A regularization strategy for discontinuity when modelling coupled water and heat flow in freezing unsaturated soil

Abstract

Freezing tightly couples the water and heat flow. In most porous media, the interface between liquid and frozen water is not sharp and a slushy zone is present. There are two distinct approaches for mathematical modelling of freezing. It is the equilibrium approach which allows an instant freezing under given conditions and non-equilibrium approach where a specific timing in the freezing process is considered. In this contribution, we specifically target the equilibrium approach.The key mathematical model for the equilibrium approach is the Clausius–Clapeyron equation, which allows the derivation of a soil freezing curve relating temperature to pressure head. Implementing freezing soil accurately is not a straight-forward. Using the Clausius–Clapeyron equation creates a discontinuity in the freezing rate and latent heat at the freezing point. Little attention has been paid to the adequate description of the numerical treatment of this phenomenon and to the computational challenges that it poses. Numerical approximation of this discontinuous system is prone to spurious oscillations. In this contribution, we show the application of regularization of the discontinuous term. This treatment successfully stabilizes the computation and can remove oscillations. To avoid over-regularization, we present here a minimax strategy to determine optimal regularization parameters. We further compare an over-regularized setup with the non-equilibrium approach, where we show that a successful regularization is equivalent to incorporating a minimal timing to the freezing process. Finally – for validating our implementation and computational approach – experimental laboratory data of the volumetric water content and temperature profiles from previously published soil freezing experiments were represented here with the optimally regulized equilibrium approach.