Gardner's relation

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Gardner's relation, or Gardner's equation, named after Gerald 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:

ρ = α V p β {\displaystyle \rho =\alpha V_{p}^{\beta }} {\displaystyle \rho =\alpha V_{p}^{\beta }}

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

ρ = 0.23 V p 0.25 , {\displaystyle \rho =0.23V_{p}^{0.25},} {\displaystyle \rho =0.23V_{p}^{0.25},}

where the unit of V p {\displaystyle V_{p}} {\displaystyle V_{p}} is feet/s.

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

ρ = 0.31 V p 0.25 . {\displaystyle \rho =0.31V_{p}^{0.25}.} {\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 α {\displaystyle \alpha } {\displaystyle \alpha } and β {\displaystyle \beta } {\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.

References

  1. Gardner, G.H.F.; Gardner L.W.; Gregory A.R. (1974). "Formation velocity and density -- the diagnostic basics for stratigraphic traps" (PDF). Geophysics. 39: 770–780. Bibcode:1974Geop...39..770G. doi:10.1190/1.1440465. Archived from the original (PDF) on 2017-08-09. Retrieved 2010-03-07.