Application of Backus average to simulate acoustic logs in thinly bedded formations

Abstract

Slowness logs acquired in layered formations are not only affected by spatial averaging associated with the borehole acoustic tool. Layers with thicknesses smaller than the acoustic wavelength can cause measurable effects on the associated wave propagation phenomena. While spatial averaging functions can be used to model tool averaging effects, computer-intensive numerical methods such as finite differences must be used to simulate slowness logs across formations with thin layers. We adopted Backus averaging as a faster alternative to model borehole slownesses when layer thicknesses are smaller than the acoustic wavelength (i.e., in the long-wavelength limit). Using synthetic models and numerical simulations via finite-element and finite-difference methods, we have determined that borehole slownesses of a stack of horizontal layers first approach the average slowness of the individual layers. However, as the layer thickness decreases, sonic slownesses approach the slowness of a homogeneous medium with elastic properties obtained from the Backus average. Therefore, to model acoustic logs acquired in layered formations, we first approximate thin layers as a single homogeneous layer with stiffness coefficients calculated using the Backus average. Next, we apply a spatial averaging function to reproduce the spatial averaging effect inherent to borehole acoustic tools. Results indicate that the latter method is accurate and efficient for fast modeling borehole slownesses of formations with thin layers that are isotropic and intrinsically vertical transversely isotropic. The fast simulation method decreases computation time by at least a factor of 10 and yields slowness logs with a relative error below 2% compared with finite-difference numerical simulations. We also determine that the moving Backus average that is typically applied to upscale acoustic logs for seismic applications is not accurate to model borehole acoustic logs acquired across thinly layered formations.