Simple numerical strategies to model freezing in variably-saturated soil with the standard finite element method

An accurate representation of freezing and thawing in soil covers many applications including simulation of land surface processes, hydrology, and degrading permafrost. Freezing and thawing tightly couple water and heat flow, where temperature and temperature gradients influence the water flow and phase changes, and water content and flow influence the heat transport. In most porous media, the interface between liquid and frozen water is not sharp and a slushy zone is present. A common observation of freezing soil is water accumulation towards the freezing front due to Cryosuction. A mathematical model can be derived using the Clausius-Clapeyron equation, which allows the derivation of a soil freezing curve relating temperature to pressure head. This is based on the assumption that soil freezing is similar to soil drying.

Many models still lack features such as Cryosuction. We believe that this may be due to numerical issues that model developers face with their current solver and discretization setup. Implementing freezing soil accurately is not straight-forward. Using the Clausius-Clapeyron creates a discontinuity in the freezing rate and latent heat at the freezing point and little attention has been paid to the adequate description of their numerical treatment and computational challenges. Discretizing this discontinuous system with standard finite element methods (standard Galerkin type) can cause spurious oscillations because the standard finite element method uses continuous base/shape functions that are incapable of handling discontinuity of any kind within an element. Similarly, standard finite difference methods are also not capable of handling discontinuities. In this contribution, we present the application of regularization of the discontinuous term, which allows the use of the standard finite element method. We implemented the model in the open-source code base DRUtES (www.drutes.org). We verify this approach on synthetic and various real freezing soil column experiments conducted by Jame (1977) and Mizoguchi (1990).

Jame, Y.-W., Norum, D.I., 1980. Heat and mass transfer in a freezing unsaturated porous medium. Water Resources Research 16, 811–819. https://doi.org/10.1029/WR016i004p00811

Mizoguchi, M., 1990. Water, heat and salt transport in freezing soil. sensible and latent heat flow in a partially frozen unsaturated soil. University of Tokyo.