Gardner's relation

Gardner's relation, or Gardner's equation, named after G. H. F. Gardner and L. W. Gardner, is an empirically derived equation that relates seismic P-wave velocity to the bulk density of the lithology in which the wave travels. The equation reads:


 * $$\rho = \alpha V_p^{\beta}$$

where $$\rho $$ is bulk density given in g/cm3, $$V_p$$ is P-wave velocity given in ft/s, and $$\alpha$$ and $$\beta$$ are empirically derived constants that depend on the geology. Gardner et al. proposed that one can obtain a good fit by taking $$\alpha = 0.23$$ and $$\beta = 0.25$$. Assuming this, the equation is reduced to:


 * $$\rho = 0.23 V_p^{0.25},$$

where the unit of $$ V_p$$ is feet/s.

If $$V_p$$ is measured in m/s, $$\alpha = 0.31$$ and the equation is:


 * $$\rho = 0.31 V_p^{0.25}.$$

This equation is very popular in petroleum exploration because it can provide information about the lithology from interval velocities obtained from seismic data. The constants $$\alpha$$ and $$\beta$$ are usually calibrated from sonic and density well log information but in the absence of these, Gardner's constants are a good approximation.