Earth Resources Laboratory :: Abstracts

Elastic Wave Propagation Along a Borehole in Anisotropic Medium

Karl J. Ellefsen

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 1, 1990 in partial fulfillment of the requirements for the degree of Doctor of Science

Abstract

In the first part of this thesis several applications of perturbation theory are developed to study normal mode propagation along a borehole. This theory is used to relate first order perturbations in frequency, wave number, elastic moduli, densities and location of interfaces. Although the perturbation equation is derived for a general model with many fluid and solid layers which have cross-sectional shape, the equation is applied to a two-layer model consisting of a fluid-filled borehole through a transversely isotropic solid (with its symmetry axis parallel to the borehole). Because analytical expressions for the displacements exist for this particular model, the terms in the perturbation equation simplify greatly. Formulas are derived to calculate 1) phase velocities for a model with slight, general anisotropy, 2) partial derivatives of either the wavenumber or frequency with respect to either an elastic modulus or density, 3) group velocity, and 4) phase velocities for a model with a slightly irregular borehole. These formulas are applicable also to models with an isotropic solid because it is a special case of transversely isotropic solid.

In the second part, the effects of anisotropy upon elastic wave propagation 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. (The phase and group velocities obtained with the perturbation method indicating that both are correctly formulated and implemented.) 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 a orthorhombic model for which the borehole is parallel to the intersection of two symmetry planes. The normal modes for these two cases show several effects which do not exist when the solid is isotropic or transversely isotropic with its symmetry axis parallel to the borehole:

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.)
The two quasi-flexural waves have different phase and group velocities; the differences are greatest at low frequencies and diminish as the frequency increases. In general, the two quasi-screw waves behave similarly.
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.
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.
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 fast qS-wave.
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.
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.
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 qS-waves.

(In this list, the qS-waves are those plane 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.

In the last part, a method to estimate c66 which is a shear modulus of a transversely isotropic formation (with its symmetry axis parallel to the borehole), is developed and tested. The inversion for c66 is based upon a cost function which has three terms: a measure of misfit between the observed and predicted wavenumbers, a measure of the misfit between the current estimate for c66 and the initial guess of its value, and penalty functions which constrain the estimate c66 to physically acceptable values. The inversion is applied to synthetic data for fast and slow formations, and the estimates for c66 are within 5% of their correct values and are moderately well resolved. When the inversion was applied to field data, the estimates for c66 were significantly higher than the values for c44 in a zone with low permeability and high clay content. The percentage of S-wave anisotropy ranged from 5 to 20%.