A finite-difference time-domain (FDTD) formulation with perfectly matched layer (PML) enables analysis of elastic-wave propagation in a fluid-filled borehole in an arbitrarily anisotropic formation. The FDTD formulation yields synthetic waveforms at an array of receivers produced by a monopole or dipole source placed on the borehole axis. Synthetic waveforms are then processed by a modified matrix pencil algorithm to isolate both nondispersive and dispersive arrivals in the wavetrain. The processing algorithm used in this study extracts phase slownesses of plane waves that agree well with the corresponding phase slownesses calculated for propagation along an arbitrary direction in an anisotropic formation. The tube-wave phase velocity obtained from the zero-frequency intercept of the Stoneley dispersion compares well with the analytical results for deviated wellbores in both fast and slow transversely isotropic (TI) formations. Good agreement is also obtained between the low-frequencyasymptotes of borehole flexural dispersion and the corresponding shear-wave velocities from a numerically exact solution of Kelvin-Christoffel equations for plane-wave velocities in anisotropic formations. Numerical results indicate that the Stoneley dispersion changes by a rather small amount, whereas the dipole flexural dispersions exhibit larger changes with wellbore deviations. The influence of a sonic tool structure on borehole elastic-waves can be described by an equivalent heavy-fluid column placed concentrically with the borehole axis. The effect of a heavy-fluid column on the borehole flexural mode is larger in fast than in slow formations. However, the Stoneley dispersion at low frequencies is affected by the presence of the tool structure in both the fast and slow formations. The present study confirms that the two orthogonal dipole flexural dispersions are nearly parallel to each other in slow formations and nonintersecting in fast formations, even in deviated wellbores and in the presence of a sonic tool structure described by a heavy-fluid column.
The elastic wave propagation in two-dimensional phononic crystal at low frequencies and the anisotropy of effective velocity
