# The Relationship Between the Compressional Wave **Velocity** of Saturated Porous Carbonate Rocks and Density: Theory and Application

**Velocity**

## Abstract

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

*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.*

**velocity**AAPG Datapages/Search and Discovery Article #90332 © 2018 AAPG International Conference and Exhibition, Cape Town, South Africa, November 4-11, 2018