The mathematical representation of freezing and thawing processes in variably-saturated, non-deformable soils

Recently, there has been a revival in the development of models simulating coupled heat and water transport in cold regions. These models represent significant advances in our ability to simulate the sensitivity of permafrost environments to future climate change. However, there are considerable differences in model formulations arising from the diverse backgrounds of researchers and practitioners in this field. The variability in existing model formulations warrants a review and synthesis of the underlying theory to demonstrate the implicit assumptions and limitations of a particular approach. This contribution examines various forms of the Clapeyron equation, the relationship between the soil moisture curve and soil freezing curve, and processes for developing soil freezing curves and hydraulic conductivity models for partially frozen soils. Where applicable, results from recent laboratory tests are presented to demonstrate the validity of existing theoretical formulations. Identified variations in model formulations form the basis for briefly comparing and contrasting existing models. Several unresolved questions are addressed to highlight the need for further research in this rapidly expanding field.