1887
Volume 61 Number 1
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

The dynamic response of a semi‐infinite fluid‐filled borehole embedded in an elastic half‐space under a concentrated normal surface load is analysed in the long‐wavelength limit. The solution of the problem is obtained with integral transforms in the form of a double integral with respect to the slowness and frequency. The partial P‐ and SVwave responses are further transformed to path integrals along Cagniard paths in the complex slowness plane. Unlike the traditional Cagniard‐de Hoop technique based on the Laplace transform of time dependence, this paper is based on the Fourier transform. The tube‐wave response is presented as a causal integral over a slowness range. The resultant representation in the time‐domain is suitable for the numerical evaluation of the complete response in the fluid‐filled borehole, especially at large distances.

Asymptotic analysis of seismic phases arising in the borehole is performed on the basis of the obtained solution. The complete asymptotic wavefield consists in P and SVwaves, the Rayleigh wave and the low‐frequency Stoneley (tube) wave. Pressure synthetics obtained by the use of the asymptotic formulas are shown to be in good agreement with straightforward calculations.

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References

  1. AbramowitzM. and StegunI.1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series‐55. ISBN 0486612724.
  2. AchenbachJ.D.1973. Wave propagation in elastic solids . North‐Holland Publishing Co., Amsterdam & London . ISBN 0720403251.
    [Google Scholar]
  3. AkiK. and RichardsP.G.2002. Quantitative Seismology , 2nd edition. University Science Books. CA . ISBN 0935702962.
    [Google Scholar]
  4. BokovP.M. and IonovA.M.2002. Tube‐wave propagation in a fluid‐filled borehole generated by a single point force applied to the surrounding formation. Journal of the Acoustical Society of America 112, 2634–2644.
    [Google Scholar]
  5. BouchonM. and SchmittD.P.1989. Full‐wave acoustic logging in an irregular borehole. Geophysics 54, 758–765.
    [Google Scholar]
  6. BurridgeR. and KostekS.1993. Tube waves, seismic waves and effective sources. Wave Motion 18, 163–210.
    [Google Scholar]
  7. CagniardL.1962. Reflection and refraction of progressive seismic waves . Translation by E.A.Flinn and C.H.Dix . McGraw‐Hill Co., NY .
    [Google Scholar]
  8. ChengC.H., ToksözM.N. and WillisM.E.1982. Determination of in situ attenuation from full waveform acoustic logs. Journal of Geophysical Research 87, 5477–5484.
    [Google Scholar]
  9. ChengC.H., ZhangJ. and BurnsD.R.1987. Effects of in‐situ permeability of Stoneley (tube) waves in a borehole. Geophysics 52, 1279–1289.
    [Google Scholar]
  10. De HoopA.T.1960. A modification of Cagniard's method for solving seismic pulse problems. Applied Science Research B8, 349–356.
    [Google Scholar]
  11. EwingW.M., JardetzkyW.S. and PressF.1957. Elastic waves in layered media . McGraw‐Hill Co., NY . ISBN 0070198608.
    [Google Scholar]
  12. FalkJ., TessmerE. and GajewskiD.1996. Tube wave modeling by the finite‐difference method with varying grid spacing. Pure and Applied Geophysics 148, 77–93.
    [Google Scholar]
  13. GrayA. and MathewsG.B.1931. Bessel functions and their applications to physics , 2nd edition. London . ISBN 1603860452.
    [Google Scholar]
  14. HornbyB.E., JohnsonD.L., WinklerK.W. and PlumbR.A.1989. Fracture evaluation using reflected Stoneley‐wave arrivals. Geophysics 54, 1274–1288.
    [Google Scholar]
  15. IonovA.M. and MaximovG.A.1996. Propagation of tube waves generated by an external source in layered permeable rocks. Geophysical Journal International 124, 888–906.
    [Google Scholar]
  16. IonovA.M. and MaximovG.A.1999. Excitation of a tube wave in a borehole by an external seismic source. Acoustical Physics 45, 311–320.
    [Google Scholar]
  17. KrauklisP.V., MolotkovL.A. and KrauklisA.P.2007. The tube wave generated by a point source located outside a borehole. Journal of Mathematical Sciences 142, 2576–2588.
    [Google Scholar]
  18. KurkjianA.L. and ChangS.K.1986. Acoustic multipole sources in fluid‐filled boreholes. Geophysics 51, 148–163.
    [Google Scholar]
  19. KurkjianA.L., CoatesR.T., WhiteJ.E. and SchmidtH.1994. Finite‐difference and frequency‐wavenumber modeling of seismic monopole sources and receivers in fluid‐filled boreholes. Geophysics 59, 1053–1064.
    [Google Scholar]
  20. LeeM.W.1986. Low‐frequency radiation from point sources in a fluid‐filled borehole. Geophysics 51, 1801–1807.
    [Google Scholar]
  21. LeeM.W.1987. Particle displacement on the wall of a borehole from incident plane wave. Geophysics 52, 1290–1296.
    [Google Scholar]
  22. LeeM.W. and BalchA.H.1982. Theoretical seismic wave radiation from a fluid‐filled borehole. Geophysics 47, 1308–1314.
    [Google Scholar]
  23. LiY.D., RabbelW. and WangR.1994. Investigation of permeable fracture zones by tube wave analysis. Geophysical Journal International 116, 739–753.
    [Google Scholar]
  24. MathewsJ. and WalkerR.L.1970. Mathematical Methods of Physics , 2nd edition. Addison‐Wesley Publishing Co, Inc, New York‐Amsterdam . ISBN 0805370021.
    [Google Scholar]
  25. MelroseR.B.1995. Geometric scattering theory. Stanford Lectures: Distinguished Visiting Lecturers in Mathematics . Cambridge University Press. ISBN 0521498104.
    [Google Scholar]
  26. MeredithJ.A., ToksözM.N. and ChengC.H.1993. Secondary shear waves from source boreholes. Geophysical Prospecting 41, 287–312.
    [Google Scholar]
  27. MiklowitzJ.1978. The Theory of Elastic Waves and Waveguides . North‐Holland Publishing Co., Amsterdam . ISBN 0720405513.
    [Google Scholar]
  28. PekerisC.L.1955. The seismic surface pulse. Proceedings of the National Academy of Sciences41, 469–480.
  29. PengC., ChengC.H. and ToksözM.N.1993. Borehole effects on downhole seismic measurements. Geophysical Prospecting 41, 883–912.
    [Google Scholar]
  30. PengC., ChengC.H. and ToksözM.N.1994.Cased borehole effects on downhole seismic measurements. Geophysical Prospecting 42, 777–811.
    [Google Scholar]
  31. PengC., LeeJ.M. and ToksözM.N.1996.Pressure in a fluid‐filled borehole caused by a seismic source in stratified media. Geophysics 61, 43–55.
    [Google Scholar]
  32. PoularikasA.D.
    (ed.) 2000. The Transforms and Applications Handbook , 2nd edition. CRC Press. ISBN 0849385954.
    [Google Scholar]
  33. SchoenbergM.1986. Fluid and solid motion in the neighborhood of a fluid‐filled borehole due to the passage of a low‐frequency elastic plane wave. Geophysics 51, 1191–1205.
    [Google Scholar]
  34. SeriffA.J. and SriramK.P.1991. P‐SV reflection moveouts for transversely isotropic media with a vertical symmetry axis. Geophysics 56, 1271–1274.
    [Google Scholar]
  35. TadeuA.J.B. and SantosP.F.A.2001. 3‐D wave propagation in fluid‐filled irregular boreholes in elastic formations. Soil dynamics and earthquake engineering 21, 499–517
    [Google Scholar]
  36. TangX.M. and ChengC.H.1993. Borehole Stoneley wave propagation across permeable structures. Geophysical Prospecting 41, 165–187.
    [Google Scholar]
  37. WatsonG.A.1944. Treatise on the Theory of Bessel Functions , 2nd edition. Cambridge University Press. ISBN 0521093821.
    [Google Scholar]
  38. WhiteJ.E.1983. Underground Sound: Application of Seismic Waves . Elsevier Science Publishing Co. ISBN 0444421394.
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): Borehole seismics; Stoneley wave; Wave propagation

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