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  • Letter
  • Open Access

Universal scaling for the permeability of random packs of overlapping and nonoverlapping particles

Jérémie Vasseur1,*, Fabian B. Wadsworth2, Eloïse Bretagne2, and Donald B. Dingwell1

  • 1Earth and Environmental Science, Ludwig-Maximilians-Universität, Theresienstrasse 41, 80333 Munich, Germany
  • 2Department of Earth Sciences, Science Laboratories, Durham University, Durham DL1 3LE, United Kingdom

  • *j.vasseur@lmu.de
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Phys. Rev. E 105, L043301 – Published 20 April, 2022

DOI: https://doi.org/10.1103/PhysRevE.105.L043301

Abstract

Constraining fluid permeability in porous media is central to a wide range of theoretical, industrial, and natural processes. In this Letter, we validate a scaling for fluid permeability in random and lattice packs of spheres and show that the permeability of packs of both hard and overlapping spheres of any sphere size or size distribution collapse to a universal curve across all porosity ϕ in the range of ϕc<ϕ<1, where ϕc is the percolation threshold. We use this universality to demonstrate that permeability can be predicted using percolation theory at ϕc<ϕ≲0.30, Kozeny-Carman models at 0.30≲ϕ≲0.40, and dilute expansions of Stokes theory for lattice models at ϕ≳0.40. This result leads us to conclude that the inverse specific surface area, rather than an effective sphere size or pore size is a universal controlling length scale for hydraulic properties of packs of spheres. Finally, we extend this result to predict the permeability for some packs of concave nonspherical particles.

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References (34)

  1. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Vol. 16 (Springer, New York, 2002).
  2. C. F. Berg, Permeability description by characteristic length, tortuosity, constriction and porosity, Transp. Porous Media 103, 381 (2014).
  3. R. A. Lopes and A. M. Segadães, Microstructure, permeability and mechanical behaviour of ceramic foams, Mater. Sci. Eng. A 209, 149 (1996).
  4. F. B. Wadsworth, J. Vasseur, B. Scheu, J. E. Kendrick, Y. Lavallée, and D. B. Dingwell, Universal scaling of fluid permeability during volcanic welding and sediment diagenesis, Geology 44, 219 (2016).
  5. N. S. Martys, S. Torquato, and D. P. Bentz, Universal scaling of fluid permeability for sphere packings, Phys. Rev. E 50, 403 (1994).
  6. D. L. Johnson, J. Koplik, and R. Dashen, Theory of dynamic permeability and tortuosity in fluid saturated porous media, J. Fluid Mech. 176, 379 (1987).
  7. M. Röding, Z. Ma, and S. Torquato, Predicting permeability via statistical learning on higher-order microstructural information, Sci. Rep. 10, 15239 (2020).
  8. A. S. Sangani and A. Acrivos, Slow flow through a periodic array of spheres, Int. J. Multiphase Flow 8, 343 (1982).
  9. P. C. Carman, Fluid flow through granular beds, AIChE J. 15, 150 (1937).
  10. S. Torquato and B. Lu, Rigorous bounds on the fluid permeability: Effect of polydispersivity in grain size, Phys. Fluids A Fluid Dyn. 2, 487 (1990).
  11. A. Nabovati, E. W. Llewellin, and A. C. M. Sousa, A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice boltzmann method, Compos. Part A Appl. Sci. Manuf. 40, 860 (2009).
  12. S. Torquato, Flow in random porous media: Mathematical formulation, variational principles, and rigorous bounds, J. Fluid Mech. 206, 25 (1989).
  13. A. E. Scheidegger, The Physics of Flow through Porous Media (Macmillan, New York, 1960) .
  14. S. P. Neuman, Theoretical derivation of darcy's law, Acta Mech. 25, 153 (1977).
  15. S. Feng, B. I. Halperin, and P. N. Sen, Transport properties of continuum systems near the percolation threshold, Phys. Rev. B 35, 197 (1987).
  16. J. Vasseur and F. B. Wadsworth, Sphere models for pore geometry and fluid permeability in heterogeneous magmas, Bull. Volcanol. 79, 77 (2017).
  17. F. B. Wadsworth, J. Vasseur, E. W. Llewellin, K. J. Dobson, M. Colombier, F. W. von Aulock, J. L. Fife, S. Wiesmaier, K.-U. Hess, B. Scheu, Y. Lavallée, and D. B. Dingwell, Topological inversions in coalescing granular media control fluid-flow regimes, Phys. Rev. E 96, 033113 (2017).
  18. F. Boyer, É. Guazzelli, and O. Pouliquen, Unifying Suspension and Granular Rheology, Phys. Rev. Lett. 107, 188301 (2011).
  19. S. Torquato, T. M. Truskett, and P. G. Debenedetti, Is Random Close Packing of Spheres Well Defined?, Phys. Rev. Lett. 84, 2064 (2000).
  20. H. Hasimoto, On the periodic fundamental solutions of the stokes equations and their application to viscous flow past a cubic array of spheres, J. Fluid Mech. 5, 317 (1959).
  21. J. Vasseur, F. B. Wadsworth, J. P. Coumans, and D. B. Dingwell, Permeability of packs of polydisperse hard spheres, Phys. Rev. E 103, 062613 (2021) .
  22. F. B. Wadsworth, J. Vasseur, E. W. Llewellin, R. J. Brown, H. Tuffen, J. E. Gardner, J. E. Kendrick, Y. Lavallée, K. J. Dobson, M. J. Heap, D. B. Dingwell, K.-U. Hess, J. Schauroth, F. W. von Aulock, A. R. L. Kushnir, and F. Marone, A model for permeability evolution during volcanic welding, J. Volcanol. Geotherm. Res. 409, 107118 (2021).
  23. J. Kertesz, Percolation of holes between overlapping spheres: Monte carlo calculation of the critical volume fraction, J. Phys. Lett. 42, 393 (1981).
  24. M. D. Rintoul, Precise determination of the void percolation threshold for two distributions of overlapping spheres, Phys. Rev. E 62, 68 (2000).
  25. E. Ghossein and M. Lévesque, A fully automated numerical tool for a comprehensive validation of homogenization models and its application to spherical particles reinforced composites, Int. J. Solids Struct. 49, 1387 (2012).
  26. J. Vasseur, F. B. Wadsworth, and D. B. Dingwell, Permeability of polydisperse magma foam, Geology 48, 536 (2020).
  27. E. W. Llewellin, LBflow: An extensible lattice boltzmann framework for the simulation of geophysical flows. part i: Theory and implementation, Comput. Geosci. 36, 115 (2010).
  28. E. W. Llewellin, LBflow: An extensible lattice boltzmann framework for the simulation of geophysical flows. part ii: Usage and validation, Comput. Geosci. 36, 123 (2010).
  29. M. Matyka, Z. Koza, J. Gołembiewski, M. Kostur, and M. Januszewski, Anisotropy of flow in stochastically generated porous media, Phys. Rev. E 88, 023018 (2013) .
  30. W. E. Lorensen and H. E. Cline, Marching cubes: A high resolution 3d surface construction algorithm., Comput. Graph. 21, 163 (1987).
  31. T. Lewiner, H. Lopes, A. Vieira, and G. Tavares, Efficient implementation of marching cubes' cases with topological guarantees, J. Graph. Tools 8, 1 (2003).
  32. L. Liu, Z. Li, Y. Jiao, and S. Li, Maximally dense random packings of cubes and cuboids via a novel inverse packing method, Soft Matter 13, 748 (2017).
  33. T. Bourbie and B. Zinszner, Hydraulic and acoustic properties as a function of porosity in fontainebleau sandstone, J. Geophys. Res.: Solid Earth 90, 11524 (1985).
  34. P. E. Øren and S. Bakke, Process based reconstruction of sandstones and prediction of transport properties, Transp. Porous Media 46, 311 (2002).

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Information

Phys. Rev. E 105, L043301– Published 20 April, 2022
  • Received 12 January 2022
  • Accepted 8 March 2022
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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