Acoustic waveforms have been simulated for monopole, dipole, and quadrupole sources in a prestressed formation. The prestressed medium is modeled as an unstressed medium with the stress effect replaced by effective elastic moduli. The effective, unstressed medium is acoustically anisotropic. Because the medium is inhomogeneous due to the local stress concentrations around the borehole, the anisotropy tensor varies in space. To avoid the use of a full anisotropy tensor, the effective medium is assumed to be transversely isotropic (TI) with a symmetry axis parallel to the minimum stress direction. The elastic moduli of the TI medium are determined from the shear velocities of the unstressed medium using an empirical relation between static stress and shear velocities. The effective medium is then modeled with the conventional finite-difference time-domain method. The effective-medium approach successfully predicts the known first-order effects of stress; that is, shear-wave splitting for a monopole source and dispersion crossover for cross-dipole sources. It also shows that the dipole wave-dispersion crossover frequency decreases linearly with increasing borehole radius.
With the effective medium model, we study the interaction of quadrupole waves excited and recorded in a vertical well with static formation stress. It is shown that the quadrupole wave in a prestressed formation splits into a fast wave and a slow wave. The velocities of the waves are similar to those of crossed-dipole waves in the same formation. Both waves are faster than quadrupole waveforms without stress. Moreover, the wave splitting is observed for different tool orientations in the borehole and different borehole sizes. Therefore, the wave splitting may be used as an indication of static formation stress if the medium without stress is isotropic.