The tricky Physics of freezing soils

 Soil freezing is one of the most complex physical processes affecting the Earth's hydrological cycle, yet it remains poorly understood despite its critical importance for climate modeling, agriculture, and infrastructure.  This presentation aims to introduce the intricate thermodynamic relationships governing frozen soil behavior and introduces innovative numerical solutions that are revolutionizing how we model these processes. A video on these slides can be found here. 

Modified from Lu and Godt, 2012. Find the presentation by clicking here

"Permafrost is not ice." This seemingly simple observation reflects a profound understanding that frozen soil represents a complex multiphase system where air, biota, liquid water, ice, and soil particles coexist in dynamic equilibrium. The traditional view of soil freezing as a simple phase transition grossly oversimplifies the physics involved.

The presentation emphasizes that proper soil freezing models must account for three fundamental thermodynamic potentials: temperature (or its inverse in non-equilibrium thermodynamics), pressure exerted by the system on the environment, and chemical potential. Each of these potentials drives different aspects of the freezing process and their interactions determine the overall system behavior.

The energy conservation equation reveals the intimate coupling between heat transfer and mass transfer during freezing. When soil freezes, the energy budget becomes strongly coupled to the mass budget and phase transitions, creating a system where small changes in one variable can cascade through the entire soil column. This coupling is mathematically expressed through terms that include both temperature gradients and mass flux divergence, highlighting why traditional approaches that treat heat and water transport separately often fail.

One of the most important insights from recent research concerns how water actually freezes in soil pores. Due to freezing point depression effects, the largest pores freeze first. This sequential freezing process means that as temperature drops, progressively smaller pores freeze, each at different temperatures determined by the complex interplay of solute concentration, pore geometry, and surface tension effects.

The research identifies several mechanisms controlling freezing point depression: the Gibbs-Thomson effect (curvature-induced freezing point depression), solute presence, ice nucleation kinetics, and interactions with pore boundaries. These processes combine to create soil freezing characteristic curves that show unfrozen water content decreasing gradually with temperature rather than exhibiting the sharp transition seen in pure water.

An educated guess in understanding soil freezing comes from recognizing its mathematical similarity to soil drying. The "freezing = drying hypothesis" suggests that during freezing, the effective chemical potential is determined only by liquid water, not the total water content. This insight allows researchers to use established soil water retention theory, such as the van Genuchten or Kosugi models, to predict freezing behavior.

This analogy proves particularly powerful because it enables the use of well-established relationships like Mualem's theory for predicting hydraulic conductivity as a function of unfrozen water content. The result is a unified framework where soil freezing can be modeled using modified versions of the Richardson-Richards equation, the standard equation for unsaturated soil water flow.

The mathematical complexity of coupled heat and water transport in freezing soil creates severe numerical challenges. The governing equations become highly nonlinear, with hydraulic capacity functions that exhibit sharp peaks near the freezing point. Traditional Newton methods fail to converge when solving these systems, leading to computational failures or unphysical results.

The breakthrough solution presented is the nested Newton-Casulli-Zanolli (NCZ) algorithm, which decomposes the nonlinear problem using Jordan decomposition. This approach separates the sharp nonlinear functions into monotonic components that can be solved iteratively. The NCZ algorithm dramatically outperforms traditional Newton methods, allowing stable solutions with large time steps while maintaining energy conservation.

The researchers have implemented these theoretical advances in WHETGEO-1D, a sophisticated modeling framework built on object-oriented programming principles. Unlike traditional procedural codes that hardwire specific equations, WHETGEO uses abstract interfaces and factory patterns to allow flexible combination of different soil water retention curves, hydraulic conductivity functions, and energy budget formulations.

This design philosophy, built on the OMS3 framework, enables rapid model evolution and prevents the "screwdriver problem" where having only one tool leads to seeing every problem as a screw. The modular architecture allows researchers to easily substitute different physical theories while maintaining the same robust numerical solver.

Field applications demonstrate WHETGEO's capabilities across multiple scales and conditions. The model successfully simulates complex scenarios including infiltration events that bring thermal energy deep into soil columns, surface energy exchanges during diurnal cycles, and seasonal freeze-thaw cycles. Comparison studies show that including phase change effects significantly alters predicted soil behavior, with frozen periods exhibiting markedly different hydraulic properties than unfrozen conditions.

The model's efficiency allows simulation of multi-year periods with time steps of hours or days, making it practical for long-term climate studies. This computational efficiency, combined with rigorous energy conservation, makes WHETGEO suitable for integration into larger Earth system models. WHETGEO  is an open source software distributed under the GPL 3.0 license. For learning its use, please browse the GEOframe 2022 Summer School slides and videos.

Bibliography

Amankwah, S. K., A. M. Ireson, C. Maulé, R. Brannen, and S. A. Mathias. 2021. "A Model for the Soil Freezing Characteristic Curve That Represents the Dominant Role of Salt Exclusion." Water Resources Research 57 (8). https://doi.org/10.1029/2021wr030070.

Casulli, Vincenzo, and P. Zanolli. 2010. "A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form." SIAM Journal of Scientific Computing 32 (4): 2225–73.

Dall'Amico, M., S. Endrizzi, S. Gruber, and R. Rigon. 2011. "A Robust and Energy-Conserving Model of Freezing Variably-Saturated Soil." The Cryosphere 5 (2): 469–84. https://doi.org/10.5194/tc-5-469-2011.

Devoie, Élise G., Stephan Gruber, and Jeffrey M. McKenzie. 2022. "A Repository of Measured Soil Freezing Characteristic Curves: 1921 to 2021." Earth System Science Data 14 (7): 3365–77. https://doi.org/10.5194/essd-14-3365-2022.

Groot, Sybren Ruurds de, and Peter Mazur. 1984. Non-Equilibrium Thermodynamics. New York, NY: Dover Publications.

Kosugi, K. 1999. "General Model for Unsaturated Hydraulic Conductivity for Soils with Lognormal Pore-size Distribution." Soil Science Society of America Journal 63 (2): 270–77. https://doi.org/10.2136/sssaj1999.03615995006300020003x.

Lunardini, V. J. 1985. "Freezing Soil Phase Change Occurring over Finite Temperature Difference." Proceedings 4th International Offshore Mechanics Arctic Engineering Symposium. ASM.

Muskat, M., and M. W. Meres. 1936. "The Flow of Heterogeneous Fluids through Porous Media." Physics 7 (September): 346–63. https://doi.org/10.1063/1.1745403.

Tubini, Niccolò. 2021. "Theoretical and Numerical Tools for Studying the Critical Zone from Plot to Catchments." Ph.D., Università degli Studi di Trento. https://iris.unitn.it/retrieve/handle/11572/319821/498093.

Tubini, Niccolò, Stephan Gruber, and Riccardo Rigon. 2021. "A Method for Solving Heat Transfer with Phase Change in Ice or Soil That Allows for Large Time Steps While Guaranteeing Energy Conservation." The Cryosphere 15 (6): 2541–68. https://doi.org/10.5194/tc-15-2541-2021.

Tubini, Niccolò, and Riccardo Rigon. 2022. "Implementing the Water, HEat and Transport Model in GEOframe (WHETGEO-1D v.1.0): Algorithms, Informatics, Design Patterns, Open Science Features, and 1D Deployment." Geoscientific Model Development 15 (1): 75–104. https://doi.org/10.5194/gmd-15-75-2022.

Zhang, Chao, Lingyun Gou, Shaojie Hu, and Ning Lu. 2022. "A Thermodynamic Formulation of Water Potential in Soil." Water Resources Research 58 (9). https://doi.org/10.1029/2022wr032369.

Zhang, Lianhai, Chengsong Yang, Dayan Wang, Peng Zhang, and Yida Zhang. 2022. "Freezing Point Depression of Soil Water Depending on Its Non-Uniform Nature in Pore Water Pressure." Geoderma 412 (115724): 115724. https://doi.org/10.1016/j.geoderma.2022.115724.

Freezing soil requires new algorithms

Assuming you have got the physics right and you wrote the right equations, which is not given for granted when you deal with freezing soils, you have to solve the equations. This paper, deals with this last topic: given the freezing soil equation it implements a new algorithm to solve it. This algorithm was invented by Casulli and Zanolli in their 2010 paper. They called it nested Newton, we renamed it NCZ from Newton-Casulli-Zanolli. It was implemented for solving Richards equation which present a spiky term called hydraulic capacity, that poses serous challenges to the convergence of the solver. We extended here to a new equation with the same type of terms. In fact all equations that involve phase transitions have terms of this type. 

To someone a new algorithm for integration of some equation can seem a minor achievement but, while in some type of simulation, the numerical errors of traditional methods can be somewhat constrained, in most of the simulation they do not and tend to increase up to a point that any prediction either quantitative or qualitative becomes useless. Obviously our case has an enormous effect when dealing with simulations of permafrost areas under the threat of climate change. If this introduction makes you curious, you can find the the preprint at The Cryosphere page, by clicking on the Figure above.

Reference

Casulli, Vincenzo, and ZANOLLI. 2010. “A Nested Newton-Type Algorithm for Finite Colume Methods Solving Richards’ Equation in Mixed Form.” SIAM Journal of Scientific Computing 32 (4): 2225–73.

Tubini, Niccolò, Stephan Gruber, and Riccardo Rigon. “A Method for Solving Heat Transfer with Phase Change in Ice or Soil That Allows for Large Time Steps While Guaranteeing Energy Conservation.”

Freezing-thawing processes studying with numerical models

This is the presentation given by Niccolò Tubini at the Carleton University last October 6th. Niccolò in his Master Thesis is working for a new implementation of the theory of freezing and thawing already covered by Matteo Dall'Amico in his Ph.D. Thesis and in Dall'Amico et al., 2011.

This work is part of the GEOtop project and its new implementations (see also here). The intention is to use the new numerical method implemented by Casulli and Zanolli (2010, 2011) of the extended Richards equation (see also the video here). The new method promise to be faster than the older one, more stable, and implemented for unstructured grids, while at present, GEOtop 2.0 uses a structured grid. Hopefully, the new development will be made in OMS3. 

References

Casulli, V., & Zanolli, P (2010). A nested newton-type algorithm for finite colume methods solving Richards' equation in mixed form. SIAM J. SCI. Comput., 32(4), 2225–2273.

Casulli, V., & Zanolli, P. (2012). Iterative solutions of mildly non linear systems, Journal of Computational and Applied Mathematics, 236(16), 3937–3947. http://doi.org/10.1016/j.cam.2012.02.042

Dall'Amico, M., Endrizzi, S., Gruber, S., & Rigon, R. (2011). A robust and energy-conserving model of freezing variably-saturated soil. The Cryosphere, 469–484. http://doi.org/10.5194/tc-5-469-2011

Ubiquitous Diffusion

These are the slides for the lecture I gave to Michael Dumbser' class on Environmental Modelling. I tried to show the non linear diffusion equations that can be found in analysing the water and energy budget of the soil-snow (with freezing soil) continuum. In practice I used the material from my class in hydrology (the whole stuff here) and other material from GEOtop's talks, and the presentation is, at the moment, in Italian and English.

Incidentally two of these equations present  discontinuities due to phase transitions and the three of them require special numerical methods to be integrated. Here I suggest that a good method could be the Nested Newton one, introduced recently by Casulli and Zanolli (for integrating Richards), and before by  Brugnano and Casulli (for integrating Boussinesq equation).
Below you can find also the audio (in Italian) of the lecture: Richards equation (21.9 Mb); Frozen Soil (18.4 Mb); Snow (7.1 Mb). I gave longer presentations on Richards equation, in Todini Symposium (here), and at the summer School on Landslide Modelling in Praia a Mare (here).
An update. A new treatment of part of this matter is given by Niccolo Tubini in his Master Thesis. The slides he used in the 2017 lecture are here.

Essential References

L. Brugnano and V. Casulli, Iterative solution of piecewise linear systems and applications
to flows in porous media, SIAM J. Sci. Comput., 31 (2009), pp. 1858–1873.

Casulli, V., and Zanolli, P., A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form, SIAM J. Sci. Comput., 32(4), 2255–2273,  Volume 32, Issue 4, 2010.

Cordano E., and Rigon R., A mass-conservative method for the integration of the two-dimensional groundwater (Boussinesq) equation,  Water Resour. Res., 49, doi:10.1002/wrcr.20072, 2013.

Dall'Amico, Matteo  (2011), Coupled Water and Heat Transfer in Permafrost Modeling. Ph.D. Thesis

Dall’Amico, M.; Endrizzi, S., Gruber, S; and Rigon, R. (2011), An energy-conserving model of freezing variably-saturated soil, The Cryosphere.

Endrizzi S., Gruber S., Dall’Amico M., Rigon R., GEOtop 2.0.: Simulating the combined energy and water balance at and below the land surface accounting for soil freezing, snow cover and terrain effects, Geosci. Model Dev., 2015

GEOtop 2.0 at AGU 2013 - II - The Cryosphere

In this marathon I am doing at this Fall AGU Meeting, I am also giving a second talk about GEOtop 2.0. But this time I talk about the simulations of the snow modelling and the soil freezing.
The presentation is mainly based on the work initiated with Stefano Endrizzi and Matteo Dall'Amico thesis, and subsequently pursed together with Stephan Gruber of Zurich University, now at Carleton University. The two reference papers are Dall'Amico et al. 2011 and Endrizzi et al., 2013, cited in the talk, but also the work in Gubler et al. 2013 is extremely relevant for all the testing it performed on the models

.

So clicking on the image, as usual, you will gain access to the presentation. While, in the first post I took the occasion for adding the References of GEOtop, in this case I just collect the main presentations on the topic that you can find below.

GEOtop relevant presentation

GEOtop 2008

Especially important to understand GEOtop history (see also these post: I and II).

GEOtop, the making of version 1.45  (Summer school on Environmental Dynamics, 2011)

summarised concepts already present in GEOtop 2008 presentation.

GEOtop, the snow modelling (now actually obsolete ... but a good reading for the general concepts)

GEOtop, landslides and other stuff

Some more philosophical arguments on modelling

GEOtop and Alpine ecohydrology

Beyond and side by side with numerics

The 23rd of april was quite busy for me. Early in the morning I gave a tak of acouple of hours to the students of Numerical Analysis.  My goal was actually to try to capture their interest about the topics I cover in my research, which I considere complementary to good numerics.

For supporting this idea, I divided my presentation in two parts. The first under the motto: the right numerics, for the right equations. There, I showed how Richards equation can be modified to account for transition to saturated conditions and  for freezing soil. This last part has been largely derived from the work of Matteo Dall'Amico (here his Ph.D. thesis) and the subsequent paper on the Cryosphere journal. With the benefit of hindsight, I can tell that I could have been much more clear on the physics of the problem, but it was just doing the presentation that I realised it.

The second part is dedicated to justify the rational of what I called "The Geoframe project", which is an integrated system for doing hydrology by computer, completely open source, and which has a deployment in the JGrass-NewAGE system. Click on the painting above  to see the presentation. 

New GEOtop presentations given at the Summer School on Surface Hydrology in Marsico Nuovo

Thank you to Salvatore Manfreda for having organized this event which I found fruitful and interesting. General Information on the summer School can be found here. What it is reported below are just the sequence of my seminars: actually a five parts seminar given in four hours.

First hour covers the motivation behind GEOtop, and its structure in terms of grids, equations, and boundary conditions.

The second hour covers the snow modeling (almost all of it, excluding snow compaction). Its equations and boundary conditions.

The third and fourth hours cover Richards equation (above), its extension to deal with saturated and freezing soils, and some material regarding landslide modeling with (and without) GEOtop (below).

The material is far from being complete. But it is better to have them now, a little broken, than never. The Authors ask people using this material to cite the appropriate GEOtop papers appeared in Journal of Hydrometeorology, Hydrological Processeses, and recently on The Cryosphere. Soon the user manual of GEOtop will be available. Continue to watch the blog !

A robust and energy-conserving model of freezing variably-saturated soil

We worked a lot for this paper, where we implemented the freezing = drying theory of frozen soil. The theoretical work behind it open the way to generalizations that we will pursue in the next months. Here it is the abstract:


Phenomena involving frozen soil or rock are important in many natural systems and, as a consequence, there is a great interest in the modeling of their behavior. Few models exist that describe this process for both saturated and unsaturated soil and in conditions of freezing and thawing, as the energy equation shows strongly non-linear characteristics and is often difficult to handle with normal methods of iterative integration. Therefore in this paper we propose a method for solving the energy equation in freezing soil. The solver is linked with the solution of Richards equation, and is able to approximate water movement in unsaturated soils and near the liquid-solid phase transition. A globally-convergent Newton method has been implemented to achieve robust convergence of this scheme. The method is tested by comparison with an analytical solution to the Stefan problem and by comparison with experimental data derived from the literature.

The paper and the previous discussion paper can be freely downloaded at The Cryosphere site

For who is interested, a nice companion of the paper is the reading of Matteo Dall'Amico thesis, where there is some other remarkable material, and an introduction to equilibrium Thermodynamics that can make happy those who never really understood the notation used by thermodynamicists. Matteo and I made an effort to derive the basic thermodynamics from some postulates (derived from Callen's book), and use the normal algebra (the one we learn at the beginning of our undergraduate studies): I think we were quite successful, and for us Thermodynamics is not anymore the "a dismal swamp of obscurity", as C. Truesdal said. Also the books by Garbrect-and -Bohren, and Muller-and-Weiss were very useful to achieve our results.

Please take the time to read the reviews which added very informed and beautiful literature, and knowledge to the first paper.

References

Bohren C. F., and Albrecth B. A., Atmospheric Thermodynamics, Oxford University Press, 1998

Callen, H. B., Thermodynamics and an Introduction to Thermostatistics, J. Wiley and Sons, 1985

Muller, I., and Weiss, Entropy and Energy: A Universal Competition, Springer, 2005

Truesdall, C., The tragicomical history of thermodynamics 1822-1854, Springer Verlag, 1980 (the sentence is at page 6)

For who interested in thermodynamics, I found also another couple of good books:

Ganguly A., Thermodynamics in Earth and Planetary Sciences, Springer, 2010

Zdunkowski W., and Bott, A., Thermodynamics of the Atmosphere: A Course in Theoretical Meteorology, Cambridge University Press, 2004