Archive of Applied Mechanics

, Volume 85, Issue 8, pp 1043–1054 | Cite as

Finite difference calculations of permeability in large domains in a wide porosity range

  • Maria Osorno
  • David Uribe
  • Oscar E. Ruiz
  • Holger Steeb


Determining effective hydraulic, thermal, mechanical and electrical properties of porous materials by means of classical physical experiments is often time-consuming and expensive. Thus, accurate numerical calculations of material properties are of increasing interest in geophysical, manufacturing, bio-mechanical and environmental applications, among other fields. Characteristic material properties (e.g. intrinsic permeability, thermal conductivity and elastic moduli) depend on morphological details on the porescale such as shape and size of pores and pore throats or cracks. To obtain reliable predictions of these properties it is necessary to perform numerical analyses of sufficiently large unit cells. Such representative volume elements require optimized numerical simulation techniques. Current state-of-the-art simulation tools to calculate effective permeabilities of porous materials are based on various methods, e.g. lattice Boltzmann, finite volumes or explicit jump Stokes methods. All approaches still have limitations in the maximum size of the simulation domain. In response to these deficits of the well-established methods we propose an efficient and reliable numerical method which allows to calculate intrinsic permeabilities directly from voxel-based data obtained from 3D imaging techniques like X-ray microtomography. We present a modelling framework based on a parallel finite differences solver, allowing the calculation of large domains with relative low computing requirements (i.e. desktop computers). The presented method is validated in a diverse selection of materials, obtaining accurate results for a large range of porosities, wider than the ranges previously reported. Ongoing work includes the estimation of other effective properties of porous media.


Effective permeability Porous materials Digital rock physics 

List of symbols

\(\mathbf {u}\)

Fluid velocity on porescale (m/s)


Sphere diameter (m)

\(k^\mathfrak {s}\)

Intrinsic permeability (\(\hbox {m}^2\))


Pressure (Pa)


Radius of capillary tube (m)


RVE-scale Reynolds number (–)


Characteristic size of the investigated RVE domain (–)


Volume-averaged velocity (m/s)

\(\Delta p\)

Pressure drop in the medium (Pa/m)

\(\eta \)

Effective dynamic viscosity of the fluid (Pa s)

\(\rho \)

Density of the fluid (\(\hbox {kg}/\hbox {m}^3\))

\(\phi \)

Porosity of the material (–)

\({\varOmega }\)

Domain of investigated material in \(\mathbb {R}^3\)



The present work was supported by Ruhr-University Bochum, Germany, the CAD CAM CAE Laboratory EAFIT, Colombia, and the Colombian Administration for Science and Technology (Colciencias). M. Osorno thanks Colciencias and the programme Jovenes investigadores.


  1. 1.
    Dvorkin, J., Derzhi, N., Diaz, E., Fang, Q.: Relevance of computational rock physics. Geophysics 76(5), E141–E153 (2011)CrossRefGoogle Scholar
  2. 2.
    Cowin, S.C., Cardoso, L.: Difficulties arising from different definitions of tortuosity for wave propagation in saturated poroelastic media models. Z. Angew. Math. Mech. 11, 1–11 (2013)Google Scholar
  3. 3.
    Fredrich, J.T., DiGiovanni, A.A., Noble, D.R.: Predicting macroscopic transport properties using microscopic image data. J. Geophys. Res. Solid Earth 111(B3) (2006). doi: 10.1029/2005JB003774
  4. 4.
    Yang, A., Miller, C.T., Turcoliver, L.D.: Simulation of correlated and uncorrelated packing of random size spheres. Phys. Rev. E 53(2), 1516 (1996)CrossRefGoogle Scholar
  5. 5.
    Gerbaux, O., Buyens, F., Mourzenko, V.V., Memponteil, A., Vabre, A., Thovert, J.F., Adler, P.M.: Transport properties of real metallic foams. J. Colloid Interface Sci. 342(1), 155–165 (2010)CrossRefGoogle Scholar
  6. 6.
    Wiegmann, A.: Computation of the Permeability of Porous Materials from Their Microstructure by FFF-Stokes. Bericht des Fraunhofer-Institut für Techno- und Wirtschaftsmathematik, Fraunhofer (ITWM) (2007)Google Scholar
  7. 7.
    Xu, W., Zhang, H., Yang, Z., Zhang, J.: Numerical investigation on the flow characteristics and permeability of three-dimensional reticulated foam materials. Chem. Eng. J. 140(1), 562–569 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Boomsma, K., Poulikakos, D., Ventikos, Y.: Simulations of flow through open cell metal foams using an idealized periodic cell structure. Int. J. Heat Fluid Flow 24(6), 825–834 (2003)CrossRefGoogle Scholar
  9. 9.
    Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer, R., Timonen, J.: Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Phys. Rev. E 66(1), 016702 (2002)CrossRefGoogle Scholar
  10. 10.
    Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerjic, T., Saenger, E.H., Sainf, R., Saxenac, N., Rickera, S., Wiegmann, A., Zhanf, X.: Digital rock physics benchmarks—Part I: imaging and segmentation. Comput. Geosci. 50, 25–32 (2013)CrossRefGoogle Scholar
  11. 11.
    Andrä, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerjic, T., Saenger, E.H., Sainf, R., Saxenac, N., Rickera, S., Wiegmann, A., Zhanf, X.: Digital rock physics benchmarks—Part II: computing effective properties proposed paper for computers & geoscience special issue. Comput. Geosci. 50, 33–43 (2013)CrossRefGoogle Scholar
  12. 12.
    Petrasch, J., Meier, F., Friess, H., Steinfeld, A.: Tomography based determination of permeability, Dupuit–Forchheimer coefficient, and interfacial heat transfer coefficient in reticulate porous ceramics. Int. J. Heat Fluid Flow 29(1), 315–326 (2008)CrossRefGoogle Scholar
  13. 13.
    Saenger, E.H., Uribe, D., Jänicke, R., Ruiz, O., Steeb, H.: Digital material laboratory: wave propagation effects in open-cell aluminium foams. Int. J. Eng. Sci. 58, 115–123 (2012)CrossRefzbMATHGoogle Scholar
  14. 14.
    Saenger, E.H., Enzmann, F., Keehm, Y., Steeb, H.: Digital rock physics: effect of fluid viscosity on effective elastic properties. J. Appl. Geophys. 74(4), 236–241 (2011)CrossRefGoogle Scholar
  15. 15.
    Nagarajan, S., Lele, S.K., Ferziger, J.H.: A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191(2), 392–419 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Young, D.: Iterative methods for solving partial difference equations of elliptic type. Trans. Am. Math. Soc. 76(1), 92–111 (1954)CrossRefzbMATHGoogle Scholar
  17. 17.
    Young, D.: Convergence properties of the symmetric and unsymmetric successive overrelaxation methods and related methods. Math. Comput. 24(112), 793–807 (1970)CrossRefGoogle Scholar
  18. 18.
    Kranzlmüller, D., Kacsuk, P., Dongarra, J., Volkert, J.: Recent Advances in Parallel Virtual Machine and Message Passing Interface. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    Rauber, T., Rünger, G.: Parallel Programming: For Multicore and Cluster Systems. Springer, Berlin (2010)CrossRefGoogle Scholar
  20. 20.
    Darcy, H.: Les fontaines publiques de la ville de Dijon. V. Dalmont, Paris (1856)Google Scholar
  21. 21.
    Carman, P.C.: Permeability of saturated sands, soils and clays. J. Agric. Sci. 29, 262–273 (1939)CrossRefGoogle Scholar
  22. 22.
    Rumpf, H.C.H., Gupte, A.R.: Einflüsse der Porosität und Korngrößenverteilung im Widerstandsgesetz der Porenströmung. Chem. Ing. Tech. 43(6), 367–375 (2004)CrossRefGoogle Scholar
  23. 23.
    Paek, J.W., Kang, B.H., Kim, S.Y., Hyun, J.M.: Effective thermal conductivity and permeability of aluminum foam materials. Int. J. Thermophys. 21(2), 453–464 (2000)CrossRefGoogle Scholar
  24. 24.
    Medraj, M., Baril, E., Loya, V., Lefebvre, L.-P.: The effect of microstructure on the permeability of metallic foams. J. Mater. Sci. 42, 4372–4383 (2007)CrossRefGoogle Scholar
  25. 25.
    Baril, E., Mostafid, A., Lefebvre, L.-P., Medraj, M.: Experimental demonstration of entrance/exit effects on the permeability measurements of porous materials. Adv. Eng. Mater. 10(9), 889–894 (2008)CrossRefGoogle Scholar
  26. 26.
    Tidwell, V.C., Wilson, J.L.: Permeability upscaling measured on a block of Berea Sandstone: results and interpretation. Math. Geol. 31(7), 749–769 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Maria Osorno
    • 1
    • 2
  • David Uribe
    • 1
    • 2
  • Oscar E. Ruiz
    • 1
  • Holger Steeb
    • 2
  1. 1.Laboratorio de CAD/CAM/CAEUniversidad EAFITMedellinColombia
  2. 2.Institute of MechanicsRuhr-University BochumBochumGermany

Personalised recommendations