Work in progress
In Sion I could show only a few slides of each step. This post collects the whole series Statistical physics of soil water (version 2.5.1): the decks, and for each one the Real Book that writes out every step, the storyboard and the notebook that computes its figures. The series is built on the preprint arXiv:2607.09416 and two companion papers, one on the Chapman–Enskog reduction and one on water age.
The idea in a few lines. Richards' equation is usually postulated: mass conservation, the Darcy–Buckingham law, and two curves fitted to data. The series derives it, as the Navier–Stokes equations are derived from the kinetic theory of gases. The state of soil water is the occupancy g(r, x, t), the filled fraction of each class of pores. Water moves among pore classes inside a volume and between neighbouring volumes, which gives a kinetic equation with two currents. When redistribution among the pores is fast with respect to the forcing (small Damköhler number), a Chapman–Enskog expansion returns Richards' equation: the retention curve at order zero, the conductivity at order one. Where the Damköhler number is not small, Richards' equation fails, and that is where dual permeability, preferential flow and hysteresis come from.
Deck 0. The whole story
A talk built on three ideas: the state is a function, Richards' equation is a limit, the age of water is written in the pores. An appendix gives two slides for each deck.
→ Slides · Real Book · Storyboard · Notebook
Part I. The pieces: geometry, potential, state
Deck 1. The nature of hydraulic conductivity. K(θ) from a network of pores: Hagen–Poiseuille in each pore, connectivity between classes, a cutoff fixed by suction. Burdine and Mualem are two bounds of one network; K vanishes at percolation.
→ Slides · Real Book · Storyboard · Notebook
Deck 2. Micro potential and macro potential. Capillary, adsorptive, osmotic and gravitational terms; the potential between two pores and between places. Suction is the level of the largest filled pore, defined only at equilibrium.
→ Slides · Real Book · Storyboard · Notebook
Deck 3. The occupancy as the state. Many occupancies share one water content; this degeneracy is where history dependence comes from. How the occupancy could be measured.
→ Slides · Real Book · Storyboard · Notebook
Deck 3b. Equilibrium occupancies. Gravity, films, solutes and temperature, one at a time, and what each does to the equilibrium step and to a measured retention curve.
→ Slides · Real Book · Storyboard · Notebook
Part II. From the pore to Richards' equation
Deck 4. The kinetic continuum equation. The budget of one pore class in one volume, the exchange between volumes, the limit to a point: one equation, two currents.
→ Slides · Real Book · Storyboard · Notebook
Deck 5. Mass conservation and the forbidden operation. Integrating over pore sizes gives mass conservation, exact and empty; integrating first and then looking for a flux law in θ is the shortcut of the classical theory.
→ Slides · Real Book · Storyboard · Notebook
Deck 6. The right procedure. Two clocks and their ratio, the Damköhler number; the shape of the state is slaved and eliminated.
→ Slides · Real Book · Storyboard · Notebook
Deck 7. Free energy and the Rayleighian. A Fermi–Dirac occupancy minimises the free energy; the kinetic equation descends it.
→ Slides · Real Book · Storyboard · Notebook
Deck 8. The Chapman–Enskog development. The expansion term by term: Richards' equation as a limit, and the band of conditions where it does not hold.
→ Slides · Real Book · Storyboard · Notebook
Deck 9. The spectrum. The conductivity as a sum over relaxation modes; the gap closes at percolation; Burdine and Mualem return as trial functions.
→ Slides · Real Book · Storyboard · Notebook
Part III. Structure and memory
Deck 10. Soil structure and dual permeability. A bimodal pore population splits the spectrum into a fast and a slow band: dual permeability derived, with the exchange coefficient computed.
→ Slides · Real Book · Storyboard · Notebook
Deck 11. Hysteresis. Wetting and drying do not commute: hysteresis as a holonomy, a loop that does not close.
→ Slides · Real Book · Storyboard · Notebook
A first look at water age
Deck 12A. Volume and flux. Tracers see the pores weighted by volume, the water balance by flux: why tracers find preferential flow everywhere while water balances close with Richards' equation.
→ Slides · Real Book · Storyboard
The rest of the work on water age and travel times (StorAge Selection functions derived from the occupancy, the age contrast between storage and discharge, and the lectures Travel times, kinetic edition) is not yet mature, and will have a post of its own.
Related. The MOOC on the hydrology of soil and the Lausanne lectures for the classical theory; the Sion lectures for the short path through it.
© 2026 R. Rigon, CC BY 4.0 (cover paintings excluded). Notebooks need numpy, scipy and matplotlib.
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