Model of soil freezing and thawing

In many practical applications, heat conduction is the dominant mode of energy transfer in a ground material. Within certain assumptions (Kudryavtsev, 1978; Andersland and Anderson, 1978) the soil temperature $T, [{}^{\circ}C]$ can be simulated by a 1-D heat equation with phase change (Carslaw and Jaeger, 1959):

C{\frac{\partial{}}{\partial{t}}}T(x,t)+ L{\frac{\partial{...
\ \ \ \ x\in[0,l], \ \ \ \ t\in[0,\tau].
\end{displaymath} (1)


The quantities $C{=}C(T,x)$$[Jm^{-3}K^{-1}]$ and$\lambda{=}\lambda(T,x)$$[Wm^{-1}K^{-1}]$ represents the volumetric heat capacity and thermal conductivity of soil, respectively; $L~[Jm^{-3}]$ is the volumetric latent heat of fusion of water, and $\theta$ is the volumetric water content. The heat equation (1) is supplemented by initial temperature distribution $T(x,0){=}T_0(x)$, and boundary conditions at the ground surface $x{=}0$ and at the depth $l$. We use the Dirichlet boundary conditions, i.e. $T(0,t){=}T_u(t)$$T(l,t){=}T_l(t)$. Here, $T_0(x)$ is the temperature at $x\in[0,l]$ at time $t{=}0$; $T_u$ and $T_l$ are observed temperatures at the ground surface and at the depth $l$, respectively.

One of the commonly used measures of liquid water in the freezing soil is the volumetric unfrozen water content (Anderson and Morgenstern, 1973; Osterkamp and Romanovsky, 1997; Watanabe and Mizoguchi, 2002; Williams, 1967). There are many approximations to $\theta$ in the fully saturated soil (Galushkin, 1997; Lunardini, 1987). The most common approximations are associated with power or exponential functions. Based on our positive experience in Romanovsky and Osterkamp (2000), we parameterize $\theta$ by a power function $\theta(T){=}a\vert T\vert^{-b}; a,b{>}0$ for $T{<}T_*{<}0^{\circ}C$ (Lovell, 1957). The constant $T_*$ is called the freezing point depression. In thawed soils ($T{>}T_*$), the amount of water in the saturated soil is equal to the soil porosity $\eta$. Therefore, we assume that

\theta(T,x)=\eta(x)\phi(T,x), \ \ \ \phi(T,x)=\left\{
...{b(x)}\vert T\vert^{-b(x)}, & T<T_*\\
\end{displaymath} (2)


where $\phi$ represents the liquid pore water fraction. For example, small values of $b$ describe the liquid water content in fine-grained soils, whereas large values of $b$ are related to coarse-grained materials in which almost all water freezes at the temperature $T_*$.

We adopt the parametrization of thermal properties proposed by (de Vries, 1963; Sass et al., 1971) with some modifications. We express thermal conductivity $\lambda$ of the soil and its volumetric heat capacity $C$ as

C=C_f(1-\phi)+C_t\phi, \ \ \ \ \ \ \ \lambda=\lambda_f^{1-\phi}\lambda_t^{\phi},
\end{displaymath} (3)


where $C_f$ and $C_t$ are the effective volumetric heat capacities, respectively, and $\lambda_f$ and $\lambda_t$ are the effective thermal conductivities of soil for frozen and thawed states, respectively. For most soils, seasonal deformation of the soil skeleton is negligible, and hence temporal variations in the total soil porosity, $\eta$, for each horizon are insignificant. Therefore, the thawed and frozen thermal conductivities for the fully saturated soil are obtained from

\lambda_f{=}\lambda_s^{1{-}\eta}\lambda_i^{\eta}, \ \ \
...a\big){+}C_i\eta,\ \ \ C_t{=}C_s\big(1{-}\eta\big){+}C_w\eta.
\end{displaymath} (4)


where subscripts $i$, $l$, and $s$ mark heat capacity $C$, thermal conductivity $\lambda$ for ice at $0^{\circ}C$, liquid water at $0^{\circ}C$ and solid soil particles, respectively. Combining formulas in (4), we derive that


\lambda_t{=}\lambda_f\Big[\frac{\lambda_l}{\lambda_i}\Big]^{\eta}, \ \ \ \ \ \ \


Hence, the thermal properties $\lambda$ and $C$ are expressed by only four variables such as $\lambda_f$, $C_f$, $\eta$, and $\phi$.

Evaporation from the ground surface and from within the upper organic layer can cause partial saturation of upper soil horizons (Kane et al., 2001; Hinzman et al., 1991). Therefore, formulae (4) need not hold in the presence of live vegetation and within organic soil layers, and possibly not within organically enriched mineral soil (Romanovsky and Osterkamp, 1997). Besides organic and organically enriched mineral soil layers, there are other horizons, all of which can have distinctive physical properties, texture and mineral composition. We assume that there are several horizons, namely: an organic layer, an organically enriched mineral soil layer, and a series of mineral soil layers. We assume that physical and thermal properties do not vary within each horizon, and hence $\lambda_f(x)$, $C_f(x)$, $\eta(x),T_*(x),b(x)$ can be assumed to be constants within each soil horizon:

\lambda_f(x){=}\lambda_f^{(i)}, \ \ C_f(x){=}C_f^{(i)}, \ ...
...x){=}\eta^{(i)}, \ \
b(x){=}b^{(i)}, \ \ T_*(x){=}T_*^{(i)},
\end{displaymath} (5)


where the index $i$ marks the index of the soil layer. Table 1 shows a typical soil horizon geometry and the commonly occurring ranges for the porosity $\eta$, thermal conductivity $\lambda_f$ and the coefficients $b$, $T_*$ parameterizing the unfrozen water content.

Table 1: A range of thermal properties for common soil types at the North Slope, Alaska.
Layer Thermal conductivity, $\lambda_f$ Porosity, $\eta$ $b$ in (2) $T_*$ in (2)
Mineral-organic mixture $[0.7,1.8]$ $[0.2,0.6]$ $[0.5,0.8]$ $[-0.05,-0.01]$
Mineral soil(silt) $[1.3,2.4]$ $[0.2,0.4]$ $[0.5,0.7]$ $[-0.1,-0.01]$
Mineral soil(gravel) $[2.5,3.5]$ $[0.2,0.4]$ $[0.5,0.7]$ $[-0.1,-0.01]$
Mineral soil(shale) $[1.0,2.0]$ $[0.1,0.3]$ $[0.5,0.7]$ $[-0.1,-0.01]$