Effects Of Anisotropy Upon The Normal Modes

Abstract

The effects of anisotropy upon elastic wave propagation along a fluid-filled cylindrical borehole are determined. The wave equation is solved in the frequency-wavenumber domain with a variational method, and the solution yields the phase velocities, group velocities, pressures, and displacements for the normal modes. These properties are studied for two cases: a transversely isotropic model for which the borehole has several different orientations with respect to the symmetry axis and an orthorhombic model for which the borehole is parallel to the intersection of two symmetry planes. The normal modes for these two cases show several significant effects which do not exist when the solid is isotropic or transversely isotropic with its symmetry axis parallel to the borehole: 1. The phase velocities for the quasi-pseudo-Rayleigh, both quasi-flexural, and both quasi-screw waves do not exceed the phase velocity of the slowest qS-wave. (The phase velocities of the leaky modes, which were not investigated, will exceed this threshold. ) 2. The two quasi-flexural waves have different phase and group velocities; the differences are greatest at low frequencies and diminish as the frequency increases. The two quasi-screw waves behave similarly. 3. The greater the difference between the phase velocities of the qS-waves, the greater the difference between the phase velocities of the quasi-flexural waves at all frequencies. The two quasi-screw waves behave similarly. 4. Near the limiting qS-wave velocity, the difference between the phase velocities of the two quasi-flexural waves is greater than that for the two quasi-screw waves. 48 Ellefsen et al. 5. For the slow quasi-flexural wave, the particle displacements in the plane perpendicular to the borehole, when viewed together, are aligned with the polarization of the slow qS-wave. 6. For the fast quasi-flexural wave, the particle displacements in the plane perpendicular to the borehole, when viewed together, are aligned with the polarization of the fast qS-wave. 7. For the slow quasi-screw wave, the particle displacements in the plane perpendicular to the borehole, when viewed together, are aligned along two mutually perpendicular directions which are rotated 45° with respect to the polarizations of both qS-waves. 8. For the fast quasi-screw wave, the particle displacements in the plane perpendicular to the borehole, when viewed together, are aligned along two mutually perpendicular directions which are parallel with the polarizations of both qSwaves.

(In this list, the qS-waves are those plane waves whose wavenumber vectors are parallel to the borehole.) Despite these significant effects, the general characteristics of the phase and group velocities, pressures, and displacements are similar (but not identical) to those that would exist if the solid were isotropic or transversely isotropic with its symmetry axis parallel to the borehole. This result is expected because the models are only slightly anisotropic.