# Dictionary:Gardner’s equation

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Term Gardner's equation gard’ n∂r Geophysical Reference Series Robert E. Sheriff http://dx.doi.org/10.1190/1.9781560802969 978-1-56080-118-4 SEG Online Store

Gardner's equation, or Gardner's relation, 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:

${\displaystyle \rho =\alpha V_{p}^{\beta }}$

where ${\displaystyle \rho }$ is bulk density given in g/cm3, ${\displaystyle V_{p}}$ is P-wave velocity given in ft/s, and ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are empirically derived constants that depend on the geology. Gardner et al. proposed that one can obtain a good fit by taking ${\displaystyle \alpha =0.23}$ and ${\displaystyle \beta =0.25}$.[1] Assuming this, the equation is reduced to:

${\displaystyle \rho =0.23V_{p}^{0.25}.}$

If ${\displaystyle V_{p}}$ is measured in m/s, ${\displaystyle \alpha =0.31}$ and the equation is:

${\displaystyle \rho =0.31V_{p}^{0.25}.}$

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 ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are usually calibrated from sonic and density well log information but in the absence of these, Gardner's constants are a good approximation.

The empirical relationship that density is proportional to the 1/4 power of P-wave velocity:

${\displaystyle \rho =aV_{\mathrm {P} }^{\frac {1}{4}},}$

where a is 0.31 when V is in m/s, 0.23 when in ft/s[1].

## References

1. Gardner, G.H.F.; Gardner L.W. & Gregory A.R. (1974). "Formation velocity and density -- the diagnostic basics for stratigraphic traps". Geophysics 39: 770–780. doi:10.1190/1.1440465.
2. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
3. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
4. Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.