The Stiff-Sand Line
Learning objectives
- Draw the stiff-sand trend as a modified Hashin-Shtrikman upper bound between the same two end members as the soft-sand line
- See that the two lines share both endpoints but the stiff line rides above the soft line everywhere between
- Read the gap: at porosity 0.30 the frame is 3.05 GPa soft against 5.99 stiff, and at 0.20 it is 5.74 against 13.78
- Read the stiff line as a sand stiffened by early diagenesis, and the two lines together as brackets on a sand's possible frame
The Same Ends, the Other Road
The soft-sand line took the two anchored end members, the Hertz-Mindlin pack at the critical porosity and the solid mineral at zero porosity, and joined them along the lower bound. The stiff-sand model joins the very same two points along the upper bound instead, the modified Hashin-Shtrikman connection that uses the stiff mineral as the material knitting the frame together. Nothing about the endpoints changes. At both lines are the mineral, GPa; at both are the Hertz-Mindlin pack, GPa. What changes is the road between them, and the upper road runs high.
How Wide the Gap Is
Lay the two curves on one plot and the space between them is the story. At a porosity of 0.30 the soft line gives a dry-frame bulk modulus of 3.05 GPa while the stiff line gives 5.99, nearly double. At 0.20 the split is wider still, 5.74 against 13.78, the stiff frame close to two and a half times the soft one at the same porosity. The two curves pin the same endpoints and fan apart in between, widest in the middle of the range where a sand has room to be either loosely friable or firmly knit. Any clean sand of a given porosity, whatever its history, has a dry frame that falls somewhere in that band, on the lower edge if it is friable, on the upper edge if it is stiff, and somewhere inside if it is partway between.
What Rides the Upper Edge
The stiff line is a geologic claim of its own. It describes a sand that carries more stiffness than packing alone would give, because early diagenesis has begun to stiffen the grain contacts, a touch of cement or pressure solution welding the grains a little without yet filling much pore space. Such a sand holds high porosity but is already firmer than a friable pack, so it rides near the upper bound. The soft and stiff lines are therefore not rival models to choose between; they are the two edges of where a clean sand can plausibly sit, and reading a measured sand against both is what turns a modulus into a statement about the rock. There is, though, one way a sand can be stiffer than even the upper bound suggests at high porosity, when cement lands precisely at the contacts and stiffens them out of all proportion to its volume. That near-vertical effect is the subject of the next section.
References
- Dvorkin, J., & Nur, A. (1996). Elasticity of high-porosity sandstones: Theory for two North Sea data sets. Geophysics, 61(5), 1363-1370.
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.