“…The simulation of soil freezing and thawing processes is an active research topic for cold climate region due to their important roles in climate change, active layer and permafrost depths, hydrologic regimes, and cold region infrastructure sustainability (Hjort et al, 2022;Matzner & Borken, 2008;Painter et al, 2012Painter et al, , 2016Walvoord & Kurylyk, 2016). Various numerical models were developed to solve the coupled heat and moisture dynamics of frozen soil (Harlan, 1973;Mu & Ladanyi, 1987;Painter & Karra, 2014;Sheshukov & Nieber, 2011;Taylor & Luthin, 1978;Tubini et al, 2021;Westermann et al, 2016Westermann et al, , 2023Zhang et al, 2007). In this study, we adopt the solution for the saturated frozen soil from Sheshukov and Egorov (2002).…”