A closed-form model of saline soil freezing curve developed using symbolic regression and its numerical implementation

Abstract

The soil freezing characteristic curve (SFCC) plays a crucial role in modeling coupled water-heat-salt transport in saline frozen soils. However, existing SFCC models are either empirical, lacking the ability to capture dynamic salt transport, or theoretically based but too complex for numerical implementation due to intricate formulations or the requirement for iterative algorithms and many parameters. To overcome these limitations, this study employs symbolic regression to derive a closed-form SFCC model, trained on synthetic data generated by a theoretical model grounded in the pore water freezing theory. The proposed model performs well in reflecting theoretical results across various soil types and initial water contents, providing a concise yet physically meaningful representation of the pore solution freezing mechanisms. The reliability of the model is demonstrated by comparison with 26 sets of experimental SFCC data. Furthermore, the model is simple in form and is successfully applied to simulate the coupled water-heat-salt transport during one-dimensional freezing. The simulations reasonably capture frost heave, temperature evolution, and water content variation under different initial and boundary conditions, in agreement with experimental results. The findings indicate that increasing boundary salinity elevates internal salt concentration and depresses the local freezing point, leading to an upward shift (closer to the cold end) of the freezing front. Meanwhile, ice formation causes salt accumulation at the freezing front. The proposed closed-form SFCC model allows for time-varying salt concentration in soils to be incorporated into fully coupled moisture and temperature fields, offering both strong theoretical grounding and promising potential for numerical applications.