Gardner Equation Calculator: Bulk Density from P-Wave Velocity

Density logs get skipped more often than any other log. Bad hole, budget cuts, tool failures, old wells that predate the tool entirely: the reasons pile up, and yet every synthetic seismogram, impedance estimate, and overburden calculation wants a density. The Gardner equation is the industry sixty-year-old answer to that gap. Fit by Gardner, Gardner, and Gregory at Amoco in 1974 to a large set of brine-saturated laboratory and log measurements, it ties bulk density to P-wave velocity with one compact power law:

rho = 0.23 x Vp^0.25 (Vp in ft/s, rho in g/cc), or equivalently rho = 1.741 x Vp^0.25 with Vp in km/s.

Why one curve works for most rocks

The quarter-power exponent means density is a very gentle function of velocity: double the velocity and density rises by only about 19 percent. That gentleness is exactly why a single average curve can serve shale, sandstone, limestone, and dolomite at once. On a crossplot the individual lithologies form recognizable clusters slightly above or below the average line, and refined per-lithology coefficients exist for when precision matters, but the classic single-curve version stays within a few percent for the great majority of water-bearing sedimentary rocks. The calculator runs the relation both directions, so you can also back out the velocity implied by a known density.

Worked example

• A sandstone interval logs Vp = 10,000 ft/s and the density curve is washed out
• 10,000^0.25 = 10, so rho = 0.23 x 10 = 2.30 g/cc
• Acoustic impedance follows directly: 2.30 x 10,000 = 23,000 g/cc x ft/s
• That impedance feeds a synthetic seismogram without ever touching the bad density curve

Where Gardner will lie to you

The relation assumes brine-filled sedimentary rock, and each word in that phrase is a warning label. Gas lowers Vp dramatically while barely touching density, so applying Gardner in a gas sand predicts a density far too low; that mismatch is not a nuisance, it is actually the physics behind bright-spot amplitudes. Salt is fast but light (about 4,500 m/s at only 2.16 g/cc), so Gardner overestimates its density badly, which matters enormously for subsalt pore pressure work. Coal is slow and very light, anhydrite dense and fast off-trend. The practical rule from decades of use: Gardner is a fine default in shale-sand-carbonate sections, and every one of its documented failures corresponds to a lithology you can usually spot on other logs first.

Related Calculators and Reference

The density this predicts feeds straight into the acoustic impedance calculator and from there the reflection coefficient calculator. Velocity conversions between interval, average, and RMS flavors live in the seismic velocity converter. For the original context, the SEG Wiki page on the Gardner equation summarizes the 1974 paper and its coefficients.