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

Go to part II 

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.

Does exist ink bottle effect ? A little survey on hydraulic hysteresis

Soil physics textbooks often illustrate hysteresis with a simple diagram of a pore shaped like an old ink bottle: a wide body connected to a narrow neck. The story is familiar: during drainage, water is trapped in the wide body because the narrow neck controls the suction at which the pore can empty; during wetting, the pore fills at a lower suction than it empties. The ink-bottle model has been the standard pedagogical explanation for soil water hysteresis for more than half a century.

There is, however, a flaw in this reasoning. In the original version of this post I asked the reader to find it. The answer, which I develop below, is that the ink-bottle model explains Haines jumps but not macroscopic hysteresis — and the distinction matters profoundly for how we think about unsaturated flow.

The ink-bottle mechanism describes what happens at a single pore: a meniscus passing through a constriction undergoes an abrupt rearrangement — a Haines jump (Haines, 1930). These jumps are real, experimentally observable, and dissipate energy at the pore scale. But they are microscopic events. A single pore filling or emptying in a jump tells us nothing about the macroscopic retention curve.

Macroscopic hysteresis — the fact that the drying curve sits above the wetting curve in the θ(ψ) plane — requires that the entire filling distribution (which pores are full and which are empty) differ between wetting and drying at the same total water content. That is a network-scale phenomenon, not a single-pore phenomenon. The ink bottle, being a local geometric feature, cannot explain it.The most compelling evidence comes from lattice Boltzmann simulations by Hosseini, Kumar, and Delenne (2024; arXiv preprint 2022). They systematically eliminated every mechanism that has traditionally been proposed to explain hysteresis: 

• Contact angle hysteresis kept constant throughout wetting and drying
• The ink-bottle effect — eliminated by using simultaneous drainage/injection throughout the domain rather than boundary flow, preventing water from flowing in opposite directions
• Air entrapment — eliminated by the same scheme
• Soil fabric changes — grain positions fixed

Despite eliminating all of these, hysteresis persisted. What they found instead was a fundamental asymmetry in how the two fluid phases expand within the pore network:

• During wetting, new liquid zones appear throughout the gas phase in the form of capillary bridges. Many bridges expand simultaneously and coalesce, filling pores from smallest to largest. This is a multi-site nucleation process.
• During drying, new gas zones cannot spontaneously nucleate within the liquid phase — that would require cavitation. Only the existing gas cluster connected to the boundary can expand, and it is constrained by the pore openings surrounding it. This is a single-cluster expansion process.

The asymmetry is not local geometry (bottle necks) but global topology (many simultaneous nucleation sites versus single connected cluster). It is a property of the network, not of individual pores.I f hysteresis comes from topological asymmetry rather than local geometry, then several foundational questions in soil physics need rethinking:

The classical view treats the drying and wetting retention curves as two distinct equilibrium relationships. But if the asymmetry is topological — if it is about which pores the system can reach rather than about multiple thermodynamic minima — then perhaps there is only one equilibrium at each water content, and what we call “hysteresis” is the inability of the system to reach that equilibrium along certain paths. If so, the water retention curve is not a constitutive law but a trajectory through a configuration space, and hysteresis is a kinematic phenomenon, not a thermodynamic one.This distinction has consequences. If there is a unique equilibrium, then the relevant question is not “what is the equilibrium pressure at this water content?” but rather “how does the system approach equilibrium, and what prevents it from getting there?”
The ink-bottle picture treats wetting and drying as the same process run in opposite directions. But the Hosseini result shows they are qualitatively different: multi-site nucleation versus single-cluster expansion. This asymmetry suggests that wetting and drying should be described by different operators — not by a single reversible process with a sign change. And if we go further: in the field, drying is not a single process at all. At least four distinct mechanisms remove water from soil:

• Capillary drainage: gas invades from the boundary through the connected gas cluster, emptying the largest accessible pores first.
• Stage I evaporation (atmosphere-limited): removes water from small surface pores, where the vapor pressure is highest.
• Stage II evaporation (diffusion-limited): vapor diffusion through a dry surface layer can empty pores that are hydraulically disconnected from the gas-phase boundary — it bypasses the topological constraint.
• Root uptake: plants extract water through the root–soil interface, accessing a pore topology entirely unrelated to the gas-phase connectivity.

Each mechanism creates a different pore-emptying sequence. A soil dried by evaporation and then subjected to root uptake reaches a different internal configuration than one dried by roots first and then by evaporation. This suggests a richer non-commutativity than just “wetting versus drying” — the different drying mechanisms themselves may not commute. 

The Hosseini result already changes how we should think about several practical questions:

The body/throat distinction is unnecessary. If hysteresis does not come from the ink-bottle geometry, then the elaborate pore network models that distinguish “pore bodies” from “pore throats” are solving the wrong problem. What matters is the connectivity structure C(r, r’) — which pore classes are connected — not whether a given pore segment is a “body” or a “throat.”

Laboratory retention curves are insufficient for field predictions. Lab curves are measured under quasi-static capillary drainage — a single drying mechanism, at near-zero forcing rate. They capture none of the kinetic effects, none of the multiple drying mechanisms, and none of the rate dependence that dominate under field conditions. The gap between lab and field measurements may not be experimental error but a genuine physical discrepancy that no amount of careful laboratory work can eliminate.

Scanning curves require new models. The Mualem (1974) independent domain model — still the standard for predicting scanning curves — assumes that each pore class fills and empties independently. The topological asymmetry found by Hosseini implies that accessibility is history-dependent: which pores can participate in the current filling/emptying step depends on which pores are already full. This correlation is precisely what the independent domain assumption neglects.

Related posts: Minkowski functionals (a way to track water movement) | Five Paradoxes of Soil Hydrology | Minkowski functionals: Critical Limitations

References for further reading:

• Celia, M. A., Reeves, P. C., & Ferrand, L. A. (1995). Recent advances in pore scale models for multiphase flow in porous media. *Reviews of Geophysics*, 33(S2), 1049-1057.
• Haines, W. B. (1930). Studies in the physical properties of soil. V. The hysteresis effect in capillary properties, and the modes of moisture distribution associated therewith. J. Agric. Sci., 20(1), 97–116.
• Hassanizadeh, S. M., & Gray, W. G. (1993). Thermodynamic basis of capillary pressure in porous media. *Water Resources Research*, 29(10), 3389-3405.
• Hosseini, R., Kumar, K., and Delenne, J.-Y. (2024). Investigating the source of hysteresis in the soil–water characteristic curve using the multiphase lattice Boltzmann method. Acta Geotechnica, 19, 7577–7601.
• Lehmann, P., Assouline, S., and Or, D. (2008). Characteristic lengths affecting evaporative drying of porous media. Phys. Rev. E, 77, 056309.
• Lu, N., & Likos, W. J. (2004). Unsaturated Soil Mechanics. Wiley.
• Mualem, Y. (1974). A conceptual model of hysteresis. Water Resour. Res., 10, 514–520.
• Or, D., & Tuller, M. (1999). Liquid retention and interfacial area in variably saturated porous media: Upscaling from single-pore to sample-scale model. Water Resources Research, 35(12), 3591-3606.
• Tuller, M., Or, D., & Dudley, L. M. (1999). Adsorption and capillary condensation in porous media: Liquid retention and interfacial configurations in angular pores. Water Resources Research, 35(7), 1949-1964.
• Tuller, M., Or, D., & Hillel, D. (2004). Retention of water in soil and the soil water characteristic curve. Encyclopedia of Soils in the Environment, 4, 278-289.
• Mualem, Y. (1984). A modified dependent-domain theory of hysteresis. Soil Science, 137(5), 283-291.

The Tricky Energy Budget of Freezing Soil: A Thermodynamic Framework for Understanding Phase Changes

Freezing soil presents unique challenges in understanding the coupled mass and energy dynamics within the Earth’s critical zone. This paper presents a comprehensive thermodynamic framework for analyzing phase transitions in soil-water-ice systems. We present a unified treatment based on non-equilibrium
thermodynamics, where temperature, pressure, and chemical potential act as primary driving forces. The framework accounts for freezing point depression through mechanisms including the Gibbs-Thomson effect, solute presence, ice nucleation, and surface interactions. We demonstrate how upscaling from pore-
scale thermodynamics to the Darcy scale introduces theoretical challenges in determining phase transformation rates and flux laws. The sequential freezing process, governed by water energetic states in different pore sizes, creates complex interplay between capillary forces and phase changes essential for modeling

https://mosaicworks.com/gallery/fineart/permafrost/

This paper is an evolution of the talk given last summer at PanAm Unsat 2025 conference that you can find here. Official, preprint will be soon available.  Final accepted paper here. 

After the feedback of the LinkedIn post

 Following my LinkedIn post, I've gained around fifty new followers and received several inquiries about the PhD and postdoc positions. While many CVs show promise, they often lack the specific background we're seeking. To be clear: we value genuine interest and appreciation of our research over purely academic credentials, but this commitment must be demonstrated, not just stated.

Before reaching out, I strongly encourage prospective candidates to enroll (free of charge) in our Winter School [info here and here]. The Winter School has already begun, but a motivated PhD candidate should be able to catch up on the material independently and address any gaps. If difficulties arise, you're welcome to ask questions, but the ability to work through challenges autonomously is precisely what we're looking for.

No pain, no gain!

P.S. - Although the Winter School could not available at the time of your application, please take time to explore our materials and understand our research approach. We appreciate candidates who show commitment through their preparation.

During interviews, we typically explore:

• Your relevant skills for this specific project
• Your programming capabilities and experience
• Your motivation for not applying the Winter School (if you did not apply to any)
• Your understanding of GEOframe and our research
• Your availability and current commitments
• Technical aspects specific to the project
• The skills and expertise you've highlighted in your application (specificity is valued over generic statements)

A double failure is a giant failure and probably a sin of pride.I see now that I was wrong/ it was hubris all along

I've shared various posts about the STRADIVARI project over recent months. Now that both versions, submitted to the ERC Advanced Grant and FIS3 programs, were not selected for funding, I feel free to upload them to my collection of unsuccessful proposals. The panel evaluations, when they arrive, will provide valuable learning opportunities.

I'm fully aware these projects were ambitious, perhaps overly so. I sinned of pride. Yet I would be far more troubled if the panels judged them as "incremental" science, which would reveal a superficial analysis. While science advances steadily, hydrological science in particular suffers from persistent methodological weaknesses in model construction that prevent us from addressing fundamental questions properly. Too many simulation-based papers rely on flawed algorithms, imprecise conceptualizations, vague process descriptions, equations applied beyond their validity ranges,  missing feedbacks between coupled processes, imprecise and not well controlled data inputs (especially when going global).

The root cause lies in how models are conceptualized and built. Until our community recognizes this fundamental issue, we cannot progress toward answering the discipline's core questions. The current modeling paradigm perpetuates a cycle where narrative sophistication masks conceptual and  physical/mathematical  inadequacy.

When one attempts serious methodological critique, it's too easily dismissed as contrarianism, nastiness, or the rant of an angry dog rather than engaged with substantively. Flawed or myopic established approaches routinely appear in top journals without appropriate analysis, often relying on cherry-picked validation through calibration exercises. Let's be clear: many models do not work properly, and their results lack accountability. They are built on fragile foundations and superficial verification, yet their publication convinces project reviewers that fundamental problems have been solved when they haven't. This reaches its apex in global models where hyperresolution is married with hyperignorance (Beven et al., 2015), and in ancillary sciences studying the "Total Environment," where results on water quantities, fluxes and quality are presented as established fact when they remain deeply uncertain. 

To Cosma Shalizi's sharp observation (adapted to our context): the public discussion of hydrological models and processes often becomes "polluted by maniacal cultists with obscure ties to decadent plutocrats." I am obviously kidding. However, today there are many people  ".. who think they can go from the definition of conditional probability, via Harry Potter fanfic, to prophesying that an AI god will judge the quick and the dead, and condemn those who hindered the coming of the Last Day to the everlasting simulated-but-still-painful fire." I fight against this but I am, maybe, obsolete. 

This resistance and blindness to foundational critique, choosing institutional legacy over rational evaluation, ultimately impedes scientific progress. The STRADIVARI projects represented an attempt to break this cycle through component-based, rigorously coupled Earth system modeling, allowing hypothesis testing and cooperative work. Whether deemed too ambitious or not, the underlying questions it raised remain urgent and unresolved (and I am not the only one to write it. Voices are many).

Here the projects, useful to learn how to write a project differently:

• FIS3 project
• ERC PART 1
• ERC PART 2
• Timeline

Oh Notation ! About notation in Transit Time literature

What written here is nothing new. Is just what already present in previous papers and post, just said in a different manner to put a bridge among what people write in literature and what we think should be written.

 Time Variables and Their Definitions

The foundation of travel time theory rests on two fundamental time variables: $T$ (transit time) and $t$ (actual or clock time). Transit time is defined as the time a water parcel takes to cross a control volume or domain, expressed mathematically as $T = t_{ex} - t_i$, where $t_{ex}$ is the exit time and $t_i$ is the injection (entry) time.

If we observe the system at time $t$ and have measured or estimated the transit time $T$, we are effectively positioned at the outlet of the domain, considering water parcels exiting at that moment. These exiting parcels entered the domain at various previous times $t_i$, extending theoretically back to $-\infty$, meaning $T$ ranges from 0 (for parcels entering at time $t$ and exiting instantaneously) to $\infty$ (for parcels that entered in the distant past). While this represents an idealization—since the distant past is practically unknowable—it provides a useful theoretical framework. Please also observe that the injection time $t_i$ is a discrete variable, since the rainfall happens at discrete time steps and the same happens to any transported material, even if it is usually approximated as a continuous one. (Note:We cannot  distinguish ages from data at a very fine resolution. Therefore blurring the concept could not be wrong).  

The Transit Time Distribution

Transit time follows a probability distribution that can be conceptualized as the distribution of a random variable $T$. This distribution is time-dependent and conditional on the observation time $t$. The proper notation is $p_Q(T|t)$, though the literature often uses $p_Q(T,t)$, which can be misleading as it suggests a bivariate distribution when it's actually a conditional one. Some authors use $\overset{\leftarrow}{p}_Q(T,t)$ or place a backward arrow above $p$ to indicate that this probability refers to past events—hence the term "backward probability" or "backward transit time distribution."

As noted in Benettin et al. (2022)'s comprehensive review on transit time estimation and your previous AboutHydrology posts on residence time approaches, this backward-looking perspective is crucial for understanding catchment memory and the age composition of streamflow. In literature $p_Q(T|t)$ is treated as a continuous variable, but for the reason $t_i$ is a discrete variable, it is a discrete time distribution.  

The Complexity of Transit Time as a Variable

Using $T$ as the primary variable introduces conceptual complexity. Since $T = t - t_i$ with $t$ being the observation time (the conditioning variable), $T$ inherently depends on both the current time and the injection time $t_i$. The injection time $t_i$ is the true independent variable in this framework.

This distinction has important mathematical consequences: when expressed in terms of $t_i$, the mass conservation law remains an ordinary differential equation, but when formulated using $T$, it becomes a partial differential equation—a transformation that significantly complicates the mathematical treatment, as discussed in Botter et al.'s (2011) work on the master equation.

Residence Time Distribution

A distinct but related concept is the residence time distribution, denoted as $p_S(T_r|t)$, where the residence time at observation time $t$ is $T_r = t - t_i$. Unlike transit time, residence time considers all water parcels currently within the domain, not just those exiting. We assume we can label parcels by their age (their injection time $t_i$).

This distribution also looks backward from the current time $t$ and is often represented in literature as $p_S(T,t)$ or $\overset{\leftarrow}{p}_S(T,t)$, potentially causing confusion with transit time notation. Crucially, this distribution characterizes the age composition of water currently stored in the domain at time $t$—it makes no predictions about the future but provides a snapshot of the past up to the present moment.

The Link Between Distributions: StorAge Selection Functions

While $p_Q$ refers to water exiting the domain and $p_S$ refers to water stored within it, these distributions are naturally related—what exits must have been stored. The relationship between these probabilities is mediated by the StorAge Selection (SAS) function, denoted as $\omega(T,t)$, which describes how water of different ages is preferentially selected for discharge.

The SAS framework, as elaborated in recent work including studies on the contribution of groundwater to catchment travel time distributions, provides:

$$p_Q(T|t) = \omega(T,t) \cdot p_S(T|t)$$

The Special Case of Complete Mixing

There exists a special case where residence time and transit time distributions coincide: when parcels exiting the domain are uniformly sampled from the population of ages within the domain. This represents the "complete mixing" or "random sampling" assumption, where $\omega(T,t) = 1$ for all $T$.

Under this condition:

• Transit time and residence time collapse in a unique variable and the distributions become equal: $p_Q(T|t) = p_S(T|t)$
• All mass conservation equations simplify and become linear in the probability distributions
• The system behavior resembles that of a well-mixed reactor

As discussed in my posts about celerity versus velocity and the travel time problem, this simplification, while mathematically convenient, rarely holds in real catchments where preferential flow paths and incomplete mixing dominate the hydrological response.

Here's the complete reference list including your own AboutHydrology posts on the topic:

References

Published Literature

Benettin, P., Rodriguez, N. B., Sprenger, M., Kim, M., Klaus, J., Harman, C. J., van der Velde, Y., et al. (2022). Transit Time Estimation in Catchments: Recent Developments and Future Directions. Water Resources Research, 58(11). https://doi.org/10.1029/2022wr033096

Botter, G., Bertuzzo, E., & Rinaldo, A. (2010). Transport in the hydrologic response: Travel time distributions, soil moisture dynamics, and the old water paradox. Water Resources Research, 46, W03514. doi:10.1029/2009WR008371

Botter, G., Bertuzzo, E., & Rinaldo, A. (2011). Catchment residence and travel time distributions: The master equation. Geophysical Research Letters, 38, L11403. doi:10.1029/2011GL047666

Botter, G. (2012). Catchment mixing processes and travel time distributions. Water Resources Research, 48, W05545. doi:10.1029/2011WR011160

Comola, F., Schaefli, B., Rinaldo, A., & Lehning, M. (2015). Thermodynamics in the hydrologic response: Travel time formulation and application to Alpine catchments. Water Resources Research, 51(2), 1671-1687. doi:10.1002/2014WR016228

Cornaton, F., & Perrochet, P. (2006). Groundwater age, life expectancy and transit time distributions in advective-dispersive systems: 1. Generalized reservoir theory. Advances in Water Resources, 29(9), 1267-1291. doi:10.1016/j.advwatres.2005.10.009

McDonnell, J. J., et al. (2010). How old is the water? Open questions in catchment transit time conceptualization, modelling and analysis. Hydrological Processes, 24(12), 1745-1754.

McGuire, K. J., & McDonnell, J. J. (2006). A review and evaluation of catchment transit time modelling. Journal of Hydrology, 330, 543-563.

Niemi, A. J. (1977). Residence time distribution of variable flow processes. International Journal of Applied Radiation and Isotopes, 28, 855-860.

Rigon, Riccardo, Marialaura Bancheri, and Timothy R. Green. 2016. “Age-Ranked Hydrological Budgets and a Travel Time Description of Catchment Hydrology.” Hydrology and Earth System Sciences 20 (12): 4929–47. https://doi.org/10.5194/hess-20-4929-2016.

Rigon, Riccardo, and Marialaura Bancheri. 2021. “On the Relations between the Hydrological Dynamical Systems of Water Budget, Travel Time, Response Time and Tracer Concentrations.” Hydrological Processes 35 (1). https://doi.org/10.1002/hyp.14007.

Rinaldo, A., & Rodriguez-Iturbe, I. (1996). Geomorphological theory of the hydrologic response. Hydrological Processes, 10, 803-829.

Rinaldo, A., Beven, K. J., Bertuzzo, E., Nicotina, L., Davies, J., Fiori, A., Russo, D., & Botter, G. (2011). Catchment travel time distributions and water flow in soils. Water Resources Research, 47, W07537. doi:10.1029/2011WR010478

van der Velde, Y., Torfs, P. J. J. F., van der Zee, S. E. A. T. M., & Uijlenhoet, R. (2012). Quantifying catchment-scale mixing and its effect on time-varying travel time distributions. Water Resources Research, 48, W06536. doi:10.1029/2011WR011310

AboutHydrology Blog Posts

Rigon, R. (2014). Residence time approaches to the hydrological budgets. AboutHydrology Blog. http://abouthydrology.blogspot.com/2014/06/residence-time-approaches-to.html

Rigon, R. (2016). Celerity versus velocity and the travel time problem. AboutHydrology Blog. http://abouthydrology.blogspot.com/2016/06/celerity-vs-velocity.html

Rigon, R. (2023). A Commentary on transit (travel) times theory. AboutHydrology Blog. http://abouthydrology.blogspot.com/2023/01/a-commentary-on-transit-travel-times.html

Rigon, R. (2025). Transit Time and Residence Time Distributions: Fundamental Time Variables in Catchment Hydrology. AboutHydrology Blog. [This post]

Alps Are Losing Snow

Seasonal snowpack is a key component of the mountain cryosphere, acting as a vital natural reservoir that regulates runoff downstream in snowfed basins. In mid- and low-elevation mountain regions such as the European Alps, snow processes, such as accumulation and ablation, are highly sensitive to climate change, having direct implications for hydrological forecasting and water availability.

This study provides the first comprehensive long-term (1991-2021) analysis of snow water equivalent changes in the Po River District, Italy, one of Europe’s second most climate sensitive regions. Our findings show stark elevation-dependent changes in snow water storage and duration with profound and immediate implications for water security and climate adaptation.

Using a high-resolution (500m, daily) dataset from 1991-2021 (Dall'Amico et al., 2025), we observed two primary findings: First, we observed a profound loss of snow volume and decrease in duration below 2000 meters, with some low-elevation bands losing over 30% of their total snow-water storage. In contrast, high-elevation zones (>2500 m) are experiencing increased accumulation, but a continued shortened snow season. However, the increase in snow water storage at high elevations requires careful interpretation due to methodological constraints and systematic overestimation of high elevation precipitation by ERA5. Second, we show that this shorter snow season is not just an artifact of earlier spring melt, but is primarily driven by a delayed onset of snow accumulation in early winter. 

These elevation-dependent changes and loss of the seasonal snowpack highlight a fundamental shift in the hydrological regime of the Po River Basin, with significant implications for the timing and volume of runoff and the future availability of water in the region.Therefore, the Po River Basin is moving from a stable to a more volatile system.

Note: To know more, read our preprint.

References:

Dall’Amico, M., Tasin, S., Di Paolo, F. et al. 30-years (1991-2021) Snow Water Equivalent Dataset in the Po River District, Italy. Sci Data 12, 374 (2025). https://doi.org/10.1038/s41597-025-04633-5