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 given in g/cc, $$V_\mathrm{P}$$ 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:


 * $$\rho = 0.23 V_\mathrm{P}^{0.25} .$$

This 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
Sometimes you need to estimate density from slowness, if V is in ft/s and &rho; is in g/cc:


 * $$V_\mathrm{P} = 360 \rho^4 \ $$

These need converting to SI units!