Blackwell Publishing Journal Backfiles 1879-2005
Previous studies of radiation from point sources in fluid-filled boreholes have most often been based on far-field, stationary phase analysis. In these papers, the explicit contribution of the borehole itself acting as a waveguide has not been properly considered, with a few exceptions. In general, these studies accurately describe S-wave radiation in high-velocity rocks such as granites and limestones and P-wave radiation in most rocks, and experiments have confirmed this. However, tube waves directly influence the external wavefield and in fact create a shear-wave ‘wake’ outside the borehole due to constructive interference of tube-wave emission if a velocity condition is met. This constructive interference or wake is generated when the tube-wave velocity is greater than the shear-wave velocity. When this happens, a tube-wave complex pole invalidates the mathematical assumptions for stationary phase analysis and the stationary phase predictions do not agree with experimentally derived radiation patterns. Shales at shallow depths and other soft sediments characteristically have tube-wave velocities greater than shear-wave velocities. Because the tube-wave is of relatively high amplitude compared to body waves generated directly by the source, these secondary shear waves can be the highest amplitude arrivals on receiver arrays.The shape and properties of these secondary shear waves are calculated and shown to have identical properties to Mach waves of aerodynamics and seismology. For instance, these waves are geometrically conical and the aperture of the cone and the moveout velocity can be calculated. This paper also demonstrates the important effect that casing has on the Mach waves and provides predictions about when these waves are likely to be observed. Finally, evidence of Mach waves in data sets is examined and it is shown how these waves have been confused with receiver borehole tube waves.It is possible, though rare, that the tube-wave velocity of the borehole is greater than the compressional-wave velocity of the surrounding medium. In this case secondary compressional or compressional Mach waves would be generated although this problem is not addressed here.
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