One-dimensional freezing of nonheaving unsaturated soils: Model formulation and similarity solution

Freezing of unsaturated soils is associated with the formation of a moving freezing zone and liquid water flow toward the zone. An equilibrium thermodynamic formulation of coupled flow and heat transport in variably saturated partially frozen porous media is developed and a self-similar solution is derived for the case of a semi-infinite horizontal porous media column with a constant freezing temperature on one boundary. Solutions to the self-similar equations are derived using a Runge-Kutta solution procedure. The solution is found to yield two possible modes distinguished by zones composed of different combinations of ice, liquid water, and air. One of the modes contains three zones: a frozen zone (WI) with just ice and liquid water; a transition zone (AWI) with ice, liquid water, and air; and an unsaturated zone (AW) with liquid water and air. The second mode contains only the WI zone and the AW zone. It is found that the WI zone is a quintessential part of the solution. The AWI zone is found to exist when the advancement of the freezing zone is relatively fast, while it is absent when the zone advances slowly. Predictions of ice saturation and liquid water saturation with the self-similar solution are compared to published experimental data. Pore pressure is calculated as a linear combination of ice pressure and liquid water pressure, and the calculated figures are used to provide a condition for model limitation in the case of incipient ice lens formation. The developed similarity solution provides insight into the mechanics of liquid water movement and pore filling with ice and the conditions for incipient heaving.