Critical Porosity

Part 2, Part 2: Bounds and Mixtures

Learning objectives

State Nur's observation that sediments stop being load-bearing near a critical porosity of about 0.40 for sandstones
Write the modified-Voigt dry frame that softens linearly from the mineral point to zero at the critical porosity
Explain what happens beyond the critical porosity, where the sediment becomes a suspension on the lower bound
See how this one observation turns the bounds into a usable first frame model, as Part 0's quickRock already did

Where a Sand Stops Being a Rock

The bounds bracket a rock but do not say where along the porosity axis its behavior changes character. Nur's observation supplies the missing landmark. As you add porosity to a clean sand, its frame softens, and at a critical porosity the grains can no longer form a load-bearing network at all: the sediment stops being a rock and becomes a suspension of grains in fluid. For sandstones this threshold sits near 0.40, typically in the range 0.36 to 0.40; chalks, cracked basalts, and other materials have their own critical porosities, some much higher. It is an empirical fact about how grains pack and support load, not a derived constant, but it is remarkably consistent within a lithology.

The Modified-Voigt Frame

That single number organizes the whole porosity axis. Below the critical porosity $\phi_c$ c the dry frame stiffens from zero back up to the mineral as porosity falls, and the simplest description that respects the two endpoints is a straight line: the dry modulus runs from the mineral value at zero porosity down to zero at $\phi_c$ c. This is the modified-Voigt dry frame the kernel implements, $K_{dry} = K_{min}\left(1 - \dfrac{\phi}{\phi_c}\right)$ dry=Kminleft(1−dfracphiphicright), with the same form for the shear modulus. For a clean quartz sand, $K_{min} = 36.6$ min=36.6 GPa, so the model reads $K_{dry} = 18.3$ dry=18.3 GPa at twenty percent porosity and falls to zero as porosity approaches forty percent. It is a modified Voigt bound because it connects the mineral point not to the other mineral but to the critical-porosity point where the frame vanishes.

Beyond the Threshold

Past the critical porosity the picture switches. There is no frame left to carry load, so the sediment is a suspension, and its modulus is no longer on the modified-Voigt line but on the lower bound, the Reuss value of Wood's equation for grains in fluid. The figure draws both branches: the modified-Voigt line descending from the mineral point to $\phi_c$ c, and the suspension region beyond it living on the floor, with a slider for $\phi_c$ c so you can watch the load-bearing branch stretch or shrink as the critical porosity changes with lithology. The two branches meet at $\phi_c$ c, which is exactly where a sand hands its stiffness over to its pore fluid.

This one empirical observation is what turns the bounds from a bracket into a usable model. The bounds told you the stiffest and softest a composition could be; critical porosity locates the load-bearing branch inside that bracket and gives you a first, honest dry frame with a single lithology parameter. Part 0's quickRock used it quietly to make the course thesis tangible, softening the mineral toward $\phi_c$ c before adding fluid. And that is where Part 2 closes and Part 3 opens: the bounds bracket the answer, critical porosity locates the frame, and the one ingredient every saturated rock still needs is its fluid. The next part gives the pore fluid real properties at reservoir conditions, through the Batzle-Wang relations.

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.