AboutHydrology

Friday, February 6, 2026

The Hydrological Modelling 2026 Class

Work in progress

Hydrology is a fascinating field because water is essential for life and human activities. It is fundamentally the Physics of the Hydrological Cycle, yet it is deeply interconnected with biochemical processes and geology due to water's crucial role in ecosystems. Here a brief introduction from a National Geographics post. A companion page is available for the laboratory exercises, where you can find all the necessary materials for hands-on practice.

The lab material is here.

Classes and Related Materials

Available Resources

Storyboards – A summary of the lecture, usually in Italian.
Whiteboard – A detailed explanation of a specific topic, presented using Notability on an iPad.
Slides – Commented in English.
Videos – Commentary on the slides, typically recorded during lectures with no editing (as post-production would be too time-consuming).
- 2025 Videos are available on a Vimeo Showcase [link here].
Additional Information & References – For those eager to explore more, supplementary details and references are provided in italics.

Class Schedule & Materials

📅 24 February 2025 – Introduction to the Course and Hydrology

Syllabus (Vimeo 2026)
A very short introduction to hydrology (Vimeo 2025)

Vimeo 2024,Vimeo2022, Vimeo2022,Vimeo2021, YouTube2020,YouTube 2019

Mass & Energy budgets (Vimeo 2025)

Vimeo 2024,Vimeo2023, Vimeo2022,Vimeo2021, YouTube 2019,YouTube2020

A short Lab introductions. Go to the installation page or (look at here for a short video summary)

🔎 Complementary Reference:

Blöschl, Günter. 2022. Flood Generation: Process Patterns from the Raindrop to the Ocean. DOI: 10.5194/hess-2022-2.

Sunday, January 25, 2026

Leveraging LLMs in Hydrological Research: A Personal Workflow and Supporting Literature

In the rapidly shifting landscape of scientific research, Large Language Models (LLMs) have emerged as more than just productivity tools; they are powerful engines for augmenting human creativity. As a hydrologist navigating complex phenomena like percolation theory and soil water dynamics, I have developed a hybrid workflow that integrates a suite of LLMs—including Gemini, Grok, ChatGPT, and Claude—into my daily practice.

This post outlines my approach, refined through cross-model interaction, and contextualizes it within recent literature. My goal is to demonstrate how we can exploit these "information reservoirs" with ease while maintaining the rigorous standards required by the physical sciences.

The Workflow: From Intuition to Iteration

My process is built on the principle that LLMs should act as collaborative aids, not replacements. This ensures human oversight counters the inherent risks of .

1. Conceptualization and Targeted Drafting

Rather than starting with exhaustive preliminary reading, I begin by formalizing concepts that have been "simmering" in my mind. I draft notes with highly specific questions. Precision is the key: the more targeted the query, the more useful the response. For example, I might prompt a model to bridge adjacent fields, such as connecting Minkowski functionals to percolation theory—a link I conceive independently but use AI to flesh out and expand.

2. The Multi-Model Reasoning Loop

Once a draft is mature, I submit it to an ensemble of models to leverage their unique strengths (e.g., Claude’s nuanced reasoning, Gemini’s expansive context window, or Grok’s real-time information access):

Structural Refinement: Improving the logical flow and hierarchy of arguments.
Cross-Verification: I use a multi-model approach, asking ChatGPT to critique the mathematical derivations provided by Gemini, or vice versa.
Fact-Checking: Any discrepancies identified between models are looped back for revision until the logic holds across all platforms.

3. Traditional Validation

The AI output is never the final word. I validate the substance through traditional scientific means:

Deep reading of primary peer-reviewed literature.
Direct discussion with colleagues.
Implementation of toy models or full-scale simulations to empirically verify the AI-assisted hypotheses.

4. Use impressions

The use of these tools has significantly boosted my search productivity and provides real comfort through the assistance they offer. While large language models (LLMs) often capture the general ideas and broad directions quite effectively, they frequently make errors in equations and produce interpretations that can feel hallucinated—mixed among many correct suggestions.

Nevertheless, because LLMs draw from a vastly wider knowledge base than any individual can access, they enable rapid retrieval of relevant information that would otherwise take weeks (or longer) to uncover through manual searching. On balance, I consider the overall impact positive.

Recent versions of several LLMs now provide accurate references in the vast majority of cases (often 99%), which represents a major advantage. I routinely retrieve and verify each cited source one by one, then store them in my personal papers organizer. As a valuable side effect, LLMs demonstrate strong recall of older literature and can point directly to the original sources of key ideas—helping improve the accuracy and integrity of citations.

The Environmental "Metabolism" of Research

A recurring theme in my work is the tension between the "easy exploitation" of information and the environmental cost. To provide a concrete example, I have estimated the footprint required to produce a substantive research commentary, such as my recent work on the Richards Equation.

Metric	Per Standard Query	Per Reasoning Query	Total for Research Cycle (est. 15-20 interactions)
Energy	~0.34 Wh	4.3 – 33.6 Wh	~110 - 250 Wh
Carbon ($CO_2e$)	~0.15 g	~1.5 - 12.0 g	~25 - 50 g
Water Withdrawal	~0.26 ml	~3.5 - 25.0 ml	~150 - 500 ml

Contextualizing the Impact:

Energy: The ~150 Wh consumed is roughly equivalent to leaving a 10W LED bulb on for 15 hours.
Water: At the upper end, the research for a single deep-dive commentary "drinks" about 500ml of water (one standard bottle), used for cooling data centers.
Carbon: 50g of
$CO_2$
is equivalent to driving a gasoline car for approximately 250 meters.

References

1. LLM4SR: A Survey on Large Language Models for Scientific Research Authors: Various (survey paper) arXiv preprint arXiv:2501.04306, 2025 https://arxiv.org/abs/2501.04306

2. Exploring the role of large language models in the scientific method: from hypothesis to discovery npj Artificial Intelligence (Nature Portfolio), 2025 https://www.nature.com/articles/s44387-025-00019-5

3. A Survey of Human-AI Collaboration for Scientific Discovery \Preprints.org, 2026 https://www.preprints.org/manuscript/202601.0405/v1

4. Scientific Discoveries by LLM Agents OpenReview (conference/journal paper) https://openreview.net/pdf?id=fxL6eFPsd1

5. How Much Energy Do LLMs Consume? Unveiling the Power Behind AI ADASCI Blog / systematic reviews on LLM energy consumption Approximate figure: GPT-3 training ~1,287 MWh https://adasci.org/blog/how-much-energy-do-llms-consume-unveiling-the-power-behind-ai

6. Embracing large language model (LLM) technologies in hydrology research IOPscience / Environmental Research: Infrastructure and Sustainability, recent https://iopscience.iop.org/article/10.1088/3033-4942/addd43

7. Large Language Models as Calibration Agents in Hydrological Modeling Geophysical Research Letters (AGU Journals), 2025 https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2025GL120043

Musical Coda

Monday, January 12, 2026

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.

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

Wednesday, January 7, 2026

Various resources on Evapotranspiration, as we treat it

Following our session on Evapotranspiration, I am listing some relevant resources beyond what we have already shared. Videos from previous GEOframe Schools are available here, and recordings from this year's school will be posted soon.

Core Reading

The most mature presentation of the theory, particularly for the latter part, is D'Amato and Rigon (2025a). A brief Claude-generated summary is available here. The paper addresses more complex cases than those presented at the School, but the first part—the —is complete and clear.

For a more extensive treatment of this material, I recommend starting with the theses by and .

https://water.usgs.gov/edu/gallery/evaporation-fog.html

The Resistance Model for ET

An explanation of the can be found in The Marvelous Physics of Plants, beginning at slide 33. Unfortunately, the accompanying video is in Italian.

Differentiating Soil Behavior from Transpiration

Further lectures covering the differentiation of soil behavior from transpiration are available in the BACI course materials. The advantage of this resource is that the blog page contains step-by-step references to Bottazzi's thesis.

For Those Who Want to Go Deeper

The tool to use is , which is fully documented in Concetta D'Amato's thesis and in D'Amato et al. (2025b). A Claude-generated summary of this paper is also available on the shared page.

References

Bottazzi, M. 2020. “Transpiration Theory and the Prospero Component of GEOframe.” Supervisors: Rigon and G. Bertoldi. Ph.D., DICAM. https://paperpile.com/shared/sTnYXIdFURLC65TntLUZqUA
D’Amato, Concetta. 2024. Supervisor: Riccardo Rigon, “Exploring the Soil-Plant-Atmosphere Continuum: Advancements, Integrated Modeling and Ecohydrological Insights.” Ph.D., Università di Trento. https://paperpile.com/shared/sPLgpgiM8Tt~egbfVu~2ISg
D'Amato, C., and Rigon, R. (2025a). A big leaf model with layer soil water uptake for tower-scale evapotranspiration simulations. Ecohydrology, e2748. https://doi.org/10.1002/eco.2748
D'Amato, C., Tubini, N., and Rigon, R. (2025b). GEOSPACE 1.0: An integrated numerical model for the global water and energy balance. Geoscientific Model Development, 18, 1041–1062. https://doi.org/10.5194/gmd-18-1041-2025

Musical coda

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 as a mathematically complete framework for geometric characterization, grounded in the . 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 .

The Hadwiger theorem states that any continuous, motion-invariant, and additive functional defined on 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 property ensures that measurements don't depend on how we orient or position our coordinate system, while 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. stable under downward gravity would drain immediately under upward gravity. 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 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

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.