Time delays in geophysical consolidation problems

The tectonic response of the lithosphere to loads applied over a period of years is one of the few relatively direct ways of measuring lithospheric mechanical properties. We discuss here a method for estimating gross permeability of shallow lithosphere if such a lithosphere can be modeled as a Biot solid. Induced seismicity at artificial lakes sometimes lags the history of lake filling. Clearly this indicates the anomalous load takes some finite time to create a stress increment over the tectonic regime associated with the lake. Such delays may result from diffusion into inhomogeneous regimes, but intuitively it seems that the Biot consolidation theory ought to contain the physics required to produce delayed response in the simplest model, an isotropic half-space with arbitrary vertical layering. The response of such a half-space can be calculated most quickly from matrix solutions of first-order differential equations. We explore here a consistent formulation for the physics of the problem and examine the relation between the rate of diffusion of changes on the boundary, the geometry of the boundary change, and the physical properties of the material. The resulting formulas can be used to estimate probable delays in response of the physical system. Unfortunately, the values of physical properties required to make such estimates are hard to obtain.