A three-dimensional finite difference simulation of sonic logging

Abstract

A three-dimensional finite-difference (FD) method is used to simulate sonic wave propagation in a borehole with an inhomogeneous solid formation. The second-order FD scheme solves the first-order elastic wave equations with central differencing in both space and time via staggered grids. Liao’s boundary condition is used to reduce artificial reflections from the finite computational domain. In the staggered grids, sources have to be implemented at the discrete center in order to maintain the appropriate symmetry in an axisymmetric borehole environment. The FD scheme is validated for multipole sources in three special media: (i) a homogeneous medium; (ii) a homogeneous formation with a fluid-filled borehole; and (iii) a horizontally layered formation. The staircase approximation of a circular borehole introduces little error in dipole wave fields, although it causes a noticeable phase velocity error in the monopole Stoneley wave. This error has been drastically reduced by using a material averaging scheme and finer grids. Numerical examples show that the FD scheme can accurately model 3-D elastic wave propagation in complex borehole environments.