A numerical investigation of head waves and leaky modes in fluid- filled boreholes

Although synthetic borehole seismograms can be computed for a wide range of borehole conditions, the physical nature of shear and compressional head waves in fluid-filled boreholes is poorly understood. This paper presents a series of numerical experiments designed to explain the physical mechanisms controlling head-wave propagation in boreholes. These calculations demonstrate the existence of compressional normal modes equivalent to shear normal modes, or pseudo-Rayleigh waves, with sequential cutoff frequencies spaced between the cutoff frequencies for the shear normal modes. Major contributions to head-wave spectra occur in discrete peaks at frequencies just below mode cutoff for both compressional and shear modes. This result is confirmed by calculations with synthetic waveforms at frequencies corresponding to mode cutoff, and by branch-cut integrals designed to yield independent spectra for the compressional mode. For soft formations where shear velocity falls below acoustic velocity in the borehole fluid, leaky compressional normal modes attain properties similar to those of shear normal modes in the case of hard rock. In the limit of vanishing S-wave velocity, this result is formally related to a fluid-fluid waveguide with undamped compressional normal modes. Synthetic waveforms demonstrate that high-amplitude arrivals, traveling at velocities less than the acoustic velocity of the borehole fluid and at frequencies above a few kilohertz, represent the Airy phase of the compressional mode and are not a tube wave. Comparison of synthetic waveforms with waveforms obtained in soft sea sediments indicates that the predicted Airy phase arrivals are present in the experimental data.