Earth:Gardner's relation

Gardner's relation, or Gardner's equation, named after G. H. F. Gardner and L. W. Gardner, is an empirically derived equation that relates seismic P-wave velocity to the bulk density of the lithology in which the wave travels. The equation reads:

[math]\displaystyle{ \rho = \alpha V_p^{\beta} }[/math]

where [math]\displaystyle{ \rho }[/math] is bulk density given in g/cm³, [math]\displaystyle{ V_p }[/math] is P-wave velocity given in ft/s, and [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are empirically derived constants that depend on the geology. Gardner et al. proposed that one can obtain a good fit by taking [math]\displaystyle{ \alpha = 0.23 }[/math] and [math]\displaystyle{ \beta = 0.25 }[/math]. Assuming this, the equation is reduced to:

[math]\displaystyle{ \rho = 0.23 V_p^{0.25}, }[/math]

where the unit of [math]\displaystyle{ V_p }[/math] is feet/s.

If [math]\displaystyle{ V_p }[/math] is measured in m/s, [math]\displaystyle{ \alpha = 0.31 }[/math] and the equation is:

[math]\displaystyle{ \rho = 0.31 V_p^{0.25}. }[/math]

This equation is very popular in petroleum exploration because it can provide information about the lithology from interval velocities obtained from seismic data. The constants [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are usually calibrated from sonic and density well log information but in the absence of these, Gardner's constants are a good approximation.