A Rock as a Grain Pack

Grains Touching at Points

A sandstone is not a solid block with holes drilled in it. It is a heap of grains resting against one another, and a passing wave travels through the rock only where two grains actually touch. Those contacts are small, and each grain has only a handful of them. Everything about how stiff the dry frame is, how hard it resists being squeezed or sheared, is decided at those tiny contact patches. This is the reverse of how a mineral behaves. Quartz has a bulk modulus of 36.6 GPa because it is a continuous crystal with load paths everywhere; a loose quartz sand can have a dry frame near 2 GPa, because the wave never crosses continuous quartz, it threads through a sparse web of grain-to-grain contacts. Same mineral, a frame nearly twenty times softer, and the whole difference is the geometry of touching.

How Many Contacts, and How Hard Pressed

Two things set the stiffness of that web. The first is the coordination number $C$, the average number of contacts each grain shares with its neighbours. A well-packed sand puts more neighbours against each grain than a loose one does, and more load paths make a stiffer frame. Murphy (1982) gives a common estimate of $C$ from porosity alone: at a porosity $\phi$ of 0.36 it returns about 9.6 contacts per grain, at a tighter 0.30 about 11.1, and at a loose 0.40 only about 8.6. Fewer grains crowded in, meaning higher porosity, means fewer contacts and a softer pack. The second knob is the effective pressure pressing the pack together. Squeeze the grains harder and every contact flattens and widens, carrying more load, so the whole frame stiffens. Part 5.2 turns that pressure dependence into an exact law.

Why It Has to Be Modeled

Put the two knobs together and it becomes plain why a granular frame cannot be read from a reference table. Its modulus is not a property of quartz; it is a property of an arrangement, fixed by the porosity, the sorting, the packing, and the pressure, and a mineral table knows none of those. Two sands of identical mineralogy and identical porosity can carry different frames when one is better sorted or more compacted than the other. So the dry frame that Part 4 quietly demanded as an input, and there recovered only by inverting Gassmann on an existing log, now has to be built from the pack itself. The next section starts building it. Hertz-Mindlin theory takes the grain, the coordination number, the critical porosity, and the effective pressure, and returns the modulus of the random dense pack that anchors every granular model in this part.

References

• Murphy, W. F. (1982). Effects of microstructure and pore fluids on the acoustic properties of granular sedimentary materials (PhD thesis). Stanford University.
• Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.