| Popis: |
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 elastic modulus, M, and density, ρ. 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, where the velocity is directly proportional to approximately the 4th power of density as stated by Gardner's numerical approximation. To clarify the apparent contrast between theory and observations, a new expression for the elastic modulus, M, is derived 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, which subsequently results with the observed dependence of velocity on the 4th power of density. In addition, Gardner's equation is modified to accurately obtain the velocity over range of densities (from 1 g/cm3 to around 3 g/cm3). The findings are tested on real velocity and density well-log data. The results validate the derived expression of the elastic modulus as well as the generalized form of Gardner's equation. |
