Spatial sensitivity functions for rapid numerical simulation of borehole sonic measurements in vertical wells

Abstract

Borehole sonic measurements are routinely used to measure dynamic elastic properties of rock formations surrounding a wellbore. However, shoulder-bed effects, mud-filtrate invasion, and near-wellbore damage often influence the measurements and bias the interpretations. Inversion-based methods can reduce the influence of complex geometric conditions in the estimation of formation properties from sonic logs, but they require fast-forward modeling algorithms. We have developed new spatial sensitivity functions for rapid modeling of borehole sonic measurements, which were equivalent to the Green’s functions of borehole modal wave measurements. An adaptive finite-element (FE) method was used to calculate 1D axial (vertical) and 2D (axial-radial) sensitivity functions, which combined the dependence of sonic logs on the wave mode, frequency, formation properties, and logging-instrument geometry. Semianalytical formulations were also used to calculate the sensitivity functions, which involved modeling the wave propagation through an effective layered medium. The 1D axial sensitivity functions were also extended to quantify propagation properties of nondispersive modes in a borehole. Calculated semianalytical sensitivity coefficients were efficient for rapid simulation of modal frequency dispersion in the presence of a logging tool. Simulated sonic logging measurements across synthetic thinly bedded and invaded formations were compared with numerical simulations obtained with the finite-element method. Our results confirmed the efficiency, reliability, and accuracy of the approximate sonic simulation method. The maximum relative error of the rapid simulation method was consistently less than 4%, with only 2% of the central processing unit time and 2% of the memory usage compared with FE simulations.