Richards as a limit: a derivation of Richards' equation from the Continuum Kinetic soil Equation

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 content. Not approximately. Identically, both zero, at the same θ. That coincidence is the subject of this post: it is the point where a macroscopic constitutive law stops existing, and the theory says so by itself.

In July I posted the first of the two papers — the kinetic theory of the pore-occupancy g(r) — and promised the companion "shortly." Here it is:

Richards' equation as a hydrodynamic limit: Chapman–Enskog reduction of the continuum kinetic equation for unsaturated soil water
R. Rigon, https://arxiv.org/abs/2607.17358v2, 24 pages, 1 figure, 8 appendices, with numerical Supplemental Material.

It goes to Physical Review E, like its companion. The first paper said: θ is not enough, the state is g(r). This one says the other half, and it is the half that makes the first one respectable.

The argument in one sentence

Richards' equation is not an assumption of soil physics. It is a theorem — the solvability condition of a kinetic equation whose redistribution operator has exactly one invariant.

That is the whole paper. Everything below is a consequence.

Two limits that hydrology has always taken together

The reason Richards' equation has never been derived, only motivated, is that the passage from pores to fields conflates two entirely different operations. The paper's first move is simply to take them apart.

• The spatial limit, ε = L/Λ → 0, shrinks the representative volume to a point. It is pure kinematics: it says nothing whatsoever about time scales. What comes out is a closed continuum kinetic equation, ∂g/∂t + ∇·F = 𝒞[g], with F a pore-resolved flux that is still entangled — a transport coefficient and a driving gradient multiplied together inside one kernel, inseparable.
• The temporal limit is where the physics is. Redistribution among pores is fast; the forcing is slow; their ratio is the Damköhler number Da. When Da ≪ 1 the soil is pinned near local equilibrium, and one can expand — a Chapman–Enskog reduction, exactly as one passes from Boltzmann to Navier–Stokes.

Keeping them apart is what makes the structure visible, and it is why the paper can be honest about where each classical assumption enters.

The one operator everything depends on

Linearise the redistribution operator about equilibrium and you get 𝒥, and then the whole reduction is a statement about 𝒥. Three properties, and nothing else, do all the work:

1. 𝒥 is self-adjoint — in the mass inner product, ⟨u,v⟩ = ∫ u v f dr. And here is the small piece of algebra I find most satisfying in the paper: self-adjointness and conservation of water are literally the same statement. Detailed balance K(r,r′) f(r) = K(r′,r) f(r′) holds identically because the mobility is symmetric, and that single fact gives you both.
2. Its kernel is one-dimensional. There is exactly one thing redistribution cannot change: water. The Boltzmann gas has five invariants and therefore five macroscopic equations. Viscous pore flow has one invariant, and therefore one macroscopic equation.
3. It has a spectral gap — a slowest relaxation rate λ₁ > 0 — which is what makes the expansion asymptotic at all.

Then the derivation is almost mechanical. Take the f-moment of the first-order equation: redistribution drops out (that is the null space), and what remains is a condition on the source. That condition — the Fredholm alternative, the unglamorous requirement that the first-order correction should exist at all — is mass conservation. It is Richards' equation.

Five things that fall out, which I did not put in

1. Hydraulic conductivity is a transport coefficient. K is not a constitutive input. It is the first-order Chapman–Enskog coefficient, K = φ⟨κ|𝒥⁻¹|S⟩ — the exact structural counterpart of viscosity in the kinetic theory of gases. Which means, among other things, that K is a property of the medium's relaxation spectrum and is independent of the forcing, in the same sense that viscosity does not depend on the shear rate you apply.

2. The classical formulas are approximations of that coefficient. Take the mean field of it and you get back the standard Mualem–Burdine integral. Resum the serial paths — the fact that large pores must push through small-pore bottlenecks — and out comes Mualem's heterogeneity penalty exp(−4σ²), not as an empirical factor but as a Neumann series that collapses to a harmonic mean. I did not expect that to work as cleanly as it did.

3. Dual-permeability models are derived, not posited. This is the result with the widest practical reach. Give the operator a bimodal pore-size distribution and its relaxation spectrum splits into two bands separated by a gap. Apply Chapman–Enskog within each band, keep the slow cross-band relaxation, and two coupled Richards equations fall out — Gerke–van Genuchten, with Weiler's IN3M as the three-band case and mobile–immobile as the limit where one band is conductively dead. And the exchange coefficient Γ_w, which everybody fits, is computed from the cross-band connectivity. What was a modelling choice becomes a theorem with a formula.

4. The theory predicts its own breakdown — by two different routes. Raise the forcing and Da → 1: sharp fronts, the expansion fails, and you are in the preferential-flow regime the first paper described. But lower the water content and something else happens: at the percolation threshold the spectral gap closes, ‖𝒥⁻¹‖ diverges, and the closed conductivity ceases to exist. These are genuinely two different exits from the Richards regime — one by fast forcing, one by loss of connectivity — and the theory locates both. That is the figure at the top: the gap and the permeability going to zero together.

5. Hysteresis and dynamic capillarity have an operator address. The non-commutativity [𝒲,𝒟] ≠ 0 from the first paper turns out, at the operator level, to be the curvature of the projection onto local equilibrium — the same object that carries dynamic capillarity. I am not claiming to have solved hysteresis. I am claiming to know where in the mathematics it lives.

The part I am least able to hide behind

Point 4 above is the paper's most exposed claim, so it is the one I made numerical. The Supplement does three parameter-free computations, on soils I can name:

• A loam (unimodal, median 8 μm, draining around −1.9 m). Diagonalise the operator: one exact invariant (λ₀ ≈ 10⁻¹⁷), and the modes turn out to be localized by pore radius, each relaxing at the local rate of its own radius, to within 1%. They are not standing waves on the band — I had assumed they were, wrote it into an early draft, and the numerics said no. Ordering the modes by rate is ordering the pores by size: slow modes on small pores.
• The same loam, drained on a 24³ pore network. Gap and permeability vanish together at θ_c ≈ 3.5×10⁻³. Below it, no spanning cluster: the water is there, it simply cannot go anywhere.
• A structured soil (matrix at 4 μm plus macropores at 40 μm). The spectrum splits into two bands with a slow, sign-changing mode across the split — the exchange mode, appearing on its own. And projecting the operator onto the two bands gives an exchange rate that converges to the true one as the bands decouple. The split radius the operator produces falls at ψ ≈ −1 m ≈ −10 kPa, which is where soil physicists have been drawing the macropore boundary by hand for decades.

Where this connects

Nothing here replaces anything. Capillary-bundle models are the diagonal limit; Mualem and Burdine are the mean field; critical-path analysis is the spectral limit near θ_c; Gerke–van Genuchten is the two-band projection. The classical results are not overturned — they are located, each one identified as a particular approximation to a single operator inversion. That is the most useful thing a derivation can do for a field: not to declare the old results wrong, but to say precisely what they are approximations of, and therefore when they will fail.

What I am not claiming

Again, better said by me than to me.

This is a formal Chapman–Enskog reduction, in the sense the phrase carries in kinetic theory: the expansion is organised in powers of Da and closed order by order, but I do not prove convergence, and the higher-order remainders are not bounded rigorously. The closures come from the companion paper and are physical, not derived from molecular dynamics. The numerics are on synthetic networks, not on imaged soils. And the linearisation that makes 𝒥 an operator at all is exactly that — a linearisation, with the nonlinearity pushed into successive sources.

There is also one honest limitation inside the numerics that I have written into the Supplement rather than left for a referee to find: away from the threshold, the spectral gap of a growing cluster is increasingly dominated by its size rather than its connectivity, so the correlation between gap and permeability is only meaningful near θ_c. What is unambiguous — and all the argument needs — is that they vanish together, and that is exact rather than statistical.

Materials

• Paper: https://arxiv.org/abs/2607.09416
• Companion paper: arXiv:2607.09416 — the kinetic theory this reduces
• The first post: If not Richards, what else ?
• Numerical Supplement (included in the main file) + Jupyter notebook: every number and figure above is reproducible; the notebook runs top to bottom in a few seconds.
• "The Real Book": a companion document I wrote for myself and then decided to keep — the entire derivation worked step by step, blackboard style, nothing skipped, with boxes reminding the reader of the linear algebra (Fredholm alternative, pseudo-inverse, graph Laplacians, Rayleigh quotients) and a glossary. If the paper looks forbidding, start there. It is the gentlest way in.
• On the same line of the Real Book, I also provide a little rehearsal on Linear Algebra and Linear operators, whose knowledge is necessary to the understanding of the paper calculations. 

As always: comments, objections and counterexamples are welcome — especially the counterexamples. This paper makes a falsifiable structural claim (Γ_w computable from connectivity, gap closure at θ_c) and I would rather find out early.

Water In Soil - A MOOC

I'm happy to share that my new MOOC, SOIL – The Hydrology of Soil, is now live on the University of Trento's MOOC platform. It's free, open, and self-paced, and it's aimed at anyone who wants to properly understand how water moves through unsaturated soil — not just as a set of formulas to memorize, but as a coherent chain of physical reasoning.

What the course is about

The course covers the hydrology of unsaturated soils and builds up, step by step, the mathematical tools needed to describe water flow through them — culminating in the Richards equation, the cornerstone of unsaturated flow theory.

It's organized into five chapters, each combining short video episodes, further readings, hands-on activities, and self-assessment quizzes:

1. What is Soil — the basic quantities used to describe water content and structure in soil.
2. The Energy of Water in Soil — how water's energy is distributed, capillary pressure, and the construction of soil-water retention curves.
3. Darcy and Buckingham's Laws — from Darcy's law in saturated soil to Buckingham's extension for unsaturated conditions, and how hydraulic conductivity is derived.
4. Soil Water Budget — the mass budget, conservation laws, and the derivation of Richards' equation itself (plus its groundwater counterpart).
5. Solving Richards' Equation and Its Limits — pedotransfer functions, numerical solution strategies, macropores, and where the classical theory breaks down.

What you'll be able to do by the end

By the end of the course, you should be able to explain the physical meaning of the water retention curve and the Richards equation and the reasoning that connects them, solve simple unsaturated flow problems by choosing the right constitutive relationships, and critically evaluate when the Richards-equation framework is — and isn't — a valid description of what's happening in real soil.

Who it's for

If you work in hydrology, agronomy, environmental engineering, or soil science — or you're a student who wants to go beyond a black-box use of Richards' equation — this course is built for you. The course can be seen as an introduction to the theory of the WHETGEO model.  WHETGEO provides an open source tool for solving 1D and 2D Richards equation. 

Try it

The course is free to enroll: mooc.unitn.it/course/view.php?id=32

If not Richards, what else ?

Two soils with the same water content θ are not in the same hydraulic state. The pore-occupancy g(r) — the fraction of pores of radius r that are water-filled — distinguishes them, while θ, being an integral of g, cannot. The navy step is the reference equilibrium g_eq = H(r* − r): water fills the small pores first. Everything else on the plot is a state that Richards' equation is blind to.

In May I posted the talk I gave at EGU 2026 in Vienna, and promised the two papers “in a couple of weeks after EGU.” It took a little longer than that — it always does — but the first one is now public:

The Statistical Physics of Unsaturated Soil Water: kinetic theory and non-commutative pore-water dynamics
R. Rigon, arXiv:2607.09416 [cond-mat.stat-mech], 22 pages, 9 figures, 2 appendices.

It is going to be submitted to Physical Review E. The companion paper — the Chapman–Enskog derivation that recovers Richards' equation as a hydrodynamic limit — follows shortly, and I will post it here when it lands.

The argument in one sentence

Unchanged from the talk, and worth repeating because everything else is a consequence of it:

Richards' equation is not wrong; it is the equilibrium limit of a deeper kinetic theory — in the same sense that Navier–Stokes is the hydrodynamic limit of Boltzmann's equation for a gas.

Mario Putti asked me, twenty years ago, “if not Richards, what else?” This is my attempt at an answer, and it arrives only after many years spent trying to solve Richards' equation properly — first with GEOtop, later with WHETGEO. You have to take an equation seriously for a long time before you earn the right to say what it is missing.

What the theory actually says

The state variable is not θ. It is the pore-occupancy g(r, x, t): the fraction of pores of radius r that are water-filled at position x and time t. Water content is recovered as a moment of it, θ[g] = φ ∫ g(r) f(r) dr — which is precisely the point: θ is an integral of g, so it throws information away. Two soils with the same θ, one wetted by rain (which fills pores by areal exposure, favouring the large ones) and one drained to the same θ (which empties the large ones first), are in genuinely different hydraulic states. They will conduct water differently, and they will respond to the next rainfall differently. Richards' equation cannot see the difference. That is the figure above, and that is the whole motivation.

The theory is built by passing through three scales, and I think this is the part hydrologists will find easiest to trust, because each step is ordinary physics:

• Microscale. A single water transfer between two pores is set by a Hagen–Poiseuille rate and driven by the difference of pore chemical potentials Φ(r, r′) — capillary and gravitational here, but open to adsorptive, osmotic, or thermal refinement without touching the structure of the theory.
• Mesoscale. Averaging over a representative volume gives a master equation — a gain–loss (Boltzmann-type) kinetic equation whose terms relax the occupancy toward its equilibrium, with a connectivity kernel C(r, r′) that encodes which pores can actually talk to which.
• Macroscale. A Chapman–Enskog reduction gives back Richards' equation in the quasi-static limit Da → 0.

Everything the theory needs as input is a geometric property of the pore network — measurable from micro-CT. Nothing is calibrated against macroscopic hydrological data. I want to be blunt about how unusual that is, and how exposed it leaves me: the theory makes parameter-free predictions, and parameter-free predictions can be wrong in public.

Four things that fall out, which I did not put in

This is the part I care about. These were not assumptions; they are consequences.

1. Matric potential and hydraulic conductivity exist only in the limit. ψ and K are not primitive quantities of the theory. They emerge at Da → 0, and K is derived from the connectivity kernel rather than postulated. Below the percolation threshold, K vanishes — not as a fitting choice, but because the water phase stops spanning the medium. Field capacity gets a geometric meaning: θ_FC ≈ θ_c.

2. Hysteresis is geometry, not memory. It is the holonomy of a forcing bundle — a geometric phase, arising from the non-commutativity [W, D] ≠ 0 of the wetting and drying operators. Wetting fills by areal exposure; drying empties by capillary ordering; the two operations do not commute, so a closed loop in the forcing does not return you to where you started. Independent-domain and Preisach models posit bistable pores and reproduce the loop. Here the loop is derived, and it comes with a falsifiable prediction: the loop area scales as H ∼ I² with the forcing intensity. Domain models are rate-independent and predict no such thing. That is a clean experimental discriminant, and I would very much like someone to go and measure it.

3. Preferential flow is not a separate process. It is what the same equation does when Da > 1. The molecular-chaos (Stosszahlansatz) closure that underlies the kinetic equation fails exactly when pore occupancies become correlated near the percolation threshold — and that correlated, channelized regime is fingering and preferential flow. So the Richards / preferential-flow dichotomy dissolves into a continuous, Da-controlled crossover. We do not need two domains and a phenomenological exchange term; we need one equation and an honest look at its Damköhler number.

4. Out of the quasi-static limit, g(r) is irreducible. No single scalar — not θ, not ψ — is a complete description. And, as I discovered while revising the manuscript (a lesson in the value of being asked a hard question at the right moment): this is true even at equilibrium. With gravity present in a finite volume, the equilibrium occupancy is not the sharp step H(r* − r) at all; it is a smeared step, because a large pore low in the profile can stay filled while a smaller pore higher up has already drained. Each pore holds water within its own Jurin rise. The retention curve — the last place where the classical scalar picture was supposed to be exact — is not exact either.

Where this connects

The framework absorbs rather than replaces. Capillary-bundle models are its diagonal limit; critical-path models are its spectral limit; Hassanizadeh–Gray is a thermodynamically consistent extension, here resolved pore-class by pore-class; phase-field methods are gradient flow on a free energy, here with explicit network connectivity; dual-permeability models are the Da > 1 regime, without the phenomenology. The same machinery, with capillary pressure replaced by freezing-point depression, is the freezing-soil problem I have worked on with Niccolò Tubini and John Mohd Wani.

This is not a parallel universe to Richards. It contains it.

What I am not claiming

I would rather say this myself than have it said to me. The paper is a construction, not a rigorous reduction from molecular dynamics: the closures are posited on physical grounds and judged by their consequences. The full kinetic equation has not yet been solved on a real soil — the numerics live in the companion paper and in the supplementary demonstrations. And the single most obvious next step is also the hardest and the most interesting one:

directly observing g(r).

Micro-CT can see it. Nobody, as far as I know, has yet used it to test a kinetic theory of soil water. If you work with imaging of pore-scale water and this sounds like a collaboration, write to me.

Materials

• Paper: arXiv:2607.09416
• The EGU 2026 talk (slides, storyboard, and the notebooks behind the figures): the May post — still the gentlest way in, if the paper looks forbidding.
• Code: will be added on GitHub; the OpenPNM notebooks that generate the supporting figures are already in the OSF repository linked from the talk.
• The companion paper: "Richards as a Limit"  where Richards equation is derived from the main general equation. Here.
• My MOOC (Massive Open Online Course) on Water in Soil. This, especially the part related to the energy of water in soil, can be though as an introduction to these more advanced topics. 

Comments, objections, and counterexamples are all welcome — especially the counterexamples. A theory that cannot be attacked is not saying anything.

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

Five Paradoxes of Soil Hydrology (Observations that quietly undermine equilibrium soil physics)

Unsaturated flow theory is one of the cornerstones of hydrology. For nearly a century, the Richards equation has provided a mathematical framework for describing how water moves through partially saturated soils. At its core lies a powerful simplification: the hydraulic state of soil can be described by water content alone.

Yet decades of experiments tell a different story.

Across laboratories, field sites, and scales, soil water exhibits behaviors that contradict this assumption in systematic ways. These contradictions have become known, implicitly if not always explicitly, as paradoxes of vadose zone hydrology. They persist not because of experimental error, but because they expose limits in the classical conceptual model.

Below, we review five such paradoxes that continue to shape how hydrologists think about unsaturated flow.

---

1. Hysteresis

The same water content, different hydraulic states

Observation

During wetting, soils follow a different relationship between water content and matric potential than during drying. Hydraulic conductivity likewise differs between wetting and drying paths, even at identical water content.

Why this is paradoxical

Classical theory assumes a unique retention curve and unique conductivity function. Hysteresis directly violates this assumption and implies that soil retains memory of its past.

---

2. Rate Dependence

Why infiltration speed changes soil properties

Observation

Fast infiltration experiments routinely yield hydraulic conductivities several times larger than values obtained under slow, quasi-static conditions, even in the same soil.

Why this is paradoxical

Hydraulic conductivity is assumed to be a material property. If that were true, it should not depend on how quickly water is applied.

---

3. Scale Dependence

Why field conductivities exceed laboratory values

Observation

Field-scale saturated or near-saturated hydraulic conductivities are often one to two orders of magnitude larger than laboratory measurements on the same soil material. The discrepancy increases with measurement scale.

Why this is paradoxical

If conductivity is intrinsic to the soil, it should not depend on the size of the experiment.

---

4. Persistence of Compaction Effects

Why soils don’t recover hydraulically

Observation

Mechanical compaction reduces hydraulic conductivity dramatically. Even after bulk density and porosity appear to recover, conductivity often remains suppressed for years.

Why this is paradoxical

If conductivity depends primarily on porosity, it should recover once porosity does.

---

5. Non-Commutativity of Wetting and Drying

Why the order of processes matters

Observation

Wetting followed by drying does not lead to the same hydraulic state as drying followed by wetting, even if final water content is identical.

Why this is paradoxical

In classical physics, state variables are path-independent. Soil water violates this expectation.

---

A Shared Message from Five Paradoxes

Each paradox has often been addressed with a separate modeling fix—hysteresis rules, dynamic conductivity, macropore domains, or empirical memory terms. Taken together, however, they point to a single conclusion:

Water content alone is insufficient to describe the hydraulic state of soil.

Soils exhibit memory, path dependence, and sensitivity to forcing because internal processes do not instantaneously equilibrate.

---

Why Hydrologists Should Care

These paradoxes affect:

• infiltration and runoff prediction

• groundwater recharge estimates

• irrigation efficiency assessments

• land–surface and Earth system models

• transfer of parameters from lab to field

They remind us that the vadose zone is not a passive filter but a dynamic system with internal states and history.

---

References

Haines, W. B. (1930). Studies in the physical properties of soil: V. The hysteresis effect. Journal of Agricultural Science, 20(1), 97–116. DOI: 10.1017/S002185960008864X

Mualem, Y. (1974). A conceptual model of hysteresis. Water Resources Research, 10(3), 514–520. DOI: 10.1029/WR010i003p00514

Smiles, D. E., Vachaud, G., & Vauclin, M. (1971). A test of the uniqueness of the soil moisture characteristic during transient, nonhysteretic flow. Soil Science Society of America Journal, 35, 534–539. DOI: 10.2136/sssaj1971.03615995003500040007x

Beven, K., & Germann, P. (1982). Macropores and water flow in soils. Water Resources Research, 18(5), 1311–1325. DOI: 10.1029/WR018i005p01311

Hamza, M. A., & Anderson, W. K. (2005). Soil compaction in cropping systems: A review. Soil and Tillage Research, 82, 121–145. DOI: 10.1016/j.still.2004.08.009

Philip, J. R. (1964). Similarity hypothesis for capillary hysteresis. Soil Science, 97(3), 155–164. DOI: 10.1097/00010694-196403000-00001

Kool, J. B., & Parker, J. C. (1987). Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties. Water Resources Research, 23(1), 105–114. DOI: 10.1029/WR023i001p00105

Musical Coda

Minkowski functionals: Critical Limitations and Future Directions for Minkowski Functionals in Soil Hydrology

Go to Part I Go to Part III

 Revisiting the Hadwiger Theorem

In Part 1, we introduced Minkowski functionals as a mathematically complete framework for geometric characterization, grounded in the Hadwiger theorem. Let's examine this completeness claim more carefully, because understanding what it means—and what it doesn't mean, is crucial for applying these tools appropriately in soil hydrology.

The Hadwiger theorem states that any continuous, motion-invariant, and additive functional defined on convex bodies in n-dimensional Euclidean space can be expressed as a linear combination of exactly (n+1) Minkowski functionals. In three dimensions: volume, surface area, integral mean curvature, and Euler characteristic—no more, no fewer.

The theorem's elegance lies in its completeness for static geometric description. The motion-invariance property ensures that measurements don't depend on how we orient or position our coordinate system, while additivity means that measuring two separate objects equals the sum of measuring them individually (accounting correctly for any overlap). These four functionals capture all the geometric information that can be extracted from a spatial pattern in a coordinate-independent, additive way.

For soil hydrologists, the Hadwiger theorem provides mathematical assurance: when we use these four functionals to characterize water distribution, we're working with a complete geometric description under these mathematical constraints. There are no "hidden" geometric properties we're missing that satisfy motion-invariance and additivity.

But, and this is crucia, soil hydrology involves much more than static, isotropic geometry.

Critical Limitations: When Geometry Is Not Enough

While the Hadwiger theorem guarantees that Minkowski functionals provide a complete geometric characterization, we must recognize fundamental limitations when applying these tools to soil hydrology. The mathematical elegance should not obscure the physical complexity of water movement in soils.

The Problem of Motion-Invariance

The "motion-invariance" in Hadwiger's theorem means that geometric measurements don't depend on coordinate system orientation—rotate your sample, and the Minkowski functionals remain unchanged. This is a property of how we measure geometry, not a statement about physical processes.

But water flow in soils is profoundly direction-dependent:

• Gravity creates fundamental vertical asymmetry
• Upward capillary rise versus downward drainage follow completely different dynamics
• Preferential flow paths depend on structural orientation
• Hydraulic conductivity is often anisotropic due to soil structure (layering, aggregation, root channels)

Minkowski functionals, being isotropic measures, cannot capture this directional information. Consider a simple thought experiment: imagine a water configuration in soil where gravity acts downward. Now imagine the identical geometric configuration but with gravity acting upward. The Minkowski functionals would be identical, same volume, surface area, curvature, connectivity. Yet the hydraulic behavior would be completely different. Pendant drops stable under downward gravity would drain immediately under upward gravity. Perched water tables would behave entirely differently.

This is not a flaw in the mathematics, motion-invariance is precisely what makes Minkowski functionals so powerful as geometric descriptors. But it highlights that geometry alone cannot determine hydraulic behavior in the presence of directional forcing like gravity.

Discontinuities and Dynamic Transitions

Water movement in porous media is characterized by discontinuous events:

• Haines jumps during drainage, where menisci suddenly snap across pore throats
• Snap-off phenomena, where water films pinch off creating isolated ganglia
• Burst-like pore filling during infiltration
• Rapid redistribution events following connectivity changes

These are not smooth, continuous processes, they involve abrupt topological changes where clusters suddenly merge or disconnect, interfaces jump across geometrically controlled barriers, and connectivity changes instantaneously. Energy is dissipated in these events, creating irreversibility that contributes to hysteresis.

While Minkowski functionals can detect these transitions (for instance, through sudden jumps in the Euler characteristic when clusters merge), they describe only the "before" and "after" geometric states. They don't capture the dynamics of the transition:

• The velocity fields during rapid interface motion
• Pressure gradients and their relaxation
• Energy dissipation during Haines jumps
• Timescales of geometric evolution
• Inertial effects during rapid events

The "continuity" property in Hadwiger's theorem refers to mathematical smoothness of the measure itself (small geometric changes produce small functional changes), not to the physical continuity of water distribution or flow processes. Real soil hydrology involves fundamental discontinuities that static geometry cannot capture.

History Dependence and Hysteresis

Perhaps most critically, Minkowski functionals provide only a static snapshot of geometry at a particular moment. They cannot inherently encode how the system arrived at that configuration, the drainage versus imbibition history that creates hysteresis.

Consider two soil samples at the same water content θ = 0.4. Sample A reached this state through drainage from saturation, while Sample B reached it through imbibition from dry conditions. They might even have identical Minkowski functionals, same volume (M₀ = 0.4 by definition), possibly similar surface areas, curvatures, and connectivity if the pore structure allows different geometric configurations at the same saturation.

Yet their hydraulic behavior will differ dramatically:

• Different capillary pressures (water retention hysteresis)
• Different hydraulic conductivities (conductivity hysteresis)
• Different contact angles at interfaces
• Different stability under perturbations

The drainage sample likely has pendant drops and isolated clusters in large pores, trapped by capillary barriers at pore throats. The imbibition sample probably has water preferentially filling small pores and coating surfaces as films. These geometric differences might be detectable by Minkowski functionals, but the reason for the difference, the process history, is not encoded in the geometry itself.

Hysteresis is fundamentally about path-dependence: the relationship between state variables depends on the history of transitions. Geometry alone, no matter how completely characterized, cannot capture this temporal causality without additional information about the process history.

What's Missing from Static Geometry

To fully characterize soil hydraulic behavior, we need information beyond what static geometry provides:

1. Flow directionality: Which pathways are active for flow in specific directions under gravity or pressure gradients? Two pore networks might have identical connectivity topology but completely different vertical versus horizontal permeability.

2. Dynamic connectivity: Not just whether clusters are geometrically connected, but how efficiently they transport water. This involves:

• Tortuosity and path length distributions
• Bottleneck locations and their sizes
• Dead-end pores that contribute to storage but not transport
• Interface mobility and contact line pinning

3. Process history: The sequence of wetting/drying events that determines:

• Current interfacial configurations
• Contact angles (which depend on whether surfaces are advancing or receding)
• Which pores are filled or empty at a given capillary pressure
• The distribution of trapped air or water ganglia

4. Temporal evolution:

• Rates of geometric change and interface motion
• Relaxation times toward equilibrium configurations
• Characteristic timescales for different processes (capillary equilibration, gravity drainage, diffusive redistribution)
• Dynamic versus quasi-static flow regimes

5. Energetic states: Not just geometric configuration but:

• The energy landscape and barriers between different configurations
• Metastable states and their stability
• Energy dissipation during transitions
• The thermodynamic distance from equilibrium

6. Mechanical coupling:

• Soil deformation affecting pore geometry
• Swelling and shrinkage during wetting/drying
• Crack formation and closure
• Aggregate structural changes

Go to Part III where we discuss specifically M3

Comprehensive Bibliography

Hadwiger Theorem and Integral Geometry Foundations

• Hadwiger, H. (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin.
• Hadwiger, H. (1959). Normale Körper im euklidischen Raum und ihre topologischen und metrischen Eigenschaften. Mathematische Zeitschrift, 71(1), 124-140.
• Klain, D. A., & Rota, G. C. (1997). Introduction to Geometric Probability. Cambridge University Press.
• Schneider, R., & Weil, W. (2008). Stochastic and Integral Geometry. Springer-Verlag, Berlin.
• Alesker, S. (1999). Continuous rotation invariant valuations on convex sets. Annals of Mathematics, 149(3), 977-1005.

Foundational Works on Minkowski Functionals

• Mecke, K. R. (1998). Integral geometry in statistical physics. International Journal of Modern Physics B, 12(09), 861-899.
• Mecke, K. R. (2000). Additivity, convexity, and beyond: applications of Minkowski functionals in statistical physics. In Statistical Physics and Spatial Statistics (pp. 111-184). Springer, Berlin.
• Schröder-Turk, G. E., et al. (2011). Minkowski tensor shape analysis of cellular, granular and porous structures. Advanced Materials, 23(22-23), 2535-2543.
• Michielsen, K., & De Raedt, H. (2001). Integral-geometry morphological image analysis. Physics Reports, 347(6), 461-538.
• Mantz, H., et al. (2008). Utilizing Minkowski functionals for image analysis: a marching square algorithm. Journal of Statistical Mechanics: Theory and Experiment, 2008(12), P12015.

Applications to Porous Media

• Arns, C. H., et al. (2002). Computation of linear elastic properties from microtomographic images: Methodology and agreement between theory and experiment. Geophysics, 67(5), 1396-1405.
• Mecke, K. R., & Arns, C. H. (2005). Fluids in porous media: a morphometric approach. Journal of Physics: Condensed Matter, 17(9), S503-S534.
• Armstrong, R. T., et al. (2016). Beyond Darcy's law: The role of phase topology and ganglion dynamics for two-fluid flow. Physical Review E, 94(4), 043113.
• Hilfer, R., & Manwart, C. (2001). Permeability and conductivity for reconstruction models of porous media. Physical Review E, 64(2), 021304.
• Thovert, J. F., et al. (2001). Grain reconstruction of porous media: application to a Bentheim sandstone. Physical Review E, 63(6), 061307.
• Arns, C. H., et al. (2005). Accurate estimation of transport properties from microtomographic images. Geophysical Research Letters, 28(17), 3361-3364.

Soil Science and Hydrology Applications

• Vogel, H. J., & Roth, K. (2001). Quantitative morphology and network representation of soil pore structure. Advances in Water Resources, 24(3-4), 233-242.
• Vogel, H. J., et al. (2010). Quantification of soil structure based on Minkowski functions. Computers & Geosciences, 36(10), 1236-1245.
• Schlüter, S., et al. (2014). Image processing of multiphase images obtained via X-ray microtomography: a review. Water Resources Research, 50(4), 3615-3639.
• Peth, S., et al. (2008). Three-dimensional quantification of intra-aggregate pore-space features using synchrotron-radiation-based microtomography. Soil Science Society of America Journal, 72(4), 897-907.
• Cousin, I., et al. (1996). Three-dimensional analysis of a loamy-clay soil using pore and solid chord distributions. European Journal of Soil Science, 47(4), 439-452.
• Perret, J., et al. (1999). Three-dimensional quantification of macropore networks in undisturbed soil cores. Soil Science Society of America Journal, 63(6), 1530-1543.

Topology, Connectivity, and Percolation

• Hunt, A. G., & Sahimi, M. (2017). Flow, transport, and reaction in porous media: Percolation scaling, critical-path analysis, and effective medium approximation. Reviews of Geophysics, 55(4), 993-1078.
• Vogel, H. J. (1997). Morphological determination of pore connectivity as a function of pore size using serial sections. European Journal of Soil Science, 48(3), 365-377.
• Vogel, H. J. (2000). A numerical experiment on pore size, pore connectivity, water retention, permeability, and solute transport using network models. European Journal of Soil Science, 51(1), 99-105.
• Mecke, K. R. (1994). Integral geometry in statistical physics. International Journal of Modern Physics B, 12(09), 861-899.
• Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer Science & Business Media.
• Rintoul, M. D., & Torquato, S. (1997). Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model. Journal of Physics A: Mathematical and General, 30(16), L585.

Hysteresis, Non-Equilibrium, and Dynamic Processes

• Berg, S., et al. (2013). Real-time 3D imaging of Haines jumps in porous media flow. Proceedings of the National Academy of Sciences, 110(10), 3755-3759.
• Armstrong, R. T., & Berg, S. (2013). Interfacial velocities and capillary pressure gradients during Haines jumps. Physical Review E, 88(4), 043010.
• McClure, J. E., et al. (2018). Geometric state function for two-fluid flow in porous media. Physical Review Fluids, 3(8), 084306.
• Schlüter, S., et al. (2016). Pore-scale displacement mechanisms as a source of hysteresis for two-phase flow in porous media. Water Resources Research, 52(3), 2194-2205.
• Armstrong, R. T., et al. (2014). Linking pore-scale interfacial curvature to column-scale capillary pressure. Advances in Water Resources, 46, 55-62.
• Schlüter, S., et al. (2017). Time scales of relaxation dynamics during transient conditions in two-phase flow. Water Resources Research, 53(6), 4709-4724.
• Rücker, M., et al. (2015). From connected pathway flow to ganglion dynamics. Geophysical Research Letters, 42(10), 3888-3894.

Curvature, Interfacial Area, and Thermodynamics

• Hilpert, M., & Miller, C. T. (2001). Pore-morphology-based simulation of drainage in totally wetting porous media. Advances in Water Resources, 24(3-4), 243-255.
• McClure, J. E., et al. (2016). Influence of phase connectivity on the relationship among capillary pressure, fluid saturation, and interfacial area in two-fluid-phase porous medium systems. Physical Review E, 94(3), 033102.
• Porter, M. L., et al. (2009). Measurement and prediction of the relationship between capillary pressure, saturation, and interfacial area in a NAPL-water-glass bead system. Water Resources Research, 45(8), W08402.
• Joekar-Niasar, V., & Hassanizadeh, S. M. (2012). Analysis of fundamentals of two-phase flow in porous media using dynamic pore-network models: A review. Critical Reviews in Environmental Science and Technology, 42(18), 1895-1976.
• Hassanizadeh, S. M., & Gray, W. G. (1993). Thermodynamic basis of capillary pressure in porous media. Water Resources Research, 29(10), 3389-3405.
• Niessner, J., & Hassanizadeh, S. M. (2008). A model for two-phase flow in porous media including fluid-fluid interfacial area. Water Resources Research, 44(8), W08439.

Computational Methods and Image Analysis

• Ohser, J., & Schladitz, K. (2009). 3D Images of Materials Structures: Processing and Analysis. Wiley-VCH.
• Legland, D., et al. (2016). MorphoLibJ: integrated library and plugins for mathematical morphology with ImageJ. Bioinformatics, 32(22), 3532-3534.
• Schladitz, K., et al. (2006). Design of acoustic trim based on geometric modeling and flow simulation for non-woven. Computational Materials Science, 38(1), 56-66.
• Lindquist, W. B., & Venkatarangan, A. (1999). Investigating 3D geometry of porous media from high resolution images. Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, 24(7), 593-599.
• Lindquist, W. B., et al. (2000). Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones. Journal of Geophysical Research: Solid Earth, 105(B9), 21509-21527.

Stochastic Reconstruction and Multiscale Analysis

• Karsanina, M. V., & Gerke, K. M. (2018). Hierarchical optimization: Fast and robust multiscale stochastic reconstructions with rescaled correlation functions. Physical Review Letters, 121(26), 265501.
• Gerke, K. M., et al. (2019). Improving watershed-based pore-network extraction method using maximum inscribed ball pore-body positioning. Advances in Water Resources, 140, 103576.
• Yeong, C. L. Y., & Torquato, S. (1998). Reconstructing random media. Physical Review E, 57(1), 495-506.
• Hilfer, R. (1991). Geometric and dielectric characterization of porous media. Physical Review B, 44(1), 60-75.
• Jiao, Y., et al. (2007). Modeling heterogeneous materials via two-point correlation functions: Basic principles. Physical Review E, 76(3), 031110.
• Tahmasebi, P., & Sahimi, M. (2012). Reconstruction of three-dimensional porous media using a single thin section. Physical Review E, 85(6), 066709.

Upscaling and Effective Properties

• Wildenschild, D., & Sheppard, A. P. (2013). X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Advances in Water Resources, 51, 217-246.
• Blunt, M. J., et al. (2013). Pore-scale imaging and modelling. Advances in Water Resources, 51, 197-216.
• Costanza-Robinson, M. S., et al. (2008). X-ray microtomography determination of air-water interfacial area-water saturation relationships in sandy porous media. Environmental Science & Technology, 42(7), 2949-2956.
• Arns, C. H., et al. (2001). Cross-property correlations and permeability estimation in sandstone. Physical Review E, 72(4), 046304.
• Knackstedt, M. A., et al. (2001). Percolation properties of the three-dimensional pore space in rocks. Physical Review E, 64(5), 056302.

Recent Developments and Advanced Topics

• Lin, Q., et al. (2018). Minimal surfaces in porous media: Pore-scale imaging of multiphase flow in an altered-wettability Bentheimer sandstone. Physical Review E, 99(6), 063105.
• Rabbani, A., et al. (2021). Review of data science trends and issues in porous media research with a focus on image-based techniques. Water Resources Research, 57(3), e2020WR028597.
• Bultreys, T., et al. (2016). Fast laboratory-based micro-computed tomography for pore-scale research: Illustrative experiments and perspectives on the future. Advances in Water Resources, 95, 341-351.
• Schlüter, S., et al. (2020). Time scales of relaxation dynamics during transient conditions in two-phase flow. Water Resources Research, 56(4), e2019WR025815.
• Singh, K., et al. (2017). Dynamics of snap-off and pore-filling events during two-phase fluid flow in permeable media. Scientific Reports, 7(1), 5192.
• Andrew, M., et al. (2014). Pore-scale imaging of geological carbon dioxide storage under in situ conditions. Geophysical Research Letters, 41(15), 5347-5354.

Persistent Homology and Advanced Topology

• Edelsbrunner, H., & Harer, J. (2008). Persistent homology—a survey. Contemporary Mathematics, 453, 257-282.
• Robins, V., et al. (2011). Percolating length scales from topological persistence analysis of micro-CT images of porous materials. Water Resources Research, 52(1), 315-329.
• Kramár, M., et al. (2013). Quantifying force networks in particulate systems. Physica D: Nonlinear Phenomena, 283, 37-55.

Multiphase Flow and Interface Dynamics

• Raeini, A. Q., et al. (2014). Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. Journal of Computational Physics, 231(17), 5653-5668.
• Ferrari, A., & Lunati, I. (2013). Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy. Advances in Water Resources, 57, 19-31.
• Zacharoudiou, I., et al. (2017). The impact of drainage displacement patterns and Haines jumps on CO2 storage efficiency. Scientific Reports, 8(1), 15561.

Minkowsky functionals (a way to track water movement in soil)

Go to part II 

Hysteresis and non equilibrium in unsaturated soil flow

Introduction

How do we truly characterize the spatial distribution of water in soil? Beyond simple metrics like water content or saturation, the geometry and topology of water distribution carry crucial information about soil hydraulic behavior. This is where Minkowski functionals offer a powerful mathematical framework, one that has been largely under-explored in soil hydrology despite its rich potential.

Minkowski functionals are mathematical measures that completely characterize the morphology of spatial patterns in Euclidean space. Originally developed in integral geometry, they provide a set of scalar descriptors that capture essential geometric and topological properties of spatial structures. In the context of soil hydrology, they offer a sophisticated way to quantify how water phases are distributed, connected, and structured within the pore space.

What Are Minkowski Functionals?

For a three-dimensional body or pattern, there are four Minkowski functionals, each capturing different geometric properties:

• M₀ (Volume): The total volume occupied by the phase of interest (e.g., water)
• M₁ (Surface Area): The total surface area of the interface between phases (e.g., water-air interface, water-soil interface)
• M₂ (Mean Breadth/Integral Mean Curvature): Related to the total mean curvature of the surface, capturing how "curved" the interface is
• M₃ (Euler Characteristic): A topological invariant that counts the number of connected components minus the number of handles (tunnels) plus the number of cavities

These functionals are additive, motion-invariant, and continuous, properties that make them particularly useful for analyzing complex spatial patterns. Importantly, they form a complete set of geometric measures under certain mathematical conditions, though they remain informative even for the non-convex structures found in porous media.

The Hadwiger Theorem: Why These Four?

The answer lies in a deep mathematical result called the Hadwiger theorem, proven by Hugo Hadwiger in 1957. This fundamental theorem in integral geometry states that any continuous, motion-invariant, and additive functional (called a valuation) defined on convex bodies in n-dimensional Euclidean space can be expressed as a linear combination of exactly (n+1) Minkowski functionals.

In three dimensions, this means these four functionals—volume, surface area, integral mean curvature, and Euler characteristic—form a complete basis for geometric description. There are no "missing" geometric properties that satisfy these natural mathematical requirements. This completeness distinguishes Minkowski functionals from ad-hoc geometric measures and provides theoretical assurance that we're capturing all the geometric information available in a coordinate-independent, additive framework.

The Euler Characteristic: Topology Meets Hydrology

The Euler characteristic (χ = M₃) deserves special attention in soil hydrology. For a three-dimensional pattern:

χ = N₀ - N₁ + N₂

where N₀ is the number of connected water clusters, N₁ is the number of tunnels or loops through the water phase, and N₂ is the number of isolated cavities within the water.

This topological descriptor reveals critical information about hydraulic connectivity. A high positive χ suggests many isolated water clusters (poor connectivity), while negative values indicate a well-connected network with many redundant pathways. This directly relates to hydraulic conductivity and capillary connectivity, fundamental properties governing water flow.

Consider a simple example: at high saturation during imbibition, water forms a continuous network with many interconnected pathways (negative χ). As drainage proceeds, this network fragments into increasingly isolated clusters, and χ increases, eventually becoming positive. The point where χ crosses zero marks a fundamental topological transition—from a connected network to a collection of isolated features.

Applications to Soil Water Dynamics

1. Characterizing Drainage and Imbibition Paths

During drainage, water typically fragments from a well-connected network into increasingly isolated clusters. The Euler characteristic tracks this transition: starting negative (connected network) and becoming positive (isolated clusters) as saturation decreases. The rate of change dχ/dθ could identify critical thresholds where major topological transitions occur, perhaps corresponding to air entry values or percolation thresholds.

During imbibition, the reverse process occurs, but hysteresis means the path differs. At the same water content, drainage configurations might show more isolated clusters while imbibition shows more connected films and wedges. Minkowski functionals could quantify these path-dependent differences, providing geometric signatures of hysteretic behavior beyond traditional water retention curves.

Imagine tracking all four functionals simultaneously during a drainage-imbibition cycle. We'd see not just how much water is present (M₀), but how its surface area (M₁), curvature distribution (M₂), and connectivity (M₃) evolve differently along drainage versus imbibition paths. These geometric trajectories could reveal fundamental aspects of hysteretic mechanisms.

2. Linking Pore Structure to Hydraulic Properties

The mean breadth (M₂) relates to interfacial curvature, which directly connects to capillary pressure via the Young-Laplace equation:

Pc = γ(1/r₁ + 1/r₂)

where γ is surface tension and r₁, r₂ are the principal radii of curvature. Tracking M₂ as a function of water content provides information about the distribution of capillary pressures in the system—essentially a geometric interpretation of the water retention curve.

The surface area functional (M₁) quantifies the extent of water-air interfaces, which is crucial for understanding interfacial phenomena, evaporation dynamics, and the energetics of water distribution. During evaporation, for instance, M₁ determines the total interfacial area available for vapor transport, while changes in M₂ reflect how the geometry of menisci evolves as drying proceeds.

Recent research has shown that interfacial area is not uniquely determined by water content and capillary pressure alone, it exhibits hysteresis and depends on flow history. Minkowski functionals provide tools to quantify this additional complexity.

3. Non-Equilibrium States and Hysteresis

One of the most intriguing applications is tracking non-equilibrium water distributions. During rapid infiltration or redistribution, water occupies configurations that differ from equilibrium states at the same water content. Minkowski functionals could distinguish these transient states by their geometric signatures.

For instance, pendant drops trapped during rapid drainage versus uniform film coatings during slow imbibition might have similar water contents but dramatically different Euler characteristics (many isolated clusters versus one connected film) and surface areas. This geometric information could inform models that go beyond equilibrium assumptions.

Consider infiltration into initially dry soil: water advances as a wetting front, creating fingering patterns or preferential flow paths depending on initial conditions and infiltration rate. The evolving Minkowski functionals during this transient process could reveal when and how the system transitions from non-equilibrium invasion patterns to more uniform, equilibrium-like distributions.

4. Scale-Dependent Analysis

By computing Minkowski functionals at different scales (through morphological operations like erosion and dilation), we can examine how geometric properties change across scales. This multiscale analysis could reveal how local pore-scale water distribution relates to effective continuum-scale hydraulic properties—a crucial link for upscaling.

For example, at fine scales we might observe highly fragmented water distributions with positive χ, but coarse-graining could reveal that these fragments form a connected network at larger scales (negative χ). This scale-dependent connectivity has direct implications for how we define effective hydraulic conductivity and for understanding the scale-dependence of hydraulic properties.

The technique of morphological operations—systematically growing or shrinking phases—allows us to explore the "thickness distribution" of water features. Thin films coating particles might disappear at modest coarse-graining, while thicker wedges and pore-body water persist. Tracking how Minkowski functionals change with scale provides a geometric signature of this hierarchical structure.

Relating to Hydraulic Models

The real power emerges when we connect these geometric descriptors to physically based models. Several promising directions include:

Connectivity-based conductivity: Using χ to parameterize how hydraulic conductivity depends not just on water content but on the topological structure of water distribution. A well-connected network (negative χ) should conduct much better than isolated clusters (positive χ) at the same saturation.

Capillary pressure distributions: Relating M₂ to the distribution of capillary pressures in the system, potentially informing multi-scale or dual-porosity models where different geometric domains have different characteristic pressures.

Interfacial area in evaporation: Incorporating M₁ into evaporation models, where the rate of water loss depends on the available interfacial area for vapor diffusion.

Geometric state variables: Developing constitutive relations where hydraulic properties are functions not just of water content, but of the complete set of Minkowski functionals, creating geometry-informed models that capture hysteretic and non-equilibrium behavior.

This could lead to a new class of hydraulic models where geometric descriptors serve as state variables alongside traditional quantities like water content and pressure. The challenge is developing these relationships in ways that are both physically meaningful and practically implementable.

Looking Ahead

Minkowski functionals provide a rigorous, mathematically complete framework for geometric characterization of water distribution in soils. They offer quantitative descriptors that capture volume, surface area, curvature, and connectivity—fundamental aspects of spatial organization that determine hydraulic behavior.

For soil hydrologists, these tools open new possibilities for understanding hysteresis, characterizing non-equilibrium states, and linking pore-scale geometry to continuum-scale properties. As imaging technologies advance and computational methods mature, geometry-based approaches may become increasingly central to how we model and predict water movement in soils.

However, geometry is only part of the story. In Part 2 of this series, we'll examine critical limitations of purely geometric approaches and explore how static spatial descriptors must be augmented with dynamic, directional, and historical information to fully capture the complexity of soil hydraulic processes.

Selected References 

Foundations and Theory

• Hadwiger, H. (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin.
• Mecke, K. R. (2000). Additivity, convexity, and beyond: applications of Minkowski functionals in statistical physics. In Statistical Physics and Spatial Statistics (pp. 111-184). Springer, Berlin.
• Schröder-Turk, G. E., et al. (2011). Minkowski tensor shape analysis of cellular, granular and porous structures. Advanced Materials, 23(22-23), 2535-2543.

Soil and Porous Media Applications

• Vogel, H. J., & Roth, K. (2001). Quantitative morphology and network representation of soil pore structure. Advances in Water Resources, 24(3-4), 233-242.
• Vogel, H. J., et al. (2010). Quantification of soil structure based on Minkowski functions. Computers & Geosciences, 36(10), 1236-1245.
• Schlüter, S., et al. (2014). Image processing of multiphase images obtained via X-ray microtomography: a review. Water Resources Research, 50(4), 3615-3639.

Connectivity and Topology

• Mecke, K. R., & Arns, C. H. (2005). Fluids in porous media: a morphometric approach. Journal of Physics: Condensed Matter, 17(9), S503-S534.
• Armstrong, R. T., et al. (2016). Beyond Darcy's law: The role of phase topology and ganglion dynamics for two-fluid flow. Physical Review E, 94(4), 043113.