On the relationship between compressional wave velocity of saturated porous rocks and density : theory and application

Abstract

Understanding the velocity of the compressional waves travelling through rocks is essential for the purposes of applied geophysics in such areas as groundwater and hydrocarbon exploration. The wave velocity is defined theoretically by the Newton-Laplace equation, which relates the wave velocity, V, to the square root of the ratio of the rock's elastic modulus, M, and its density, [rho] (Bourvie et al., 1987). Therefore, the equation indicates that the velocity is inversely proportional to density. However, the in-situ field measurements and laboratory experiments of compressional wave velocity through different rocks show otherwise. In other words, the velocity is directly proportional to approximately the 4th power of density as stated by Gardner (Gardner et al., 1974). This thesis investigates the inconsistency between theory and observations regarding the relationship between velocity and density of saturated porous rocks. The inconsistency is clarified by deriving a new expression for the elastic modulus, M, using Wyllie's time average equation and the Newton-Laplace equation. The new derived expression of the elastic modulus, M, provides dependence of M on density to approximately the 9th power. In addition, Gardner's equation is modified to accurately obtain the velocity over the entire range of densities (from 1.00 g/cm³ to around 3.00 g/cm³) and porosity (from 0% to 100%). The end of this thesis is an application of the previous outcomes with real data sets, where the results validate the derived expression of the elastic modulus as well as the generalized form of Gardner's equation.