Acoustic waves in pressurized boreholes in formations with triaxial stresses

A finite-difference 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 borehole. Equations of motion for elastic waves in prestressed materials are derived from rotationally invariant equations of nonlinear elasticity. These equations describe the influence of borehole hydrostatic (mud) pressure as well as formation stresses on acoustic waves produced by either a monopole or dipole transmitter placed on the borehole axis. The synthetic waveforms are processed by a modified matrix pencil algorithm for isolating both dispersive and non-dispersive arrivals in the wavetrain. Computational results show that a difference in the maximum and minimum horizontal stresses causes dipole dispersion crossovers that can be used as an indicator of stress-induced anisotropy dominating the data. On the other hand, an increase in the overburden stress causes both the Stoneley and dipole dispersions to shift toward higher velocities by varying amounts at various frequencies. Any increase in the mud pressure introduces an altered annulus surrounding the borehole surface. This annulus exhibits radial variations in the near-wellbore radial and hoop stresses. Both the Stoneley and dipole dispersions show increasing velocities at higher frequencies. Changes in borehole acoustic wave velocities caused by a change in any one of the formation principal stresses yield frequency-dependent overburden, maximum horizontal, and minimum horizontal stress coefficients of velocities. These stress coefficients of velocities can be used to invert measured changes in borehole dispersions at various depths for corresponding changes in formation stresses.