Both might have similar C(r→r') statistics, but their hydraulic behavior under non-equilibrium conditions would be dramatically different.
The Euler characteristic χ captures this higher-order topology . But even χ is incomplete—it's a single scalar. The full story requires the Betti numbers as functions of pore size :
$$\beta_1(r_{\min}) = \text{number of loops using pores } r \geq r_{\min}$$
This size-resolved topology tells us which pore size classes contribute to network connectivity and robustness.
Persistent Homology: A New FrontierRecent work in computational topology has introduced persistent homology to porous media analysis (Jiang et al., 2018; Moon et al., 2019). This technique tracks how Betti numbers evolve as you "fill" the pore space from small to large pores:Birth : The pore size at which a topological feature (component, loop, void) first appears
Death : The pore size at which it disappears (merges with another feature)
Persistence : Death - Birth (a measure of robustness)
Features with high persistence are geometrically significant; those with low persistence are noise.
For soil hydrology, this means:
During Drainage (Decreasing Saturation):Large loops "die" first (they contain large pore bodies that empty early)
Small loops persist longer (they're held by capillarity in small throats)
The transition χ = 0 often marks the
percolation threshold where the network fragments
During Imbibition (Increasing Saturation):Small pores fill first, creating many disconnected clusters (high β₀)
As larger pores fill, clusters merge (β₀ decreases, β₁ increases as loops form)
Air becomes trapped in loops that cannot drain (high β₁ of the air phase)
Connection to Permeability: Beyond Kozeny-CarmanScholz et al. (2012) demonstrated experimentally that
permeability scales with the Euler characteristic:
$$k \propto |\chi|^\alpha \cdot f(\phi)$$
This is remarkable because it's independent of the percolation threshold —unlike Katz-Thompson and other models that require identifying a critical pore size.
Why does this work? Because χ directly encodes:Connectivity (via β₀ - β₁): How many pathways are available?
Network topology: Tree-like (low β₁) vs. loop-rich (high β₁) structures
Liu et al. (2017) extended this to 3D and found that void ratio must also be included. But the fundamental insight remains: topology, not just geometry, controls flow .
Application to Non-Equilibrium InfiltrationThis brings us to the central challenge: developing a
pore-network theory that replaces Richards' equation. Richards assumes:Local equilibrium : θ(ψ) and K(ψ) are unique, path-independent functions
Instantaneous capillary-gravity balance : No dynamic lag
Continuum description : Pore-scale structure doesn't matter
All three assumptions break down during rapid infiltration, preferential flow, and hysteretic cycling. A pore-network approach can address these limitations, but only if we properly account for topology .
The Role of Loops in Non-Equilibrium DynamicsFor each pore i with saturation S_i(t):
$$\frac{\partial S_i}{\partial t} = \frac{1}{V_i} \sum_{j \in \mathcal{N}(i)} Q_{ij}(S_i, S_j, \psi_i, \psi_j, \text{Topology})$$
The flow Q_ij depends not just on local states (S_i, S_j, ψ_i, ψ_j) but also on: Whether (i,j) is part of a loop : If yes, alternative paths exist—the flow can redistribute when one path becomes unfavorable
The saturation state of loops containing (i,j) : A fully saturated loop behaves differently from a partially saturated one
Contact line pinning : In rough or angular pores, interfaces can be pinned by geometry, creating metastable states
This requires tracking:
$$\beta_1^{\text{filled}}(t) = \text{number of loops with all pores filled at time } t$$
As infiltration progresses, new loops become active. During drainage, loops trap water. The dynamics are fundamentally topology-dependent .
The Missing MeasurementCurrent persistent homology applications to porous media compute β₁(r_min) globally. But for infiltration dynamics, we need:
$$\beta_1(r_{\min}, r_{\max}, S) = \text{loops using pores in } [r_{\min}, r_{\max}] \text{ at saturation } S$$
This would tell us:
- Which pore size classes create redundant pathways?
- At what saturation does the network transition from tree-like (β₁ ≈ 0) to loop-rich (β₁ >> 0)?
- How does the air phase topology evolve during imbibition?
This is a critical gap in current literature. While C(r→r') and χ(r) are both measured, the explicit loop statistics connecting them—particularly as functions of saturation—remain uncharacterized.
Practical ImplicationsHysteresis is fundamentally about loops. The ink-bottle effect, snap-off during imbibition, and air entrapment all require topological understanding. Empirical hysteresis models (like Scott 1983 or Luckner et al. 1989) lack physical basis. A topology-based approach could predict hysteresis from pore structure alone.
In macroporous or structured soils, flow doesn't occur uniformly—it follows preferred pathways. These pathways are defined by topology: which loops provide the path of least resistance? Traditional continuum models cannot represent this; pore networks can, if we track β₁(r,S).
During infiltration, trapped air creates additional resistance. But how much air gets trapped? Where? This depends entirely on loop structure. High β₁ in large pores means air can be trapped in those loops even after the matrix is saturated.
Upscaling:Can we derive effective Richards-like equations from pore-network topology? Perhaps. If we can relate:
$$K(S) = K_{\text{sat}} \cdot f(S, \chi(S), \beta_1(S), \ldots)$$
then topology provides the missing link between pore structure and continuum behavior.
Methodological ChallengesImplementing this vision requires solving several problems:
Current algorithms compute Betti numbers for geometric complexes. We need algorithms that track topology by pore size class and by saturation state . This is "attributed" persistent homology—a frontier in computational topology.
During infiltration, topology evolves. We need:
$$\beta_1(r, t) = \text{loops at pore size } r \text{ and time } t$$
This requires combining persistent homology with time-dependent network analysis—an open problem.
A representative elementary volume (REV) for structured soil might contain 10⁹ pores. Current persistent homology algorithms scale poorly beyond 10⁶ elements. Optimizations are needed.
We need dynamic X-ray CT experiments that track topology during infiltration/drainage cycles. Some pioneering work exists (Armstrong & Berg 2013; Schlüter et al. 2016), but systematic topology analysis is lacking.
The Path ForwardThe integration of algebraic topology into soil hydrology is still in its infancy. The foundational work exists:
- Vogel & Roth (2001) : Introduced χ(r) for pore connectivity
- Scholz et al. (2012) : Connected χ to permeability
- Jiang et al. (2018), Moon et al. (2019) : Applied persistent homology to predict flow properties
- Lucas et al. (2020) : Multi-scale connectivity analysis
But critical gaps remain:
Loop statistics L_n(r₁,...,r_n) connecting C(r→r') to β₁(r)
Size-resolved persistent homology for two-phase flow
Dynamic topology tracking during infiltration
Topology-dependent constitutive relations for pore-network models
Filling these gaps would enable a truly mechanistic alternative to Richards' equation—one that captures non-equilibrium dynamics, hysteresis, and preferential flow from first principles.
Topology as the Fourth DimensionWe often think of soil characterization in three dimensions:Size (pore size distribution)
Shape (surface area, curvature)
Space (spatial correlations)
But there's a fourth dimension, one that geometry alone cannot capture: Topology .
M₃ and its decomposition into Betti numbers provide access to this dimension. They tell us not just what pores exist, but how they connect —and that connectivity determines everything about non-equilibrium flow.
As we move beyond Richards' equation toward pore-network theories of infiltration, topology will be as important as geometry. The Euler characteristic is our window into this hidden structure—but we're only beginning to look through it.
References
