Theory and equations

Coupled water, vapor and heat flow

With Vapor flow switched on, liquid water and water vapor move under gradients of the pressure head and of the temperature Philip and de Vries (1957); Saito et al. (2006). The water flow equation becomes

∂θ_T/∂t = ∂/∂x [ K_Lh (∂h/∂x + cos α) + K_LT ∂T/∂x + K_vh ∂h/∂x + K_vT ∂T/∂x ] − S

Symbol	Meaning
θ_T	total water content, θ_T = θ_L + θ_v, the liquid water content plus the water vapor content expressed as an equivalent water content
K_Lh	isothermal hydraulic conductivity of the liquid phase [L/T] (the conductivity K(h) of the hydraulic model)
K_LT	thermal hydraulic conductivity of the liquid phase [L²/K/T] (temperature dependence of the surface tension)
K_vh	isothermal vapor hydraulic conductivity [L/T]
K_vT	thermal vapor hydraulic conductivity [L²/K/T]

The vapor conductivities follow from the vapor diffusivity in the soil air, the saturated vapor density and the relative humidity h_r = exp(h M g / (R T)) of the soil air (M molecular weight of water, g gravitational acceleration, R gas constant). The thermal vapor flux is multiplied by the enhancement factor η(θ) of Cass et al. (1984), which accounts for the higher thermal gradients in the air-filled pores and liquid islands (Further options: switchable).

The heat transport equation includes the transport of latent heat by the vapor flux and the convection of sensible heat by liquid water and vapor (heat transport); water flow and heat transport are solved sequentially in every time step. The surface boundary condition is usually the surface energy balance.

As in HYDRUS, the capacity term of the vapor content contains the unit conversion in a form that is exact for metres; with cm or mm the results differ slightly (e.g. 3.4 % in the cumulative surface flux of the HYDRUS example EnBal2b when changing from cm/d to m/h). Rusteau keeps the HYDRUS expression so that its results match HYDRUS 4.17.

Literature cited in this topic

Cass, A., G. S. Campbell, and T. L. Jones (1984). Enhancement of thermal water vapor diffusion in soil. Soil Science Society of America Journal 48(1): 25–32. doi:10.2136/sssaj1984.03615995004800010005x
Philip, J. R., and D. A. de Vries (1957). Moisture movement in porous materials under temperature gradients. Transactions, American Geophysical Union 38(2): 222–232. doi:10.1029/TR038i002p00222
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

All sources: References.