Applications of perturbation theory to acoustic logging

To study the normal modes generated during acoustic logging, we have developed a method to calculate (1) phase velocities when the formation has slight, general anisotropy, (2) partial derivatives of either the wave number or frequency with respect to either an elastic modulus or density, (3) group velocities, and (4) phase velocities when the cross section of the borehole is slightly irregular. The method, which is based upon perturbation theory, relates first‐order perturbations in frequency, wave number, elastic moduli, densities, and locations of interfaces for a general model with many fluid and solid layers which have any cross‐sectional shape and any type of anisotropy. To demonstrate the use of this method, formulas for the four applications are developed for a relatively simple model consisting of a fluid‐filled borehole through a transversely isotropic solid with its symmetry axis parallel to the borehole, and some sample calculations are performed. Using these examples, formulas for more complicated borehole models that may be appropriate for some field situations could readily be developed. The significance of this work is that three applications can be used to study the normal modes when the borehole environment is complicated and the other can be used in an inversion for formation properties.