The Relationship Between the Compressional Wave Velocity of Saturated Porous Carbonate Rocks and Density: Theory and Application
Understanding the wave velocity through rocks is essential for the purposes of applied geophysics in such areas as groundwater and hydrocarbon exploration. Theoretically, the wave velocity is defined by the Newton-Laplace equation. It 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. 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 observed by Gardner. This inconsistency is caused by the interrelationship within the Newton-Laplace equation as the elastic modulus also depends on density. Another observation is the fact that Wyllie’s time average equation also satisfies the 4th power dependence of velocity on density. As a result, a new expression for the elastic modulus is derived using Wyllie’s equation and the Newton-Laplace equation. The new equation is a function of the properties of the rock components such as the matrix velocity and density, fluid velocity and density, and the bulk density or porosity. This dependence is numerically approximated by , which is obtained using Wyllie’s numerical approximation and the Newton-Laplace equation. In addition, Gardner’s equation is modified to accurately obtain the velocity over a range of densities (from 1 g/cm3 to around 3 g/cm3). The findings are validated by applying the new expression to field data using velocity and density well logs for carbonate rocks as well as the volume fraction of each mineral within the formations. The root mean square error (rmse) between the measured velocity and calculated velocity is low, which validates the derived expression. The results of this work provide a new model to calculate the elastic modulus for carbonate rocks knowing only the properties of the matrix, the fluid and the bulk density.
AAPG Datapages/Search and Discovery Article #90332 © 2018 AAPG International Conference and Exhibition, Cape Town, South Africa, November 4-11, 2018