The Hill Average

Part 2, Part 2: Bounds and Mixtures

Learning objectives

Compute the Hill average as the arithmetic mean of the Voigt and Reuss bounds
State honestly what the Hill average is: a practical estimate, not a bound and not derived physics
Explain why it is used for the mineral modulus of a mixed solid, as this course's mineralMix does
Judge when Hill is trustworthy and when it is a guess, from the width of the Voigt-Reuss gap

Split the Difference

The Voigt and Reuss bounds bracket the answer but often leave a wide gap, and practical work needs one number to carry forward. Hill's move is disarmingly simple: take the midpoint. The Voigt-Reuss-Hill average is $M_{VRH} = \tfrac{1}{2}\left(M_V + M_R\right)$ VRH=tfrac12left(MV+MRright), the arithmetic mean of the two bounds, applied separately to $K$ and to $G$ . For the fifty-fifty quartz-clay solid the bounds are $K$ from 26.61 to 28.75 and $G$ from 11.89 to 25.93 GPa, so the Hill estimates are $K = 27.68$ and $G = 18.91$ GPa. That is the number most workflows use when they need the modulus of a mixed mineral.

Be Honest About What It Is

Hill's average earns its keep, but it must be labelled correctly. It is not a bound: real rocks can and do sit above or below the midpoint. It is not derived physics: no arrangement of grains is guaranteed to produce exactly the mean of the two extremes. It is an estimate, justified by the observation that many well-mixed solids happen to sit near the middle of their envelope, and by the practical fact that the midpoint minimizes the worst-case error when you have no texture information at all. This course uses it deliberately and narrowly: the mineral modulus of a mixed solid, the quartz-clay matrix point that every later frame and fluid model builds on, is a Hill average. Using it for the mineral is defensible because the solid grains are genuinely well mixed and, crucially, because their moduli are not wildly far apart.

When to Trust It

The figure shows the Hill line running down the center of the Voigt-Reuss envelope for a mineral pair you pick, with the gap drawn explicitly. That gap is the whole story. Choose two similar minerals, quartz and feldspar, and the bulk bounds nearly coincide, from about 37.04 to 37.05 GPa, so the Hill value is excellent and the choice of averaging rule barely matters. Choose a stiff and a soft phase, quartz and clay in shear, and the bounds gape from 11.89 to 25.93 GPa: the Hill value of 18.91 is now a guess wearing a suit, a single number papering over a fourteen-GPa ignorance about texture. The rule is therefore to read the gap before trusting the midpoint. A narrow envelope makes Hill a fact; a wide one makes it a placeholder that a better model, or real data, should replace.

Hill splits the difference, but it does not narrow the difference. To genuinely tighten the bracket without inventing texture, we need the sharpest bounds physics allows on a mixture of unknown geometry. Those are the Hashin-Shtrikman bounds, the star of this part, and the next section.

References

Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Hill, R. (1952). The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society. Section A, 65(5), 349-354.