Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-07T17:28:08.435Z Has data issue: false hasContentIssue false
  • English
  • Français

An improved macroscale model for gas slip flow in porous media

Published online by Cambridge University Press:  16 September 2016

Didier Lasseux ,
Francisco J. Valdés Parada  and
Mark L. Porter
Didier Lasseux*
Affiliation:
CNRS, (Univ. Bordeaux, IPB, ENSAM) - I2M, UMR5295, Esplanade des Arts et Métiers, 33405 Talence, CEDEX, France
Francisco J. Valdés Parada
Affiliation:
Universidad Autónoma Metropolitana-Iztapalapa, Departamento de Ingeniería de Procesos e Hidráulica, Av. San Rafael Atlixco 186, 09340 Mexico D.F., Mexico
Mark L. Porter
Affiliation:
Earth Systems Observations, MS D462, Los Alamos National Laboratory, Los Alamos, NM 85745, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We report on a refined macroscopic model for slightly compressible gas slip flow in porous media developed by upscaling the pore-scale boundary value problem. The macroscopic model is validated by comparisons with an analytic solution on a two-dimensional (2-D) ordered model structure and with direct numerical simulations on random microscale structures. The symmetry properties of the apparent slip-corrected permeability tensor in the macroscale momentum equation are analysed. Slip correction at the macroscopic scale is more accurately described if an expansion in the Knudsen number, beyond the first order considered so far, is employed at the closure level. Corrective terms beyond the first order are a signature of the curvature of solid–fluid interfaces at the pore scale that is incompletely captured by the classical first-order correction at the macroscale. With this expansion, the apparent slip-corrected permeability is shown to be the sum of the classical intrinsic permeability tensor and tensorial slip corrections at the successive orders of the Knudsen number. All the tensorial effective coefficients can be determined from intrinsic and coupled but easy-to-solve closure problems. It is further shown that the complete form of the slip boundary condition at the microscale must be considered and an important general feature of this slip condition at the different orders in the Knudsen number is highlighted. It justifies the importance of slip-flow correction terms beyond the first order in the Knudsen number in the macroscopic model and sheds more light on the physics of slip flow in the general case, especially for large porosity values. Nevertheless, this new nonlinear dependence of the apparent permeability with the Knudsen number should be further verified experimentally.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Porous media Rarefied Gas Flow: Rarefied Gas Flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 805 , 25 October 2016 , pp. 118 - 146
Check for updates
Copyright
© 2016 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

Agrawal, A. & Prabhu, S. V. 2008 Survey on measurement of tangential momentum accommodation coefficient. J. Vac. Sci. Technol. A 26, 634–645.Google 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.Google Scholar
Barber, R. W., Sun, Y., Gu, X. J. & Emerson, D. R. 2004 Isothermal slip flow over curved surfaces. Vacuum 76, 73–81.Google Scholar
Barrère, J., Gipouloux, O. & Whitaker, S. 1992 On the closure problem for Darcy’s law. Trans. Porous Med. 7, 209–222.Google Scholar
Bruschke, M. V. & Advani, S. G. 1993 Flow of generalized Newtonian fluids across a periodic array of cylinders. J. Rheol. 37, 479–498.Google Scholar
Cai, C., Sun, Q. & Boyd, I. D. 2007 Gas flows in microchannels and microtubes. J. Fluid Mech. 589, 305–314.Google Scholar
Chai, Z., Lu, J., Shi, B. & Guo, Z. 2011 Gas slippage effect on the permeability of circular cylinders in a square array. Intl J. Heat Mass Transfer 54, 3009–3014.Google Scholar
Chmielewski, R. D. & Goren, S. L. 1972 Aerosol filtration with slip flow. Environ. Sci. Technol. 6 (13), 1101–1105.Google Scholar
Cowling, T. G. 1950 Molecules in Motion, chap. IV. Hutchinson.Google Scholar
Darabi, H., Ettehad, A., Javadpour, F. & Sepehrnoori, K. 2012 Gas flow in ultra-tight shale strata. J. Fluid Mech. 710, 641–658.Google Scholar
deSocio, L. M. & Marino, L. 2006 Gas flow in a permeable medium. J. Fluid Mech. 557, 119–133.Google Scholar
Einzel, D., Panzer, P. & Liu, M. 1990 Boundary condition for fluid flow: curved or rough surfaces. Phys. Rev. Lett. 64 (19), 2269–2272.Google Scholar
Fishman, M. & Hetsroni, G. 2005 Viscosity and slip velocity in gas flow in microchannels. Phys. Fluids 17, 123102.Google Scholar
Ghaddar, C. K. 1995 On the permeability of unidirectional fibrous media: a parallel computational approach. Phys. Fluids 7 (11), 2563–2586.Google Scholar
Gray, W. G. 1975 A derivation of the equations for multiphase transport. Chem. Engng Sci. 30, 229–233.Google Scholar
Harley, J. C., Huang, Y., Bau, H. H. & Zemel, J. N. 1995 Gas flow in micro-channels. J. Fluid Mech. 284, 257–274.Google Scholar
Howes, F. & Whitaker, S. 1985 The spatial averaging theorem revisited. Chem. Engng Sci. 40, 1387–1392.Google Scholar
Jannot, Y. & Lasseux, D. 2012 A new method to measure gas permeabilty of weakly permeable porous media. Rev. Sci. Instrum. 83 (1), 015113.Google Scholar
Karniadakis, G. E., Beskok, A. & Aluru, N. 2005 Microflows and nanoflows. Fundamentals and Simulation. Springer.Google Scholar
Klinkenberg, L. J. 1941 The permeability of porous media to liquids and gases. API Drilling and Production Practice pp. 200–213. Available at:https://www.onepetro.org/conferences/API/API41.Google Scholar
Kuwabara, S. 1959 The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers. J. Phys. Soc. Japan 14 (4), 527–532.Google Scholar
Lasseux, D., Jolly, P., Jannot, Y., Sauger, E. & Omnes, B. 2011 Permeability measurement of graphite compression packings. Trans. ASME J: J. Press. Vessel Technol. 133 (4), 041401.Google 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.CrossRefGoogle Scholar
Lauga, E. & Cossu, C. 2005 A note on the stability of slip channel flows. Phys. Fluids 17, 088106.Google Scholar
Lockerby, D. A., Reese, J. M., Emerson, D. R. & Barber, R. W. 2004 Velocity boundary condition at solid walls in rarefied gas calculations. Phys. Rev. E 70, 017303.Google Scholar
Maurer, J., Tabeling, P., Joseph, P. & Willaime, H. 2003 Second-order slip laws in microchannels for helium and nitrogen. Phys. Fluids 15 (9), 2613–2621.Google Scholar
Maxwell, J. C. 1879 On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond. 170, 231–256.Google Scholar
Nakashima, Y. & Watanabe, Y. 2002 Estimate of transport properties of porous media by microfocus X-ray computed tomography and random walk simulation. Water Resour. Res. 38 (12), 1272.Google Scholar
Panzer, P., Liu, M. & Einzel, D. 1992 The effect of boundary curvature on hydrodynamic fluid flow: calculation of slip length. Intl J. Mod. Phys. B 06 (20), 3251–3278.Google Scholar
Pavan, V. & Chastanet, J. 2011 Gas/solid heat transfer in gas flows under Klinkenberg conditions: comparison between the homogenization and the kinetic approaches. J. Porous Media 14, 127–148.Google Scholar
Perrier, P., Graur, I. A., Ewart, T. & Molans, J. G. 2011 Mass flow rate measurements in microtubes: From hydrodynamic to near free molecular regime. Phys. Fluids 23, 042004.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), 417–437.Google Scholar
Prodanović, M., Lindquist, W. B. & Seright, R. S. 2007 3D image-based characterization of fluid displacement in a Berea core. Adv. Water Resour. 30, 214–226.Google Scholar
Profice, S., Lasseux, D., Jannot, Y., Jebara, N. & Hamon, G. 2012 Permeability, porosity and Klinkenberg coefficient determination on crushed porous media. Petrophysics 53, 430–438.Google Scholar
Selden, N., Gimelshein, N., Gimelshein, S. & Ketsdever, A. 2009 Analysis of accommodation coefficients of noble gases on aluminum surface with an experimental/computational method. Phys. Fluids 21, 073101.Google Scholar
Shen, S., Chen, G., Crone, R. M. & Anaya-Dufresne, M. 2007 A kinetic theory based first-order slip boundary condition for gas flow. Phys. Fluids 19, 086101.Google Scholar
Skjetne, E. & Auriault, J. L. 1999 Homogenization of wall-slip gas flow through porous media. Trans. Porous Med. 36, 293–306.Google Scholar
Suetin, P. E., Porodnov, B. T., Chernjak, V. G. & Borisov, S. F. 1973 Poiseuille flow at arbitrary Knudsen numbers and tangential momentum accomodation. J. Fluid Mech. 60 (3), 581–592.Google Scholar
Truesdell, C. & Toupin, R. 1960 The Classical Field Theories. Springer.CrossRefGoogle Scholar
Whitaker, S. 1999 The Method of Volume Averaging, Theory and Applications of Transport in Porous Media, vol. 13. Kluwer.Google Scholar
Zhang, T., Zhang, P., Law, C. K. & Qi, F. 2009 CVD in weakly rarefied rotating disk flows. Chem. Vapor Depos. 15, 274–280.Google Scholar