Theory and equations

Heat transport

Neglecting the effects of water vapor diffusion, one-dimensional heat transport is described by Sophocleous (1979); Šimůnek et al. (2009):

C_p(θ) ∂T/∂t = ∂/∂x [ λ(θ) ∂T/∂x ] − C_w q ∂T/∂x − C_w S T

Symbol	Meaning
T	temperature [K or °C]
C_p, C_w	volumetric heat capacities of the porous medium and of the liquid phase [M L⁻¹ T⁻² K⁻¹]
λ(θ)	apparent thermal conductivity of the soil [M L T⁻³ K⁻¹]
q	water flux [L/T]
S	sink term of root water uptake

The first term on the right is conduction, the second convection by flowing water, the third the heat removed with the root water uptake. With vapor flow the equation contains the latent heat of the vapor flux as well Saito et al. (2006).

Heat capacity

Following de Vries (1963):

C_p(θ) = C_n θ_n + C_o θ_o + C_w θ + C_v a_v

with the volume fractions θ_n (solid phase), θ_o (organic matter), θ (water) and a_v (air), and the corresponding volumetric heat capacities (C_n = 1.92, C_o = 2.51, C_w = 4.18 MJ/m³/K; the air term is small).

Thermal conductivity

The apparent thermal conductivity combines the thermal conductivity of the soil without flow, λ₀(θ), with the macrodispersivity, a linear function of the water flux de Marsily (1986):

λ(θ) = λ₀(θ) + β_t C_w |q|

with the longitudinal thermal dispersivity β_t (λL). λ₀ follows Chung and Horton (1987):

λ₀(θ) = b₁ + b₂ θ + b₃ θ^0.5

with the empirical coefficients b₁, b₂, b₃; for sand 0.228, −2.406, 4.909, for loam 0.243, 0.393, 1.534 and for clay −0.197, −0.962, 2.521 W/m/K. Alternatively the function of Campbell (1985) computes λ₀ from the volume fractions of quartz, other minerals and clay.

Boundary conditions

At the surface and the bottom, the temperature (first type) or the heat flux (third type) is prescribed; the bottom can also have a zero gradient. The heat flux condition carries the heat of the water crossing the boundary:

−λ ∂T/∂x + C_w q T = C_w q T₀

A sinusoidal daily surface temperature T(t) = T̄ + A sin(2π t/p − 7π/12) can be prescribed (maximum at 13:00), or the surface temperature follows from the surface energy balance.

Numerical solution

The heat transport equation is solved with Galerkin finite elements and the Crank–Nicolson scheme, like the solute transport, after the water flow of each time step.

Literature cited in this topic

Campbell, G. S. (1985). Soil Physics with BASIC: Transport Models for Soil-Plant Systems. Elsevier, Amsterdam, 150 pp.
Chung, S.-O., and R. Horton (1987). Soil heat and water flow with a partial surface mulch. Water Resources Research 23(12): 2175–2186. doi:10.1029/WR023i012p02175
de Marsily, G. (1986). Quantitative Hydrogeology: Groundwater Hydrology for Engineers. Academic Press, London, 440 pp.
de Vries, D. A. (1963). Thermal properties of soils. In: W. R. van Wijk (ed.), Physics of Plant Environment, North-Holland, Amsterdam, pp. 210–235.
Saito, H., J. Šimůnek, and B. P. Mohanty (2006). Numerical analysis of coupled water, vapor, and heat transport in the vadose zone. Vadose Zone Journal 5(2): 784–800. doi:10.2136/vzj2006.0007
Sophocleous, M. (1979). Analysis of water and heat flow in unsaturated-saturated porous media. Water Resources Research 15(5): 1195–1206. doi:10.1029/WR015i005p01195
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten (2009). The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.08. HYDRUS Software Series 3, Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, 330 pp. https://www.pc-progress.com/en/Default.aspx?h1d-description

All sources: References.