Multipole acoustic waveforms in fluid‐filled boreholes in biaxially stressed formations: A finite‐difference method

Abstract

A finite‐difference method is developed to simulate elastic wave propagation in a borehole surrounded by a biaxially stressed solid formation. The linear elastic formation is altered by such tectonic stresses that cause significant changes in the characteristics of wave propagation in a borehole. The 2.5-dimensional problem addressed in this work concerns the three‐dimensional wave propagation in a medium inhomogeneous in two spatial coordinates transverse to the borehole axis. A second‐order finite‐difference method is developed to solve the partial differential equations arising from a published acoustoelastic model for borehole acoustic wave propagation in prestressed formations. The algorithm is validated and applied to model both the borehole flexural and axisymmetric Stoneley waves. The computed waveforms are processed by a variation of Prony’s algorithm that yields dispersion curves for flexural waves polarized both parallel and perpendicular to the stress direction. The flexural dispersion crossover in a uniaxially stressed formation is quantitatively confirmed. The Stoneley dispersion in the presence of such stresses exhibits approximately a uniform shift toward lower slownesses over the entire bandwidth of interest. This implies that azimuthal averaging of formation stiffnesses approximately yields the same effective Stoneley stiffness at different radial positions.