# 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, and using units of g/cc, the equation is reduced to the following for a velocity log in ft/s: $\rho = 0.23\ V_\mathrm{P}^{\,0.25}\ \ \mathrm{kg}/\mathrm{m}^3$

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