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Linear and Nonlinear Elastic Wave Propagation in a Fluid-Filled Boreholeby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on April 27, 1993 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
This thesis is concerned with the propagation of waves in a fluid-filled borehole
and their interactions with a surrounding elastic formation. In particular,
we focus on the lowest order borehole mode, the Stoneley wave. The three problems
of interest here are: radiation from the borehole into an anisotropic formation;
the interaction of Stoneley waves with fluid-filled fractures; and the effects
of formation nonlinearities on the propagation of Stoneley waves.
In the first part of the thesis we develop a formalism which represents
the borehole in terms of effective sources for low-frequency radiation into
an anisotropic, slowly varying medium. The method consists of introducing
the ratio of borehole radius to wavelength e as a small parameter
and then obtaining an asymptotic solution in ascending powers of e.
In this way we obtain a sequence of problems which are solvable in a closed
form. The first problem is a two-dimensional static elastic problem for
the inflation of a borehole in an arbitrary anisotropic solid. The next
problem involves a one-dimensional hyperbolic system of equations for pressure
and longitudinal particle velocity in the fluid. The coefficients in this
system involve the solution to the first problem. The final step is to find
a source of seismic waves in the solid, which is equivalent to the traveling
tube wave (low-frequency limit of the Stoneley wave). That is, we replace
the solid and borehole by an intact solid and construct a moving system
of body-force dipoles concentrated along the location of the centerline
of the borehole, which generates the same seismic radiation as the propagating
tube wave in the actual borehole. By combining the body force distribution
with appropriate Green's functions we obtain the far-field radiation pattern
from a source in a borehole. Results are obtained for situations where the
tube wave is either faster or slower than the quasi-shear waves in the solid.
In the former case we identify the existence of (possibly two) Mach waves
and provide explicit solutions for the far-field radiation. The cross-well
geometry, where a source is placed in one borehole and receivers in another,
is also analysed and a solution is obtained in a form which clearly exhibits
reciprocity.
In the second part we develop analytical and finite-difference models
for studying wave propagation in boreholes surrounded by inhomogeneous elastic
media. The presence of a fluid-filled fracture intersecting the borehole
is modelled explicitly and its effects on Stoneley waves are studied. For
instance, we find that elasticity of the formation tends to increase the
reflection coefficient of Stoneley waves. On the other hand, multiple fractures
lead to an interference phenomenon which results in a decrease of the reflection
coefficient when compared to a single fracture with the same total aperture.
The effects of a washout in the presence of a fracture are shown to be negligible
at low frequencies but it strongly dominates the reflectivity at higher
frequencies. The analytical models we have developed can be used to interpret
Stoneley wave reflection data, where the effective fracture aperture can
be quantified. The first model considers the effects of borehole enlargements
(e.g., washouts) on the reflection coefficient of Stoneley waves.
By using low-frequency arguments we obtain an expression which involves
the washout volume, which can be obtained from a caliper log. Comparisons
of this model and the finite-difference solutions previously obtained are
in good agreement. Next we develop an elastic model which generalizes the
rigid formation model and correctly predicts the effective fracture aperture.
We also establish the equivalence between multiple fractures and a permeable
medium.
In the third part we study the effects of formation nonlinearities on
the propagation of Stoneley waves. We motivate the study by recognizing
that rocks are much more nonlinear than homogeneous materials. We then introduce
a formalism for studying small amplitude wave propagation in prestressed
media, and develop a perturbation model that allows the computation of changes
in Stoneley wave phase velocity as a function of borehole pressure. The
results obtained indicate that the effects are measurable and that pressurizing
the borehole is an effective method of measuring the in-situ nonlinear properties
of rocks.
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