The Soft-Sand Line

Part 5, Part 5: Granular Rocks and Contact Models

Learning objectives

Draw the friable-sand trend as a modified Hashin-Shtrikman lower bound between the Hertz-Mindlin pack and the solid mineral
Confirm the two endpoints are exact: the mineral at zero porosity and the Hertz-Mindlin frame at the critical porosity
Read the curve for a quartz sand: K near 5.74 GPa at porosity 0.20 and near 3.05 GPa at 0.30
Recognize this as the trend of a clean sand that lost porosity by sorting and compaction, with no cement

Joining the Two End Members

Hertz-Mindlin gave one point, the frame of a freshly packed sand at the critical porosity $\phi_c$ c. A real sand is rarely at exactly that porosity; it has usually lost some pore space and sits at a lower $\phi$ . To get its frame we need a curve that runs from the pack at $\phi_c$ c down to the solid mineral at $\phi = 0$ , and the soft-sand model supplies it. It connects those two known end members with a modified Hashin-Shtrikman lower bound: the softest frame consistent with a mixture of the two, treating the loose pack as one phase and the solid mineral as the other. The word lower is the whole physical claim. This is the least stiff a sand of this porosity can plausibly be, the frame of a grainy rock with nothing helping the contacts.

Exact at the Ends, a Lower Bound Between

The construction is pinned exactly where we know the answer. At $\phi = 0$ there is no pore space and the frame is the mineral itself, $K = 36.6$ GPa for quartz. At $\phi = \phi_c = 0.36$ c=0.36 the frame is the Hertz-Mindlin pack, $K = 2.05$ GPa. Between them the bulk modulus follows the lower-bound interpolation, and for the quartz sand it reads $K = 3.05$ GPa at a porosity of 0.30, 4.18 at 0.25, and 5.74 at 0.20. Each step of porosity lost stiffens the frame, and it stiffens gently, because the lower bound is the soft road between the pack and the mineral. The shear modulus follows the same shape on its own bound, so the model returns a complete dry frame, both $K$ and $\mu$ , at every porosity in the range.

What Kind of Rock This Is

The soft-sand line is not an arbitrary curve; it is a geologic claim. It describes a clean sand that lost porosity the gentle way, by better sorting and mechanical compaction, the grains rearranging and pressing closer without anything gluing them. A friable, unconsolidated sand, the kind that returns as loose grains on the drill floor, plots here. That is why the model is often called the friable-sand or unconsolidated-sand line. It matters that this is a lower bound, because a sand can be stiffer than this at the same porosity if diagenesis has begun to work on it. The very next section takes the same two endpoints and connects them along the upper bound instead, drawing the stiffest a sand of that porosity can be, and the gap between the two lines becomes the range in which real sands live.

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.