2.1 Heat Conduction Equation

If we neglect the anisotropy of the soil and assume that horizontal gradients of all relevant physical quantities are small compared to the vertical gradients, the heat conduction problem can be reduced to one dimension. By further neglecting radiation and all forms of convection, the vertical heat flux can be described by Fourier's Law

$${q_{h}}(z,t) = - {k_{h,app}}(z,t) {\partial}_{z} T(z,t)$$ (2.1)

where $T$ is the soil temperature, $q_{h}$ is the flux of thermal energy and $k_{h,app}=k_{h}+k_{h,pseudo}$ is the apparent thermal conductivity consisting of the macroscopic thermal conductivity and a contribution to account for the transfer of latent heat by processes which are not purely conductive, such as latent heat transfer. Substituting Fourier's Law fourier into the equation of continuity for thermal energy yields

$${\partial}_{t} u(z,t) = {\partial}_{z} \left( k_{h,app}(z,t) {\partial}_{z}T(z,t) \right)$$ (2.2)

where $u$ is internal energy per unit volume. Assuming $u$ is only a function of temperature and using the definition of the volumetric heat capacity $c_{h}=\frac{\partial u}{\partial T}$, we can write Eq. (2.2) as a parabolic partial differential equation of temperature

$${c_{h}}(z,t) {\partial}_{t} T_{h}(z,t) = {\partial}_{z} \left( k_{h,app}(z,t) {\partial}_{z}T(z,t) \right)$$ (2.3)

subject to appropriate boundary conditions. Since $k_{h}$ may depend on the temperature, Eq. (2.3) is generally a non-linear equation. Assuming $c_{h}$ and $k_{h,app}$ are independent of depth, time and temperature, Eq. (2.3) becomes

$$\partial_{t} T_{h}(z,t) = D_{h,app} \partial_{z}^2 T(z,t)$$ (2.4)

where $D_{h,app}=\frac{k_{h,app}}{c_{h}}$ is the apparent thermal diffusivity.