Water freezing characteristics in granular soils: Insights from pore-scale simulations
Graphical abstract
Introduction
Shallow soil layer freezing and thawing play important roles in the annual hydrologic cycle of systems in cold regions because the degree to which soil freezes determines the infiltration rate of precipitated water or snowmelt, and thus affects the partitioning of water between surface runoff and infiltration (Hayashi, 2013). Understanding the fundamental processes governing the freeze-thaw dynamics in near surface soils and its impact on soils' hydraulic and transport properties are important for both hydrological and hydrogeological assessments including assessment of engineering designs, forecasting the migration of nutrients/contaminant leaching through frozen soil layers, and appropriately parameterizing soil freeze-thaw relationships for fully-integrated surface water and groundwater models such as HydroGeoSphere (Aquanty, 2019). Additionally, understanding the details of these processes at the pore-scale will facilitate the development of upscaling methodologies such that soil freezing processes can be effectively represented in local to large scale numerical models (e.g., Clausnitzer and Mirnyy 2016; Grenier et al. 2018; Berg and Sudicky, 2019; Langford et al., 2019). While several fundamental studies have been published on soil freezing processes and the hydraulic properties of frozen soils (e.g. Williams, 1964; Koopmans and Miller, 1966; Black and Tice, 1989; Spaans and Baker, 1996; Andersland et al., 1996; Stähli et al., 1999; Watanabe and Flury, 2008; Kurylyk and Watanabe, 2013), there still remain several fundamental questions that haven't been fully answered. For example, it not clearly known how partial freezing of soil could influence retention of unfrozen water; or, Is permeability saturation relation of unfrozen water in partially frozen soil different from that under partially drained condition? What role ice nucleation could play in soil freezing, and how important it could be in the natural shallow groundwater systems? How different is the transport of solutes (e.g. nutrient or dissolved contaminants) through partially frozen soil from their transport in partially saturated soil?
While analogies between the soil moisture curve (SMC; the relation between soil water content and water pressure in unsaturated conditions, sometimes also referred to as water retention curve) and the soil freezing curve (SFC; the relation between unfrozen water content and soil temperature in partially frozen soils) have been proposed (e.g. Williams, 1964; Koopmans and Miller, 1966; Black and Tice, 1989; Kurylyk and Watanabe,2013), the role of pore structure and connectivity in their similarity remains unclear. For a given soil type it is well known that shape of the SMC is determined by its pore-scale properties such as pore size distribution, pore connectivity, and aspect ratio (as the ratio of pore body size to that of connected pore throat); however, it is not clear if the SFC has the same sensitivity and functionality to these pore-scale properties. In addition, given that the distribution of water in the pore space determines the relative permeability-saturation (kr-S) relationship for the water phase, the similarities and differences of the kr-S relation of unfrozen water in partially frozen soils to that of water in partially saturated soils have remained uninvestigated.
The primary mechanisms controlling hysteresis observed in SFCs between freezing and thawing conditions are uncertain. Hysteresis in the SFC is observed as a difference in the unfrozen moisture content of the soil at a given temperature and depends on whether the soil is in freezing (decreasing temperature) or thawing (increasing temperature) conditions. This hysteretic behavior has been documented in several laboratory and field studies (e.g. Black and Tice, 1989; Spaans and Baker, 1996; Quinton and Blatzer, 2013), however, very few studies have attempted to explain the mechanisms for SFC hysteresis in granular soils where capillarity is dominant. In a study on gas hydrate growth in pore space, Anderson et al. (2009) explained the hysteresis in hydrate growth and desiccation curves with the ink-bottle effect. Tian et al. (2014) referred to the arguments of Anderson et al. (2009) and extended it to explain hysteresis in SFCs in soil. It should be noted that while there are similarities between gas hydrate growth and ice growth in pore space (i.e., both follow the Gibbs-Thomson equation for solid phase thermodynamic stability) there are fundamental differences between them. Gas hydrate is not exclusively formed out of water, and requires CH4 and/or CO2 for its growth. This means gas hydrate cannot start growing from the middle of a porous medium, and its growth must be piston-like, starting from the edges of the system and spreading into its center. In contrast, ice growth can occur anywhere in liquid water of the pore space (as ice can be formed out of water and without a need for a secondary molecule) during freezing. This means that thermodynamics determines the ice-water interface, and not the fluid transport (Wettlaufer and Worster, 2006). As a result, ice growth or melting in a given pore is not necessarily controlled by the connectivity of the pore to a continuous ice phase and thus is not subject to the effects of pore connectivity. Consequently, the ink-bottle effect on freezing and melting is negligible. In addition, Tien et al. (2014) argued that the freezing point depression during freezing is twice that in thawing (in identical unfrozen water contents) and used it as an evidence to support their explanations for SFC hysteresis; however, the results of Black and Tice (1989) and Spaans and Barker (1996) doesn't support this, which necessitates considering the possibility of other explanations. In another study, Spaans and Baker (1996) note that the hysteresis in SFC is similar to the hysteresis in SMC; however, comparisons of causal mechanisms in each case were not discussed.
Another potential mechanism that may play a role in SFC hysteresis is hysteresis of the water-ice contact angle. Hysteresis of the water contact angle for advancing and receding conditions has been vastly documented for multiphase systems (e.g., Morrow, 1975; Bikkina 2011; Andrew et al. 2014; Khishvand et al. 2016; Gharedaghloo and Price 2017), and it is a well-established point that a hysteresis in contact angle causes a hysteresis in the capillary pressure and soil moisture curves (e.g. Zhou, 2013; Ruspini et al. 2017; Sun et al., 2020). Such hysteresis and variability in the contact angle at the front of the solidifying phase (at its interface with the unsolidified liquid phase) has been observed and discussed before (Grosse et al. 1997; Wei et al. 2004). However, it has not been directly used to describe the hysteresis in soil freezing and thawing curves. It is worthwhile to assess whether the hysteresis of SFC can be explained by hysteresis in water-ice contact angles. This will provide insight for further investigations using high-resolution laboratory imaging techniques.
During soil freezing, when the frost front establishes and grows in the vadose zone, liquid water potential drops in the frozen zone causing a pressure gradient that drives liquid water to move from unfrozen depths to the freezing front (Hayashi, 2013); this is associated with an increase in total water content in the frozen portion of the vadose zone (Quinton and Blatzer, 2013), potentially providing water for additional freezing and leading to increased ice saturation. Ice saturation antecedent to snowmelt events significantly influences infiltration of meltwater (Iwata et al., 2010) and partitioning of meltwater between surface runoff and subsurface flow (Kane and Stein, 1983). An improved understanding of the upward migration of water during freezing could inform forecasts of pre-melt ice saturation, and support estimations of surface/subsurface water partitioning ratios.
Pore-scale numerical studies in porous media have been previously conducted to investigate other related phenomena, e.g. the relationship of pore size and pore connectivity to processes and properties including two-phase and three-phase flow (e.g. Ioannidis et al., 1993; Blunt, 2001; Valvatne and Blunt, 2004; Vogel et al., 2005; Culligan et al., 2006; Raeini et al., 2012; Blunt et al., 2013; Berg et al. 2016), solute transport (e.g. Bijeljic et al., 2004; 2013; Mehmani et al., 2012; Gharedaghloo et al., 2018), biofilm growth (von der Schulenburg et al., 2009; Peszynska et al., 2016), and phase change (Duval et al., 2004). Despite the advances in pore-scale simulation techniques, the micro-scale studies of soil freezing have been limited to the bundle of capillaries model (e.g. Watanabe and Flury, 2008; Lebeau and Konrad, 2012; Ming et al., 2020). Due to the complex nature of soil freezing/thawing, the simplifying assumptions used in the bundle of capillaries modelling approach are not necessarily valid for water freezing process in porous media. For example, these models lack micro-scale pore morphologies including pore bodies and pore throats, and connection between individual pores; capillary water and ice cannot be present simultaneously inside an individual pore. Additionally, the bundle of capillaries approach is over parameterized (e.g., tortuosity, hydraulic conductivity, and pore size distribution parameters) in comparison to realistic 3D pore space models which explicitly represent these features in their pore space representations and do not require such fitting parameters. Furthermore, bundle of tubes models cannot evaluate the possible causes of hysteresis in soil freezing.
Given the significant challenges associated with experimentally isolating the individual mechanisms that govern soil freezing (both in the field and the laboratory), we propose that pore-scale simulations of water freezing and water drainage with 3D pore space models helps addressing some of the abovementioned points. For example, with the 3D pore-scale simulations, the similarities between SFC and SMC, and between kr-S relations of soils undergoing freezing or drainage can be evaluated. Considering the uncertainty in the causes of SFC hysteresis and the difficulty in conducting pore-scale experiments, pore-scale numerical simulations may serve as an informative ‘laboratory’ for hypothesis testing and can be used to explore the effect of contact angle hysteresis on SFC hysteresis. Furthermore, it provides the ability to evaluate variations in water retention characteristics after a soil partially freezes, clarifying whether ice growth in pore space enhances or reduces the retention of liquid water.
With the use of pore-scale simulations the objectives of this study are: 1) to assess the analogies between SMC and SFC and kr-S relations of water in drainage and freezing conditions; 2) to study the influence of water-ice contact angle hysteresis on hysteresis of SFC; and, 3) to explore the influences of ice growth in the pore space on unfrozen water retention in the partially frozen soils. To our knowledge, this is the first study that proposes an algorithm and a model for simulating ice nucleation and ice growth in realistic 3D pore spaces. Here, for the first time, independent water drainage and water freezing simulation results are validated against an empirically validated phase equilibrium equation. The methods and workflow presented in this manuscript can also be extended to explore other phenomena including nutrients leaching, solute transport, and spilled non-aqueous phase liquids (NAPLs) percolation in frozen porous media.
Section snippets
Theory
The curvature at the interface of unfrozen water and frozen water (denoted as ice hereafter) causes a decrease in the water's freezing temperature. This is known as the Gibbs-Thomson effect (Wettlaufer and Worster, 2006) and in porous media it causes the water in micropores and pore corners (that necessitates a smaller water-ice interface curvature) to freeze in cooler temperatures in comparison to the water in the macropores. Eq. (1) describes this process (Liu et al., 2003) relating radius of
Water drainage
The modelling approach was verified by simulating water drainage for the 3 sand packs and comparing the simulated capillary pressure-saturation curves to previously published relations for similar sand packs in the literature. Fig. 1 illustrates the simulated (black and grey curves) capillary pressure-water saturation (Pc-S) curves (which is equivalent of SMC) and compares them to data measured for sand packs and sands available in literature (Brooks and Corey, 1964; Likos and Jaafer, 2013;
Conclusion
In this study we develop a pore-scale simulator capable of reproducing soil freeze/thaw and water drainage processes. This simulator was verified by simulating SFC and SMC for sand packs and synthetic silicate soil by illustrating that the adjusted phase equilibrium equation can calculate SMC out of SFC (and vice versa); this agrees with the results of previous studies. In the simulations, water drainage is controlled by the throat radii and requires connection of the air phase to one side of
CRediT authorship contribution statement
Behrad Gharedaghloo: Conceptualization, Methodology, Software, Validation, Visualization, Writing - original draft, Writing - review & editing, Resources. Steven J. Berg: Writing - review & editing, Resources, Supervision, Writing - review & editing, Resources. Edward A. Sudicky: Writing - review & editing, Resources, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank all anonymous reviewers; their comments and suggestions significantly improved this work.
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