Gardner's relation
# 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 $g/cm^3,$ $V_{p}$ 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:

    $\rho =0.23V_{p}^{0.25},$

where the unit of $V_{p}$ is feet/s.

If $V_{p}$ is measured in m/s, $\alpha =0.31$ and the equation is:

    $\rho =0.31V_{p}^{0.25}.$

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.

