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Borehole Wave Propagation in Isotropic and Anisotropic Media: Three-Dimensional Finite Difference Approachby Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on February, 1994 in partial fulfillment of the requirements for the degree of Doctor of Philosophy ABSTRACT
In this thesis we develop a three-dimensional method to simulate wave propagation
in an isotropic as well as an anisotropic medium. The wave equation is formulated
into the first-order hyperbolic equations by using velocity and stress. They
are discretized on a staggered grid. The three-dimensional finite difference
time domain scheme is second-order accurate in time and fourth-order accurate
in space. The grid dispersion and anisotropy are analyzed and the stable condition
of the scheme is obtained. Higdon's absorbing boundary condition is discussed
and generalized to the anisotropic medium. The scheme provides realistic 3-D
wave propagation simulation by the use of a parallel computer.
The finite difference method is first tested in homogeneous media. The finite
difference results agree excellently with the analytic solutions of a point
explosion source in the acoustic medium and a point force source in elastic
and transversely isotropic media. The finite difference method accurately
models not only the far field P and S waves, but also the near field term.
The method is then tested in a fluid-filled borehole surrounded by a homogeneous
elastic formation. The finite difference results are in good agreement with
the discrete wavenumber solutions for both monopole and dipole sources in
the hard as well as the soft formations. These tests also show the good performance
of Higdon's absorbing boundary in isotropic and anisotropic media. It
not only works for the body waves but also for the guided waves.
The 3-D finite difference time domain method is applied to fluid-filled
borehole wave propagation problems in isotropic and anisotropic formations.
The effects of the off-centered sources, the elliptic borehole, and the tilted
layer on acoustic logs are investigated for the isotropic formation. The finite
difference synthetics are compared with ultrasonic laboratory measurements
in a scaled borehole in an orthorhombic phenolite solid. Both monopole and
dipole logs agree well. In the anisotropic formation the different borehole
orientations are considered for monopole and dipole logs. Due to shear wave
anisotropy, there are shear-pseudo-Rayleigh wave arrivals on the monopole
log between the P and Stoneley waves in the phenolite formation. Anisotropy
can also cause shear wave splitting on the dipole log.
Field data sets collected by an array monopole acoustic logging tool and
a shear wave logging tool were processed and interpreted. The P- and S-wave
velocities of the formation are determined by threshold detection with cross-correlation
correction from the full waveform and the shear wave log, respectively. The
extended Prony's method is used to estimate the borehole Stoneley wave
phase velocity and attenuation as a function of frequency. The formation between
the depths of 2950 and 3150 ft can be described as an isotropic elastic medium.
The inverted Vs from the Stoneley wave phase velocity is in excellent
agreement with the shear wave log results. The formation between the depths
of 3715 and 3780 ft is a porous, permeable and anisotropic medium. The shear
wave velocity anisotropy is about 10% to 20%, and the symmetry axis
is perpendicular to the borehole axis. The disagreement between estimated
permeabilities from low frequency Stoneley wave velocity and attenuation data
are in good agreement with the core measurements. Also, it is shown that the
formation permeability is not the primary cause of the discrepancy between
the shear wave velocity inverted from the Stoneley wave and measured by the
shear wave logs. The 3-D finite difference synthetics in the anisotropic formation
confirm that the discrepancy can be explained as shear wave anisotropy.
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