Abstract

A perturbation method of solution is an efficient way of analyzing elastic wave propagation along a borehole in anisotropic formations. The perturbation model allows us to calculate changes in the modal dispersion curves caused by the differences in elastic constants between the anisotropic formation of interest and a reference, or unperturbed, isotropic formation. The equivalent isotropic constants in the reference formation are obtained from the appropriate compressional-and shear-wave velocities for the selected propagation and polarization directions of the flexural mode. This choice of the unperturbed solution means that the required perturbation is minimal, resulting in enhanced accuracy of the perturbed solution. Computational results are presented for the dispersion curves of borehole flexural waves in a transversely isotropic (TI) formation as a function of borehole deviation from the TI symmetry axis. In addition, radial distributions of displacement and stress fields associated with the flexural wave are obtained as a function of frequency. These provide qualitative information on the radial depth of investigation with flexural wave logging. The flexural wave excitation function is a measure of the energy that a source converts to flexural motion. We deduce an expression for the flexural wave excitation and show that its bandlimited characteristic is influenced by both the borehole diameter and formation parameters. From the dispersion curves and excitation functions, we can compute the flexural waveforms caused by a dipole source with arbitrary orientation in the borehole. In the numerical computations, we have used the unperturbed mode shapes for an equivalent isotropic medium together with the perturbed dispersion relations caused by the formation anisotropy.

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