Abouthydrology

In Part 1 we built the toolkit on finite matrices: a Laplacian-like operator with \(\ker = \mathrm{span}\{\mathbf{1}\}\), the Fredholm alternative as the source of macroscopic equations, and the pseudo-inverse as the source of transport coefficients. Now we let the matrix indices become continuous and watch the same algebra, verbatim, derive Richards' equation. Thi...

This is the first of two posts written as a gentle companion to our recent work on the kinetic theory of unsaturated soil water and, in particular, to the derivation of Richards' equation as a Chapman–Enskog hydrodynamic limit. Before touching any soil physics, I want to isolate the linear algebra that makes the derivation work. It turns out to be the linear algebr...

Two quantities that have no business being related: the spectral gap of a pore network — a purely structural number, computed from the connectivity, with no flow solved anywhere — and the Stokes permeability of the same network, computed by actually solving viscous flow under a pressure drop. Drain the network step by step and they vanish at the same water ...