Formation nonlinear constants from sonic measurements at two borehole pressures

Granular structure and microcracks in rocks cause large nonlinearities in the constitutive relations that result in the stress dependence of acoustic-wave velocities. The nonlinear constitutive relations of isotropic materials are described in terms of two linear and three nonlinear elastic constants. For nonhyperelastic materials such as rocks, these constants are defined in terms of strain derivatives of stresses for either the load or unload cycle. Acoustic waveforms at an array of receivers recorded at two different borehole pressures can be used to estimate two of the three formation nonlinear constants. Processing of these time waveforms produced by a monopole or dipole source yields the Stoneley or flexural dispersions, respectively. The differences in the Stoneley and flexural dispersions caused by a known change in the borehole pressure are then utilized in a multifrequency inversion model that yields two of the three independent nonlinear constants of the formation. These two nonlinear constants, c (sub 144) and c (sub 155) , are sufficient to calculate the difference between the maximum and minimum stresses in the azimuthal plane from the dipole anisotropy in the fast and slow shear-wave velocities. In addition, these two formation nonlinear constants are also sufficient to estimate the stress derivatives, delta (rho (sub 0) V (super 2) (sub 12) )/delta S and delta (rho (sub 0) V (super 2) (sub 13) )/delta S for shear-wave propagation normal to the uniaxial stress direction in a cylindrical rock sample of the same material as that of the in-situ formation rock. Here rho (sub 0) is the formation mass density in the reference state; V (sub IJ) denotes plane-wave velocity for propagation along the X (sub I) -direction and polarization along the X (sub J) -direction; and S is the uniaxial stress magnitude applied normal to the propagation direction. Generally, a positive derivative indicates that the rock sample would stiffen with increasing uniaxial stress and a negative derivative indicates that it would soften.