A physical model for shear‐wave velocity prediction 1

The clay‐sand mixture model of Xu and White is shown to simulate observed relationships between S‐wave velocity (or transit time), porosity and clay content. In general, neither S‐wave velocity nor S‐wave transit time is a linear function of porosity and clay content. For practical purposes, clay content is approximated by shale volume in well‐log applications. In principle, the model can predict S‐wave velocity from lithology and any pair of P‐wave velocity, porosity and shale volume. Although the predictions should be the same if all measurements are error free, comparison of predictions with laboratory and logging measurements show that predictions using P‐wave velocity are the most reliable. The robust relationship between S‐ and P‐wave velocities is due to the fact that both are similarly affected by porosity, clay content and lithology. Moreover, errors in the measured P‐wave velocity are normally smaller than those in porosity and shale volume, both of which are subject to errors introduced by imperfect models and imperfect parameters when estimated from logs. Because the model evaluates the bulk and shear moduli of the dry rock frame by a combination of Kuster and Toksöz’ theory and differential effective medium theory, using pore aspect ratios to characterize the compliances of the sand and clay components, the relationship between P‐ and S‐wave velocities is explicit and consistent. Consequently the model sidesteps problems and assumptions that arise from the lack of knowledge of these moduli when applying Gassmann's theory to this relationship, making it a very flexible tool for investigating how the v P ‐ v s relationship is affected by lithology, porosity, clay content and water saturation. Numerical results from the model are confirmed by laboratory and logging data and demonstrate, for example, how the presence of gas has a more pronounced effect on P‐wave velocity in shaly sands than in less compliant cleaner sandstones.