Continuum Mechanics and Thermodynamics

, Volume 24, Issue 4–6, pp 403–416 | Cite as

Acoustics of two-component porous materials with anisotropic tortuosity

  • Bettina AlbersEmail author
  • Krzysztof Wilmanski
Original Article


The paper is devoted to the analysis of monochromatic waves in two-component poroelastic materials described by a Biot-like model whose stress–strain relations are isotropic but the permeability is anisotropic. This anisotropy is induced by the anisotropy of the tortuosity which is given by a second order symmetric tensor. This is a new feature of the model while in earlier papers only isotropic permeabilities were considered. We show that this new model describes four modes of propagation. For our special choice of orientation of the direction of propagation these are two pseudo longitudinal modes P1 and P2, one pseudo transversal mode S2 and one transversal mode S1. The latter becomes also pseudo transversal in the general case of anisotropy. We analyze the speeds of propagation and the attenuation of these waves as well as the polarization properties in dependence on the orientation of the principal directions of the tortuosity. We indicate the practical importance of different shear (transversal) modes of propagation in a possible new nondestructive test of geophysical materials.


Anisotropy of tortuosity Acoustic waves Poroelastic media 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albers, B.: Modeling and numerical analysis of wave propagation in saturated and partially saturated porous media. Habilitation thesis, Veröffentlichungen des Grundbauinstitutes der Technischen Universität Berlin, vol. 48, Shaker, Aachen (2010)Google Scholar
  2. 2.
    Albers B., Wilmanski K.: Modeling acoustic waves in saturated poroelastic media. J. Eng. Mech. ASCE. 131, 974–985 (2005)CrossRefGoogle Scholar
  3. 3.
    Arnold, J.: Mobile NMR for rock porosity and permeability, PhD-thesis, RWTH Aachen (2007)Google Scholar
  4. 4.
    Bear J.: Dynamics of fluids in porous media. Dover, New York (1991)Google Scholar
  5. 5.
    Bear J., Bachmat Y.: Introduction to modeling of transport phenomena in porous media. Kluwer, Dordrecht (1991)zbMATHGoogle Scholar
  6. 6.
    Carman P.C.: Flow of gases through porous media. Butterworth, London (1956)zbMATHGoogle Scholar
  7. 7.
    Cowin S.C.: Bone poroelasticity. J. Biomech. 32(3), 217–238 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cowin S.C., Cardoso L.: Fabric dependence of wave propagation in anisotropic porous media. Biomech. Model Mechanobiol. 10, 39–65 (2011)CrossRefGoogle Scholar
  9. 9.
    Epstein N.: On tortuosity and the tortuosity factor in flow and diffusion through porous media. Chem. Eng. Sci. 44(3), 777–779 (1989)CrossRefGoogle Scholar
  10. 10.
    Ferrandon J.: Le lois de l’ecoulement de filtration. Génie Civil. 125(2), 24–28 (1948)Google Scholar
  11. 11.
    Johnson W.E., Breston J.N.: Directional permeability measurements on oil sandstone from various states. Prod. Mon. 14(4), 10–19 (1951)Google Scholar
  12. 12.
    Kozeny J.: Ü ber kapillare Leitung des Wassers im Boden (Aufstieg, Versickerung und Anwendung auf Bewässerung). Sber. Akad. Wiss. Wien. 136(Abt. IIa), 271–306 (1927)Google Scholar
  13. 13.
    Masutani Y., Aoki S., Abe O., Hayashi N., Otomo K.: MR diffusion tensor imaging: recent advance and new techniques for diffusion tensor visualization. Eur. J. Radiol. 46, 53–66 (2003)CrossRefGoogle Scholar
  14. 14.
    Rusakov D.A., Kullmann D.M.: Geometric and viscous components of the tortuosity of the extracellular space in the brain. Proc. Natl. Acad. Sci. USA Neurobiol. 95, 8975–8990 (1998)ADSCrossRefGoogle Scholar
  15. 15.
    Scheidegger A.E.: Directional permeability of porous media to homogeneous fluids. Geofis. Pur. Appl. Milano 28, 75–90 (1954)ADSCrossRefGoogle Scholar
  16. 16.
    Stoll R.C.: Sediment acoustics, vol. 26 of Lecture Notes in Earth Sciences. Springer, New York (1989)Google Scholar
  17. 17.
    Taylor W.D., Hsu E., Rama Krishnan K.R., MacFall J.R.: Diffusion tensor imaging: background, potential, and utility in psychiatric research. Biol. Psychiatry. 55, 201–207 (2004)CrossRefGoogle Scholar
  18. 18.
    Wang R., Pavlin T., Rosen M.S., Mair R.W., Cory D.G., Walsworth R.L.: Xenon NMR measurements of permeability and tortuosity in reservoir rocks. Magn. Res. Im 23(2), 329–331 (2005)CrossRefGoogle Scholar
  19. 19.
    Wilmanski K., Albers B.: Acoustic waves in porous solid-fluid mixtures. In: Hutter, K., Kirchner, N. (eds) Dynamic response of granular and porous materials under large and catastrophic deformations., pp. 285–313. Springer, Berlin (2003)Google Scholar
  20. 20.
    Wilmanski K.: A few remarks on Biot’s model and linear acoustics of poroelastic saturated materials. Soil Dyn. Earthq. Eng. 26(6–7), 509–536 (2006)CrossRefGoogle Scholar
  21. 21.
    Wilmanski K.: Continuum thermodynamics. Part I. Foundations. World Scientific, New Jersey (2008)zbMATHGoogle Scholar
  22. 22.
    Wilmanski K.: Permeability, tortuosity, and attenuation of waves in porous materials. Civ. Environ. Eng. Res. Zielona Gora. 5, 9–52 (2010)Google Scholar
  23. 23.
    Wilmanski, K.: Monochromatic waves in saturated porous materials with anisotropic permeability, ZAMM (submitted for publication) (2011)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.TU BerlinBerlinGermany
  2. 2.TU BerlinBerlinGermany
  3. 3.ROSE School PaviaPaviaItaly

Personalised recommendations