Skip to main content Accessibility help
×
Home

Theory of dynamic permeability and tortuosity in fluid-saturated porous media

  • David Linton Johnson (a1), Joel Koplik (a1) and Roger Dashen (a1)

Abstract

We consider the response of a Newtonian fluid, saturating the pore space of a rigid isotropic porous medium, subjected to an infinitesimal oscillatory pressure gradient across the sample. We derive the analytic properties of the linear response function as well as the high- and low-frequency limits. In so doing we present a new and well-defined parameter Λ, which enters the high-frequency limit, characteristic of dynamically connected pore sizes. Using these results we construct a simple model for the response in terms of the exact high- and low-frequency parameters; the model is very successful when compared with direct numerical simulations on large lattices with randomly varying tube radii. We demonstrate the relevance of these results to the acoustic properties of non-rigid porous media, and we show how the dynamic permeability/tortuosity can be measured using superfluid 4He as the pore fluid. We derive the expected response in the case that the internal walls of the pore space are fractal in character.

Copyright

References

Hide All
Achiam, Y. & Bergman, D. 1974 Hydrodynamic theory of fourth sound in clamped conditions. J. Low Temp. Phys. 15, 559576.
Attenborough, K. 1983 Acoustical characteristics of rigid fibrous absorbents and granular materials. J. Acoust. Soc. Am. 73, 785799.
Auriault, J.-L., Borne, L. & Chambon, R. 1985 Dynamics of porous saturated media, checking of the generalized law of Darcy. J. Acoust. Soc. Am. 77, 16411650.
Avnir, D., Farin, D. & Pfeiffer, P. 1984 Molecular fractal surfaces. Nature 308, 261263.
Baker, S. 1985 Measurement of the Biot structural factor dL for sintered bronze spheres. In Proc. IEEE Ultrasonics Symposium (to be published).
Baker, S. 1986 Ph.D. thesis, Dept of Physics, UCLA.
Bedford, A., Costley, R. D. & Stern, M. 1984 On the drag and virtual mass coefficients in Biot's equations. J. Acoust. Soc. Am. 76, 18041809.
Bergman, D. J. 1979 Dielectric constant of a simple cubic array of identical spheres. J. Phys. C: Solid State Phys. 12, 49474960
Bergman, D. J., Halperin, B. I. & Hohenberg, P. C. 1975 Hydrodynamic theory of fourth sound in a moving superfluid. Phys. Rev. B 11, 42534263.
Biot, M. A. 1956a Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28, 168178.
Biot, M. A. 1956b Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 28, 179191.
Biot, M. A. 1962a Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 14821498.
Biot, M. A. 1962b Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34, 12541264.
Biot, M. A. & Willis, D. G. 1957 The elastic coefficients of the theory of consolidation. Trans. ASME E: J. Appl. Mech. 24, 594601
Brown, R. J. S. 1980 Connection between formation factor for electrical resistivity and fluid-solid coupling factor in Biot's equations for acoustic waves in fluid-filled media. Geophys. 45, 12691275.
Chandler, R. N. 1981 Transient streaming potential measurements on fluid-saturated porous structures: an experimental verification of Biot's slow wave in the quasi-static limit. J. Acoust. Soc. Am. 70, 116121.
Chandler, R. N. & Johnson, D. L. 1981 The equivalence of quasi-static flow in fluid-saturated porous media and Biot's slow wave in the limit of zero frequency. J. Appl. Phys. 52, 33913395.
Frohlich, H. 1949 Theory of Dielectrics; Dielectric Constant and Dielectric Loss. Clarendon.
Jayasinghe, D. A. P., Letelier, M. & Leutheusser, H. J. 1974 Frequency dependent friction in oscillatory laminar pipe flow. Intl J. Mech. Sci. 16, 819827.
Johnson, D. L. 1980 Equivalence between fourth sound and liquid He II at low temperatures and the Biot slow wave in consolidated porous media. Appl. Phys. Lett. 37, 10651067; ibid. 38, 827 (E).
Johnson, D. L. 1982 Elastodynamics of gels. J. Chem. Phys. 77, 15311539.
Johnson, D. L. 1986 Recent developments in the acoustic properties of porous media. Frontiers of Physical Acoustics, Proc. Enrico Fermi Summer School, Varenna, Italy. Elsevier.
Johnson, D. L. & Plona, T. J. 1982 Acoustic slow waves and the consolidation transition. J. Acoust. Soc. Am. 72, 556565.
Johnson, D. L., Plona, T. J., Scala, C., Pasierb, F. & Kojima, H. 1982 Tortuosity and acoustic slow waves. Phys. Rev. Lett. 49, 18401844.
Johnson, D. L. & Sen, P. N. 1981 Multiple scattering of acoustic waves with application to the index of refraction of fourth sound. Phys. Rev. B 24, 24862496.
Katz, A. J. & Thompson, A. H. 1985 Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54, 13251328.
Kirkpatrick, S. 1973 Percolation and conduction. Rev. Mod. Phys. 45, 574588.
Koplik, J. 1981 On the effective medium theory of random linear networks. J. Phys. C: Solid State Phys. 14, 48214837
Kriss, M. 1969 Size effects in liquid helium II as measured by fourth sound and the attenuation of fourth sound. Ph.D. thesis, Department of Physics, UCLA
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Pergamon.
Landau, L. D. & Lifshitz, E. M. 1960 Electrodynamics of Continuous Media. Pergamon.
Mandelbrot, B. 1982 The Fractal Geometry of Nature. Freeman.
Pines, D. 1964 In Elementary Excitations in Solids, Lecture Notes and Supplements in Physics (ed. J. D. Jackson & D. Pines), p. 123 ff. Benjamin.
Pollack, G. L. & Pellam, J. R. 1965 Wave-mode modification in liquid helium with partially clamped normal fluid. Phys. Rev. 137, A1676A1684.
Putterman, S. J. 1974 Superfluid Hydrodynamics. North-Holland Elsevier.
Rudnick, I. 1976 Physical acoustics at UCLA in the study of superfluid helium. In New Directions in Physical Acoustics. Proc. Enrico Fermi Summer School, Course LXIII, p. 112. Academic.
Scheidegger, A. E. 1974 Physics of Flow Through Porous Media. University of Toronto Press.
Shapiro, K. A. & Rudnick, I. 1965 Experimental determination of the fourth sound velocity in helium II. Phys. Rev. 137, A1383A1391.
Singer, D., Pasierb, F. Ruel, R. & Kojima, H. 1984 Multiple scattering of second sound in superfluid He II-filled porous medium. Phys. Rev. B 30, 29092912.
Stoll, R. D. 1974 Acoustic waves in saturated sediments. In Physics of Sound in Marine Sediments (ed. L. Hampton). Plenum.
Tam, W. Y. & Ahlers, G. 1985 Damping of fourth sound in 4He due to normal fluid flow. J. Low Temp. Phys. 58, 497512.
Tilley, D. R. & Tilley, J. 1974 Superfluidity and Superconductivity. Wiley.
Weichert, M. & Meinhold-Heerlein, L. 1970 Theoretical studies of the propagation of sound in narrow channels filled with He II. I. The dispersion relations of fourth sound and of the fifth wave mode. J. Low Temp. Phys. 1, 273.
Weichert, M. & Passing, R. 1982 Observation of a new thermal wave in a planar superfluid helium layer. Phys. Rev. B 26, 61146122.
Wong, P.-Z., Koplik, J. & Tomanic, J. 1984 Phys. Rev. B 30, 66066614.
Zwikker, C. & Kosten, C. W. 1949 Sound Absorbing Materials. Elsevier.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Theory of dynamic permeability and tortuosity in fluid-saturated porous media

  • David Linton Johnson (a1), Joel Koplik (a1) and Roger Dashen (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed