Hi

Not sure if this helps but frost depths are a common topic in research when looking at district heating networks in regions with seasonal freezing. I know there is a research group looking at this at the university of Luleå in Sweden (you can look up papers by Youen Pericault, he was a PhD and although I don't know if he still works there, you might be able to find other people from that research group). Good luck!

In my opinion, one of the best models to accurately simulate frost depths (provided you have the meteorological and subsurface data to drive it) is the SHAW model (https://www.ars.usda.gov/pacific-west-area/boise-id/watershed-management-research/docs/shaw-model/). It solves the surface energy and water balances as well as the coupled heat and water transport equations in a 1-D soil profile. It's free and quite easy to use. Hope this helps.

Thank you! Well, I know that USDA's SHAW model can simulate frost depth. I have tried to run the model. However, the results were not very good for saline soil or near saturated soil and too many soil and meteorological data were used to drive the model. Other models like Hydrus- 1d freezing- thawing module, CoupModel, et al. can also simulate frost depth in seasonal frozen soil, which need to further study. I just want to find a emperical or simple mathematical model that use less parameters (i.e., Stefan model) can also determine the frost depth better. As far as I know, it's quite difficult. Aaron Mohammed

Thanks for your suggestion! I'll check the literatures that you have supported to me.@ Ellika Rose Taveres-Cachat Ellika Rose Taveres-Cachat

It would seem that the dynamics involved in "frost heaves" in pavement is based on the pressure created by the expansion of the water in the soil from freezing no matter at what depth it occurs. I was modeling this for an insurance claim 35-years ago because the concrete wall of a swimming pool collapsed during a winter maintenance in which a very hard freeze occurred and the pool had been drained. The pour water expands by about 9-percent at the point of freezing (regardless of salinity which only varies the temperature of freezing). The pressure from only the thin layer where the freezing is occurring can be predicted and it is relatively large, and although it is in all directions in the soil, the unfrozen soil below is still compressible whereas the frozen soil above is rigid while the soil on all sides is experiencing the same expansion. The compressibility of the soil below can be modeled, and the resultant forces are pushed up causing the heave in whatever is above. Simple lab tests can be used to predict the compressibility of the soils in question with different moister contents and a curve developed (back in the 70's I had a soils lab where I could do the modeling.) Working in Missouri 9-years ago with a rural county which suffered what they called "frost boils", or soft spots (failure of the subbase) in their roads. These seemed to occur at predictable points in the road where surface water or a spring near or adjacent to the road was causing increased saturation of the subbase and base. We found that the solution was deepening the construction of the base course to below the anticipated frost depth with very larger stones that would drain away all of the excess pour water. You can see the mechanics (or the dynamics) of this, but the results were nearly 100-percent in the reduction of the frost problems.

Thank you all the same！@James Jatko James Jatko

Lately, there is a model STEMMUS-FT can mimic the dynamic change of soil ice content profile, which turns out the contribution of vapor flux is equivalently important as the liquid flux.

Article Liquid-Vapor-Air Flow in the Frozen Soil

Abstract

Accurate representing freeze‐thaw (FT) process is of great importance in cold region hydrology and climate studies. With the STEMMUS‐FT model (Simultaneous Transfer of Energy, Mass and Momentum in Unsaturated Soil), we investigated the coupled water and heat transfer in the variably saturated frozen soil and the mechanisms of water phase change along with both evaporation and FT process, at a typical meadow ecosystem on the Tibetan Plateau. The STEMMUS‐FT showed its capability of depicting the simultaneous movement of soil moisture and heat flow in frozen soil. The comparison of different parameterizations of soil thermal conductivity indicated that the de Vries parameterization performed better than others in reproducing the hydrothermal dynamics of frozen soils. The analysis of water/vapor fluxes indicated that both the liquid water and vapor fluxes move upward to the freezing front and highlighted the crucial role of vapor flow during soil FT cycles as it connects the water/vapor transfer beneath the freezing front and above the evaporation front. The liquid/vapor advective fluxes make a negligible contribution to the total mass transfer. Nevertheless, the interactive effect of soil ice and air can be found on the spatial and temporal variations of advective fluxes in frozen soils.