Single-well shear-wave imaging using a dipole source-receiver system is an important application for detecting geological structures away from the borehole. This development allows for determining the azimuth information of the structures. Existing analyses, however, focus on the data received at the borehole axis and use the elastic reciprocity theorem to model the borehole radiation and recording. We extend the existing analyses to model the radiation, reflection, and the recording response of the borehole for azimuthally spaced receivers off the borehole axis. By treating the mirror image of the borehole source with respect to the reflector plane as a virtual source, the borehole reception problem is shown to be equivalent to the response of the borehole to the spherical wave incidence from the virtual source, which can be solved using the cylindrical-wave expansion method. An asymptotic solution using the steepest decent method is obtained if the virtual source is far from the borehole. The analytical solution allows us to analyze the borehole response for azimuthally spaced off-axis receivers. The analysis results agree well with those from 3D finite-difference simulations. With this analysis, one can further model the multi-component shear-wave reflection data from the cross-dipole acoustic tool and study the azimuthal variation characteristics of the data. The results show that, while the data characteristics are dominated by those of a dipole, non-dipole responses due to the off-axis reception can be observed, the magnitude of the responses depending on the off-axis distance and frequency and on the formation elasticity. The non-dipole response characteristics have the potential to resolve the 180°-ambiguity problem in the azimuth determination for the dipole shear-wave imaging. The findings, therefore, provide new information to the shear-wave reflection imaging analysis and development.
Reverberation time and audibility in phased geometrical acoustics using plane or spherical wave reflection coefficients
