Abstract:
We have studied the velocity dispersion of guided waves in transversely isotropic formations. Theoretical velocity dispersion curves were calculated with elastic constants
based on laboratory and field measurements and compared to dispersion curves for
isotropic formations having the same vertical po and S-wave velocities. The symmetry
axis for the transverse isotropy was parallel to the borehole. The differences between
the phase velocities for the transversely isotropic and isotropic formations depend on
the type of wave, its frequency, and the amount of anisotropy, and can be as high as
7 to 10 percent.
The changes in the phase velocity due to changes in the elastic constants of the
formation (c[subscript 11], C[subscript 1]3, C[subscript 33], C[subscript 44], and C[subscript 66]) and the bulk modulus of the borehole fluid (⋋) vary with frequency. In a hard formation, the tube wave's velocity is sensitive to C[subscript 66] at low frequencies, to C[subscript 44] at high frequencies, and to ⋋ at all frequencies. The pseudo-Rayleigh wave is affected by C[subscript 44] near its cutoff frequency and by ⋋ at high frequencies. The flexural wave, which is generated by a shear wave logging tool, is similarly affected by C[subscript 44] at low frequencies and by ⋋ at high frequencies. As the formation becomes soft, the effect of the elastic constants upon the phase velocity gradually changes. Like a hard formation, the tube wave's velocities in a moderately soft formation are primarily affected by C[subscript 66] and ⋋ at low frequencies, but the influence of C[subscript 44] is much greater at high frequencies.
Since array processing methods can accurately estimate the velocity dispersion of
the guided waves over a wide range of frequencies, some elastic constants can be estimated. In a hard formation, the refracted P- and S-wave velocities uniquely determine
C[subscript 33] and C[subscript 44], and an inversion can be used to estimate C[subscript 66] and ⋋. In a moderately soft formation, the refracted P-wave velocity determines C[subscript 33], the flexural wave from the shear wave logging tool determines C[subscript 44], and the tube wave's velocity dispersion can be
used to estimate C[subscript 66] and ⋋.