← All summaries · 02 Effective Stress

Elastic deformation = the recompression range

When effective stress increases but stays below the preconsolidation stress σ′pc (the historic maximum the sediment has "felt" before), grains shift only slightly within their existing packing. Remove the load and they spring back. This recoverable deformation is governed by the elastic skeletal specific storage:

\[ S_{ske} = \gamma_w\,\alpha_{sk(e)} \qquad \Delta b = S_{ke}\,\Delta h \] \(S_{ske}\) = elastic skeletal specific storage (1/L) · \(\alpha_{sk(e)}\) = elastic skeletal compressibility · \(S_{ke} = S_{ske}\,b_0\) = elastic skeletal storativity of a bed of thickness \(b_0\)

The compaction of a bed equals its storativity times the head change. In the elastic range the same coefficient applies to loading and unloading, so a head cycle that returns to its starting point leaves no net compaction.

It's the storage you already know

Aquifer storativity has two parts: water expansion/compression and skeletal compression. The skeletal part, in its elastic range, is Sske. So elastic compaction is just the visible, vertical expression of releasing water from elastic storage.

Typical values: elastic skeletal specific storage is small — on the order of 1×10⁻⁶ to 1×10⁻⁵ ft⁻¹ for aquifer materials. Real but modest.

Interactive: a seasonal head cycle in the elastic range

Inputs

Peak seasonal compaction (per cycle) in
Net compaction after full cycles≈ 0 (recoverable)

Compaction shown in inches for a single representative interbed; the elastic loop closes because the same Ske applies on the way up and down.

Figure 1. Top: head (blue) and the resulting elastic compaction (orange) over several seasons — compaction tracks head with no lag and returns exactly to its starting value each year. Bottom: the same data as a stress–strain path. Because loading and unloading follow the same line, the path is reversible and encloses no area — the signature of purely elastic behavior. Contrast this with the permanent offset that appears once stress crosses σ′pc (pages 04–05).
Where elastic behavior holds — and where it stops
Stays elastic

Above the historic low

If seasonal drawdown never pushes effective stress past σ′pc — i.e., heads never fall below the historic low — deformation is recoverable and net subsidence is negligible over the long run.

At the threshold

The preconsolidation stress

σ′pc records the deepest the skeleton has ever been loaded. Reach it and the response changes character: the stiff elastic spring gives way to soft, permanent virgin compression (page 04).

Goes inelastic

Past the threshold

Beyond σ′pc, compressibility jumps by 1–2 orders of magnitude and the deformation no longer rebounds. The same head decline now produces far more — and permanent — subsidence.

Why elastic rebound looks "free" — but isn't quite

In aquifers managed within the elastic range, water levels and the land surface oscillate seasonally and recover. Extensometer records from such sites show clean, closed annual loops (Riley 1969). This is the target condition for sustainable management.

The catch: even small inelastic "leakage" each year accumulates, and thick aquitards equilibrate so slowly (page 06) that an apparently elastic seasonal record can hide slow permanent compaction at depth.

The two skeletal storativities

The whole of subsidence theory turns on comparing two coefficients:

\[ S_{skv} / S_{ske} \approx 10\text{–}100 \] \(S_{ske}\) = elastic (recoverable) · \(S_{skv}\) = inelastic (permanent, page 04)

Elastic storage is what you get back. Inelastic storage is what you spend permanently. Telling them apart in field records is the subject of page 07.

Key references

  1. Jacob, C.E. (1940). On the flow of water in an elastic artesian aquifer. Transactions, AGU 21(2): 574–586.
  2. Riley, F.S. (1969). Analysis of borehole extensometer data from central California. In Land Subsidence, IAHS Publication 89, Vol. 2, p. 423–431.
  3. Helm, D.C. (1975). One-dimensional simulation of aquifer system compaction near Pixley, California: 1. Constant parameters. Water Resources Research 11(3): 465–478.
  4. 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.
  5. Domenico, P.A. & Schwartz, F.W. (1998). Physical and Chemical Hydrogeology (2nd ed.). Wiley (Ch. 8–9, deformation and storage).