Poroelasticity pp 475-571 | Cite as


  • Alexander H.-D. Cheng
Part of the Theory and Applications of Transport in Porous Media book series (TATP, volume 27)


In the preceding chapters we have been dealing with poroelastic theories and problems under the assumptions similar to elastostatics; that is, at any instant of a loading, the poroelastic body is at a state of static equilibrium. In other words, for a body of any size, finite or infinitesimal, the summation of all forces, including surface and body forces, must equal to zero, \(\sum \vec{F} = 0\), such that there is no acceleration created by the imbalance of forces. This, however, does not mean that there is no motion. One of the characteristics of poroelastic body is that its deformation is time-dependent, giving the impression of a creeping-like motion, even if the applied load is constant in time. This transient behavior is the consequence of a fluid phase. Fluid has no shear strength to resist shear deformation, but has a viscosity that resists the rate of shear deformation. Hence the force equilibrium of a fluid can be accompanied by motion. So even without considering the acceleration caused by force imbalance, the poroelastic body is not exactly static, and the poroelastic theory presented in the preceding chapters can be called a quasi-static theory. When a force is rapidly applied, such as by an explosion in the air, by the impact of a solid body, or due to the slippage of a fault, the inertial effect, that is, the right hand side of Newton’s second law of motion \(\vec{F} = m\vec{a}\), cannot be neglected. A dynamic theory should be introduced. When the inertial effect is considered in a continuum body that is compressible, a wave phenomenon results. Particularly, the stress caused by the applied force is not instantly felt throughout the body—it has a finite speed of propagation. Sound propagation in the air as a wave phenomenon was recognized by philosophers and scientists as early as Aristotle (384–322 BC), and then by Galileo Galilei (1564–1642) (Imelda and Subramaniam, Phys Educ 42(2):173–179, 2007). In fact, Aristotle already recognized that sound is a longitudinal wave when he wrote “(the air) is set in motion …by contraction or expansion or compression” (Barnes J (ed), The complete works of Aristotle. Revised Oxford translation, vol 1. Princeton University Press, Princeton, 1984, 1256pp). This is indicative of the definition of a longitudinal wave, in which the particle motion is parallel to the wave propagation direction. In a solid, elastic medium, there exist two types of waves: in addition to the longitudinal, or compressional wave, there also exists a transverse wave, also called a shear wave, in which the particle motion is perpendicular to the wave propagation direction. These two waves propagate at different wave speed. The historical development of elastic wave theory, or elastodynamics, is well summarized in Love (A treatise on the mathematical theory of elasticity, vol 1. Cambridge University Press, Cambridge/New York, 1892, 354pp). In this chapter we are interested in the wave propagation in porous medium, or the theory of poroelastodynamics. In such medium, a third wave, called the second compressional wave, is observed, due to the existence of two phases, a solid and a fluid. The reasoning and theoretical demonstration of such waves were first presented by Yackov Frenkel (J Phys USSR 13(4):230–241, 1944. Republished, J Eng Mech ASCE 131(9):879–887, 2005) (see Sect. F.13 for a biography). Frenkel’s work was motivated by the field observation of Ivanov (Doklady Akademii Nauk SSSR 24(1):42–45, 1939; Izvestiya Akademii Nauk SSSR, Ser. Geogr. Geofiz 5:699–727, 1940), who discovered the so-called seismoelectric effect of the second kind (E-effect) generated by underground explosion—when a seismic wave is generated by an explosion, electric potential differences can be observed between electrodes situated at different distances from the source of the waves. Based on the continuum mechanics theory, Frenkel demonstrated that in a fluid infiltrated isotropic porous medium, in addition to a longitudinal and a shear wave, there existed a second longitudinal wave characterized by the out-of-phase movement between solid and fluid. He then showed that in the presence of electrolytes in liquids, electric current was generated due to the relative movement between the phases. The alternating directions of the electrical current in turn generate an electromagnetic wave. However, as quoted in the prologue of the chapter, after the proclamation of the discovery of a second wave, Frenkel did not further pursue its characteristics. Twelve years later, citing Frenkel’s original contribution, Biot (J Acoust Soc Am 28(2):168–178, 1956; J Acoust Soc Am 28(2):179–191, 1956) re-derived the theory of wave propagation in porous medium. Biot not only demonstrated the existence of the waves, but also presented the wave speeds. Particularly, it was shown that the second compressional wave is highly dissipative, and propagates at a much lower speed than the first compressional wave; hence they are respectively called the slow wave and the fast wave. In the higher frequency range, Biot also introduced a physical model of capillary flow in parallel plates or tubes, to account for the viscous-inertial attenuation. He also discovered a characteristic frequency at which the attenuation reaches its maximum. Biot’s model became immensely popular and the second wave has largely been referred to as the Biot second wave.


  1. 1.
    Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, Amsterdam, 439ppzbMATHGoogle Scholar
  2. 2.
    Achenbach JD (2003) Reciprocity in elastodynamics. Cambridge University Press, Cambridge/New York, 255ppzbMATHGoogle Scholar
  3. 3.
    Abramowitz M, Stegun IA (1965) Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Dover, New York, 1046ppzbMATHGoogle Scholar
  4. 4.
    Aris R (1990) Vectors, tensors and the basic equations of fluid mechanics. Dover, New York, 320ppzbMATHGoogle Scholar
  5. 5.
    Attenborough K (1982) Acoustical characteristics of porous materials. Phys Rep—Rev Sect Phys Lett 82(3):179–227Google Scholar
  6. 6.
    Attenborough K (1987) On the acoustic slow wave in air-filled granular media. J Acoust Soc Am 81(1):93–102MathSciNetCrossRefGoogle Scholar
  7. 7.
    Auriault JL (1980) Dynamic behavior of a porous medium saturated by a Newtonian fluid. Int J Eng Sci 18(6):775–785zbMATHCrossRefGoogle Scholar
  8. 8.
    Auriault JL, Borne L, Chambon R (1985) Dynamics of porous saturated media, checking of the generalized law of Darcy. J Acoust Soc Am 77(5):1641–1650zbMATHCrossRefGoogle Scholar
  9. 9.
    Aznárez JJ, Maeso O, Domínguez J (2006) BE analysis of bottom sediments in dynamic fluid-structure interaction problems. Eng Anal Bound Elem 30(2):124–136zbMATHCrossRefGoogle Scholar
  10. 10.
    Badiey M, Cheng AHD, Mu YK (1998) From geology to geoacoustics—evaluation of Biot-Stoll sound speed and attenuation for shallow water acoustics. J Acoust Soc Am 103(1):309–320CrossRefGoogle Scholar
  11. 11.
    Badiey M, Jaya I, Cheng AHD (1994) Propagator matrix for plane wave reflection from inhomogeneous anisotropic poroelastic seafloor. J Comput Acoust 2(1):11–27CrossRefGoogle Scholar
  12. 12.
    Badiey M, Jaya I, Cheng AHD (1994) A shallow water acoustic/geoacoustic experiment near the New Jersey Atlantic Generating Station site. J Acoust Soc Am 96(6):3593–3604CrossRefGoogle Scholar
  13. 13.
    Barnes J (ed) (1984) The complete works of Aristotle. Revised Oxford translation, vol 1. Princeton University Press, Princeton, 1256ppGoogle Scholar
  14. 14.
    Basset AB (1888) Treatise on hydrodynamics, vol 2. Deighton, Bell, London, 328ppzbMATHGoogle Scholar
  15. 15.
    Berryman JG (1980) Confirmation of Biot’s theory. Appl Phys Lett 37(4):382–384MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bessel FW (1826) Untersuchungen über die Länge des einfachen Sekundenpendels (Studies on the length of the simple seconds pendulum). In: Abhandlungen der Königliche Akademie der Wissenschaften zu Berlin (Proceedings of the Royal Academy of Sciences in Berlin), BerlinGoogle Scholar
  17. 17.
    Biot MA (1955) Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys Rev 97(6):1463–1469MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Biot MA (1956) Thermoelasticity and irreversible thermodynamics. J Appl Phys 27(3):240–253MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. 1. Low-frequency range. J Acoust Soc Am 28(2):168–178MathSciNetCrossRefGoogle Scholar
  20. 20.
    Biot MA (1956) Theory of propagation of elastic waves in a fluid-saturated porous solid. 2. Higher frequency range. J Acoust Soc Am 28(2):179–191MathSciNetCrossRefGoogle Scholar
  21. 21.
    Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33(4):1482–1498MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Biot MA (1962) Generalized theory of acoustic propagation in porous dissipative media. J Acoust Soc Am 34(9):1254–1264MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bonnet G (1987) Basic singular solutions for a poroelastic medium in the dynamic range. J Acoust Soc Am 82(5):1758–1762CrossRefGoogle Scholar
  24. 24.
    Bonnet G, Auriault JL (1985) Dynamics of saturated and deformable porous media: homogenization theory and determination of the solid-liquid coupling coefficients. In: Boccara N, Daoud ZM (eds) Physics of finely divided matter, Proc. Winter School. Springer, Les Houches, pp 306–316CrossRefGoogle Scholar
  25. 25.
    Bourbié T, Coussy O, Zinszner B (1987) Acoustics of porous media. Editions Technip, Paris, 334ppGoogle Scholar
  26. 26.
    Boutin C, Bonnet G, Bard PY (1987) Green functions and associated sources in infinite and stratified poroelastic media. Geophys J R Astron Soc 90(3):521–550CrossRefGoogle Scholar
  27. 27.
    Bowen RM (1982) Compressible porous media models by use of the theory of mixtures. Int J Eng Sci 20(6):697–735zbMATHCrossRefGoogle Scholar
  28. 28.
    Bowen RM, Lockett RR (1983) Inertial effects in poroelasticity. J Appl Mech, ASME 50(2):334–342zbMATHCrossRefGoogle Scholar
  29. 29.
    Boyle FA, Chotiros NP (1992) Experimental detection of a slow acoustic wave in sediment at shallow grazing angles. J Acoust Soc Am 91(5):2615–2619CrossRefGoogle Scholar
  30. 30.
    Burridge R, Vargas CA (1979) Fundamental solution in dynamic poroelasticity. Geophys J R Astron Soc 58(1):61–90zbMATHCrossRefGoogle Scholar
  31. 31.
    Champoux Y, Allard JF (1991) Dynamic tortuosity and bulk modulus in air-saturated porous media. J Appl Phys 70(4):1975–1979CrossRefGoogle Scholar
  32. 32.
    Charlaix E, Kushnick AP, Stokes JP (1988) Experimental study of dynamic permeability in porous media. Phys Rev Lett 61(14):1595–1598CrossRefGoogle Scholar
  33. 33.
    Cheng AHD, Badmus T, Beskos DE (1991) Integral equation for dynamic poroelasticity in frequency domain with BEM solution. J Eng Mech ASCE 117(5):1136–1157CrossRefGoogle Scholar
  34. 34.
    Chen J (1994) Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part I: two-dimensional solution. Int J Solids Struct 31(10):1447–1490zbMATHCrossRefGoogle Scholar
  35. 35.
    Chen J (1994) Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part II: three-dimensional solution. Int J Solids Struct 31(2):169–202zbMATHCrossRefGoogle Scholar
  36. 36.
    Coussy O (2010) Mechanics and physics of porous solids. Wiley, Chichester/Hoboken, 296ppCrossRefGoogle Scholar
  37. 37.
    Dargush G, Chopra M (1996) Dynamic analysis of axisymmetric foundations on poroelastic media. J Eng Mech ASCE 122(7):623–632CrossRefGoogle Scholar
  38. 38.
    de Boer R, Ehlers W, Liu ZF (1993) One-dimensional transient wave propagation in fluid-saturated incompressible porous media. Arch Appl Mech 63(1):59–72zbMATHCrossRefGoogle Scholar
  39. 39.
    Deresiewicz H (1960) The effect of boundaries on wave propagation in a liquid-filled porous solid: I. Reflection of plane waves at a free plane boundary (non-dissipative case). Bull Seismol Soc Am 50(4):599–607MathSciNetGoogle Scholar
  40. 40.
    Deresiewicz H (1961) The effect of boundaries on wave propagation in a liquid-filled porous solid: II. Love waves in a porous layer. Bull Seismol Soc Am 51(1):51–59MathSciNetGoogle Scholar
  41. 41.
    Deresiewicz H (1962) The effect of boundaries on wave propagation in a liquid-filled porous solid: IV. Surface waves in a half-space. Bull Seismol Soc Am 52(3):627–638Google Scholar
  42. 42.
    Deresiewicz H, Rice JT (1962) The effect of boundaries on wave propagation in a liquid-filled porous solid: III. Reflection of plane waves at a free plane boundary (general case). Bull Seismol Soc Am 52(3):595–625Google Scholar
  43. 43.
    Deresiewicz H, Rice JT (1964) The effect of boundaries on wave propagation in a liquid-filled porous solid: V. Transmission across a plane interface. Bull Seismol Soc Am 54(1):409–416Google Scholar
  44. 44.
    Ding BY, Cheng AHD, Chen ZL (2013) Fundamental solutions of poroelastodynamics in frequency domain based on wave decomposition. J Appl Mech ASME 80(6):061201, 12 pCrossRefGoogle Scholar
  45. 45.
    Domenico S (1976) Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir. Geophysics 41(5):882–894CrossRefGoogle Scholar
  46. 46.
    Dominguez J (1991) An integral formulation for dynamic poroelasticity. J Appl Mech ASME 58(2):588–591zbMATHCrossRefGoogle Scholar
  47. 47.
    Dominguez J (1992) Boundary element approach for dynamic poroelastic problems. Int J Numer Methods Eng 35(2):307–324MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Domínguez J, Gallego R, Japón BR (1997) Effects of porous sediments on seismic response of concrete gravity dams. J Eng Mech ASCE 123(4):302–311CrossRefGoogle Scholar
  49. 49.
    Dvorkin J, Mavko G, Nur A (1995) Squirt flow in fully saturated rocks. Geophysics 60(1):97–107CrossRefGoogle Scholar
  50. 50.
    Dvorkin J, Nolen-Hoeksema R, Nur A (1994) The squirt-flow mechanism: macroscopic description. Geophysics 59(3):428–438CrossRefGoogle Scholar
  51. 51.
    Dvorkin J, Nur A (1993) Dynamic poroelasticity—a unified model with the squirt and the Biot mechanisms. Geophysics 58(4):524–533CrossRefGoogle Scholar
  52. 52.
    Ewing WM, Jardetsky WS, Press F (1957) Elastic waves in layered media. McGraw-Hill, New YorkzbMATHGoogle Scholar
  53. 53.
    Ferrin JL, Mikelić A (2003) Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid. Math Methods Appl Sci 26(10):831–859MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Frenkel J (1944) On the theory of seismic and seismoelectric phenomena in moist soil. J Phys USSR 13(4):230–241. Republished (2005) J Eng Mech ASCE 131(9):879–887CrossRefGoogle Scholar
  55. 55.
    Gassmann F (1951) Über die elastizität poröser medien (On elasticity of porous media) Veirteljahrsschrift der Naturforschenden Gesellschaft in Zürich, Zürich, vol 96, pp 1–23MathSciNetGoogle Scholar
  56. 56.
    Gist GA (1994) Interpreting laboratory velocity measurements in partially gas-saturated rocks. Geophysics 59(7):1100–1109CrossRefGoogle Scholar
  57. 57.
    Gist GA (1994) Fluid effects on velocity and attenuation in sandstones. J Acoust Soc Am 96(2):1158–1173CrossRefGoogle Scholar
  58. 58.
    Gregory A (1976) Fluid saturation effects on dynamic elastic properties of sedimentary rocks. Geophysics 41(5):895–921CrossRefGoogle Scholar
  59. 59.
    Halpern MR, Christiano P (1986) Response of poroelastic halfspace to steady-state harmonic surface tractions. Int J Numer Anal Methods Geomech 10(6):609–632zbMATHCrossRefGoogle Scholar
  60. 60.
    Hamilton EL (1980) Geoacoustic modeling of the sea floor. J Acoust Soc Am 68(5):1313–1340CrossRefGoogle Scholar
  61. 61.
    Hardin BO (1965) The nature of damping in sands. J Soil Mech Found Div ASCE 91:63–97Google Scholar
  62. 62.
    Hill RJ, Koch DL, Ladd AJC (2001) The first effects of fluid inertia on flows in ordered and random arrays of spheres. J Fluid Mech 448:213–241MathSciNetzbMATHGoogle Scholar
  63. 63.
    Hörmander L (1969) Linear partial differential operators. Springer, Berlin, 285ppzbMATHCrossRefGoogle Scholar
  64. 64.
    Imelda SC, Subramaniam R (2007) From Pythagoras to Sauveur: tracing the history of ideas about the nature of sound. Phys Educ 42(2):173–179CrossRefGoogle Scholar
  65. 65.
    Ionescu-Cazimir V (1964) Problem of linear coupled thermoelasticity. Theorems of reciprocity for the dynamic problem of coupled thermoelasticity. I and II. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Techniques 12:473–480, 481–488Google Scholar
  66. 66.
    Ivanov AG (1939) Effect of electrization of earth layers by elastic waves passing through them. Doklady Akademii Nauk SSSR 24(1):42–45Google Scholar
  67. 67.
    Ivanov AG (1940) The electroseismic effect of the second kind. Izvestiya Akademii Nauk SSSR, Ser. Geogr. Geofiz 5:699–727Google Scholar
  68. 68.
    Johnson DL (1980) Equivalence between 4th sound in liquid He II at low temperatures and the Biot slow wave in consolidated porous media. Appl Phys Lett 37(12):1065–1067CrossRefGoogle Scholar
  69. 69.
    Johnson DL, Koplik J, Dashen R (1987) Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J Fluid Mech 176:379–402zbMATHCrossRefGoogle Scholar
  70. 70.
    Johnson DL, Plona TJ (1982) Acoustic slow waves and the consolidation transition. J Acoust Soc Am 72(2):556–565CrossRefGoogle Scholar
  71. 71.
    Johnson DL, Plona TJ, Scala C, Pasierb F, Kojima H (1982) Tortuosity and acoustic slow waves. Phys Rev Lett 49(25):1840–1844CrossRefGoogle Scholar
  72. 72.
    Johnson DL, Sen PN (1981) Multiple scattering of acoustic waves with application to the index of refraction of 4th sound. Phys Rev B 24(5):2486–2496CrossRefGoogle Scholar
  73. 73.
    Kaynia AM, Banerjee PK (1993) Fundamental solutions of Biot’s equations of dynamic poroelasticity. Int J Eng Sci 31(5):817–830zbMATHCrossRefGoogle Scholar
  74. 74.
    Kelder O, Smeulders DMJ (1997) Observation of the Biot slow wave in water-saturated Nivelsteiner sandstone. Geophysics 62(6):1794–1796CrossRefGoogle Scholar
  75. 75.
    Kryloff N, Bogoliuboff N (1949) Introduction to non-linear mechanics. Princeton University Press, Princeton, 114ppzbMATHGoogle Scholar
  76. 76.
    Kupradze VD, Gezelia TG, Basheleishvili MO, Burchuladze TV (1979) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland, Amsterdam/New York, 929ppGoogle Scholar
  77. 77.
    Lamb H (1948) Hydrodynamics, 6th edn. Dover, New York, 738ppzbMATHGoogle Scholar
  78. 78.
    Landau LD, Lifshitz EM (1986) Theory of elasticity, vol 7 in course of theoretical physics, 3rd edn. Butterworth-Heinemann, Oxford/New York, 195ppGoogle Scholar
  79. 79.
    Landau LD, Lifshitz EM (1987) Fluid mechanics, vol 6 in course of theoretical physics, 2nd edn. Butterworth-Heinemann, Oxford/New York, 552ppGoogle Scholar
  80. 80.
    Levy T (1979) Propagation of waves in a fluid-saturated porous elastic solid. Int J Eng Sci 17:1005–1014MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Lighthill J (2001) Waves in fluids, 2nd edn. Cambridge University Press, Cambridge/New York, 524ppzbMATHGoogle Scholar
  82. 82.
    Lopatnikov SL, Cheng AHD (2004) Macroscopic Lagrangian formulation of poroelasticity with porosity dynamics. J Mech Phys Solids 52(12):2801–2839MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Love AEH (1892) A treatise on the mathematical theory of elasticity, vol 1. Cambridge University Press, Cambridge, 354ppzbMATHGoogle Scholar
  84. 84.
    Maeso O, Aznárez JJ, Domínguez J (2004) Three-dimensional models of reservoir sediment and effects on the seismic response of arch dams. Earthq Eng Struct Dyn 33(10):1103–1123CrossRefGoogle Scholar
  85. 85.
    Maeso O, Aznárez JJ, García F (2005) Dynamic impedances of piles and groups of piles in saturated soils. Comput Struct 83(10–11):769–782CrossRefGoogle Scholar
  86. 86.
    Manolis GD, Beskos DE (1989) Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech 76(1–2):89–104zbMATHCrossRefGoogle Scholar
  87. 87.
    Manolis GD, Beskos DE (1990) Errata in ‘Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity’. Acta Mech 83(3–4):223–226CrossRefGoogle Scholar
  88. 88.
    Mavko G, Jizba D (1991) Estimating grain-scale fluid effects on velocity dispersion in rocks. Geophysics 56(12):1940–1949CrossRefGoogle Scholar
  89. 89.
    Mavko G, Nur A (1975) Melt squirt in the asthenosphere. J Geophys Res 80(11):1444–1448CrossRefGoogle Scholar
  90. 90.
    Mavko G, Nur A (1979) Wave attenuation in partially saturated rocks. Geophysics 44:161–178CrossRefGoogle Scholar
  91. 91.
    Miksis MJ (1988) Effect of contact line movement on the dissipation of waves in partially saturated rocks. J Geophys Res 93:6624–6634CrossRefGoogle Scholar
  92. 92.
    Murphy W, Winkler K, Kleinberg R (1986) Acoustic relaxation in sedimentary rocks: dependence on grain contacts and fluid saturation. Geophysics 51(3):757–766CrossRefGoogle Scholar
  93. 93.
    Nigmatulin RI (1990) Dynamics of multiphase systems, vol 1 & 2. Hemisphere, New York, 878ppGoogle Scholar
  94. 94.
    Nikolaevskiy VN (1984) Mechanics of porous and cracked media (in Russian). Nedra, Moscow, 232ppGoogle Scholar
  95. 95.
    Norris AN (1985) Radiation from a point source and scattering theory in a fluid-saturated porous solid. J Acoust Soc Am 77(6):2012–2022zbMATHCrossRefGoogle Scholar
  96. 96.
    Nowacki W (1975) Dynamics problems of thermoelasticity. Noordhoff, Leyden, 456ppzbMATHGoogle Scholar
  97. 97.
    O’Connell RJ, Budiansky B (1977) Viscoelastic properties of fluid-saturated cracked solids. J Geophys Res 82(36):5719–5735CrossRefGoogle Scholar
  98. 98.
    Palmer I, Traviolia M (1980) Attenuation by squirt flow in undersaturated gas sands. Geophysics 45(12):1780–1792CrossRefGoogle Scholar
  99. 99.
    Philippacopoulos AJ (1998) Spectral Green’s dyadic for point sources in poroelastic media. J Eng Mech ASCE 124(1):24–31CrossRefGoogle Scholar
  100. 100.
    Plona TJ (1980) Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies. Appl Phys Lett 36(4):259–261CrossRefGoogle Scholar
  101. 101.
    Predeleanu M (1984) Development of boundary element method to dynamic problems for porous media. Appl Math Model 8(6):378–382zbMATHCrossRefGoogle Scholar
  102. 102.
    Pride SR, Berryman JG, Harris JM (2004) Seismic attenuation due to wave-induced flow. J Geophys Res B: Solid Earth 109(1):B01201 01201–01219Google Scholar
  103. 103.
    Sahay PN (2001) Dynamic green’s function for homogeneous and isotropic porous media. Geophys J Int 147(3):622–629CrossRefGoogle Scholar
  104. 104.
    Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, Berlin/New York, 398ppzbMATHGoogle Scholar
  105. 105.
    Sangani AS, Zhang DZ, Prosperetti A (1991) The added mass, Basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys Fluids A-Fluid Dyn 3(12):2955–2970MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Schanz M (2009) Poroelastodynamics: linear models, analytical solutions, and numerical methods. Appl Mech Rev 62(3):1–15CrossRefGoogle Scholar
  107. 107.
    Schanz M, Cheng AHD (2000) Transient wave propagation in a one-dimensional poroelastic column. Acta Mech 145(1–4):1–18zbMATHCrossRefGoogle Scholar
  108. 108.
    Schlichting H, Gersten K (2000) Boundary-layer theory, 8th edn. Springer, Berlin/New York, 826ppzbMATHCrossRefGoogle Scholar
  109. 109.
    Sen PN, Scala C, Cohen MH (1981) A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads. Geophysics 46(5):781–795CrossRefGoogle Scholar
  110. 110.
    Senjuntichai T, Rajapakse RKND (1994) Dynamic green’s functions of homogeneous poroelastic half-plane. J Eng Mech ASCE 120(11):2381–2464CrossRefGoogle Scholar
  111. 111.
    Sheng P, Zhou MY (1988) Dynamic permeability in porous media. Phys Rev Lett 61(14):1591–1594CrossRefGoogle Scholar
  112. 112.
    Stoll RD (1974) Acoustic waves in saturated sediments. In: Hampton L (ed) Physics of sound in marine sediment. Plenum, New York, pp 19–39CrossRefGoogle Scholar
  113. 113.
    Stoll RD (1977) Acoustic waves in ocean sediments. Geophysics 42(4):715–725CrossRefGoogle Scholar
  114. 114.
    Stoll RD (1989) Sediment acoustics. Springer, Berlin/New York, 155ppGoogle Scholar
  115. 115.
    Stoll RD, Bryan GM (1970) Wave attenuation in saturated sediments. J Acoust Soc Am 47(5):1440–1447CrossRefGoogle Scholar
  116. 116.
    Stoll RD, Kan TK (1981) Reflection of acoustic waves at a water-sediment interface. J Acoust Soc Am 70(1):149–156zbMATHCrossRefGoogle Scholar
  117. 117.
    Toms J, Müller TM, Ciz R, Gurevich B (2006) Comparative review of theoretical models for elastic wave attenuation and dispersion in partially saturated rocks. Soil Dyn Earthq Eng 26(6–7):548–565CrossRefGoogle Scholar
  118. 118.
    Turgut A, Yamamoto T (1988) Synthetic seismograms for marine sediments and determination of porosity and permeability. Geophysics 53(8):1056–1067CrossRefGoogle Scholar
  119. 119.
    van der Grinten JGM, van Dongen MEH, van der Kogel H (1985) A shock-tube technique for studying pore-pressure propagation in a dry and water-saturated porous medium. J Appl Phys 58(8):2937–2942CrossRefGoogle Scholar
  120. 120.
    van der Grinten JGM, van Dongen MEH, van der Kogel H (1987) Strain and pore pressure propagation in a water-saturated porous medium. J Appl Phys 62(12):4682–4687CrossRefGoogle Scholar
  121. 121.
    Wang Z, Nur A (1990) Dispersion analysis of acoustic velocities in rocks. J Acoust Soc Am 87:2384–2395CrossRefGoogle Scholar
  122. 122.
    White JE (1975) Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics 40(2):224–232CrossRefGoogle Scholar
  123. 123.
    Wilmanski K (2005) Tortuosity and objective relative accelerations in the theory of porous materials. Proc R Soc Lond, Ser A-Math Phys Sci 461(2057):1533–1561MathSciNetzbMATHCrossRefGoogle Scholar
  124. 124.
    Winkler KW (1985) Dispersion analysis of velocity and attenuation in Berea sandstone. J Geophys Res: Solid Earth 90(B8):6793–6800CrossRefGoogle Scholar
  125. 125.
    Wood AB (1941) Textbook of sound. Being an account of the physics of vibrations with special reference to recent theoretical and technical developments, 2nd edn. Bell & Sons, London, 578ppGoogle Scholar
  126. 126.
    Yamamoto T, Turgut A (1988) Acoustic wave propagation through porous media with arbitrary pore size distributions. J Acoust Soc Am 83(5):1744–1751CrossRefGoogle Scholar
  127. 127.
    Zimmerman C, Stern M (1993) Boundary element solution of 3-D wave scatter problems in a poroelastic medium. Eng Anal Bound Elem 12(4):223–240CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Alexander H.-D. Cheng
    • 1
  1. 1.University of MississippiOxfordUSA

Personalised recommendations