A finite-difference time domain (FDTD) formulation with a 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 non-dispersive and dispersive arrivals in the wavetrain. The tube wave velocity obtained from the zero frequency intercept of the Stoneley dispersion compares very well with the analytical results for a range of well deviations in both fast and slow transversely-isotropic (TI) formations. Good agreement is also obtained between low frequency asymptotes of borehole flexural dispersions and the corresponding shear wave velocities from a numerically exact solution of Kelvin-Christoffel equations for plane wave velocities in an arbitrarily anisotropic formation. Numerical results indicate that the Stoneley dispersion changes by a rather small amount, whereas dipole flexural dispersions exhibit larger changes with wellbore deviations. Even though shear wave splitting occurs in deviated wellbores, shear slowness anisotropy is less than 2% for well deviations less than 40 degrees from the TI-symmetry axis in certain types of shales, such as Austin chalk. Anisotropy induced coupling between the monopole Stoneley and dipole flexural modes is insignificant in deviated wellbores in the Austin chalk as well as in fast TIformations. The influence of an equivalent heavy-fluid column structure on borehole elastic waves is 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 all 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 non-intersecting in fast formations even in deviated wellbores and in the presence of an equivalent heavy-fluid column structure.
Most of the geophysical formations exhibit anisotropy that is characterized by transversely isotropic (TI) symmetry. Deviated drilling through the overburden shale is often required to access horizontal wells in a reservoir. It is important to distinguish between intrinsic (or structural) and stress-induced shear slowness anisotropy. Unlike intrinsic shear slowness anisotropy, large stress induced shear anisotropy in the borehole cross-sectional plane can be an indicator of the potential instability for the chosen well deviation. Crossing dipole dispersions are indicators of stress-induced shear slowness anisotropy (Sinha and Kostek, 1996). Most of the sedimentary rocks exhibit some degree of anisotropy. A horizontally layered structure exhibits transversely isotropic (TI) anisotropy with a vertical (X3-) axis of symmetry. A TI-anisotropy is also referred to as polar anisotropy with 5 independent anisotropic moduli. The well deviation is defined by rotating the TI-anisotropy axes by angle q about the X1-axis in the isotropic plane. The rotated anisotropic constants referred to the borehole axes are characterized by monoclinic symmetry with 9 anisotropic moduli.;
