Pores as Inclusions

A Different Kind of Rock

Part 5 built the dry frame of a sandstone from the outside in: grains touching at points, load running through contacts, stiffness set by how many contacts there were and how hard the pressure pressed them together. That picture works because a sand is loose, a heap of grains that only just locks into place. But a tight carbonate or a shale is not a heap of grains. It is a nearly solid mass of mineral, pierced by pores. Its load does not thread through sparse contacts; it runs through continuous crystal that happens to have holes in it. To model such a rock we stop thinking about contacts and start thinking about the holes, and the single most important thing about a hole turns out to be its shape.

The Aspect Ratio

Model each pore as an oblate spheroid, a squashed sphere, and describe its shape with one number: the aspect ratio $\alpha$, the ratio of its short axis to its long axis, thickness over diameter. At $\alpha = 1$ the pore is a perfect sphere, a round hole. As $\alpha$ falls the pore flattens: at 0.1 it is a lens, at 0.01 it is a thin crack. The claim of this part is that $\alpha$, not porosity, is the hidden variable behind the wide scatter of stiff-rock velocities. To see it, hold the porosity fixed at 10 percent, fill the pores with water, and put them in calcite. As the aspect ratio drops from 0.8 to 0.15 to 0.05 to 0.02, the P-wave velocity falls from 6.11 to 5.41 to 4.15 to 2.54 km/s. Same mineral, same fluid, same pore volume, and a velocity spread of more than 3.5 km/s, carved out by pore shape alone.

Why a Crack Is So Devastating

The reason is mechanical. A round pore is a poor stress concentrator: squeeze the rock and the sphere barely deforms, because there is solid mineral on every side to carry the load around it. A sphere at 10 percent porosity in calcite drops the bulk modulus only from 76.8 to about 59.7 GPa, and lifts the velocity barely below the solid. A thin crack is the opposite. Its two faces sit close together and nearly parallel, so the smallest compression closes it, and while it closes it carries almost no load. A tiny volume of crack removes an enormous amount of stiffness, which is why the aspect ratio of 0.02 pore collapses the same rock to Vp 2.54 km/s. Round pores are almost free; cracks are ruinous, pore for pore. That single fact, that compliance is governed by shape and not by volume, is what the rest of Part 6 turns into working models. The next section builds the first quantitative machine for it, Kuster and Toksoz, which assigns every pore shape a pair of polarization factors and assembles them into an effective modulus.

References

• Berryman, J. G. (1980). Long-wavelength propagation in composite elastic media II. Ellipsoidal inclusions. Journal of the Acoustical Society of America, 68(6), 1820-1831.
• Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.