Soil is not a solid sponge but a labyrinth of pores of various sizes. After rain, water does not fill them all at once: first the narrowest ones, then the wider ones. Physicists built a theory of this process based on a simple image — a set of communicating vessels. The flows between pores obey the laws of viscous fluid flow, and the driving force is the difference in chemical potentials. Roughly speaking, it is water's "desire" to occupy a particular pore, taking into account capillary forces and gravity.
The key turned out to be the dimensionless Damköhler number for each pore. When it is small, water manages to distribute evenly: this is the classic slow absorption described by the Richards equation. When the number is large, water darts through selected paths without having time to "look around." This gives rise to fast preferential flows, familiar to anyone who has seen a puddle disappear in minutes rather than hours after a downpour. The theory shows that these are not two different phenomena but a smooth transition governed by a single parameter.
This view makes dozens of ad hoc models unnecessary. All the necessary information about the soil is the geometry of the pore network, which can be measured with a microtomograph. There is no need to fit parameters to field data. The equation itself decides where water should flow slowly and where quickly.
as written by the authors arXiv:2607.09416
We develop a statistical-mechanical theory of water in unsaturated soil whose outcome is a continuum field equation for the pore-occupancy g(r,x,t), the fraction of pores of radius r that are water-filled at position x and time t. The theory is built across three scales: microscopic inter-pore transfers set by Hagen-Poiseuille rates and a driving potential (the difference of pore-class chemical potentials, taken in capillary-gravitational form but open to adsorptive, osmotic, or thermal refinement); a mesoscale master equation relaxing the occupancy toward the equilibrium step g_eq=H(r*-r); and, on contracting the averaging volume to a point, the continuum balance d_t g + div F = C[g] - E - T, of which everything else is a limit, a moment, or a boundary resolution. The kinetic equation is an Onsager gradient flow descending the Gibbs free energy, with an H-theorem for the isothermal unforced system and mass conservation as its zeroth moment. A single dimensionless group, the pore-resolved Damkohler number Da(r,x), organizes the behavior and unifies phenomenologies long modelled separately. A Chapman-Enskog reduction identifies Richards' equation as the quasi-static (Da->0) limit, with matric potential and hydraulic conductivity K emerging only there and K vanishing below the percolation threshold; capillary-bundle and critical-path models are its diagonal and spectral limits. Hysteresis is the holonomy of a forcing bundle, a geometric phase rather than per-pore bistability, with a falsifiable loop-area law H ~ I^2. Preferential flow is what the same equation does where Da>1, so the Richards/preferential-flow dichotomy becomes a continuous Da-controlled crossover. Out of the quasi-static limit g(r) is the irreducible state variable. All inputs are geometric properties of the pore network, measurable from micro-CT and calibrated against no macroscopic data.
The Damköhler number, which governs water behavior in soil, originally came from chemical engineering and described how fast a chemical reaction proceeds relative to the flow of matter. Now it also "works" right under our feet.