Geometry and Thermodynamics of soils

Soil physics textbooks, often illustrated with a simple diagram of a pore shaped like an old ink bottle: a wide body connected to a narrow neck. While pedagogically useful, this explanation deserves closer scrutiny, particularly when we consider what actually drives hysteresis at the pore scale.

The ink-bottle model attempts to explain why soil water retention curves differ during wetting and drying, the phenomenon we call hysteresis. The argument is straightforward:

During drainage (drying), water trapped in the wide pore body cannot escape until the matric potential is low enough to drain through the narrow neck. The neck acts as a bottleneck, controlling when water leaves. This means water remains in the pore at higher suctions than the pore body size alone would suggest. This concept pairs with the idea of feasible optimality : the system reaches equilibrium not at the minimal energy configuration, but in an intermediate state compatible with the topology of the pore network configuration. The geometry constrains what thermodynamic states are accessible.

During wetting, water enters through the narrow neck and fills the larger body once it passes through. The pore fills at a lower matric potential than it empties, creating the hysteretic loop. While this process may require time to complete, its final result approaches the minimal energy configuration allowed by the geometry.

The traditional criticism of the ink-bottle model is that it only works with very specific pore connectivity, that if you imagine soil as a simple bundle of separate capillary tubes, each pore fills and empties independently with no geometric trapping.

However, this criticism itself is based on an oversimplified geometry . Real soil pore networks are three-dimensional, interconnected structures with braided pathways where the pore radius varies continuously along each flow path. This is fundamentally different from either isolated capillaries or idealized ink-bottles.

In actual soils, as documented in the work of Or and Tuller (1999, 2004), pore spaces form complex 3D networks where:-Flow paths are not isolated tubes but interconnected channels with varying cross-sections
Pore throats and bodies alternate along any given pathway, creating natural "ink-bottle" geometries
Multiple throats converge to form larger pore spaces, then diverge again
Pore radii vary continuously rather than jumping between discrete sizes

When viewed in three dimensions, the ink-bottle configuration—wide bodies accessible through narrow throats—is not a special case but rather a generic feature of porous media structure .

This 3D perspective reveals that ink-bottle effects can indeed create non-equilibrium conditions, particularly where multiple narrow throats converge to wider pore bodies. During drainage, water trapped in these convergent zones cannot escape until the matric potential is sufficient to empty through the smallest controlling throat. During wetting, the sequence reverses but follows different pathways through the network.

Hysteresis emerges from the interplay of two distinct physical mechanisms operating together:

1. Geometric Ink-Bottle Effects (Pore Network Topology)Water distribution controlled by pore throat sizes in the connected network
Different filling/emptying sequences through the 3D braided structure
Path-dependent behavior arising from network connectivity

2. REV-Scale Non-Equilibrium (Pore-Scale Energy States)

The second mechanism operates at a different scale and through different physics:

During rapid wetting , water fills pores according to their spatial accessibility and network connectivity, not according to their equilibrium energy states. Water reaches pores based on which pathways are available through the network, creating an initial distribution that may be far from thermodynamic equilibrium. Redistribution toward equilibrium then requires a relaxation time that can be much longer than the timescale of flow. Water may be transported rapidly away from its initial position along large, well-connected pores before it has time to redistribute into smaller pores that would represent lower energy states.

Additional factors contributing to REV-scale non-equilibrium:

• Non-uniform water distribution within individual pores : Even a single pore contains water in different states
• Multiple energetic states : Water exists as capillary water, adsorbed films, and tightly-bound water with very different chemical potentials
• Chemical potential gradients : These drive slow relaxation processes that may span hours to days

These geometric and REV-scale mechanisms operate together, not in isolation. The geometric structure determines which pores can access water at a given potential, while the REV-scale thermodynamic state determines how that water distributes within and among accessible pores. Even in a single, simple cylindrical pore, water doesn't distribute uniformly during drainage and wetting:

During drainage, water doesn't simply empty from pores in a binary fashion:

• Water retreats from pore centers first as bulk capillary water drains
• Thin films remain along pore walls, held by adsorptive forces
• Water persists in corners and surface roughness features
• These films exist with chemical potentials far below that of bulk water

Macroscopically the pore may appear "empty," but significant water remains—water that's thermodynamically distinct from the bulk phase.
During subsequent wetting, water advances into these partially dry pores from a completely different initial configuration:

• Contact angles differ from those during drainage
• The progression of wetting fronts follows different pathways
• Film thickening kinetics create different water distributions

The final state depends on the entire wetting history, not just the current matric potential. Recognizing hysteresis as fundamentally a non-equilibrium phenomenon has profound implications: we need infiltration theories that don't assume local equilibrium at the REV scale .

The classical Richards equation approach assumes that at each point in the soil profile, water content and matric potential are in local thermodynamic equilibrium—related uniquely by the water retention curve. But if water distribution is inherently out of equilibrium during infiltration (both geometrically through network effects and at the REV scale through multi-timescale relaxation), this assumption breaks down precisely when we're trying to model infiltration.

A proper non-equilibrium infiltration theory must account for:

1. Kinetics of pore-scale redistribution : Water distribution lags behind the macroscopic pressure field, with different regions (capillary, adsorptive, tightly-bound) evolving on different timescales

2. Flux-dependent retention : The effective θ(ψ) relationship depends on the infiltration rate q. Higher fluxes lead to greater departures from equilibrium, with preferential filling of larger, more accessible pores

3. Adsorptive forces : The role of surface interactions in establishing water films that don't instantly equilibrate with bulk water pressure. For clay-rich soils, this can dominate the retention behavior

4. Path-dependence : The recognition that the current state depends on the entire infiltration history, not just current boundary conditions

Such a theory would treat infiltration not as instantaneous local equilibration, but as a process where pore-scale water distribution evolves dynamically, influenced by both current conditions and past states. The water retention curve becomes a trajectory in phase space rather than a unique constitutive relationship.
Understanding hysteresis as arising from both geometric network effects and REV-scale non-equilibrium fundamentally changes how we approach vadose zone hydrology:
Geometric network models capture pore connectivity and ink-bottle trapping through the 3D structure. These are essential for understanding how water accesses different regions of the pore space.
Equilibrium thermodynamic models assume retention curves uniquely describe the water state at each potential—missing both the network topology effects and the REV-scale non-equilibrium dynamics.

Non-equilibrium theories must integrate both aspects: the geometric constraints from 3D pore networks AND the REV-scale non-equilibrium within and among pores. The effective retention behavior emerges from:

• Network topology : Ink-bottle trapping in braided 3D pore structures with varying throat/body sequences
• Stochastic infiltration : Initial filling that doesn't respect equilibrium energy ordering
• Multi-timescale relaxation : Fast capillary redistribution through the network, slow adsorptive equilibration within pores
• Adsorptive forces : Creating energetically distinct water populations that relax on different timescales
• Chemical potential gradients : Driving water redistribution both between pores (through throats) and within pores (films and bulk water)
• Preferential flow : Both network connectivity (which pores are accessible) and stochastic filling (which accessible pores actually fill first)
• Rate-dependent hysteresis : Network effects persist at all rates, while REV-scale non-equilibrium increases with faster processes
• Mobile-immobile water : Geometrically trapped water in isolated pore bodies PLUS adsorbed water that equilibrates too slowly
•  Lab-field discrepancy : Laboratory measurements at equilibrium miss both network-scale connectivity effects and field-scale REV non-equilibrium dynamics

The ink-bottle effect is real and important—not as a pedagogical simplification but as a fundamental manifestation of 3D pore network geometry. When we view soil pore spaces in their true three-dimensional complexity, configurations with varying throat and body sizes along interconnected pathways are ubiquitous.

However, geometry alone cannot explain the full richness of hysteretic behavior. The complete picture requires recognizing that geometric structure and REV-scale non-equilibrium work together : network topology determines which pores can participate in water retention, while REV-scale thermodynamic processes determine how water distributes within the accessible pore space and how quickly it approaches equilibrium.

Developing rigorous infiltration theory requires capturing both the spatial constraints of pore network connectivity and the temporal evolution of REV-scale thermodynamic states. This is not about choosing between geometric or thermodynamic explanations, but understanding how pore network topology and interfacial thermodynamics interact to create the path-dependent, rate-sensitive, history-aware behavior we call hysteresis.

References for further reading:

• 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.
• Lu, N., & Likos, W. J. (2004). *Unsaturated Soil Mechanics*. Wiley.
• Hassanizadeh, S. M., & Gray, W. G. (1993). Thermodynamic basis of capillary pressure in porous media. *Water Resources Research*, 29(10), 3389-3405.
• 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.