04 — Inelastic (Permanent) Compaction Theory

Virgin compression and the e–log σ′ curve

Soil mechanics describes a clay's state by its void ratio e (volume of pores ÷ volume of solids). Plotting e against the logarithm of effective stress reveals two distinct slopes:

Below the preconsolidation stress σ′_pc: the gentle recompression slope C_r — elastic, recoverable.
Above σ′_pc: the steep virgin compression slope C_c — inelastic, permanent.

\[ \Delta e = -C_c\,\log_{10}\!\left(\frac{\sigma'_f}{\sigma'_0}\right) \quad(\text{virgin}) \] \(C_c\) = compression index · \(C_r\) = recompression index · typically \(C_c \approx 5\text{–}20 \times C_r\)

Compaction follows from the change in void ratio over the original thickness:

\[ \text{strain} = \frac{\Delta b}{b_0} = \frac{\Delta e}{1 + e_0} \]

In specific-storage terms

The same physics, in the hydrogeologist's notation, is the inelastic skeletal specific storage:

\[ S_{skv} = \gamma_w\,\alpha_{sk(v)} \] \(\alpha_{sk(v)}\) = inelastic (virgin) skeletal compressibility

And the defining inequality of subsidence science:

\[ S_{skv} \gg S_{ske} \]

A head decline that crosses into the virgin range therefore produces far more compaction — and unlike the elastic part, none of it returns when heads recover.

Interactive: load past preconsolidation, then unload

Inputs

Applied effective stress σ′ 150 psi

Preconsolidation stress σ′_pc 150 psi

Compression index C_c 0.30

Recompression index C_r 0.03

Regime at current stress	—
Void ratio (start → now)	—
Permanent Δe after full unload	—
Permanent strain (of this interbed)	— %

Figure 1. The consolidation (e–log σ′) curve. Loading (solid) follows the gentle recompression slope C_r until it reaches σ′_pc (gray line), then bends onto the steep virgin slope C_c. The dashed green line is the rebound if stress were removed — it returns along a flat C_r slope, not back up the virgin curve, leaving a permanent loss of void ratio (the horizontal gap at σ′₀). Push σ′ past σ′_pc and watch the permanent strain grow. Each new maximum stress becomes the new σ′_pc (page 05).

Why virgin compression doesn't come back

Particle rearrangement

In virgin loading, plate-like clay particles collapse from an open, "card-house" fabric into a denser, face-to-face packing. That collapse dissipates energy; reversing the stress does not reverse the structural change.

Storage is destroyed

The lost void volume is lost pore space. The aquifer system can no longer store the water it once did — the inelastic component of storativity is a one-time withdrawal from a non-renewable account.

Resetting the threshold

After virgin loading, the clay is now "preconsolidated" to the new, higher stress. Re-loading up to that point is again elastic — until the next record stress is reached and a fresh increment of permanent compaction occurs.

How big is the contrast?

Lab consolidation tests and field analyses in the San Joaquin Valley give inelastic skeletal specific storage S_skv of roughly 1×10⁻⁴ to 1×10⁻³ ft⁻¹ — versus elastic S_ske near 1×10⁻⁶ to 1×10⁻⁵ ft⁻¹ (Riley 1969; Sneed 2001). A factor of 10–100. The same foot of head decline produces 10–100× more compaction once it crosses the threshold.

The management implication

Because the inelastic response is so much larger and is permanent, the entire goal of subsidence management reduces to a single idea: keep effective stress below the preconsolidation stress — equivalently, keep heads above the critical head. That threshold is the subject of page 05.

It also means historic subsidence is not "paid back" by later recovery — a basin can stabilize, but the storage already lost is gone.

Key references

Terzaghi, K., Peck, R.B. & Mesri, G. (1996). Soil Mechanics in Engineering Practice (3rd ed.). Wiley (consolidation, C_c/C_r, preconsolidation).
Riley, F.S. (1969). Analysis of borehole extensometer data from central California. In Land Subsidence, IAHS Publication 89, Vol. 2, p. 423–431.
Helm, D.C. (1975, 1976). One-dimensional simulation of aquifer system compaction near Pixley, California: 1. Constant parameters; 2. Stress-dependent parameters. Water Resources Research 11(3): 465–478; 12(3): 375–391.
Poland, J.F. (ed.) (1984). Guidebook to studies of land subsidence due to ground-water withdrawal. UNESCO Studies and Reports in Hydrology 40.
Sneed, M. (2001). Hydraulic and mechanical properties affecting ground-water flow and aquifer-system compaction, San Joaquin Valley, California. USGS Open-File Report 01-35.