1887
Volume 66 Number 1
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

Based on knowledge of a commutative group calculation of the rock stiffness and on some geophysical assumptions, the simplest fractured medium may be regarded as a fracture embedded in an isotropic background medium, and the fracture interface can be simulated as a linear slip interface that satisfies non‐welded contact boundary conditions: the kinematic displacements are discontinuous across the interface, whereas the dynamic stresses are continuous across the interface. The finite‐difference method with boundary conditions explicitly imposed is advantageous for modelling wave propagation in fractured discontinuous media that are described by the elastic equation of motion and non‐welded contact boundary conditions. In this paper, finite‐difference schemes for horizontally, vertically, and orthogonally fractured media are derived when the fracture interfaces are aligned with the boundaries of the finite‐difference grid. The new finite‐difference schemes explicitly have an additional part that is different from the conventional second‐order finite‐difference scheme and that directly describes the contributions of the fracture to the wave equation of motion in the fractured medium. The numerical seismograms presented, to first order, show that the new finite‐difference scheme is accurate and stable and agrees well with the results of previously published finite‐difference schemes (the Coates and Schoenberg method). The results of the new finite‐difference schemes show how the amplitude of the reflection produced by the fracture varies with the fracture compliances. Later, comparisons with the reflection coefficients indicate that the reflection coefficients of the fracture are frequency dependent, whereas the reflection coefficients of the impedance contrast interface are frequency independent. In addition, the numerical seismograms show that the reflections of the fractured medium are equal to the reflections of the background medium plus the reflections of the fracture in the elastic fractured medium.

Loading

Article metrics loading...

/content/journals/10.1111/1365-2478.12536
2017-06-16
2024-03-29
Loading full text...

Full text loading...

References

  1. AkiK. and RichardsP.G.1980. Quantitative Seismology: Theory and Methods, Vol. 1, W.H. Freeman and Company.
    [Google Scholar]
  2. AuldB.1973. Acoustic Fields and Waves in Solids. New York: John Wiley & Sons.
    [Google Scholar]
  3. BackusG.E.1962. Long‐wave elastic anisotropy produced by horizontal layering. Journal of Geophysical Research67, 4427–4440.
    [Google Scholar]
  4. BakulinA., GrechkaV. and TsvankinI.2000a. Estimation of fracture parameters from reflection seismic data—Part I: HTI model due to a single fracture set. Geophysics65(6), 1788–1802.
    [Google Scholar]
  5. BakulinA., GrechkaV. and TsvankinI.2000b. Estimation of fracture parameters from reflection seismic data—Part II: fractured models with orthorhombic symmetry. Geophysics65(6), 1803–1817.
    [Google Scholar]
  6. BakulinA.GrechkaV. and TsvankinI.2000c. Estimation of fracture parameters from reflection seismic data—Part III: fractured models with monoclinic symmetry. Geophysics65(6), 1818–1830.
    [Google Scholar]
  7. CarcioneJ.M., PicottiS., CavalliniF. and SantosJ.E.2012. Numerical test of the Schoenberg–Muir theory. Geophysics77, C27–C35.
    [Google Scholar]
  8. ChaisriS. and KrebesE.S.2000. Exact and approximate formulas for P‐SV reflection and transmission coefficients for a nonwelded contact interface. Journal of Geophysical Research105, 28045–28054.
    [Google Scholar]
  9. ChengJ. and KangW.2013. Simulating propagation of separated wave modes in general anisotropic media. Part I: qP‐wave propagators. Geophysics79(1), C1–C18.
    [Google Scholar]
  10. CoatesR.T. and SchoenbergM.1995. Finite‐difference modeling of faults and Fractures. Geophysics60, 1514–1526.
    [Google Scholar]
  11. CuiX., LinesL.R. and KrebesE.S.2013a. Numerical modeling fractures and fractal network feature‐wormhole of CHOPS. SEG Houston 2013 annual meeting.
  12. CuiX., LinesL.R. and KrebesE.S.2013b. Numerical modeling for different types of fractures. GeoConvention 2013: integration.
  13. HoodJ.1991. A simple method for decomposing fracture‐induced anisotropy. Geophysics56, 1275–1279.
    [Google Scholar]
  14. HsuC.‐J. and Schoenberg. M.1993. Elastic waves through a simulated fracture medium. Geophysics58, 964–977.
    [Google Scholar]
  15. KellyK.R., WardR.W., TreitelS. and AlfordR.M.1976. Synthetic seismograms finite‐difference approach. Geophysics41(1), 2–27.
    [Google Scholar]
  16. KornM. and StocklH.1982. Reflection and transmission of Love Channel wave at coal seam discontinuities computed with a finite difference method. Journal of Geophysics50, 171–176.
    [Google Scholar]
  17. KrügerL., OliverS., SaengerE.H., OatesS.J. and ShapiroS.A.2007. A numerical study on reflection coefficients of fractured media, Geophysics72 (4), D61–D67.
    [Google Scholar]
  18. LinesL.R., SlawinskiR. and BordingR.P.1999. Short note—A recipe for stability of finite‐difference wave‐equation computations. Geophysics64(3), 967–969.
    [Google Scholar]
  19. LinesL.R. and NewrickR.T.2004. Fundamentals of Geophysical Interpretation. Tulsa, OK: Society of Exploration Geophysicists Publication.
    [Google Scholar]
  20. ManningP.M.2008. Techniques to enhance the accuracy and efficiency of finite difference modelling for the propagation of elastic waves. PhD thesis, University of Calgary, Canada.
  21. MusgraveM.J.P.1970. Crystal Acoustics. Holden‐Day.
  22. NicholsD., MuirF. and SchoenbergM.1989. Elastic properties of rocks with multiple fracture sets. 59th Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, 471–474.
  23. Popovici, A.M., SturzuI. and MoserT.J.2015. High resolution diffraction imaging of small scale fractures in shale and carbonate reservoirs: 14th international congress of the Brazilian Geophysical Society, 782–787. Read more: http://library.seg.org/doi/full/10.1190tle35010086.1.
  24. Pyrak‐NolteL.J., MyerL.R. and CookN.G.W.1990. Anisotropy in seismic velocities and amplitudes from multiple parallel fractures. Journal of Geophysical Research95(B7), 11345–11358.
    [Google Scholar]
  25. SaengerE.H. and ShapiroS.A.2002. Effective velocities in fractured media: a numerical study using the rotated staggered finite‐difference grid. Geophysical Prospecting50, 183–194.
    [Google Scholar]
  26. SayersC.M., TaleghaniA.D. and AdachiJ.2009. The effect of mineralization on the ratio of normal to tangential compliance of fractures. Geophysical Prospecting57(3), 439–446.
    [Google Scholar]
  27. SchoenbergM.1980. Elastic wave behavior across linear slip interfaces. Journal of the Acoustical Society of America68, 1516–1521.
    [Google Scholar]
  28. SchoenbergM. and DoumaJ.1988. Elastic wave propagation in media with parallel fractures and aligned cracks. Geophysical Prospecting36, 571–590.
    [Google Scholar]
  29. SchoenbergM. and MuirF.1989. A calculus for finely layered anisotropic media. Geophysics54, 581–589.
    [Google Scholar]
  30. SchoenbergM. and ProtázioJ.1992. ‘Zoeppritz’ rationalized and generalized to anisotropy. Journal of Seismic Exploration1, 125–144.
    [Google Scholar]
  31. SilvestrovI., RedaB. and EvgenyL.2016. Poststack diffraction imaging using reverse‐time migration. Geophysical Prospecting64, 129–142.
    [Google Scholar]
  32. SlawinskiR.A. and KrebesE.S.2002a. Finite‐difference modeling of SH‐wave propagation in nonwelded contact media. Geophysics67, 1656–1663.
    [Google Scholar]
  33. SlawinskiR.A. and KrebesE.S.2002b. The homogeneous finite difference formulation of the P‐SV wave equation of motion. Studia Geophysica et Geodaetica46, 731–751.
    [Google Scholar]
  34. SlawinskiR.A.1999. Finite‐difference modeling of seismic wave propagation in fractured media. PhD thesis, University of Calgary, Canada.
  35. ThomsenL.1986. Weak elastic anisotropy. Geophysics51(10), 1954–1966.
    [Google Scholar]
  36. TsvankinI.1997. Anisotropic parameters and P‐wave velocity for orthorhombic media. Geophysics62, 1292–1309.
    [Google Scholar]
  37. VirieuxJ.1986. P‐SV wave propagation in heterogeneous media: velocity–stress finite‐difference method. Geophysics51(4), 889–901.
    [Google Scholar]
  38. VlastosS., LiuE., MainI.G. and LiX.‐Y.2003. Numerical simulation of wave propagation in media with discrete distributions of fractures: effects of fracture sizes and spatial distributions. Geophysical Journal International152, 649–668.
    [Google Scholar]
  39. ZhangJ.2005. Elastic wave modeling in fractured media with an explicit approachGeophysics70, T75–T85.
    [Google Scholar]
  40. ZhangJ. and GaoH.2009. Elastic wave modelling in 3‐D fractured media: an explicit approach. Geophysical Journal International177, 1233–1241.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1111/1365-2478.12536
Loading
/content/journals/10.1111/1365-2478.12536
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): Finite‐difference modelling; Fracture; Mathematical formulation

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