An approximation for the Xu-White velocity model

In 1995, S. Xu and R. E. White described a method for estimating compressional and shear-wave velocities of shaley sandstones from porosity and shale content. Their model was able to predict the effect of increasing clay content on compressional-wave velocity observed in laboratory measurements. A key step in the Xu-White method estimates dry rock bulk and shear moduli for the sand/shale mixture. This step is performed numerically by applying the differential effective medium method to the Kuster-Toksoz equations for ellipsoidal pores. This step is computationally intensive. Using reasonable assumptions about dry rock elastic properties, this step can be replaced with a set of approximations for dry rock bulk and shear moduli. Numerical experiments show an extremely close match between velocities obtained with these approximations and velocities computed with the differential effective medium method. These approximations simplify the application of the Xu-White method, and make the method computationally more efficient. They also provide insight into the Xu-White method. For example, these approximations show how the Xu-White model is related to the critical porosity model.