# Gardner's equation

Gardner's equation is an empirical equation that relates P-wave velocity to bulk density. It is a pseudo-velocity relationship commonly used in estimating sonic or density logs when only one of them is available (both are required for a synthetic when performing a well tie).

Gardner showed that[1]:

${\displaystyle \rho =\alpha V_{\mathrm {P} }^{\beta }}$

where ${\displaystyle \rho }$ is bulk density, ${\displaystyle V_{\mathrm {P} }}$ is P-wave velocity 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 estimate of density in g/cc, given velocity in ft/s, by taking ${\displaystyle \alpha =0.23}$ and ${\displaystyle \beta =0.25}$.

Assuming this, and using units of g/cc, the equation is reduced to the following for a velocity log in ft/s:

${\displaystyle \rho =0.23\ V_{\mathrm {P} }^{\,0.25}\ \ \mathrm {kg} /\mathrm {m} ^{3}}$

If ${\displaystyle V_{\mathrm {P} }}$ is measured in m/s and you want density in kg/m3, then ${\displaystyle \alpha =310}$ and the equation is:

${\displaystyle \rho =310\ V_{\mathrm {P} }^{\,0.25}\ \ \mathrm {kg} /\mathrm {m} ^{3}}$

The equation is very popular in hydrocarbon 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.

## Inverse Gardner equation from density in g/cc

Sometimes you need to estimate density from velocity, if ${\displaystyle V_{\mathrm {P} }}$ is in ft/s and ${\displaystyle \rho }$ is in g/cc:

${\displaystyle V_{\mathrm {P} }=357\rho ^{4}\ }$

Or, if velocity is in m/s:

${\displaystyle V_{\mathrm {P} }=108\rho ^{4}\ }$

If ${\displaystyle \rho }$ is in kg/m3, the factors are much smaller: ${\displaystyle 3.57\times 10^{-10}}$ and ${\displaystyle 1.08\times 10^{-10}}$ respectively.