Bounds as Diagnostics

Part 2, Part 2: Bounds and Mixtures

Learning objectives

Read where a rock sits between the bounds as a direct measure of its texture
Associate the upper bound with load-bearing, cemented, stiff-frame rocks and the lower bound with grain-supported, compliant sediments
Use the normalized position between the bounds as a texture meter by interpolating between them
Trace a quartz-water mix and see the upper bound stay stiff while the lower collapses toward the suspension floor

The Gap Is the Message

So far the width of the envelope has looked like a nuisance, the uncertainty texture leaves behind. Turn it around and it becomes the most useful thing the bounds provide. The bounds do not know the texture, but the rock does, and where a measured rock plots inside the envelope reports that texture directly. A point riding near the upper bound is stiff for its porosity: its solid is arranged into a connected, load-bearing frame, the signature of cementation and grain contacts that carry stress. A point sagging toward the lower bound is soft for its porosity: its solid is grain-supported at best, or barely supported at all, the signature of loose, compliant, uncemented sediment. Between them the position is continuous, and it reads as a continuous measure of how load-bearing the frame has become.

A Texture Meter

Make that quantitative. For a measured modulus $M$ between the lower bound $M_-$ − and the upper bound $M_+$ +, the normalized position $d = \dfrac{M - M_-}{M_+ - M_-}$ −M+−M− runs from zero at the lower bound to one at the upper. That single number, obtained by interpolating between the bounds, is a texture meter: near zero means suspension-like or barely consolidated, near one means fully cemented and stiff-framed, and the middle grades between them. This is the working heart of the whole bounds family. It needs no new physics beyond the bounds you already have, only the recognition that the fractional distance between them carries the texture that composition omitted.

Quartz and Water

The figure makes it concrete with a quartz-and-water system across porosity, the simplest stand-in for a clean sand losing its frame. Take thirty percent water. The Hashin-Shtrikman upper bound stays stiff, near 23 GPa in bulk modulus, the modulus a well-cemented sand of that porosity could reach. The lower bound has collapsed toward the suspension floor, near 8 GPa, which is exactly the Reuss value of Wood's equation for that mix. The envelope between them is enormous, and that is the point: at a given porosity the same quartz and water can be anything from a stiff cemented sandstone plotting high, to a loose grain pack plotting low, to a true suspension sitting on the floor of the lower bound. The example rocks drawn on the figure, a cemented sandstone near the top, a loose sand well down, a mud on the floor, all share composition and porosity and differ only in where texture places them between the bounds.

Reading texture from the bounds is powerful, but it still leaves the porosity axis unstructured: the bounds run from zero porosity to pure fluid with nothing to mark where a sand stops behaving like a load-bearing rock at all. That threshold is a real, observable feature of sediments, and the next section, the part's closer, gives it a name: critical porosity.

References

Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Nur, A., Mavko, G., Dvorkin, J., & Galmudi, D. (1998). Critical porosity: A key to relating physical properties to porosity in rocks. The Leading Edge, 17(3), 357-362.