# 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:

$\rho =\alpha V_{\mathrm {P} }^{\beta }$ where $\rho$ is bulk density, $V_{\mathrm {P} }$ is P-wave velocity and $\alpha$ and $\beta$ 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 $\alpha =0.23$ and $\beta =0.25$ . Assuming this, the equation is reduced to:

$\rho =230\ V_{\mathrm {P} }^{\,0.25}\ \ \mathrm {kg} /\mathrm {m} ^{3}$ If $V_{p}$ is measured in m/s, $\alpha =0.31$ and the equation is:

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

## Inverse Gardner equation from density in g/cc

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

$V_{\mathrm {P} }=357\rho ^{4}\$ Or, if velocity is in m/s:

$V_{\mathrm {P} }=108\rho ^{4}\$ If $\rho$ is in kg/m3, the factors are much smaller: $3.57\times 10^{-10}$ and $1.08\times 10^{-10}$ respectively.