The Statistical physics of unsaturated soil water: kinetic theory and non commutative pore water dynamics

I am giving this talk at the EGU General Assembly 2026 in Vienna last week, in the Hydrological Sciences division. The argument, in a single sentence: Richards' equation is not wrong, but it is the equilibrium limit of a deeper kinetic theory — in the same sense that the Navier–Stokes equations are the hydrodynamic limit of the Boltzmann equation for a gas. Mario Putti twenty years ago once asked me, "if not Richards, what else?"; this is my attempt at an answer that arrives after year dedicated to properly solve Richards equation, before with GEOtop and later with WHETGEO. 

The core object is a filling distribution g(r, x, t) : ℝ⁺ → [0, 1] that gives the volume fraction of pores of radius r that are water-filled at position x and time t. Theta is recovered as θ[g] = φ ∫₀^∞ g(r) f(r) dr. Hysteresis becomes the non-commutativity [W, D] ≠ 0 of the wetting and drying operators — geometry, not memory. Richards' equation is recovered as the small Damköhler limit Da → 0, with K(ψ) emerging as a derived transport coefficient built from the connectivity kernel C(r, r') rather than being postulated.

Materials

• Slides (PDF)  the deck I'll use in the presentation.
• Storyboard (DOCX)  the slide-by-slide reading guide, in five columns: spoken text, visual content, speaker notes, mounting comments. Useful if you want to present the same material yourself, or if you just want to follow along with what I actually said.
• Extended version of the slides — give me a few days — an annotated version with the full speaker text, more references, and the bits I had to cut for time.

Notebooks

These are the Jupyter notebooks I used to generate some of the figures in the slides, plus a few that produce supporting evidence in the supplementary material of the upcoming PRE papers. All run on top of OpenPNM 3.x and a small custom Y–L percolation code.

• Hysteresis_SWRC.ipynb — drainage and wetting branches in the (ψ, S_e) plane on a 3D pore network, with internal scanning curves. The figure on slide 9 of the talk comes from here. The notebook also documents an algorithmic artifact near the air-entry value (the missing air-trapping term during imbibition) — which is honest enough that I left it in.
• OpenPNM_Da_overshoot.ipynb — non-equilibrium overshoot in (θ, ⟨r⟩) and the universality crossover when the pore-size distribution becomes bimodal, governed by the Bhattacharyya overlap of the two modes.
• Percolation_K_threshold.ipynb — the percolation scaling K ∝ (θ − θ_c)^t with t ≈ 2, with finite-size scaling on three lattice sizes.
• subsection_pnm_mapping.tex — a short LaTeX subsection on how a two-tier pore-network maps onto the kinetic theory through a bimodal f(r) and a block-structured C(r, r'). Background reading for the OpenPNM notebooks.

Please find them zipped at this link.

Two upcoming papers

The full theoretical development is in two manuscripts, going to arXiv soon and submitted thereafter to Physical Review E --- give me a couple of weeks after EGU26:

1. The Statistical Physics of Unsaturated Soil Water: kinetic theory and non-commutative pore-water dynamics — the long paper. Builds the kinetic equation from the network thermodynamics, identifies the Onsager–Rayleigh gradient-flow structure, and proves that hysteresis is a geometric property of the configuration bundle (not a memory effect).
2. Richards' equation as a hydrodynamic limit: Chapman–Enskog derivation from the kinetic equation for unsaturated soil water — the short companion. Walks through the Chapman–Enskog expansion that recovers Richards' equation in the Da → 0 limit, with K(ψ) derived from the connectivity kernel.

Where this connects

The framework absorbs and extends a number of existing approaches that have been circling the same physics from different angles:

• Mixed-form Richards as the Da → 0 limit, with K(ψ) derived rather than postulated.
• Hassanizadeh–Gray as a thermodynamically consistent extension — pore-class-resolved here.
• Phase-field methods (Cahn–Hilliard) as gradient flow on a free energy — with explicit pore-network connectivity through C(r, r').
• Lucas–Washburn and its fractal variants as the single-capillary kinetic building block of C(r, r').
• Percolation-based hillslope frameworks with Damköhler and Péclet, where macropore activation is the Da > 1 transition.
• Compressible statistical soil mechanics (Einav–Liu 2023) — same occupancy dynamics governs the (ψ, σ') coupling.
• Freezing soil thermodynamics (Rempel et al. 2023, and our own work with Wani and D'Amato) — same kinetic framework with capillary pressure replaced by freezing-point depression.

This kinetic theory is not a parallel universe to Richards. It absorbs the existing physics, and it opens new measurements — directly observing g(r) is the obvious next experimental challenge