Empirical Backbones
Learning objectives
- Define an empirical trend as a regression fitted through one dataset over a stated domain, not a law of physics
- Read Han's water-saturated shaly-sand lines: Vp and Vs both fall with porosity and with clay
- Anchor the numbers: a clean sand at porosity 0.25 gives Vp about 3.75 km/s, and each ten percent of clay costs about 0.22 km/s of Vp
- Place Wyllie and Raymer-Hunt-Gardner as older velocity backbones and know that regressions interpolate well but extrapolate treacherously
What a Regression Is, and Is Not
Parts 5 and 6 built dry frames from geometry: grain contacts, then pore shapes. Every one of those models needed a parameter that someone had to choose. This part turns to data, and the simplest thing data gives you is a trend line. An empirical backbone is a regression: someone measured velocity, porosity, and clay content on a set of real core plugs, and fitted a formula that runs through the middle of the cloud. It is a summary of that dataset, over the range that dataset covered, and nothing more. It is not a physical law. It cannot tell you why velocity falls with porosity, only that in those samples it did, by so much per unit. The distinction matters because a regression carries no warning label: the formula will happily return a number for a rock nothing like the ones it was fitted to, and that number can be badly wrong.
Han's Shaly-Sand Lines
The classic example is Han's 1986 study of water-saturated shaly sandstones at 40 MPa. His regression reads and , with velocities in km/s, the porosity, and the clay fraction. Two knobs, both pushing velocity down. A clean sand () at a porosity of 0.25 gives km/s and km/s. Add clay and both velocities slide: at the same porosity but , drops to 3.31 and to 1.82. The clay term is worth reading in isolation: each ten percent of clay subtracts about 0.22 km/s from and 0.19 from , while each ten percent of porosity subtracts about 0.69 and 0.49. Porosity is the stronger lever, but clay is far from negligible, and a model that ignored it would misread every dirty sand. Tighten the rock instead, to and , and climbs to 4.44 km/s. The intercepts, and , give 5.59 and 3.52, which are near solid quartz but not exactly it, because a regression through porous samples is not obliged to pass through the mineral point.
Older Backbones, and the Warning
Han's lines are one family among many. The Wyllie time-average and the Raymer-Hunt-Gardner equation, both taught in the Petrophysics course as ways to turn a sonic log into porosity, are the same kind of object seen from the other side: velocity-porosity regressions with a domain of validity. Read as backbones, with the course's quartz and brine, Wyllie gives km/s for a clean quartz sand at a porosity of 0.25 and Raymer gives 3.82, the two disagreeing by 0.14 km/s because they were tuned to different rocks. That spread is the whole lesson. A regression interpolates splendidly inside its cloud and extrapolates treacherously outside it: push Wyllie to very high porosity, or Han to a cemented sand at 60 MPa, and the number it returns is a straight-line guess about rocks it never saw. Use these backbones as sanity rails and starting points, the quick check that a measured velocity is in the right neighborhood, and reach for the physical models of Parts 5 and 6 when you need to know what would happen if the rock, the fluid, or the pressure changed. The next section takes one of these regressions to work, predicting the shear velocity that logs so often leave out.
References
- Han, D. (1986). Effects of porosity and clay content on acoustic properties of sandstones and unconsolidated sediments (PhD thesis). Stanford University.
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.