Sensitivity and inversion of borehole flexural dispersions

Propagation characteristics of borehole flexural waves are functions of five borehole, fluid, and formation parameters. Uncertainties in any of these model parameters may introduce errors in the estimate of formation shear wave speed. Other sources of error in the estimate may be caused by deviations from the assumed circular borehole cross-section or heterogeneity in the material properties of formation in the vicinity of a borehole from the usually assumed homogeneous properties. The influence of uncertainties in model parameters on borehole flexural dispersion has been calculated from a general perturbation model based on Hamilton's principle. A sensitivity analysis of the flexural dispersion to small variations in the model parameters shows that the formation shear speed has by far the dominant influence in a slow formation. In contrast, the flexural dispersion in a fast formation is significantly influenced by three of the five model parameters: the formation shear speed, borehole fluid compressional speed, and borehole diameter. The frequency dependence of these sensitivity functions indicates that the inversion of flexural dispersion for formation shear speed is optimal in the range of 2 to 4 kKz for a borehole of diameter 25.4 cm. The range of validity of the perturbation model has been estimated by comparing results of concentric annuli of different thicknesses and shear wave speeds different from that of the formation with those from exact numerical solutions from a modal search program. Generally, the perturbation results for altered annuli of thicknesses up to 15 cm are accurate to within 1% for shear wave speed 10% lower than that of the formation. This difference between the perturbation and exact results increases to approximately 1 to 3% for shear wave speed 20% lower than that of the formation.