Multipole borehole acoustic waveforms: Synthetic logs with beds and borehole washouts

Abstract

We give a two‐dimensional (2-D) velocity‐stress finite‐difference (vs-fd) formulation for the multipole borehole acoustic logging problem. Irregular boreholes, horizontal bedding, and axially varying alteration are encompassed by the model. Excellent agreement is obtained with transform techniques for constant radius boreholes and homogeneous formations. The method is used to generate waveforms for many positions of transmitter and receiver array relative to beds and washouts as in logging. First motion and semblance techniques are applied to the waveforms to extract formation slownesses, resulting in synthetic multipole logs which demonstrate the response of ideal multipole logging tools and signal processing to several environmental effects. For monopole, compressional, first‐motion logs, residual slowness errors remain at borehole washouts after borehole compensation. These errors increase as the effective measure point in the waveforms is moved back in time from true first motion, admitting greater interference from reflected and mode‐converted waves. Errors at bed boundaries are typically smaller than those at washouts. Stoneley slowness logs are obtained by narrow‐band filtering of low frequency monopole waveforms and application of the semblance slowness extraction algorithm STC. Similar processing applied to low frequency dipole waveforms yields flexural mode slowness logs. A small correction satisfactorily accounts for flexural mode dispersion properties, yielding formation shear slowness. Slowness errors for both Stoneley and dipole shear logs at washouts and bed boundaries are quite small, typically on the order of a few percent. Borehole compensation is of marginal benefit for both of these logs since they are based on modes of the borehole rather than head‐waves.