Cement or Sorting

Three Trends, One Template

Part 5 has produced three curves for a granular sand, and their real use appears only when they are drawn together. On one plot of dry-frame bulk modulus against porosity, lay the friable soft-sand line, the stiff-sand line, and the near-vertical contact-cement line. They all pass through the same two anchors, the mineral at zero porosity and the Hertz-Mindlin pack at the critical porosity, but between those anchors they occupy very different ground. At a porosity of 0.30 the friable line reads 3.05 GPa, the stiff line 5.99, and the contact-cement line 12.76. Those three numbers at one porosity are not a contradiction; they are a menu. A measured sand at 0.30 porosity must have a frame somewhere among them, and which one it lands near is a readout of how the rock got the way it is.

Reading a Sand's History

This is the diagnostic the whole part was built toward. Drop a measured sand onto the template, its porosity on one axis and its dry-frame modulus on the other, and read the verdict from which trend it falls on. Land on the friable line and the sand lost its porosity the gentle way, by sorting and compaction, with essentially no cement holding the grains; it is well sorted and uncemented. Land on the steep contact-cement line, high above the others, and the stiffness was bought by cement welding the grain contacts; the rock is contact cemented, a diagenetic sand. Land between the friable and stiff lines and the sand is well sorted with only a little diagenetic stiffening, neither loose nor strongly cemented. A sand at 0.30 porosity reading near 3 GPa tells one story, the same porosity reading near 12 tells the opposite one, and the seismic response that follows, through Gassmann and into the reflection, follows the frame.

The Part Closes, and the Frame Changes Shape

Trace what Part 5 built. A grain pack is stiff only at its contacts (5.1). Hertz-Mindlin fixes the modulus of that pack at the critical porosity and makes it grow as the cube root of pressure (5.2). The soft and stiff lines connect that pack to the solid mineral along the lower and upper bounds, bracketing a clean sand's possible frame (5.3, 5.4). Contact cement climbs almost vertically away from the pack when cement lands at the contacts (5.5). Together they build the dry frame for a granular rock from grains, contacts, and cement, the input Part 4's Gassmann had to borrow, now made from first principles. But every one of these models rests on the same picture: a rock as touching grains. A tight carbonate or a shale is not that. Its stiffness comes not from grain contacts but from a nearly solid mineral mass pierced by pores, and whether those pores are round or crack-like changes the modulus enormously at the same porosity. Building the frame for such a rock needs a different geometry entirely, pores with shapes rather than grains with contacts. That is the inclusion-model story of Part 6, where the dry frame is built one more way.

References

• Avseth, P., Mukerji, T., & Mavko, G. (2005). Quantitative Seismic Interpretation. Cambridge University Press.
• Dvorkin, J., & Nur, A. (1996). Elasticity of high-porosity sandstones: Theory for two North Sea data sets. Geophysics, 61(5), 1363-1370.