The Self-Consistent Medium
Learning objectives
- Read the self-consistent approximation as dropping the host privilege entirely: every phase, solid and pore, is an inclusion in the effective medium itself, solved by iteration
- Confirm the sensible-mixture behavior: quartz and clay in equal parts as spheres give K about 27.57 GPa, sitting between the Reuss 26.61 and Voigt 28.75 bounds
- See the percolation collapse: dry cracks of aspect ratio 0.02 in calcite drive K from 33.4 to 11.1 to 1.1 to 0 GPa at 2, 5, 8, and 10 percent crack porosity
- Read the difference from DEM as a physical choice about texture, not a numerical quirk: the self-consistent medium can lose connectivity while DEM never does
No Privileged Host
DEM kept the mineral as the connected host and the pores as isolated guests. The self-consistent approximation, SC, refuses to privilege anyone. In Berryman's symmetric form, every phase, the solid mineral and every pore family alike, is treated as an inclusion embedded in the effective medium itself, the very rock we are trying to find. That makes the definition circular, so it is solved by iteration: guess the effective moduli, embed each phase in that guess, recompute, and repeat until the answer stops changing. No phase is the background and no phase is the inclusion; they are all inclusions in their common product. This even-handedness is what gives SC its distinctive behavior, both the reassuring and the dramatic.
Sensible in the Middle
Start with the reassuring case. Mix quartz and clay in equal parts, both as round grains, and ask for the effective bulk modulus. SC returns 27.57 GPa. The Reuss lower bound for that mixture is 26.61 GPa and the Voigt upper bound is 28.75, so the self-consistent value lands neatly between them, exactly where a physically admissible effective modulus must sit. For ordinary solid mixtures SC behaves like a well-mannered averaging scheme, giving a single sensible number inside the bounds. Nothing here hints at what happens when the inclusions turn compliant.
Losing Percolation
Now make the inclusions cracks. Put dry cracks of aspect ratio 0.02 into calcite and raise their volume. The bulk modulus does not merely soften, it collapses: 33.4 GPa at 2 percent crack porosity, 11.1 at 5 percent, 1.1 at 8 percent, and 0 at 10 percent. The rock loses all stiffness at a finite, quite small crack density. This is percolation. In the self-consistent picture, once enough compliant cracks are present the solid can no longer be treated as a connected load path; the cracks link up through the effective medium and the mineral stops carrying stress. DEM, which by construction keeps its host connected forever, can never show this, its calcite would stay finite and positive. Neither model is wrong. They encode different beliefs about the rock's texture: DEM says the solid is a continuous skeleton that pores cannot break, SC says a dense enough population of cracks disconnects the solid. Choosing between them is choosing which rock you think you have, a cemented frame or a cracked one near failure. The next section stops asking which single texture a rock has and splits the pore space by mineralogy, building the workhorse frame for shaly sands.
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.
- O'Connell, R. J., & Budiansky, B. (1974). Seismic velocities in dry and saturated cracked solids. Journal of Geophysical Research, 79(35), 5412-5426.