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

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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.