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Elastic Wave Propagation Along a Borehole in Anisotropic Mediumby 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:
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. In general, 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.
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 fast 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 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%.
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