Modelling of surface waves in poroelastic saturated materials by means of a two component continuum

  • Bettina Albers
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 481)


These lecture notes are devoted to an overview of the modelling and the numerical analysis of surface waves in two-component saturated poroelastic media. This is an extension to the part of this book by K. Wilmanski which is primarily concerned with the classical surface waves in single component media. We use the “simple mixture model” which is a simplification of the classical Biot model for poroelastic media. Two interfaces are considered here: firstly the interface between a porous half space and a vacuum and secondly the interface between a porous halfspace and a fluid halfspace. For both problems we show how a solution can be constructed and a numerical solution of the dispersion relation can be found. We discuss the results for phase and group velocities and attenuations, and compare some of them to the high and low frequency approximations (ω → ∞, ω → 0, respectively).

For the interface porous medium/vacuum there exist in the whole range of frequencies two surface waves — a leaky Rayleigh wave and a true Stoneley wave. For the interface porous medium/fluid one more surface wave appears — a leaky Stoneley wave. For this boundary velocities and attenuations of the waves are shown in dependence on the surface permeability. The true Stoneley wave exists only in a limited range of this parameter. At the end we have a look on some results of other authors and a glance on a logical continuation of this work, namely the description of the structure and the acoustic behavior of partially saturated porous media.


Porous Medium Surface Wave Phase Velocity Rayleigh Wave Surface Permeability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. K. Aki and P. Richards. Quantitative seismology, theory and methods. W. H. Freeman and Co., 1980.Google Scholar
  2. B. Albers. Relaxation analysis and linear stability vs. adsorption in porous materials. Contin. Mech. Thermodyn., 15(1):73–95, 2003. also WIAS-Preprint Nr. 721, 2002.MATHCrossRefMathSciNetGoogle Scholar
  3. B. Albers and K. Wilmanski. Monochromatic surface waves on impermeable boundaries in two-component poroelastic media. Preprint 862, WIAS, 2003a. to appear in: Contin. Mech. Thermodyn.Google Scholar
  4. B. Albers and K. Wilmanski. On modeling acoustic waves in saturated poroelastic media. Preprint 874, WIAS, 2003b. to appear in: J. Engn. Mech.Google Scholar
  5. A. L. Anderson and L. D. Hampton. Acoustics of gas-bearing sediments. I. Background, II. Measurements and models. J. Acoust. Soc. Am., 67(6):1865–1898, 1980.CrossRefGoogle Scholar
  6. J. H. Ansell. The roots of the stoneley wave equation for solid-liquid interfaces. Pure Appl. Geophys., 94:172–188, 1972.CrossRefGoogle Scholar
  7. J. Bear. Dynamics of Fluids in Porous Media. Dover Publications, New York, 1988.Google Scholar
  8. A. Bedford and M. Stern. A model for wave propagation in gassy sediments. J. Acoust. Soc. Am., 73(2):409–417, 1983.MATHCrossRefGoogle Scholar
  9. A. Ben-Menahem and S. J. Singh. Seismic waves and sources. Springer-Verlag, New York Heidelberg Berlin, 1981.MATHGoogle Scholar
  10. J. G. Berryman, L. Thigpen, and R. C. Y. Chin. Bulk elastic wave propagation in partially saturated porous solids. J. Acoust. Soc. Am., 84(1):360–373, 1988.CrossRefGoogle Scholar
  11. T. Bourbié, O. Coussy, and B. Zinszner. Acoustics of porous media. Editions Technip, Paris, 1987.Google Scholar
  12. G. Chao, D. M. J. Smeulders, and M. E. H. van Dongen. Shock induced borehole waves in porous formations: Theory and experiments. J. Acoust. Soc. Am., 116(2), 2004.Google Scholar
  13. H. Deresiewicz. The effect of boundaries on wave propagation in a liquid filled porous solid. IV. Surface waves in a half space. Bull. Seismol. Soc. Am., 52:627–638, 1962.Google Scholar
  14. H. Deresiewicz. The effect of boundaries on wave propagation in a liquid filled porous solid. VII. Surface waves in a half space in the presence of a liquid layer. Bull. Seismol. Soc. Am., 54:425–430, 1964.Google Scholar
  15. H. Deresiewicz and R. Skalak. On uniqueness in dynamic poroelasticity. Bull. Seismol. Soc. Am., 53:783–788, 1963.Google Scholar
  16. I. Edelman. Bifurcation of the biot slow wave in a porous medium. J. Acoust. Soc. Am., 114(1):90–97, 2003.CrossRefMathSciNetGoogle Scholar
  17. I. Edelman and K. Wilmanski. Asymptotic analysis of surface waves at vacuum/porous medium and liquid/porous medium interfaces. Contin. Mech. Thermodyn., 14(1): 25–44, 2002.MATHCrossRefMathSciNetGoogle Scholar
  18. S. Feng and D. L. Johnson. High-frequency acoustic properties of a fluid/porous solid interface. I. New surface mode. J. Acoust. Soc. Am., 74(3):906–914, 1983a.MATHCrossRefGoogle Scholar
  19. S. Feng and D. L. Johnson. High-frequency acoustic properties of a fluid/porous solid interface. II. The 2D reflection Green’s function. J. Acoust. Soc. Am., 74(3):915–924, 1983b.MATHCrossRefGoogle Scholar
  20. D. G. Fredlund and H. Rahardjo. Soil Mechanics for Unsaturated Soils. John Wiley & Sons, 1993.Google Scholar
  21. P. Hess. Surface acousic waves in materials science. Physics Today, March: 42–47, 2002.Google Scholar
  22. H. Kutruff. Ultrasonics, fundamentals and applications. Elsevier, 1991.Google Scholar
  23. C. Lai. Simultaneous inversion of Rayleigh phase velocity and attenuation for near-surface site characterization. PhD thesis, Georgia Institute of Technology, 1998.Google Scholar
  24. N. G. H. Meyendorf, P. B. Nagy, and S. I. Rokhlin, editors. Nondestructive Materials Characterization. Springer, Berlin Heidelberg, 2004.Google Scholar
  25. P. B. Nagy. Acoustics and ultrasonics. In Po-zen Wong, editor, Methods in the Physics of Porous Media, pages 161–221. Academic Press, 1999.Google Scholar
  26. A. N. Norris. Stoneley-wave attenuation and dispersion in permeable formations. Geophysics, 54(3):330–341, 1989.CrossRefGoogle Scholar
  27. J. O. Owino and L. J. Jacobs. Attenuation measurements in cement-based materials using laser ultrasonics. J. Engn. Mech., 125(6):637–647, 1999.CrossRefGoogle Scholar
  28. R. A. Phinney. Propagation of leaking interface waves. Bull. Seismol. Soc. Am., 51(4):527–555, 1961.MathSciNetGoogle Scholar
  29. S. Pietruszczak and G. N. Pande. Constitutive relations for partially saturated soils containing gas inclusions. J. Geotechnical Engrg., 122(1):50–59, 1996.CrossRefGoogle Scholar
  30. G. J. Rix, C. G. Lai, and S. Foti. Simultaneous measurement of surface wave dispersion and attenuation curves. Geotechnical Testing Journal, 24(4):350–358, 2001.CrossRefGoogle Scholar
  31. J. C. Santamarina. Soils and Waves. John Wiley & Sons, 2001.Google Scholar
  32. B. A. Schrefler, L. Simoni, Li Xikui, and O. C. Zienkiewicz. Mechanics of partially saturated porous media. In C.S. Desai and G. Gioda, editors, Numerical Methods and Constitutive Modelling in Geomechanics (Udine, 1989), volume 311 of CISM Lecture Notes, pages 169–209. Springer Verlag, Wien, 1990.Google Scholar
  33. C. T. Schroder and W. R. Scott. On the complex conjugate roots of the Rayleigh equation: The leaky surface wave. J. Acoust. Soc. Am., 110(6):2867–2877, 2001.CrossRefGoogle Scholar
  34. D. M. J. Smeulders, J. P. M. de la Rosette, and M. E. H. van Dongen. Waves in partially saturated porous media. Transport in Porous Media, 9:25–37, 1992.CrossRefGoogle Scholar
  35. E. Strick. III. The pseudo-Rayleigh wave. Philos. Trans. Roy. Soc. (London), Ser. A, 251:488–522, 1959.Google Scholar
  36. I. Tolstoy, editor. Acoustics, elasticity and thermodynamics of porous media: Twenty-one papers by M. A. Biot. American Institute of Physics, 1992.Google Scholar
  37. A. Udias. Principles of Seismology. Cambridge University Press, 1999.Google Scholar
  38. I. A. Viktorov. Rayleigh and Lamb waves. Physical theory and applications. Plenum Press, New York, 1967.Google Scholar
  39. S. J. Wheeler. A conceptual model for soils containing large gas bubbles. Géotechnique, 38:389–397, 1988.CrossRefGoogle Scholar
  40. J. E. White. Underground sound. Application of seismic waves. Elsevier, Amsterdam, 1983.Google Scholar
  41. G. B. Whitham. Linear and nonlinear waves. John Wiley & Sons, New York, 1974.MATHGoogle Scholar
  42. K. Wilmanski. Waves in porous and granular materials. In K. Hutter and K. Wilmanski, editors, Kinetic and continuum theories of granular and porous media, number 400 in CISM Courses and Lectures, pages 131–186. Springer, Wien New York, 1999.Google Scholar
  43. K. Wilmanski. Some questions on material objectivity arising in models of porous materials. In M. Brocato, editor, Rational continua, classical and new, pages 149–161. Springer-Verlag, Italia Srl, Milano, 2001.Google Scholar
  44. K. Wilmanski. Propagation of sound and surface waves in porous materials. In B. Maruszewski, editor, Structured media, pages 312–326. Poznan University of Technology, Poznan, 2002.Google Scholar
  45. K. Wilmanski. Tortuosity and objective relative acceleration in the theory of porous materials. Preprint 922, WIAS, 2004. to appear in: Proc. Royal Soc. (London), Ser. A.Google Scholar
  46. K. Wilmanski. Elastic modelling of surface waves in single and multicomponent systems. In C. Lai and K. Wilmanski, editors, Surface Waves in Geomechanics, Direct and Inverse Modelling for Soils and Rocks, CISM Courses and Lectures. Springer Verlag, Wien-New York, 2005.Google Scholar
  47. K. Wilmanski and B. Albers. Acoustic waves in porous solid-fluid mixtures. In K. Hutter and N. Kirchner, editors, Dynamic response of granular and porous materials under large and catastrophic deformations, volume 11 of Lecture Notes in Applied and Computational Mechanics, pages 285–314. Springer, Berlin, 2003.Google Scholar
  48. K. W. Winkler, H.-L. Liu, and D. L. Johnson. Permeability and borehole Stoneley waves: Comparison between experiment and theory. Geophysics, 54(1):66–75, 1989.CrossRefGoogle Scholar
  49. C. J. Wisse, D. M. J. Smeulders, M. E. H. van Dongen, and G. Chao. Guided wave modes in porous cylinders: Experimental results. J. Acoust. Soc. Am., 112(3):890–895, 2002.CrossRefGoogle Scholar
  50. A. B. Wood. A Textbook of Sound. G. Bell and Sons, London, 1957.Google Scholar
  51. D. M. Wood. The behaviour of partly saturated soils: A review. Cambridge University Report, Engineering Department, 1979.Google Scholar

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© CISM, Udine 2005

Authors and Affiliations

  • Bettina Albers
    • 1
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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