1887
Volume 61, Issue 3
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

We generalize the classical theory of acoustoelasticity to the porous case (one fluid and a solid frame) and finite deformations. A unified treatment of non‐linear acoustoelasticity of finite strains in fluid‐saturated porous rocks is developed on the basis of Biot’s theory. A strain‐energy function, formed with eleven terms, combined with Biot’s kinetic and dissipation energies, yields Lagrange’s equations and consequently the wave equation of the medium. The velocities and dissipation factors of the P‐ and S‐waves are obtained as a function of the 2nd‐ and 3rd‐order elastic constants for hydrostatic and uniaxial loading. The theory yields the limit to the classical theory if the fluid is replaced with a solid with the same properties of the frame. We consider sandstone and obtain results for open‐pore jacketed and closed‐pore jacketed ‘gedanken’ experiments. Finally, we compare the theoretical results with experimental data.

Loading

Article metrics loading...

/content/journals/10.1111/j.1365-2478.2012.01091.x
2012-07-10
2024-04-28
Loading full text...

Full text loading...

References

  1. BaJ., CaoH., YaoF.C., NieJ.X. and YangH.Z.2008a. Double‐porosity rock model and squirt flow in the laboratory frequency band. Applied Geophysics 5, 261–276.
    [Google Scholar]
  2. BaJ., NieJ.X., CaoH. and YangH.Z.2008b. Mesoscopic fluid flow simulation in double‐porosity rocks. Geophysical Research Letters 35, L04303.
    [Google Scholar]
  3. BaJ., CarcioneJ.M. and NieJ.X.2011. Biot‐Rayleigh theory of wave propagation in double‐porosity media. Journal of Geophysical Research 116, B06202, doi:10.1029/2010JB008185.
    [Google Scholar]
  4. BerrymanJ.G. and ThigpenL.1985. Nonlinear and semilinear dynamic poroelasticity with microstructure. Journal of the Mechanics and Physics of Solids 33, 97–116.
    [Google Scholar]
  5. BerrymanJ.G. and WangH.F.2001. Dispersion in poroelastic systems. Physics Review 64, 011303.
    [Google Scholar]
  6. BeyerR.T.1960. Parameter of nonlinearity in fluids. Journal of the Acoustical Society of America 32, 719–721.
    [Google Scholar]
  7. BeyerR.T.1984. Nonlinear Acoustics in Fluids . Van Nostrand Reinhold, New York .
    [Google Scholar]
  8. BiotM.A.1956. Theory of propagation of elastic waves in a fluid‐saturated porous solid. I. Low‐frequency range. Journal of the Acoustical Society of America 28, 168–178.
    [Google Scholar]
  9. BiotM.A.1972. Theory of finite deformations of porous solids. Indiana Uni versity Mathematics Journal 21, 597–620.
    [Google Scholar]
  10. BiotM.A.1973. Nonlinear and semilinear rheology of porous solids. Journal of Geophysical Research 78, 4924–4937.
    [Google Scholar]
  11. BiotM.A. and WillisD.G.1957. The elastic coefficients of the theory of consolidation. Journal of Applied Mechanics 24, 594–601.
    [Google Scholar]
  12. BruggerK.1964. Thermodynamic definition of higher order elastic coefficients. Physics Review 133, A1611–A1612.
    [Google Scholar]
  13. CarcioneJ.M.2007. Wave fields in real media . Theory and Numerical Simulation of Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media . Elsevier Science, (Second edition, revised and extended).
    [Google Scholar]
  14. CarcioneJ.M. and PicottiS.2006. P‐wave seismic attenuation by slow wave diffusion: Effects of inhomogeneous properties. Geophysics 71, O1–O8.
    [Google Scholar]
  15. DaiN., VafidisA. and KanasewichE.R.1995. Wave propagation in heterogeneous, porous media: A velocity‐stress, finite‐difference method. Geophysics 60, 327–340.
    [Google Scholar]
  16. DazelO. and TournatV.2010. Nonlinear Biot waves in porous media with application to unconsolidated granular media. Journal of the Acoustical Society of America 127, 692–702.
    [Google Scholar]
  17. DonskoyD.M., KhashanahK. and MckeeT.G.1997. Nonlinear acoustic waves in porous media in the context of Biot’s theory. Journal of the Acoustical Society of America 102, 2521–2528.
    [Google Scholar]
  18. DrumhellerD.S. and BedfordA.1980. A thermomechanical theory for reacting immiscible mixtures. Archive for Rational Mechanics and Analysis 73, 257–284.
    [Google Scholar]
  19. DuttaN.C. and OdéH.1979a. Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model) ‐ Part I: Biot theory. Geophysics 44, 1777–1788.
    [Google Scholar]
  20. DuttaN.C. and OdéH.1979b. Attenuation and dispersion of compressional waves in fluid‐filled porous rocks with partial gas saturation (White model) ‐ Part II: Results. Geophysics 44, 1789–1805.
    [Google Scholar]
  21. GassmannF.1951. Uber die elastizität poröser medien. Vierteljahresschrift der Naturforschenden Gesellschaft in Zurich 96, 1–23.
    [Google Scholar]
  22. GoldbergZ.A.1961. Interaction of plane longitudinal and transverse elastic waves. Soviet Physical Acoustics 6, 306–310.
    [Google Scholar]
  23. GreenR.E.1973. Ultrasonic Investigation of Mechanical Properties, Treatise on Materials Science and Technology . Vol. 3. Academic Press, New York .
    [Google Scholar]
  24. GrinfeldM.A. and NorrisN.A.1996. Acoustoelasticity theory and applications for fluid‐saturated porous media. Journal of the Acoustical Society of America 100, 1368–1374.
    [Google Scholar]
  25. GuoM.Q., FuL.Y. and BaJ.2009. Comparison of stress‐associated coda attenuation and intrinsic attenuation from ultrasonic measurements. Geophysical Journal International , doi: 10.1111/j.1365‐246X.2009.04159.x.
    [Google Scholar]
  26. HearmonR.F.S.1953. Third‐order elastic coefficients. Acta Crystallographica 6, 331–340.
    [Google Scholar]
  27. HughesD.S. and KellyJ.L.1953. Second‐order elastic deformation of solids. Physics Review 92, 1145–1149.
    [Google Scholar]
  28. JohnsonD.L.1986. Recent Developments in the Acoustic Properties of Porous Media. Frontiers in Physical Acoustics XCIII . Edited by D.Sette , North Holland. Elsevier, New York , 255–290.
    [Google Scholar]
  29. JohnsonD.L.2001. Theory of frequency dependent acoustics in patchy‐saturated porous media. Journal of the Acoustical Society of America 110, 682–694.
    [Google Scholar]
  30. JohnsonP.A. and ShanklandT.J.1989. Nonlinear generation of elastic waves in granite and sandstone: Continuous wave and travel time observations. Journal of Geophysical Research 94, 17729–17733.
    [Google Scholar]
  31. JonesG.L. and KobettD.1963. Interaction of elastic waves in an isotropic solid. Journal of the Acoustical Society of America 35, 5–10.
    [Google Scholar]
  32. KostekS., SinhaB.K. and NorrisA.N.1993. Third‐order elastic constants for an inviscid fluid. Journal of the Acoustical Society of America 94, 3014–3017.
    [Google Scholar]
  33. MeeganG.D., JohnsonP.A., GuyerR.A. and McCallK.R.1993. Observations on nonlinear elastic wave behaviour in sandstone. Journal of the Acoustical Society of America 94, 3387–3391.
    [Google Scholar]
  34. MurnaghanF.D.1937. Finite deformations of an elastic solid. American Journal of Mathematics 59, 235–260.
    [Google Scholar]
  35. MurnaghanF.D.1951. Finite Deformation of an Elastic Solid , John Wiley & Sons, Inc., New York .
    [Google Scholar]
  36. NorrisA.N., SinhaB.K. and KostekS.1994. Acoustoelasticity of solid/fluid composite systems. Geophysical Journal International 118, 439‐446.
    [Google Scholar]
  37. PaoY.H., SachseW. and FukuokaH.1984. Acoustoelasticity and Ultrasonic Measurement of Residual Stresses. Physical Acoustics , Vol. XVII. Orlando.
    [Google Scholar]
  38. PrideS.R., BerrymanJ.G. and HarrisJ.M.2004. Seismic attenuation due to wave‐induced flow. Journal of Geophysical Research 109, B01201, doi:10.1029/2003JB002639.
    [Google Scholar]
  39. ThurtsonR.N. and BruggerK.1964. Third‐order elastic constants and the velocity of small amplitude elastic waves in homogeneously stressed media. Physics Review 133, A1604–A1610.
    [Google Scholar]
  40. ToupinP.A. and BernsteinB.1961. Sound waves in deformed perfectly elastic materials. Acoustoelastic effect. Journal of the Acoustical Society of America 33, 216–225.
    [Google Scholar]
  41. TruesdellC.1961. General and exact theory of waves in ?nite elastic strain. Archive for Rational Mechanics and Analysis 8, 263–296.
    [Google Scholar]
  42. TruesdellC.1965. Continuum Mechanics IV: Problems of Non‐linear Elasticity . Gordon & Breach, New York .
    [Google Scholar]
  43. WinklerK.W. and LiuX.1996. Measurements of third‐order elastic constants in rocks. Journal of the Acoustical Society of America 100, 1392–1398.
    [Google Scholar]
  44. WinklerK.W. and McGowanL.2004. Nonlinear acoustoelastic constants of dry and saturated rocks. Journal of Geophysical Research 109, B10204, doi:10.1029/2004JB003262.
    [Google Scholar]
  45. ZaitsevV.Y., KolpakovA.B. and NazarovV.E.1999a. Detection of acoustic pulses in river sand: Experiment. Acoustical Physics 45, 235–241.
    [Google Scholar]
  46. ZaitsevV.Y., KolpakovA.B. and NazarovV.E.1999b. Detection of acoustic pulses in river sand. Theory. Acoustical Physics 45, 347–353.
    [Google Scholar]
  47. ZaitsevV. and SasP.2004. Effect of high‐compliant porosity on variations of P‐and S‐wave velocities in dry and saturated rocks: Comparison between theory and experiment. Physical Mesomechanics 7, 37–46.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1111/j.1365-2478.2012.01091.x
Loading
/content/journals/10.1111/j.1365-2478.2012.01091.x
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): Acoustoelasticity; Non‐linearity; Porous media; Rock; Waves

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error