Applications Of Perturbation Theory To Acoustic Logging

Abstract

For guided wave propagation in boreholes, perturbation theory is used to calculate (1) the partial derivative of the wavenumber or frequency with respect to an elastic modulus or density, (2) group velocity, and (3) the effect of a borehole with a slightly irregular cross section upon the phase velocity. The method, which is developed for a fluid-filled, cylindrical borehole through a transversely isotropic formation, relates perturbations in formation properties (i.e., elastic moduli, densities, and interface locations) and wave properties (i.e., wavenumber and frequency) for guided waves with any azimuthal order number. Velocity perturbations, which are calculated for three common cross sections of irregular boreholes, show several general characteristics. The tube and pseudo-Rayleigh waves, which have no azimuthal dependence, completely smooth the effects of the irregularity making the velocity perturbation independent of the wave's orientation. The perturbations for the tube wave are small because it is a Stoneley wave, but those for the pseudo-Rayleigh wave are much larger because the borehole shape affects the multiply-reflected part of this wave. The velocity perturbations for the flexural and screw waves are similar in character to those for the pseudo-Rayleigh wave, but because these waves are directional, they can interact with the irregularity to amplify or diminish the velocity perturbation.