2.4 The Effect of Soil Moisture on the Thermal Properties of Soils

The temporal variability of the parameters governing heat conduction in soil, $c_h$ and $k_h$ is determined mainly by the soil moisture. This is due to the fact that water and air are the only soil constituents which can vary considerably on a daily basis. In this section, analytical models for $c_h$ and $k_h$ are presented to describe this dependence.

2.4.1 Heat Capacity

The macroscopic volumetric heat capacity of a soil can be calculated by summing over all constituents and phases multiplied by their respective volumetric fractions $\theta_j$:

$$c_{h} = \sum_{j=1}^n {{\theta}_{j} c_{h,j}}$$ (2.17)

Note that $c_h$ increases linearly with increasing soil moisture content $\theta_w$. Table 2.1 lists the volumetric heat capacities of some major soil constituents. Due to the small heat capacity of air, the contribution of air towards the total heat capacity may be neglected to a good approximation.

Table 2.1: Thermal properties and densities of soil materials, water and air at $10 ^{\circ}C$ according to De Vries and Afgan (1975).

Substance	$\rho$ $[{kg \over m^{3}}]$	$c_{h}$ $[{J \over {m^{3} K}}]$	$k_h$ $[{W \over {m K}}]$
Quartz	$2.66 10^3$	$2.0 10^6$	$8.8$
Clay	$2.65 10^3$	$2.0 10^6$	$2.9$
Organic matter	$1.3 10^3$	$2.5 10^6$	$0.25$
Water	$1.0 10^3$	$4.2 10^3$	$0.57$
Air (dry)	$1.25$	$1.25 10^3$	$0.025$

Figure 2.1 shows the heat capacity calculated for a sandy loam soil at Research Centre Foulum, Denmark (Schelde et al., 1998). As for all materials, the heat capacity of soils is dependent of temperature. The effect is very small over the temperature range of interest though, and can be neglected to a good approximation.

Figure 2.1: Heat capacity $c_h$ as a function of volumetric soil moisture $\theta_w$ for an upper soil layer ($5 cm$ to $15 cm$) at Foulum, Denmark (§B.1). The solid line is calculated using Eq. (2.17) and the values $\eta = 0.54$ for porosity, $\theta_q = 0.39$ for volumetric quartz content, and $\theta_c = 0.042$ for volumetric clay content of the soil.

2.4.2 Thermal Conductivity

Many models and empirical formulae to calculate the macroscopic thermal conductivity of soils have been proposed (Sepaskhah and Boersma, 1979; Kersten, 1949; Kasubuchi, 1984; De Vries and Afgan, 1975; Nakshabandi and Kohnke, 1994). A good overview including a detailed evaluation of their applicability is given by Farouki (1986).

2.4.2.1 De Vries Model

In analogy to a model developed by Maxwell^2.2, De Vries (1952a) developed a model for the macroscopic thermal conductivity of ellipsoidal soil particles in a continuous medium of water (or air)

$$k_h = \frac{\sum_{j=1}^n {\kappa_j {\theta}_j k_{h,j}}}{\sum_{j=1}^n {\kappa_j {\theta}_j}}$$ (2.18)

where $k_{h,j}$ is the thermal conductivity and ${\theta}_j$ is the volume fraction of the $j$th constituent. $\kappa_j$ is the ratio of the space average of the temperature gradient in the soil grains of kind $j$ and the space average of the temperature gradient in the water (or air). Assuming a needle like shape for soil particles, Nobre and Thomson (1993) found that

$$\kappa_j = \frac{2}{3} {\left[ 1 + \left( \frac{k_{h,j}}{k_{... ...}}{k_{h,w}} - 1 \right) \left( 1 - 2 g_j \right) \right]}^{-1}$$ (2.19)

where the $g_j$'s are the shape factors given in Tab. 2.2.

Table 2.2: Subscripts and shape factors of soil constituent according to Nobre and Thomson (1993)

Constituent	Subscript $j$	Shape Factor $g_j$
Quartz	$q$	$0.125$
Clay	$c$	$0.125$
Organic matter	$o$	$0.500$
Water	$w$	-
Air (dry)	$a$	Variable

For air enclosures, the shape factor is dependent on the volumetric water content and is deduced by linear interpolation between the value for spherical shape at saturation and a value of $0.013$ at dryness (Kimball et al., 1976). Thus,

$$g_w = 0.013 + \left( \frac{0.022}{{\theta}_{w,wilt}} + \frac{0.298}{\eta} \right) {\theta}_w$$ (2.20)

where ${\theta}_{w,wilt}$ is the wilting point moisture content, and $\eta$ is the porosity of the soil. This approximation is only valid for ${\theta}_w > {\theta}_{w,wilt}$. For very dry soils^2.3, the De Vries model may be applied using air as the continuous medium. De Vries (1952a) suggests that one should discontinue calculations with water as a continuous medium at ${\theta}_w < 0.03$ for coarse soils or at ${\theta}_w < 0.05-0.1$ for fine soils. Farouki (1986) indicates that the de Vries model gives values within $\pm 10\%$ over the applicable range of $\theta_w$.

Figure 2.2: Thermal conductivity $k_h$ as a function of volumetric soil moisture $\theta_w$ for an upper soil layer ($5 cm$ to $15 cm$) at Foulum, Denmark (§B.1). Points are field measurements using the needle probe method (Schelde et al., 1998), the solid line is calculated using the de Vries equation ($\eta = 0.54$, $\theta_q = 0.39$, $\theta_c = 0.042$), and the dashed line is calculated using the Kersten equation ($\rho_d = 1.20$, $a_1 = 1.24$, $a_2 = -0.11$, $a3 = 0.62$). The soil thermal conductivity measurements were determined in the laboratory at successively higher suction levels ($1.0$, $5.0$, $10$ and $50 kPa$) using a minimum of three soil replicates for all layers.

Figure 2.2 shows the thermal conductivity calculated using the de Vries model for a sandy loam soil at Research Centre Foulum, Denmark (Schelde et al., 1998).

Below $\theta_{w} \approx 0.1$, the De Vries model overpredicts thermal conductivities, since water can no longer be considered a continuous medium in the soil. At complete dryness, the heat flow mainly passes through the grains and has to bridge the air-filled gaps between the grains around their contact points. As with the heat capacity, temperature dependence of $k_h$ can be neglected to a first approximation.

2.4.2.2 Kersten Equation

Kersten (1949) proposes a purely empirical formula for the calculation of the thermal conductivity based on measurements for five different soils

$$k_{h} = 0.1442 \left( a_{1} \log {\theta}_{w} - a_{2} \right) 10^{a_{3} {\rho}_{d}}$$ (2.21)

where the thermal conductivity $k_{h}$ is given in $\left[ {W \over {m K}} \right]$, the dry density ${\rho}_{d}$ in $\left[ {g \over cm^3} \right]$, and $a_{1}$, $a_{2}$ and $a_{3}$ are dimensionless empirical constants. Values of $a_{1}$, $a_{2}$ and $a_{3}$ valid for unfrozen sand soils are $0.750$, $0.400$ and $0.625$, respectively. According to Farouki (1986) the equation generally applies to soils with low silt-clay content (less than about 20 %). It should ideally be applied to coarse soils with an intermediate quartz content of about 60 % of the soil solids. Kersten's equation does not apply to dry soils or to crushed rocks. Figure 2.2 shows the thermal conductivity calculated using the Kersten equation as compared to the de Vries model and field measurements. At moistures below approx.  $0.05 \frac{m^3}{m^3}$ both models are no longer applicable.

2.4.3 Thermal Diffusivity

The thermal diffusivity is defined by Eq. (2.22). It governs the temperature response of a soil to thermal perturbations.

$$D_{h} = \frac{k_h}{c_h}$$ (2.22)

Figure 2.3 shows the thermal diffusivity calculated using the De Vries and Kersten equation for a sandy loam soil at Research Centre Foulum, Denmark (Schelde et al., 1998). For mineral and loam soils, the thermal diffusivity shows a maximum value at a relatively low value of $\theta_w$.

Figure 2.3: Thermal diffusivity $c_h$ as a function of volumetric soil moisture $\theta_w$ for an upper soil layer ($5 cm$ to $15 cm$) at Foulum, Denmark (§B.1). The solid and dashed lines are calculated using the De Vries and Kersten model, respectively (Fig. 2.2). Points are calculated using field measurements of $k_h$ (see the caption to Fig. 2.2).