Clay interbeds don't compact instantly when heads drop in the adjacent aquifer. Pressure has to diffuse out through the low-permeability clay, so compaction lags by years to decades. This residual compaction is why subsidence can keep growing long after pumping is curtailed.
When the aquifer head drops, the edges of an interbed feel the new lower pressure immediately, but the interior does not. Pore pressure inside the clay equilibrates by Terzaghi's one-dimensional consolidation equation — a diffusion equation:
Compaction tracks the average dissipation of excess pore pressure across the bed — the degree of consolidation U, which rises from 0 toward 1 over a characteristic time set by how thick and how tight the clay is.
The single number that governs the lag:
Because τ scales with the square of thickness, doubling an interbed quadruples its equilibration time. A thick aquitard can take decades to centuries to fully compact.
| Consolidation coefficient cv | — ft²/d |
| Time constant τ (Tv = 1) | — yr |
| Time to 50% compaction | — yr |
| Time to 90% compaction | — yr |
| Time to 99% compaction | — yr |
If heads have been low, thick interbeds are still equilibrating. Even if pumping stopped today, the remaining excess pore pressure would continue to drain and the land would keep sinking for years — subsidence already "committed."
Extensometer records show compaction continuing during seasonal head recovery, because the deep interior of thick clays is still draining. This phase lag is itself a signature used to estimate cv and bed thickness.
Helm (1975, 1976) coupled this 1-D drainage process with stress-dependent storage to reproduce the Pixley and other San Joaquin records — the lineage behind today's MODFLOW SUB packages (page 09).
An interbed sandwiched between two aquifers drains from both faces, so its drainage path is only half its thickness (\(H = b'/2\)). A clay that drains to sand on one side and bedrock on the other drains from one face (\(H = b'\)). Since \(\tau \propto H^2\), a singly draining bed takes four times as long as the same bed draining both ways.
For thin interbeds (small τ), compaction effectively keeps pace with head changes — the "instantaneous" assumption used in simple storativity calculations. For thick aquitards (large τ), compaction is strongly transient and must be modeled with the diffusion equation. The ratio of pumping-stress timescale to τ tells you which regime you're in.
Doubly draining thin beds: minutes to months. Thick central-valley clay sequences: many decades.