Tuesday, January 6, 2026

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

Go to Part I

 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:
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:
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:
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

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.
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