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Elastic Wave Propagation and Scattering in Anisotropic Fractured Mediaby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on May 16, 1991 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
In this thesis, we develop several methods for the delineation and analysis
of anisotropic fractures in the subsurface. Our principle goal is to develop
methods to use seismic data to infer fracture properties. In turn, we relate
this information to the fluid flow properties of anisotropically fractured
media.
The first problem is to examine media with cracks oriented over a range of
directions rather than being perfectly aligned. We use this model of crack
distributions to examine both velocity and permeability variations in an anisotropically
fractured rock. The analysis begins by considering a material under a uniaxial
stress, as cracks in a rock mass subjected to a uniaxial stress will be preferentially
closed depending on the angle between the fracture normal vectors and the
direction of the applied stress. If the prestress fracture orientation distribution
is isotropic, the effective elastic properties of such a material after application
of the stress are then transversely isotropic due to the overall alignment
of the cracks still open. Velocity measurements in multiple directions are
used to invert for the probability density function describing orientations
of crack normals in such a rock. The information on fracture distribution
obtained from the velocity inversion allows an estimation of the anisotropic
permeability of the fractured rock system. Permeability estimates are based
on the number of cracks open in each direction. This approach yields a prediction
of permeability as a function of the angle from the uniaxial stress axis.
The inversion for crack orientation is applied to ultrasonic velocity measurements
on Barre granite, and permeability predictions for this sample are presented.
The inversion results are good and reproduce velocity measurements well, and
the permeability predictions show some of the expected trends. Initial comparisons
of the predictions with available permeability data, however, show deviations
suggesting that further information on partial crack closure and connectivity
of cracks should be included into the permeability model.
After considering the more general range of crack orientations considered
in the inversion procedure, we analyze the behavior of elastic waves on encountering
a fractured region which is too small for velocity variations to become apparent,
and therefore too small for observations of shear wave splitting. This new
problem is solved through the study of the scattering of elastic waves from
isolated fracture zones. When the scattering zone is much smaller than the
wavelength of an incident place background medium, the Born application allows
an estimate of the radiation pattern of elastic wave Rayleigh scattering of
both compressional and shear waves due to a perturbation of any combination
of the 21 independent elastic constants. Examination of radiation patterns
for incident shear and compressional waves shows that the shear waves are
the most sensitive to the alignment of fractures in anisotropic zones. As
the polarization of the incident S-wave ranges from perpendicular to parallel
to the fractures, the amplitude of the scattered waves goes to zero.
The calculation of scattered wavefields is extended to larger regions of
inhomogeneity by application of a Ray-Born technique. This approach applies
ray methods to the computation of Green's tensors for the background medium
and uses the Born approximation to determine the scattered wavefield from
each volume element within a discretized model of heterogeneity. Comparisons
of Ray-Born results to the complete solution for scattering from an elastic
sphere show that that method works fairly well for wavelengths on the order
of five times larger than the scale lengths typical of the heterogeneity,
but then breaks down to the failure of the Born approximation. With this restriction
in mind, the method is applied to a hypothetical layered earth model containing
a thin, laterally extensive fracture zone. The results confirm that scattering
form shear waves will give unique information on fracture orientation even
for this extended zone. On the other hand, compressional waves are more useful
in inference of nature of the fluid filling the cracks. Modeling of scattered
waves in VSP data from the Lardarello geothermal field in Italy demonstrates
the applicability of the method and suggests that at least in this locality,
anisotropic fracturing is not responsible for the observations. Analysis of
the Fresnel zones affecting reflections from the thin fracture zones responsible
for the scattering allows a delineation for regions of more intense fracturing,
information of importance for the development of geothermal resources.
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