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:

${\displaystyle \rho =\alpha V_{p}^{\beta }}$

where ${\displaystyle \rho }$ is bulk density given in g/cm³, ${\displaystyle V_{p}}$ is P-wave velocity given in ft/s, and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are empirically derived constants that depend on the geology. Gardner et al. proposed that one can obtain a good fit by taking ${\displaystyle \alpha =0.23}$ and ${\displaystyle \beta =0.25}$.^[1] Assuming this, the equation is reduced to:

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

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

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

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

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 ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are usually calibrated from sonic and density well log information but in the absence of these, Gardner's constants are a good approximation.

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