Effects of Deviated Wellbore Stresses on Compressional Headwaves and Modal Dispersions

A new 3D finite-difference time domain formulation of equations of motion for elastic waves in prestressed formations has been used to calculate synthetic waveforms at an array of receivers in a liquid-filled deviated borehole. These equations describe the influence of borehole hydrostatic pressure as well as triaxial formation stresses on elastic waves produced by either a monopole or dipole transmitter placed on the borehole axis. The synthetic waveforms are processed by a slownesstime coherence algorithm and modified matrix pencil algorithms for isolating both non-dispersive and dispersive arrivals in the wavetrain. Computational results for the formation compressional, fast-shear, and slow-shear slownesses obtained from synthetic waveforms in a wellbore with deviations of 0o, 30o, and 60o from the vertical are consistent with the plane wave velocities in homogeneously stressed rock in the far-field subject to the rotated stresses referred to the borehole measurement axes. Compressional slowness changes are primarily affected by changes in the stress along the propagation direction. In contrast, shear slowness changes are equally affected by stress changes either in the propagation or radial polarization direction. Crossing dipole dispersions are observed for all deviations and the magnitude of shear slowness anisotropy is proportional the difference of principal stresses in the borehole cross-sectional plane. A deviated wellbore with the least amount of difference between the principal stresses in the cross-sectional plane is likely to be the most stable trajectory and is characterized by the least amount of cross-dipole shear slowness anisotropy.