Skip to main content
SpringerLink
Log in

A unified theory for elastic wave propagation through porous media containing cracks—An extension of Biot’s poroelastic wave theory

  • Research Paper
  • Published:
Science China Earth Sciences Aims and scope Submit manuscript

Abstract

Rocks in earth’s crust usually contain both pores and cracks. This phenomenon significantly affects the propagation of elastic waves in earth. This study describes a unified elastic wave theory for porous rock media containing cracks. The new theory extends the classic Biot’s poroelastic wave theory to include the effects of cracks. The effect of cracks on rock’s elastic property is introduced using a crack-dependent dry bulk modulus. Another important frequency-dependent effect is the “squirt flow” phenomenon in the cracked porous rock. The analytical results of the new theory demonstrate not only reduction of elastic moduli due to cracks but also significant elastic wave attenuation and dispersion due to squirt flow. The theory shows that the effects of cracks are controlled by two most important parameters of a cracked solid: crack density and aspect ratio. An appealing feature of the new theory is its maintenance of the main characteristics of Biot’s theory, predicting the characteristics of Biot’s slow wave and the effects of permeability on elastic wave propagation. As an application example, the theory correctly simulates the change of elastic wave velocity with gas saturation in a field data set. Compared to Biot theory, the new theory has a broader application scope in the measurement of rock properties of earth’s shallow crust using seismic/acoustic waves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Eshelby J D. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc Roy Soc of London, 1957, A241, 1226: 376–396

    Article  Google Scholar 

  2. Walsh J B. The effect of cracks on the compressibility of rock. J Geophys Res, 1965, 70, 2: 381–389

    Article  Google Scholar 

  3. O’Connell R J, Budiansky B. Seismic velocities in dry and saturated cracked solids. J Geophys Res, 1974, 79: 4626–4627

    Google Scholar 

  4. Budiansky B, O’Connell R J. Elastic moduli of a cracked solid. Int J Solids Structures, 1976, 12: 81–97

    Article  Google Scholar 

  5. O’Connell R J, Budiansky B. Viscoelastic properties of fluid saturated cracked solids. J Geophys Res, 1977, 82: 5719–5736

    Article  Google Scholar 

  6. Budiansky B, O’Connell R J. Bulk dissipation in heterogeneous media. Solid Earth Geophys Geotech, 1980, 42: l–10

    Google Scholar 

  7. O’Connell R J. A viscoelastic model of anelasticity of fluid saturated porous rocks. Physics and Chemistry of Porous Media. AIP Conference Proceedings, 1984. 166–175

  8. Biot M A. Theory of propagation of elastic waves in a fluid saturated porous solid: A. Low-frequency range. J Acoust Soc Am, 1956, 28: 168–179

    Article  Google Scholar 

  9. Biot M A. Theory of propagation of elastic waves in a fluid saturated porous solid. B: Higher frequency range. J Acoust Soc Am, 1956, 28: 170–191

    Google Scholar 

  10. Biot M A. Generalized theory of acoustic propagation in porous dissipative media. J Acoust Soc Am, 1962, 34: 1254–1264

    Article  Google Scholar 

  11. Tang X M, Cheng C H. Quantitative Borehole Acoustic Methods. Amsterdam: Elsevier Science Publishing Co, 2004

    Google Scholar 

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

    Article  Google Scholar 

  13. White J E. Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics, 1975, 40: 224–232

    Article  Google Scholar 

  14. Johnson D L. Theory of frequency dependent acoustics in patchy-saturated porous media. J Acoust Soc Am, 2001, 110: 682–694

    Article  Google Scholar 

  15. Carcione J M, Helle H B, Pham N H. White’s model for wave propagation in partially saturated rocks: Comparison with poroelastic numerical experiments. Geophysics, 2003, 68: 1389–1398

    Article  Google Scholar 

  16. Hudson J A. Wave speeds and attenuation of elastic waves in material containing cracks. Geophys J R Astr Soc, 1981, 64: 133–150

    Google Scholar 

  17. Hudson J A. A higher order approximation to the wave propagation constants for a cracked solid. Geophys J R Astr Soc, 1986, 87: 265–274

    Google Scholar 

  18. Hudson J A. Overall elastic properties of isotropic materials with arbitrary distribution of circular cracks. Geophys J Int, 1990, 102: 465–469

    Article  Google Scholar 

  19. Kuster G T, Toksöz M N. Velocity and attenuation of seismic waves in two-phase media: I. Theoretical formulation. Geophysics, 1974, 39: 587–606

    Article  Google Scholar 

  20. Mavko G, Nur A. Wave attenuation in partially saturated rocks. Geophysics, 1979, 44: 161–178

    Article  Google Scholar 

  21. Pride S R, Berryman J G. Linear dynamics of double-porosity dualpermeability materials: I. Governing equations and acoustic attenuation. Phys Rev E, 2003, 68: 036603

    Article  Google Scholar 

  22. Pride S R, Berryman J G. Linear dynamics of double-porosity dualpermeability materials: II. Fluid transport equations. Phys Rev E, 2003, 68: 036604

    Article  Google Scholar 

  23. Berryman J G, Wang H F. The elastic coefficients of double-porosity models for fluid transport in jointed rock. J Geophys Res, 1995, 100(B12): 24611–24627

    Article  Google Scholar 

  24. Berryman J G. Effective medium theories for multicomponent poroelastic composites. J Eng Mech, 2006, 132: 519–531

    Article  Google Scholar 

  25. Ba J. A study on the seismic wave propagation through multicomponent poroelastic composites. Dotoral Thesis. Beijing: Tsinghua University, 2008

    Google Scholar 

  26. Ba J. Theory of wave propagation through a double-porosity medium and an experimental analysis of its seismic response. Sci Sin Phys Mech Astron, 2010, 40: 1–12

    Google Scholar 

  27. Thomsen L. Biot-consistent elastic moduli of porous rocks: Low-frequency limit. Geophysics, 1985, 50, 12: 2797–2807

    Article  Google Scholar 

  28. Mavko G, Jizba D. Estimating grain-scale fluid effects on velocity dispersion in rocks. Geophysics, 1991, 56, 12: 1940–1949

    Article  Google Scholar 

  29. Johnson D L, Koplik J, Dashen R. Theory of dynamic permeability and tortuosity in fluid saturated porous media. J Fluid Mech, 1987, 176: 379–402

    Article  Google Scholar 

  30. Brie A, Pampuri F, Marsala A F, et al. Shear sonic interpretation in gas-bearing sands. Proceedings of SPE Annual Technical Conference & Exhibition, 1995. SPE 30595: 701–710

    Google Scholar 

  31. Chen Y. Mechanical Properties of Earth’s Crustal Rocks-Basic Theory and Experimental Methods. Beijing: Seismological Press, 140–157

  32. Tang X, Cheng C H. A dynamic model for fluid flow in open borehole fractures. J Geophys Res, 1989, 94(B6): 7567–7576

    Article  Google Scholar 

  33. Cui Z W, Wang K X, Cao Z L, et al. The slow wave of BISQ model in double-porosity media. Acta Phys Sin, 2004, 53,9: 3084–3089

    Google Scholar 

  34. Garland G D. Introduction to Geophysics: Mantle, Core, and Crust. Philadelphia: W. B. Sanders Co, 1987. 117–120

    Google Scholar 

  35. Schubnel A, Benson P M, Thompson B D, et al. Quantifying damage, saturation and anisotropy in cracked rocks by inverting elastic wave velocities. Pure Appl Geophys, 2006, 163: 947–973

    Article  Google Scholar 

  36. Cheng C H, Toksoz M N. Inversion of seismic velocities for the pore aspect ratio spectrum of a rock. J Geophys Res, 1979, 84(B13): 7533–7543

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Geosciences and Technology, China University of Petroleum (East), Qingdao, 266555, China

    XiaoMing Tang

Authors
  1. XiaoMing Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XiaoMing Tang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tang, X. A unified theory for elastic wave propagation through porous media containing cracks—An extension of Biot’s poroelastic wave theory. Sci. China Earth Sci. 54, 1441–1452 (2011). https://doi.org/10.1007/s11430-011-4245-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11430-011-4245-7

Keywords

Search

Navigation