Dispersion curves of acoustoelastic borehole waves: the perturbation method with the correct formulation of the stresses around the borehole

Abstract

SUMMARY The acoustoelastic model has been widely used to investigate the influence of formation stresses on the dispersion curves of borehole waves. The analytical perturbation method (PM), the finite-difference time-domain (FDTD) and the semi-analytical finite element (SAFE) are three common-used methods to calculate the dispersion curves. However, due to different interpretations of the PM and plane strain assumptions, the obtained dispersion curves are incompatible among existing PMs, which may misguide the interpretation of formation stresses. It is therefore necessary to untangle the applicability and limitations of PM. Considering that the conventional PMs are usually inaccurate at the low frequency or inconsistent with Hamilton’s principle, we develop a revised PM to obtain the dispersion curves of borehole waves propagating along a borehole surrounded by the triaxially stressed formation assumed as a monoclinic medium. The revised PM is more accurate, reasonable and logical than existing PMs. When the formation is subjected to low stresses, our finding is of great benefit for quickly computing dispersion curves, since the revised PM is much more efficient than the FDTD method; and there are small discrepancies between the flexural dispersions obtained by the revised PM and those obtained by the FDTD method. Nevertheless, the revised PM has two limitations. The first limitation is that the revised PM cannot be used to compute the Stoneley dispersion curves, which have been validated by comparison with SAFE and FDTD methods. The second limitation is that flexural dispersion curves show significant discrepancies in the high-frequency domain when the low-stress assumption does not hold, as compared to those obtained by the FDTD method.