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Power Laws and Porous Media

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Waves with Power-Law Attenuation

Abstract

Poroelasticity or poroviscoelasticity is a rich field with complex physics-based models. It can model both compressional and shear waves in sub-bottom ocean sediments and is also used for modeling shear waves in the human body in elastography.

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Notes

  1. 1.

    Note that this equation may seemingly contradict (7.83), but \(\tilde{\rho }(\omega ) \ne \rho _c\).

References

  • R. Bachrach, J. Dvorkin, A.M. Nur, Seismic velocities and Poisson’s ratio of shallow unconsolidated sands. Geophysics 65(2), 559–564 (2000)

    Article  ADS  Google Scholar 

  • P.J. Basser, Interstitial pressure, volume, and flow during infusion into brain tissue. Microvasc. Res. 44(2), 143–165 (1992)

    Article  Google Scholar 

  • J.G. Berryman, Confirmation of Biot’s theory. Appl. Phys. Lett. 37(4), 382–384 (1980)

    Article  ADS  Google Scholar 

  • M.A. Biot, General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 23(1), 91–96 (1956a)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956b)

    Google Scholar 

  • M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 28(2), 179–191 (1956c)

    Google Scholar 

  • M.A. Biot, Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34(9A), 1254–1264 (1962)

    Article  ADS  MathSciNet  Google Scholar 

  • D. Bland, The Theory of Linear Viscoelasticity (Pergamon Press, Oxford, London, New York, Paris, 1960)

    MATH  Google Scholar 

  • M.J. Buckingham, Response to Comments on ‘Pore fluid viscosity and the wave properties of saturated granular materials including marine sediments [J. Acoust. Soc. Am. 127, 2095–2098 (2010)]’, J. Acoust. Soc. Am.127(4), 2099–2102 (2010)

    Google Scholar 

  • M.J. Buckingham, Theory of acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments. J. Acoust. Soc. Am. 102, 2579–2596 (1997)

    Article  ADS  Google Scholar 

  • M.J. Buckingham, Wave propagation, stress relaxation, and grain-to-grain shearing in saturated, unconsolidated marine sediments. J. Acoust. Soc. Am. 108(6), 2796–2815 (2000)

    Article  ADS  Google Scholar 

  • M.J. Buckingham, On pore-fluid viscosity and the wave properties of saturated granular materials including marine sediments. J. Acoust. Soc. Am. 122(3), 1486–1501 (2007)

    Article  ADS  Google Scholar 

  • M.J. Buckingham, Analysis of shear-wave attenuation in unconsolidated sands and glass beads. J. Acoust. Soc. Am. 136(5), 2478–2488 (2014)

    Article  ADS  Google Scholar 

  • Á. Cartea, D. del Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76(4), 041105–1–7 (2007)

    Google Scholar 

  • S.N. Chandrasekaran, S. Holm, A multiple relaxation interpretation of the extended Biot model (2019). Submitted for publication

    Google Scholar 

  • N.P. Chotiros, An inversion for Biot parameters in water-saturated sand. J. Acoust. Soc. Am. 112(5), 1853–1868 (2002)

    Article  ADS  Google Scholar 

  • N.P. Chotiros, Acoustics of the Seabed as a Poroelastic Medium (Springer and ASA Press, Switzerland, 2017)

    Book  Google Scholar 

  • N.P. Chotiros, M.J. Isakson, A broadband model of sandy ocean sediments: Biot-Stoll with contact squirt flow and shear drag. J. Acoust. Soc. Am. 116(4), 2011–2022 (2004)

    Article  ADS  Google Scholar 

  • N.P. Chotiros, M.J. Isakson, Shear wave attenuation and micro-fluidics in water-saturated sand and glass beads. J. Acoust. Soc. Am. 135(6), 3264–3279 (2014)

    Article  ADS  Google Scholar 

  • J. Dvorkin, A. Nur, Dynamic poroelasticity: a unified model with the squirt and the Biot mechanisms. Geophysics 58(4), 524–533 (1993)

    Article  ADS  Google Scholar 

  • M. Enelund, P. Olsson, Time domain formulation of the Biot poroelastic theory using fractional calculus. IFAC Proc. 39(11), 391–396 (2010)

    Article  Google Scholar 

  • Z.E.A. Fellah, C. Depollier, Transient acoustic wave propagation in rigid porous media: a time-domain approach. J. Acoust. Soc. Am. 107(2), 683–688 (2000)

    Article  ADS  Google Scholar 

  • Z.E.A. Fellah, J.Y. Chapelon, S. Berger, W. Lauriks, C. Depollier, Ultrasonic wave propagation in human cancellous bone: application of Biot theory. J. Acoust. Soc. Am. 116(1), 61–73 (2004)

    Article  ADS  Google Scholar 

  • M. Fellah, Z.E.A. Fellah, F. Mitri, E. Ogam, C. Depollier, Transient ultrasound propagation in porous media using Biot theory and fractional calculus: application to human cancellous bone. J. Acoust. Soc. Am. 133(4), 1867–1881 (2013)

    Article  ADS  Google Scholar 

  • R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone functions. Fract. Calc. Appl. Anal. 19(5), 1105–1160 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • J. Geertsma, D.C. Smit, Some aspects of elastic wave propagation in fluid-saturated porous solids. Geophysics 26(2), 169–181 (1961)

    Article  ADS  MathSciNet  Google Scholar 

  • I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series, and products, ed. by A. Jeffrey (Academic Press. 4th edn. by Y. V. Geronimus and M. Yu. Tseytlin 2014)

    Google Scholar 

  • A. Hanyga, Viscous dissipation and completely monotonic relaxation moduli. Rheol. Acta 44(6), 614–621 (2005)

    Article  Google Scholar 

  • S. Holm, Spring-damper equivalents of the fractional, poroelastic, and poroviscoelastic models for elastography. NMR Biomed. e3854:1–12 (2017)

    Google Scholar 

  • S. Holm, V. Pandey, Wave propagation in marine sediments expressed by fractional wave and diffusion equations, in Proceedings of the IEEE China Ocean Acoustics Symposium (COA2016) (2016)

    Google Scholar 

  • S. Holm, R. Sinkus, A unifying fractional wave equation for compressional and shear waves. J. Acoust. Soc. Am. 127, 542–548 (2010)

    Article  ADS  Google Scholar 

  • A. Hosokawa, T. Otani, Ultrasonic wave propagation in bovine cancellous bone. J. Acoust. Soc. Am. 101(1), 558–562 (1997)

    Article  ADS  Google Scholar 

  • J.M. Hovem, Marine Acoustics: The Physics of Sound in Underwater Environments (Peninsula publishing, Los Altos, CA, 2012)

    Google Scholar 

  • M. Kimura, Frame bulk modulus of porous granular marine sediments. J. Acoust. Soc. Am. 120(2), 699–710 (2006)

    Article  ADS  Google Scholar 

  • M. Kimura, Experimental validation and applications of a modified gap stiffness model for granular marine sediments. J. Acoust. Soc. Am. 123(5), 2542–2552 (2008)

    Article  ADS  Google Scholar 

  • M. Kimura, Shear wave speed dispersion and attenuation in granular marine sediments. J. Acoust. Soc. Am. 134(1), 144–155 (2013)

    Article  ADS  Google Scholar 

  • M.M. Meerschaert, A. Sikorskii, Stochastic models for fractional calculus, vol. 43 (Walter de Gruyter, Berlin, 2012)

    MATH  Google Scholar 

  • W.F. Murphy III, K.W. Winkler, R.L. Kleinberg, Acoustic relaxation in sedimentary rocks: dependence on grain contacts and fluid saturation. Geophysics 51(3), 757–766 (1986)

    Article  ADS  Google Scholar 

  • T. Nagashima, N. Tamaki, S. Matsumoto, B. Horwitz, Y. Seguchi, Biomechanics of hydrocephalus: a new theoretical model. Neurosurg 21(6), 898–904 (1987)

    Article  Google Scholar 

  • V. Pandey, S. Holm, Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations. J. Acoust. Soc. Am. 140, 4225–4236 (2016a)

    Article  ADS  Google Scholar 

  • V. Pandey, S. Holm, Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity. Phys. Rev. E 94, 032606-1–6 (2016b)

    Google Scholar 

  • T.J. Plona, Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies. Appl. Phys. Lett. 36(4), 259–261 (1980)

    Article  ADS  Google Scholar 

  • P.N. Sahay, On the Biot slow S-wave. Geophysics 73(4), N19–N33 (2008)

    Article  ADS  Google Scholar 

  • R.D. Stoll, Acoustic waves in saturated sediments, in Physics of Sound in Marine Sediments (Springer, Berlin, 1974), pp. 19–39

    Book  Google Scholar 

  • R.D. Stoll, Acoustic waves in ocean sediments. Geophysics 42(4), 715–725 (1977)

    Article  ADS  Google Scholar 

  • R.D. Stoll, G.M. Bryan, Wave attenuation in saturated sediments. J. Acoust. Soc. Am. 47(5B), 1440–1447 (1970)

    Article  ADS  Google Scholar 

  • N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction, (Springer, Berlin, 1989). Reprinted in 2012

    Google Scholar 

  • A. Turgut, An investigation of causality for Biot models by using Kramers-Krönig relations, in Shear Waves in Marine Sediments (Springer, Berlin, 1991), pp. 21–28

    Google Scholar 

  • C. Zwikker, C.W. Kosten, Sound Absorbing Materials (Elsevier, Amsterdam, 1949)

    Google Scholar 

Download references

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Holm, S. (2019). Power Laws and Porous Media. In: Waves with Power-Law Attenuation. Springer, Cham. https://doi.org/10.1007/978-3-030-14927-7_8

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