Biot consolidation theory

Remarks

The first equation is equilibrium with the poroelastic stress law, and the second is fluid mass balance with Darcy's law. Inertia is neglected. Example: one-dimensional (oedometric) consolidation of a layer with $u = (0, 0, w(z, t))$ under a constant applied vertical load. Integrating the first equation in $z$ shows that $(\lambda + 2\mu) \partial_z w - \alpha p$ equals the applied vertical total stress, so it is constant in time. So $\alpha \partial_t \partial_z w = \frac{\alpha^2}{\lambda + 2\mu} \partial_t p$, and the second equation becomes the diffusion equation below. $$\partial_t p = c \, \partial_z^2 p, \qquad c = \frac{k / \eta_f}{1/M + \alpha^2/(\lambda + 2\mu)}$$ This is Terzaghi's consolidation equation with consolidation coefficient $c$. Non-example: if the first equation is replaced by the wave equation with an inertia term $\rho \ddot u$, the system is Biot's dynamic theory of wave propagation, not the consolidation theory.