geophysics · impedance
Density from Velocity (Gardner's Relation)
ρ = a × Vp^b (SI form: a=0.31, b=0.25, Vp in m/s → ρ in g/cc)
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Inputs
ft/s
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Description
Estimates bulk density from P-wave velocity using Gardner, Gardner & Gregory's (1974) empirical power-law relation, letting a well with a sonic log but no density log still produce an acoustic impedance estimate. Provides a first-pass or gap-filling density value — it does not replace an actual density log where one exists.
Variables
| Symbol | Unit | Description |
|---|---|---|
| ρ | g/cc | Estimated Bulk Density |
| Vp | ft/s | True compressional velocity (e.g. checkshot-calibrated sonic, or 1/DT already converted from slowness) — not a raw uninverted slowness curve. |
| a | — | Gardner's multiplicative coefficient for the SI (m/s) form of the relation. Always use the SI-calibrated value (≈0.31) here regardless of the display unit system — Vp is always converted to m/s internally before this formula runs, so the imperial-Vp-calibrated 0.23 must never be entered even when working in imperial mode. |
| b | — | Power-law exponent, 0.25 in Gardner's original average-lithology fit. Local recalibration against a density log crossplot (holding or adjusting this exponent) improves accuracy over the global default. |
Assumptions
- Brine-saturated clastic or carbonate sedimentary rock — Gardner's original calibration set explicitly excludes evaporites and behaves poorly in coals.
- Vp is a true, calibrated compressional velocity (e.g. checkshot/VSP-corrected sonic), not a raw uninverted slowness curve.
- The default a/b (0.31/0.25) represent a broad regional average; local recalibration against an actual density log materially improves accuracy.
Limitations
- Major, well-documented deviations from the Gardner trend occur in anhydrite, salt, and coal — do not apply this relation across evaporite- or coal-bearing intervals without an independent lithology flag.
- A single global a/b pair can carry on the order of 0.1 g/cc mean error regionally; published local recalibrations (e.g. separate sand/shale coefficients) can reduce this substantially — treat the default as a starting point, not a calibrated result.
- Not a substitute for an actual density log where one is available — use only to fill gaps or build a first-pass impedance estimate.
Use Cases
- → Density log gap-filling: Estimate bulk density in an interval where the density log is missing, damaged, or affected by a washout, using only sonic-derived Vp.
- → Synthetic seismogram construction: Build a synthetic seismogram or acoustic impedance log for wells that have only a sonic (no density) log, feeding the estimated ρ directly into Acoustic Impedance.
- → Regional density trend QC: Cross-check a measured density log against the Gardner trend to flag intervals (e.g. gas effect, washout) where actual density deviates significantly from the expected velocity-density relationship.
Related Calculations
Region Notes
Global
Default a=0.31, b=0.25 (SI units, Vp in m/s) represent Gardner's original broad average across brine-saturated clastic and carbonate sections; always recalibrate against a local density log crossplot before relying on this for a seismic inversion workflow.
Offshore West Africa
A published local recalibration for Tertiary Niger Delta sediments found shale and sand coefficients both shifted from the 0.31 global default (roughly 0.33 for shale, 0.29 for sand, exponent held near 0.25) — a concrete illustration of how much a single global default can miss lithology-specific trends.
Permian Basin
Anhydrite stringers common in the Permian section fall well off the Gardner trend — do not apply this relation across evaporite-bearing intervals without a lithology flag from GR/PEF.
Gulf of Mexico
Gardner's original 1974 paper was framed explicitly around Gulf Coast stratigraphic-trap/bright-spot analysis; the relation performs reasonably in the unconsolidated siliciclastic sections typical of this basin, consistent with that original context.
References
Primary source
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