Gardner's equation

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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]:

\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.

External links

References

  1. Gardner, G, L Gardner & A Gregory, 1974. Formation velocity and density—the diagnostic basis for stratigraphic traps. Geophysics 39, 770–780. A PDF is available online.