Calculation Of Conical (Mach) Wave Displacement Fields Radiated By Borehole Sources In Slow Formations And Inhomogeneous Media
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1993.7 Gibson.pdf
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Author(s)
Gibson, Richard L., Jr.
Date Issued
1993
Publisher
Massachusetts Institute of Technology. Earth Resources Laboratory
Series/Report no.
Earth Resources Laboratory Industry Consortia Annual Report;1993-07
Abstract
Stationary phase solutions for the radiation patterns of borehole sources are commonly
used to study the far-field seismic wavefields produced in crosshole or reverse VSP
experiments, but they break down when the formation shear wave velocity is less than
the tube wave velocity in the source borehole. This is because the tube wave, not the
primary source, radiates the dominant shear wave signal in the form of large amplitude
conical waves, which are also called Mach waves. I model this effect by considering the
tube wave to be a moving secondary point source generated by the primary source of
acoustic energy. A discretization of the source well allows a numerical solution of the
integral equation which yields the displacement field by a general source distributed in
space and time. The time at which each point source in the discretization emits energy is
determined by the group velocity of the tube wave, while the radiation of the individual
sources is characterized by the stress field induced by the tube wave at the borehole wall. An integration along the borehole of these point sources then yields the observed Mach wave arrivals. Since this method involves the summation of shear wave ray arrivals from the many point sources along the borehole, the method is called the Ray Summation
Method (RSM). Comparison of RSM results with full waveform synthetic seismograms
computed with the discrete wavenumber method confirms the accuracy of this method.
Unlike the discrete wavenumber method, however, the use of ray tracing in the RSM
allows computation of the Mach wave arrivals for inhomogeneous layered media as well
as homogeneous models, including the waves generated by reflections of the Mach waves
at interfaces and from the reflections of the tube wave itself. The interactions of the
conical waves with interfaces can show unusual patterns of arrivals which would not be
predicted from ordinary point source behavior.
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