Inversion of borehole dispersions for formation stresses

Horizontal tectonic stresses in vertical boreholes, and overburden and tectonic stresses in horizontal wells cause stress-induced azimuthal anisotropy in shear wave logging. It has been reported that a dipole dispersion crossover is an indicator of stress-induced anisotropy dominating over other possible sources of shear anisotropy. This paper describes a method of estimating the difference between the maximum and minimum stresses in the azimuthal plane perpendicular to the borehole from the stress-induced flexural and Stoneley dispersions. This method consists of analyzing the flexural dispersions for dipole sources aligned parallel and perpendicular to the maximum azimuthal stress direction together with the Stoneley wave dispersion derived from a monopole source. In the presence of formation and borehole stresses above and beyond those in an assumed isotropic reference state, the borehole flexural and Stoneley velocity dispersions are also functions of the formation stresses and nonlinear constants. A multifrequency inversion of the flexural and Stoneley velocity dispersions over a selected frequency band is performed to determine the differential or uniaxial stress magnitude S, and the quantities Sc₁₁₁/c₆₆, Sc₁₁₂/c₆₆, and Sc ₁₂₃/c₆₆, where c₆₆ is the formation shear modulus; and c₁₁₁, c₁₁₂, and c₁₂₃ are the formation nonlinear constants in the assumed isotropic reference state. The optimal frequency band for multifrequency inversion is obtained from sensitivity analyses of flexural and Stoneley dispersions to the stress magnitude and nonlinear constants