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Detection and Characterization of In-Site Fractures in the Earth from Vertical Seismic Profiling Databy Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on September 12, 1991 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
The purpose of this thesis is to investigate the use of seismic data, specifically
hydrophone vertical seismic profiling (VSP) data, to detect and characterize
large isolated fractures in the shallow part of the earth»s crust. The research
is motivated by the observation of large amplitude Stoneley waves on hydrophone
VSP data that are generated in the borehole at the point where open fractures
intersect the borehole. Previous work by Beydoun et al. (1985) describes a
model, based on a fluid-flow mechanism, that predicts borehole Stoneley wave
to P-wave amplitude ratios vs. frequency in terms of the parameters of the
fracture (i.e., aperture and orientation). Hardin (1986), Hardin et al. (1987),
Cicerone et al. (1988), and Toks§z et al. (1991) used this model as the basis
of an inversion scheme to determine the fracture parameters. Their results
indicate that fracture aperture estimates are generally two to three orders
of magnitude lower than estimates obtained from independent information, such
as flow measurements. They attribute this discrepancy to the fact that the
model does not account for such effects as fracture stiffness and fracture
roughness.
In this thesis, an attempt is made to develop a more realistic model of borehole
Stoneley wave generation at fractures. In order to achieve this, the dynamics
of wave propagation in an infinitely long, fluid-filled fracture imbedded
in an elastic solid is studied first. Assuming that the wavelength of the
seismic wave is much larger than the fracture aperture, solution of the wave
equation under appropriate boundary conditions describes two types of waves.
The first of these is a fundamental mode Stoneley wave, analogous to the borehole
Stoneley wave (tube wave), and referred to in this work as the fracture wave.
This mode is characterized by relatively low phase velocities, which depend
on the fracture aperture, and strong dispersion, especially at low frequencies.
The second type of wave is a pseudo-Rayleigh wave, which consists of an infinite
number of modes. These waves have cut-off frequencies that are much higher
than the range of frequencies contained in the VSP data, therefore they are
not considered here. These modes, however, are important in higher-frequency
data, such as full-waveform acoustic logs.
In addition, the effects of fracture stiffness, fracture roughness, and fluid
viscosity are studied in an attempt to model the fractures more realistically.
Fracture stiffness is modeled by modifying the boundary conditions to allow
for a discontinuity in the vertical displacement while allowing the stresses
to remain continuous. The phase velocity is seen to increase nonlinearly with
increasing stiffness, and in the limit of perfect rigidity, the results are
consistent with results obtained by Tang and Cheng (1989). The displacement
and stress eigenfunctions indicate a decrease in the amplitude of the horizontal
displacement and normal stress in the fracture, as well as the expected discontinuity
in the vertical displacement. Fracture roughness is modeled as a deviation
of fracture from a perfectly plane-parallel layer to one which has a tortuous
path. The phase velocities in this case are reduced by an amount equal to
the square root of the tortuosity as defined by Johnson et al. (1982). The
horizontal displacement and normal stress in the fracture is also reduced,
and the normal stress shows a discontinuity induced by the empirical nature
of the correction made for tortuosity to the phase velocities. Fluid viscosity
is modeled by introducing a shear potential in the fluid and a constitutive
relationship that is time dependent in viscosity. This leads to a complex-valued
phase, which introduces attenuation due to viscous drag effects. The horizontal
displacement and shear stress in the fracture are characterized by the presence
of a boundary layer, made necessary by the introduction of continuous boundary
conditions.
A model of borehole Stoneley wave generation at fractures in VSP data is
then developed by coupling the fracture wave to the borehole via a pressure-continuity
boundary condition. The model differs from that developed by Beydoun et al.
(1985) conceptually: whereas the Beydoun et al. model is quasi-static in nature
and uses a fluid-flow mechanism to generate Stoneley waves in the borehole,
the model presented here utilizes a dynamic wave-propagation approach to the
development of Stoneley waves. For the simple case of a horizontal fracture
filled with an acoustic fluid (water), both models are compared to synthetic
data generated using a finite difference modeling technique for fractures
developed by Kostek (1991). While both models are in good agreement with the
finite difference data for small fracture apertures (i.e., on the order of
1 µm), the dynamic model is in much better agreement with the
finite difference data as the aperture is increased, thus verifying the hypothesis
of mode coupling as the Stoneley wave generating mechanism.
The dynamic model, like the Beydoun et al. model, predicts borehole Stoneley
wave to P-wave amplitude ratios vs. frequency as a nonlinear function of the
parameters of the fracture. In addition, when fracture stiffness, fracture
roughness, and fluid viscosity are incorporated into the model, the predicted
amplitude ratios are observed to decrease when the fracture is modeled as
being stiffer or rougher, or when the fluid is modeled as being viscous.
A nonlinear least-squares inversion scheme based on this model is developed
and applied to characterizing fractures observed in hydrophone VSP data at
three locations: Hamilton, MA, Mirror Lake, NH and Kent Cliffs, NY. These
data show large-amplitude borehole Stoneley waves were open fractures intersect
the borehole. In addition, independent estimates of the orientations of the
observed fractures is available for all three data sets from borehole televiewer
surveys, and for the Hamilton and Mirror Lake, data, fractures aperture estimates
have been derived from flow tests. Therefore, these data make excellent test
cases for the modeling and inversion.
Fracture aperture estimates obtained using this model are in better agreement
with independent information than those obtained using other models (Beydoun
et al. 1985; Hardin et al. 1987). In addition, fracture stiffness and fracture
roughness are shown to be important effects to be considered when modeling
wave propagation in in-situ fractures in the earth. However, there is indication
that there is some degree of non-uniqueness in the inversion, and there is
some degree of trade-off between aperture, stiffness, and roughness. In addition,
confidence intervals derived from a linearized error analysis are compared
to the actual data misfit. In most cases, there is good agreement between
the two, which would indicate that the model is approximately linear, at least
in the vicinity of the least-squares solution.
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