Borehole coupling at sonic frequencies

Abstract

An elastic plane wave incident on a borehole couples through the borehole wall to produce a compressional wave inside the borehole, and the effect of this coupling varies with incident angle. Imaging experiments that include a downhole source and/or hydrophones must account for this effect to achieve optimal results. In configurations pertinent to VSP surveys and cross‐well surveys at seismic frequencies, a low‐frequency source can be assumed. Recently, however, both cross‐well and single‐well imaging experiments have been undertaken at sonic frequencies (above 2000 Hz) where low‐frequency approximations are no longer valid. We have developed new formulas which are valid over all frequencies and incident angles. For both shear and compressional incident waves, we present explicit formulas to describe this dependence. Contrary to what might be expected from low‐frequency approximations, we find that at certain angles and for high enough frequencies, marked resonances in the coupling terms for both shear and compressional incidence occur. Moreover, at all frequencies, the borehole coupling term depends significantly upon incident angle. For a given incident angle, and for high enough frequencies, the coupling term becomes periodic in frequency. We give analytic expressions for these high‐frequency resonances and present a simplified high‐frequency approximation that can be computed without using Bessel or Hankel functions. Results are presented for three different formation types; very hard, medium hard, and soft. For all formation types, significant variations in amplitude as well as strong shifts in phase are apparent, as functions of both frequency and incident angle. This distortion of the wave field should be taken into account for any imaging experiment that uses high‐frequency signals.