Phenomena involving frozen soil or rock are important in many natural systems and, as a consequence, there is a great interest in the modeling of their behavior. Few models exist that describe this process for both saturated and unsaturated soil and in conditions of freezing and thawing, as the energy equation shows strongly non-linear characteristics and is often difficult to handle with normal methods of iterative integration. Therefore in this paper we propose a method for solving the energy equation in freezing soil. The solver is linked with the solution of Richards equation, and is able to approximate water movement in unsaturated soils and near the liquid-solid phase transition. A globally-convergent Newton method has been implemented to achieve robust convergence of this scheme. The method is tested by comparison with an analytical solution to the Stefan problem and by comparison with experimental data derived from the literature.

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Journal Cryosphere
Dall'Amico, M. (M.), Endrizzi, S. (S.), Gruber, S, & Rigon, R. (R.). (2011). A robust and energy-conserving model of freezing variably-saturated soil. Cryosphere, 5(2), 469–484. doi:10.5194/tc-5-469-2011