Shear slowness measurement by dispersive processing of the borehole flexural mode

In sonic well logging, the flexural mode generated by a dipole transducer allows measurement of shear slowness in slow formations, that is, formations in which the shear slowness is slower than the borehole fluid speed. The flexural mode is a dispersive borehole mode, but many current processing methods do not account for this dispersion. The approximate semblance equation is derived to explain what happens when a dispersive wave is processed nondispersively. The processing described here is a dispersive analog of the slowness-time coherence (STC) processing used commonly for array sonic waveforms. It back-propagates waveforms according to model dispersion curves and calculates semblance. It is a specialized version of the maximum likelihood (ML) and least-mean-squared-error (LMSE) estimator for formation shear slowness. The dispersive analog of STC is too slow for depth-by-depth logging but may be a helpful analytic tool. A high-speed variant, dispersive STC (DSTC), eliminates this difficulty by a new window-positioning technique and can be faster than STC. DSTC results based on data from a real-axis integration model show that the technique measures the formation shear slowness with small bias because of windowing and other arrivals. Results are presented from two wells logged with a dipole sonic tool. A log from a fast formation shows that shear slownesses measured from the flexural mode by DSTC agree with monopole results. A log from a slow formation is also presented.