Estimating a shear modulus of a transversely isotropic formation

Abstract

We have developed a method that estimates a shear modulus [Formula: see text] of a transversely isotropic formation using the tube wave generated during acoustic logging. (The symmetry axis of the anisotropy is assumed to parallel the borehole.) The inversion, which is implemented in the frequency‐wavenumber domain, is based upon a cost function that has three terms: a measure of the misfit between the observed and predicted wavenumbers of the tube wave, a measure of the misfit between the current estimate for [Formula: see text] and the most‐likely value for [Formula: see text], and penalty functions that constrain the estimate to physically acceptable values. The largest contribution to the value of the cost function ordinarily comes from the first term, indicating that the estimate for [Formula: see text] depends mostly on the data. Because the cost function only has one minimum, it can be found using standard optimization methods. The minimum is well defined indicating that the estimate for [Formula: see text] is well resolved. Estimates for [Formula: see text] from synthetic data are almost always within 1 percent of their correct value. Estimates for [Formula: see text] from field data that were collected in a formation with a high clay content are typical of transversely isotropic rocks.