1887
Volume 50, Issue 2
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

The modelling of elastic waves in fractured media with an explicit finite‐difference scheme causes instability problems on a staggered grid when the medium possesses high‐contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard‐coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P‐ and S‐wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non‐intersecting, thin, dry cracks, the numerical simulations of velocities of P‐, SV‐ and SH‐waves are in excellent agreement with the results of the modified (or differential) self‐consistent theory. On the other hand for intersecting cracks, the critical crack‐density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack‐density concept into the modified self‐consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.

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2002-11-23
2024-03-28
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References

  1. AkiK. and RichardsP.G.1980. Quantitative Seismology: Theory and Methods. W.H. Freeman & Co.
    [Google Scholar]
  2. AndrewsD.J. and Ben‐ZionY.1997. Wrinkle‐like slip pulse on a fault between different materials.Journal of Geophysical Research102, 553–571.DOI: 10.1029/1999JB900306
    [Google Scholar]
  3. BerrymanJ.G.1980. Long‐wavelength propagation in composite elastic media; i. spherical inclusions and ii. ellipsoidal inclusions.Journal of the Acoustical Society of America68, 1809–1831.
    [Google Scholar]
  4. BerrymanJ.G.1992. Single‐scattering approximations for coefficients in Biot's equations of poroelasticity.Journal of the Acoustical Society of America91, 551–571.
    [Google Scholar]
  5. BrunerW.M.1976. Comment on ``Seismic velocities in dry and saturated cracked solids'' by R.J. O'Connell and B. Budiansky.Journal of Geophysical Research81, 2573–2576.
    [Google Scholar]
  6. BudianskyB. and O'ConnellR.J.1976. Elastic moduli of a cracked solid.International Journal of Solids and Structures12, 81–97.
    [Google Scholar]
  7. ChatterjeeA., MalA. and KnopoffL.1978. Elastic moduli of a cracked solid.Journal of Geophysical Research83, 1785–1792.
    [Google Scholar]
  8. ChengC.H.1993. Crack models for a transversely isotropic medium.Journal of Geophysical Research98, 675–684.
    [Google Scholar]
  9. CraseE.1990. High‐order (space and time) finite‐difference modelling of the elastic wave equation. 60th SEG meeting, San Francisco, USA, Expanded Abstracts,987–991.
  10. DahmT. and BeckerT.1998. On the elastic and viscous properties of media containing strongly interacting in‐plane cracks.Pure and Applied Geophysics151, 1–16.
    [Google Scholar]
  11. DavisP.M. and KnopoffL.1995. The elastic modulus of media containing strongly interacting antiplane cracks.Journal of Geophysical Research100, 18253–18258.
    [Google Scholar]
  12. DoumaJ.1988. The effect of the aspect ratio on crack‐induced anisotropy.Geophysical Prospecting36, 614–632.
    [Google Scholar]
  13. EshelbyJ.D.1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems.Proceedings of the Physical Society of London, A241, 376–396.
    [Google Scholar]
  14. FrankelA. and ClaytonR.W.1986. Finite difference simulations of seismic scattering: implications for the propagation of short‐period seismic waves in the crust and models of crustal heterogeneity.Journal of Geophysical Research91, 6465–6489.
    [Google Scholar]
  15. HenyeyF.S. and PomphreyN.1982. Self‐consistent elastic moduli of a cracked solid.Geophysical Research Letters9, 903–906.
    [Google Scholar]
  16. HudsonJ.1980. Elastic moduli of a cracked solid.Mathematical Proceedings of the Cambridge Philosophical Society88, 371–384.
    [Google Scholar]
  17. HudsonJ. and KnopoffL.1989. Predicting the overall properties of composites – material with small‐seale inclusions or cracks.Pure and Applied Geophsics131, 551–576.
    [Google Scholar]
  18. IgelH., MoraP. and RiolletB.1995. Anisotropic wave propagation through finite‐difference grids.Geophysics60, 1203–1216.
    [Google Scholar]
  19. KachanovM.1992. Effective elastic properties of cracked solids: critical review of some basic concepts.Applied Mechanics Review45, 304–335.
    [Google Scholar]
  20. KarrenbachM.1995. Elastic tensor wavefields. PhD thesis, Stanford University.
    [Google Scholar]
  21. KellyK.R., WardR.W., TreitelS. and AlfordR.M.1976. Synthetic seismograms: a finite‐difference approach.Geophysics41, 2–27.
    [Google Scholar]
  22. KelnerS., BouchonM. and CoutantO.1999. Numerical simulation of the propagation of P waves in fractured media.Geophysical Journal International137, 197–206.DOI: 10.1046/j.1365-246X.1999.00784.x
    [Google Scholar]
  23. KneibG. and KernerC.1993. Accurate and efficient seismic modelling in random media.Geophysics58, 576–588.
    [Google Scholar]
  24. Kusnandi, Van BarenG., MulderW., HermanG. and Van AntwerpenV.2000. Sub‐grid finite‐difference modelling of wave propagation and diffusion in cracked media. 70th SEG meeting, Calgary, Canada, Expanded Abstracts, ST P1.2.
  25. LeRavalecM. and GuéguenY.1996. Comment on “The elastic modulus of media containing strongly interacting antiplane cracks” by Paul M. Davis and Leon Knopoff.Journal of Geophysical Research101, 25373–25375.
    [Google Scholar]
  26. LiuE., HudsonJ.A. and PointerT.2000. Equivalent medium representation of fractured rock.Journal of Geophysical Research105, 2981–3000.
    [Google Scholar]
  27. MavkoG., MukerjiT. and DvorkinJ.1998. The Rock Physics Handbook.Cambridge University Press.
    [Google Scholar]
  28. MukerjiT., BerrymanJ., MavkoG. and BergeP.1995. Differential effective medium modelling of rock elastic moduli with critica porosity constraints.Geophysical Research Letters22, 555–558.
    [Google Scholar]
  29. MuraiY., KawaharaJ. and YamashitaT.1995. Multiple scattering of SH waves in 2D elastic media with distributed cracks.Geophysical Journal International122, 925–937.
    [Google Scholar]
  30. NorrisA.N.1985. A differential scheme for the effective moduli of composites.Mechanics of Materials4, 1–16.
    [Google Scholar]
  31. NurA.1971. Effects of stress on velocity anisotropy in rocks with cracks.Journal of Geophysical Research76, 2022–2034.
    [Google Scholar]
  32. NurA.1992. Critical porosity and the seismic velocities in rocks (abstract).EOS Transactions AGU73, 66 .
    [Google Scholar]
  33. O'ConnellR.J. and BudianskyB.1974. Seismic velocities in dry and saturated cracked solids.Journal of Geophysical Research79, 5412–5426.
    [Google Scholar]
  34. O'ConnellR.J. and BudianskyB.1976. Reply.Journal of Geophysical Research81, 2577–2578.
    [Google Scholar]
  35. PeacockS. and HudsonJ.A.1990. Seismic properties of rocks with distributions of small cracks.Geophysical Journal International102, 471–484.
    [Google Scholar]
  36. RobertssonJ.O.A.1996. A numerical free‐surface condition for elastic/viscoelastic finite‐difference modelling in the presence of topography.Geophysics61, 1921–1934.
    [Google Scholar]
  37. RobinsonP.C.1983. Connectivity of fracture systems – a percolation theory approach.Journal of Physics A: Math. Gen.16, 605–614.
    [Google Scholar]
  38. RobinsonP.C.1984. Numerical calculations of critical densities for lines and planes.Journal of Physics A: Math. Gen.17, 2823–2830.
    [Google Scholar]
  39. SaengerE.H., GoldN. and ShapiroS.A.2000. Modelling the propagation of elastic waves using a modified finite‐difference grid.Wave Motion31, 77–92.DOI: 10.1016/S0165-2125(99)00023-2
    [Google Scholar]
  40. SaengerE.H. and ShapiroS.A.2000. Calculation of effective velocities in fractured media using the rotated staggered grid. 62nd EAGE conference, Glasgow, UK, Extended Abstracts, D‐34.
  41. SahimiM.1995. Flow and Transport in Porous Media and Fractured Rock. VCH, Weinheim, Germany.
    [Google Scholar]
  42. ThomsenL.1986. Weak elastic anisotropy.Geophysics51, 1954–1966.
    [Google Scholar]
  43. VirieuxJ.1986. Velocity‐stress finite‐difference method.Geophysics51, 889–901.
    [Google Scholar]
  44. WuT.T.1966. The effect of inclusion shape on the elastic moduli of a two‐phase material.International Journal of Solids and Structures2, 1–8.
    [Google Scholar]
  45. YuanF.G.1998. Lecture on Anisotropic Elasticity. Mars Mission Research Center, Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695.
    [Google Scholar]
  46. ZimmermannR.W.1991. Compressibility of Sandstones. Elsevier Science Publishing Co.
    [Google Scholar]
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