Borehole dipole and quadrupole modes in anisotropic formations

Sonic measurements while drilling are made in the presence of a drill collar (in the form of a thick steel pipe) that provides an additional path for the acoustic energy propagating from the transmitter to an array of receivers. Low-frequency asymptotes of both dipole and quadrupole modes yield the formation far-field shear slowness. Intrinsic or stress-induced anisotropy of surrounding formation in a vertical well is generally investigated by a wireline sonic tool using a borehole dipole mode that exhibits cosθ azimuthal dependence in their associated acoustic field propagating along the borehole axis. The fast and slow dipole shear polarizations are orthogonal to one another in the presence of any azimuthal anisotropy. The two flexural modes are excited by rotating the dipole transmitter by 90° from one of the primary shear polarization directions. However, borehole quadrupole modes appear to be more promising for estimating the far-field formation shear slowness in the presence of a drill collar. Formation anisotropy can also be investigated using such quadrupole modes that exhibit cos2θ azimuthal dependence in their acoustic field propagating along the borehole axis. The two canonical quadrupole modes are excited by rotating the quadrupole transmitter by 45° from one of the primary shear polarization directions. Borehole flexural dispersions exhibit a crossover in the presence of a uniaxial stress in a plane perpendicular to the borehole axis. In contrast, a perturbation analysis of quadrupole modes in the presence of a uniaxial stress does not show any such crossovers. Both the orthogonal quadrupole modes corresponding to the transmitter oriented parallel and 45° to the uniaxial stress direction asymptote to the same shear slowness at low frequencies. This shear slowness is essentially the same as the fast shear slowness measured by a dipole transmitter parallel to the uniaxial stress direction. However, there are increasing differences between the two orthogonal quadrupole mode slownesses at higher frequencies. Quadrupole waves with positive polarization parallel to the uniaxial stress direction always exhibit larger slownesses than those polarized 45° to the uniaxial stress direction. Even though stre- ss-induced anisotropy in shear slownesses at very low frequencies is not predicted the orthogonal quadrupole modes, significant differences at higher frequencies can be used as an indicator of stress-induced anisotropy magnified by the near-wellbore stress concentrations.