In our previous posts on Minkowski functionals, we established that M₀ (volume), M₁ (surface area), and M₂ (curvature) are purely geometric measures. They tell us about size, shape, and form. But M₃—the Euler characteristic—is fundamentally different. It's a topological invariant that captures something geometry alone cannot: how the pore space is connected.
Consider two soil samples with identical pore size distributions, identical surface areas, and identical mean curvatures. One could be a tree-like network where every pore has a unique path to every other pore. The other could be riddled with loops—alternative pathways that provide redundancy. The first three Minkowski functionals (M₀, M₁, M₂) might be identical, but M₃ would reveal the profound topological difference.
This distinction isn't academic. It's critical for understanding non-equilibrium infiltration, hysteresis, and the dynamics of water movement that Richards' equation cannot capture.
The Euler characteristic for a 3D pore network can be expressed in two equivalent ways:
Graph-theoretic form:
$$\chi = V - E + F$$
where V is the number of vertices (pore nodes), E is the number of edges (connections), and F is the number of faces (enclosed regions).
Topological form (via Betti numbers):
$$\chi = \beta_0 - \beta_1 + \beta_2$$
where:
- β₀ = number of connected components (separate pore clusters)
- β₁ = number of independent loops (redundant pathways)
- β₂ = number of enclosed voids (trapped air bubbles)
While Vogel and Roth (2001) pioneered the use of the Euler characteristic for soil pore characterization, the decomposition into Betti numbers reveals structure that χ alone obscures:
β₀ (Connected Components):
When you progressively drain water from soil, β₀ tracks how the water phase fragments. High β₀ means disconnected water clusters—poor hydraulic connectivity. This is crucial during drought or in the late stages of drainage.
β₁ (Independent Loops):
This is perhaps the most important quantity for non-equilibrium flow. Loops create:Alternative flow paths: If one pore throat becomes blocked (by air during drainage, or by swelling clay), water can route around it
Hysteresis mechanisms: The ink-bottle effect is fundamentally about loops—water trapped in a loop during drainage can only escape when all connecting throat pressures exceed the entry pressure
Air entrapment during imbibition: Loops trap air bubbles that persist even after the network is nominally saturated
β₂ (Enclosed Voids):
In most soils, β₂ ≈ 0 (isolated voids are rare), but in fractured or macroporous media, enclosed voids contribute to non-productive porosity and affect effective saturation.
The Critical Gap: C(r→r') vs. χ
As we discussed in earlier posts, the connectivity matrix C(r→r') describes the probability that a pore of size r connects to a pore of size r'. This is essential information—it tells us about local, pairwise connectivity. But C(r→r') is fundamentally a first-order descriptor. It cannot distinguish between:
- Linear chains : r₁ → r₂ → r₃ (tree-like, vulnerable)
- Loops : r₁ ⇄ r₂ ⇄ r₃ ⇄ r₁ (robust, redundant)
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 Frontier
Recent 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-Carman
Scholz 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 Infiltration
This 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 Dynamics
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 Frontier
Recent 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-Carman
Scholz 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 Infiltration
This 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 Dynamics
For 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 Measurement
Current 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:
Practical Implications
Hysteresis 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 Challenges
Implementing 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 Forward
The integration of algebraic topology into soil hydrology is still in its infancy. The foundational work exists:
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 Dimension
We 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
$$\frac{\partial S_i}{\partial t} = \frac{1}{V_i} \sum_{j \in \mathcal{N}(i)} Q_{ij}(S_i, S_j, \psi_i, \psi_j, \text{Topology})$$
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 Measurement
Current 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?
Practical Implications
Hysteresis 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 Challenges
Implementing 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 Forward
The 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 Dimension
We 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
- Jiang, Z., Wu, K., Couples, G., Van Dijke, M.I.J., Sorbie, K.S., and Ma, J. (2018). Pore geometry characterization by persistent homology theory. Water Resources Research , 54, 4150–4163.
- Liu, Z., Herring, A., Robins, V., and Armstrong, R.T. (2017). Prediction of permeability from Euler characteristic of 3D images. Proceedings of the International Symposium of the Society of Core Analysts , SCA2017-016.
- Lucas, M., Schlüter, S., Vogel, H.-J., and Vereecken, H. (2020). Revealing pore connectivity across scales and resolutions with X-ray CT. European Journal of Soil Science , 72, 546–560.
- Moon, C., Mitchell, S.A., Heath, J.E., and Andrew, M. (2019). Statistical inference over persistent homology predicts fluid flow in porous media. Water Resources Research , 55, 9592–9603.
- Scholz, C., Wirner, F., Götz, J., Rüde, U., Schröder-Turk, G.E., Mecke, K., and Bechinger, C. (2012). Permeability of porous materials determined from the Euler characteristic. Physical Review Letters , 109, 264504.
- Vogel, H.-J., and Roth, K. (2001). Quantitative morphology and network representation of soil pore structure. Advances in Water Resources , 24, 233–242.
