Gaseous slip flow in long microchannels. J. Microelectromech. Syst. 6 (2), 167–178.10.1109/84.585795Google Scholar
Bergström, D.2012 Rough surface generation & analysis. http://www.mysimlabs.com/surface_generation.html.Google Scholar
Berkowitz, B. 2002 Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour. 25 (8–12), 861–884.10.1016/S0309-1708(02)00042-8Google Scholar
Beskok, A. & Karniadakis, G. E. 1999 A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Therm. Engng 3 (1), 43–77.Google Scholar
Burgdorfer, A. 1959 The influence of the molecular mean free path on the performance of hydrodynamic gas lubricated bearings. Trans. ASME J. Basic Engng 81, 94–100.10.1115/1.4008375Google Scholar
Cai, C., Sun, Q. & Boyd, I. D. 2007 Gas flows in microchannels and microtubes. J. Fluid Mech. 589, 305–314.10.1017/S0022112007008178Google Scholar
Cercignani, C. 1988 The Boltzmann Equation and its Applications. Springer.10.1007/978-1-4612-1039-9Google Scholar
Christensen, H. 1970 Stochastic models for hydrodynamic lubrication of rough surfaces. In Proceedings of the Institution of Mechanical Engineers, vol. 184, pp. 1013–1026.Google Scholar
Dapp, W. B. & Müser, M. H. 2016 Fluid leakage near the percolation threshold. Sci. Rep. 6, 19513.10.1038/srep19513Google Scholar
Dongari, N., Agrawal, A. & Agrawal, A. 2007 Analytical solution of gaseous slip flow in long microchannels. Intl J. Heat Mass Transfer 50 (17–18), 3411–3421.10.1016/j.ijheatmasstransfer.2007.01.048Google Scholar
Ewart, T., Perrier, P., Graur, I. & Méolans, J. G. 2007 Tangential momemtum accommodation in microtube. Microfluid. Nanofluid. 3 (6), 689–695.10.1007/s10404-007-0158-3Google Scholar
Garcia, N. & Stoll, E. 1984 Monte Carlo calculation for electromagnetic-wave scattering from random rough surfaces. Phys. Rev. Lett. 52 (20), 1798–1801.10.1103/PhysRevLett.52.1798Google Scholar
Graur, I. A., Perrier, P., Ghozlani, G. & Molans, J. G. 2009 Measurements of tangential momentum accommodation coefficient for various gases in plane microchannel. Phys. Fluids 21, 102004.10.1063/1.3253696Google Scholar
Gray, W. G. 1975 A derivation of the equations for multi-phase transport. Chem. Engng Sci. 30 (2), 229–233.10.1016/0009-2509(75)80010-8Google Scholar
Hamrock, B. J., Schmid, S. R. & Jacobson, B. O. 2004 Fundamentals of Fluid Film Lubrication. CRC Press.10.1201/9780203021187Google Scholar
Howes, F. A. & Whitaker, S. 1985 The spatial averaging theorem revisited. Chem. Engng Sci. 40 (8), 1387–1392.10.1016/0009-2509(85)80078-6Google Scholar
Karniadakis, G. E., Beskok, A. & Aluru, N. 2005 Microflows and Nanoflows: Fundamentals and Simulation, Interdisciplinary Applied Mathematics, vol. 29. Springer.Google Scholar
Klinkenberg, L. J. 1941 The permeability of porous media to liquids and gases. In Drilling and Production Practice, pp. 200–213. American Petroleum Institute.Google Scholar
Lasseux, D. & Valdes Parada, F. J. 2017 Symmetry properties of macroscopic transport coefficients in porous media. Phys. Fluids 29 (4), 043303.10.1063/1.4979907Google Scholar
Lasseux, D., Valdes Parada, F. J., Ochoa Tapia, J. A. & Goyeau, B. 2014 A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media. Phys. Fluids 26 (5), 053102.10.1063/1.4875812Google Scholar
Lasseux, D., Valdés Parada, F. J. & Porter, M. 2016 An improved macroscale model for gas slip flow in porous media. J. Fluid Mech. 805, 118–146.10.1017/jfm.2016.562Google Scholar
Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Springer Handbook of Experimental Fluid Mechanics, pp. 1219–1240. Springer.10.1007/978-3-540-30299-5_19Google Scholar
Ledoux, Y., Lasseux, D., Favreliere, H., Samper, S. & Grandjean, J. 2011 On the dependence of static flat seal efficiency to surface defects. Int. J. Pressure Vessels Piping 88 (11–12), 518–529.10.1016/j.ijpvp.2011.06.002Google Scholar
Lee, S. H., Lough, M. F. & Jensen, C. L. 2001 Hierarchical modeling of flow in naturally fractured formations with multiple length scale. Water Resour. Res. 37 (3), 443–455.10.1029/2000WR900340Google Scholar
Lefrançois, M. 2004 Metal-to-metal seals: the innovative route in static sealing. Sealing Technol. 2004 (4), 10–13.10.1016/S1350-4789(04)00121-7Google Scholar
Letalleur, N., Plouraboué, F & Prat, M. 2002 Average flow model of rough surface lubrication: flow factors for sinusoidal surfaces. J. Tribology 124 (3), 539–546.10.1115/1.1467084Google Scholar
Loeb, L. B. 2004 The Kinetic Theory of Gases. Dover Publications.Google Scholar
Lorenz, B. & Persson, B. N. J. 2009 Leak rate of seals: comparison of theory with experiment. Eur. Phys. Lett. 86 (4), 44006.10.1209/0295-5075/86/44006Google Scholar
Marie, C. & Lasseux, D. 2007 Experimental leak-rate measurement through a static metal seal. Trans. ASME J. Fluids Engng 129 (6), 799–805.10.1115/1.2734250Google Scholar
Marie, C., Lasseux, D., Zahouani, H. & Sainsot, P. 2003 An integrated approach to characterize liquid leakage through metal contact seal. Eur. J. Mech. Environ. Eng. 48 (2), 81–86.Google Scholar
Maxwell, J. C. 1879 On stresses in rarified gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
McNenly, M. J., Gallis, M. A. & Boyd, I. D. 2005 Empirical slip and viscosity model performance for microscale gas flow. Intl J. Numer. Meth. Fluids 49, 1169–1191.10.1002/fld.1012Google Scholar
Moukalled, F., Mangani, L. & Darwish, M. 2016 The Finite Volume Method in Computational Fluid Dynamics, Fluid Mechanics and its Applications, vol. 113. Springer.10.1007/978-3-319-16874-6Google Scholar
Mourzenko, V., Thovert, J.-F. & Adler, P. 1995 Permeability of a single fracture: validity of the Reynolds equation. J. Physique II 5 (3), 465–482.Google Scholar
Patir, N. 1978 A numerical procedure for random generation of rough surfaces. Wear 47 (2), 263–277.10.1016/0043-1648(78)90157-6Google Scholar
Patir, N. & Cheng, H. S. 1978 An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. J. Lubr. Technol. 100 (1), 12–17.10.1115/1.3453103Google Scholar
Pérez-Ràfols, F., Larsson, R. & Almqvist, A. 2016 Modelling of leakage on metal-to-metal seals. Tribol. Intl 94, 421–427.10.1016/j.triboint.2015.10.003Google Scholar
Plouraboué, F., Fluckiger, F., Prat, M. & Crispel, P. 2006 Geodesic network method for flows between two rough surfaces in contact. Phys. Rev. E 73, 036305.Google Scholar
Porodnov, B. T., Suetin, P. E., Borisov, S. F. & Akinshin, V. D. 1974 Experimental investigation of rarefied gas flow in different channels. J. Fluid Mech. 64 (3), 417–438.10.1017/S0022112074002485Google Scholar
Prat, M., Plouraboué, F. & Letalleur, N. 2002 Averaged Reynolds equation for flows between rough surfaces in sliding motion. Trans. Porous Med. 48 (3), 291–313.10.1023/A:1015772525610Google Scholar
Quintard, M. & Whitaker, S. 1996 Transport in chemically and mechanically heterogeneous porous media. I: theoretical development of region-averaged equations for slightly compressible single-phase flow. Adv. Water Resour. 19 (1), 29–47.10.1016/0309-1708(95)00023-CGoogle Scholar
Shi, F., Choi, W., Lowe, M. J. S., Skelton, E. A. & Craster, R. V. 2015 The validity of Kirchhoff theory for scattering of elastic waves from rough surfaces. Proc. R. Soc. Lond. A 47 (2178), 19.Google Scholar
Skjetne, E. & Auriault, J.-L. 1999 Homogenization of wall-slip gas flow through porous media. Trans. Porous Med. 36 (3), 293–306.10.1023/A:1006572324102Google Scholar
Stout, K. J. & Blunt, L. 2013 Three Dimensional Surface Topography. Butterworth-Heinemann.Google Scholar
Stout, K. J., Blunt, L., Dong, W. P., Mainsah, E., Luo, N., Mathia, T., Sullivan, P. J. & Zahouani, H. 2000 The Development of Methods for the Characterisation of Roughness in Three Dimensions. Penton Press.Google Scholar
Stout, K. J., Davis, E. J. & Sullivan, P. J. 1990 Atlas of Machined Surfaces. Chapman and Hall.10.1007/978-94-011-7772-6Google Scholar
Szeri, A. Z. 1998 Fluid Film Lubrication. Cambridge University Press.10.1017/CBO9780511626401Google Scholar
Tripp, J. H. 1983 Surface roughness effects in hydrodynamic lubrication: the flow factor method. J. Lubr. Technol. 105 (3), 458–463.10.1115/1.3254641Google Scholar
Tsang, Y. W. & Tsang, C. F. 1987 Channel model of flow through fractured media. Water Resour. Res. 23 (3), 467–479.10.1029/WR023i003p00467Google Scholar
Vallet, C., Lasseux, D., Sainsot, P. & Zahouani, H. 2009 Real versus synthesized fractal surfaces: contact mechanics and transport properties. Tribol. Int. 42 (2), 250–259.10.1016/j.triboint.2008.06.005Google Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar
Whitaker, S. 1999 The Method of Volume Averaging, Theory and Applications of Transport in Porous Media, vol. 13. Springer.10.1007/978-94-017-3389-2Google Scholar
Zimmerman, R. W. & Bodvarsson, G. S. 1996 Hydraulic conductivity of rock fractures. Trans. Porous Med. 23 (1), 1–30.10.1007/BF00145263Google Scholar