# Numerical Simulation of Water Migration during Soil Freezing and Its Resulting Characterization
## 1 Introduction  
The phenomenon where water migrates from unfrozen to frozen zones and subsequently freezes due to \"frost suction\", leading to soil expansion, is termed frost heaving [1; 2]. To mitigate frost damage in geotechnical engineering within cold regions, it is crucial to study and predict the extent of deformation. The accuracy of frost heaving predictions hinges on the precise modeling of water migration during soil freezing [3; 4; 5; 6].  
[PERSON] was the first to demonstrate through experiments that frost heaving primarily results from moisture migration, challenging the earlier belief that it was predominantly due to the in situ freezing of pore water [7]. This revelation spurred numerous studies into the mechanisms of water migration in freezing soil, shifting the focus towards understanding hydraulic and thermal migration as well as phase changes in frozen soil. Frozen soil was compared to unsaturated soil by early researchers [8], and theorists like [PERSON] and [PERSON] attributed freezing and water migration predominantly to capillary action, formulating what is now referred to as the \"first frost heaving theory\", an early capillary model. This theory posits that capillary suction at the ice-water interface propels moisture migration infrozen soils [9; 10]. Drawing parallels between the processes of wetting, drying, freezing, and thawing in both partially frozen and unsaturated soils, [PERSON] (1973) proposed a \"hydrodynamic model\" that incorporates the effects of water flow and phase change latent heat on heat transfer [11].  
Prior to the advent of the [PERSON] hydrodynamic model, few researchers considered the coupling effects on the entire system in studies of frozen soil. [PERSON]'s work thus represents a pivotal advancement in the domain of frozen soil multi-field coupling. Subsequently, a plethora of related thermo-hydro-mechanical and hydrothermal coupling models for frozen soil have been developed. Building on [PERSON]'s foundational model, several scholars have extended its applications. Notably, [PERSON] and [PERSON] (1974, 1978, 1993) adopted the framework of the [PERSON] model and coupled the modified Richards equation with the convection-conduction heat transfer equation [12; 13; 14; 15; 16].  
[PERSON] (1978) introduced the rigid ice model to overcome the limitations of the \"first frost heating theory\", which failed to account for the development of ice lenses and water migration during freezing. He termed the intermediate area between the freezing front and the base of the warmest ice lenses as the \"freezing fringe\". This region, characterized by low water content and reduced moisture conductivity, acts as a buffer for migrating moisture as it reaches the freezing zone. Known also as the \"second frost heating theory\", the rigid ice model effectively addresses the shortcomings of the capillary theory [17; 18].  
The core of frozen soil deformation research lies in precisely delineating its water migration mechanisms. In this study, the process of pore water migration during soil freezing has been modeled, and the main mechanical representations caused by water migration: deformation, and pore water pressure were also investigated. The main goal of this study was to employ the fundamental equations of heat and mass transfer to predict these phenomena and to assess the method's feasibility and limitations.