Density Without a Log

Part 7, Part 7: Calibration, From Data to Model

Learning objectives

State why density is often missing and why impedance work cannot proceed without it
Use Gardner's relation as the universal velocity-to-density fallback when composition is unknown
Anchor the numbers: Gardner gives about 2.19 g/cc at Vp 2.5 km/s and 2.60 at 5.0 km/s
Compare Gardner against the exact mineral-fluid mix and see its clean-sand bias: 2.36 versus 2.25 g/cc for a porous quartz sand

The Third Missing Curve

Impedance is velocity times density, so every reflection and every inversion needs a density. Yet density is the third curve that old and cheap logging runs leave out, and even where it was recorded, washed-out hole makes the reading unreliable across exactly the porous zones that matter most. When density is missing you estimate it, and there are two ways, one blunt and universal, the other sharp and demanding. The blunt one is Gardner's relation, $\rho = 1.741\,V_p^{0.25}$ p0.25, with $V_p$ p in km/s and $\rho$ in g/cc. It needs nothing but the velocity you already have. Feed it $V_p = 2.5$ p=2.5 and it returns 2.19 g/cc; at 3.5 it gives 2.38; at 5.0 it gives 2.60. Faster rock is denser rock, and the quarter-power makes the climb gentle, a factor of two in velocity buying only about a fifth in density.

A Trend Through Everything

Gardner's power law is an empirical backbone of the kind the first section defined, and it is worth being precise about which cloud it was fitted to. It is an average through a great many sedimentary rocks of every sort, shales and sands and carbonates together, and shale is the most abundant of those. So Gardner is a shale-heavy average trend. That makes it a fine first guess for a mixed section and a genuinely useful fallback when you have no idea what the rock is. It also makes it biased for a specific and important case: a clean, porous sand. When you actually know the mineral, the porosity, and the fluid, you do not need a trend at all, because density is the one rock property that mixes exactly. The rock density is simply the volume average $\rho = (1 - \phi)\,\rho_{min} + \phi\,\rho_{fl}$ , no model and no fitting, just bookkeeping of what occupies each unit of volume.

Where the Two Disagree

Put numbers on the gap. Take a quartz sand at a porosity of 0.25 saturated with brine: quartz at 2.65 g/cc filling three-quarters of the volume and brine at 1.03 filling the rest gives an exact density of 2.25 g/cc. That sand has a P-wave velocity near 3.4 km/s, and Gardner evaluated there returns 2.36 g/cc. The trend reads about 0.12 g/cc too heavy for this rock, which is the shale-heavy average showing through: a clean porous sand is lighter than the general run of sediments at its velocity, and Gardner does not know that, because it averaged the sand together with the denser, slower shales around it. The lesson is a small hierarchy. When composition is unknown, Gardner is the honest fallback and you accept its bias. When you know mineral, porosity, and fluid, the exact mix beats it every time, and it beats it precisely where impedance work is most delicate, in the clean reservoir sand. With velocity, shear, and density now all recoverable, the next section stops estimating individual curves and calibrates a whole rock-physics model to a real well.

References

Gardner, G. H. F., Gardner, L. W., & Gregory, A. R. (1974). Formation velocity and density: The diagnostic basics for stratigraphic traps. Geophysics, 39(6), 770-780.
Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.