- #1

Sashatgu

$$\frac{\partial }{\partial x}(K\frac{\partial \psi}{\partial x})=\frac{\partial \theta_u}{\partial t}+\frac{\rho_i}{\rho_w}\frac{\partial \theta_i}{\partial t}(3)$$

where K is the hydraulic conductivity of soil, [m/s]; θ_i is the volumetric ice fraction, i.e., the volume of the ice in per unit volume of frozen soil - dimensionless quantity; ψ is the soil suction potential, which controls the flow of the soil water; T is the temperature, [K]; x is the position coordinate, [m]; t is the time (s), θ_u is the volumetric unfrozen water fraction - dimensionless quantity, ρ_i - ice density [kg/m^3], ρ_w- water density[kg/m^3].

I also think that this article has an error, it is located on page 990 (bottom right, equation No. 3) and page 991 (top left, equations No. and 5). I do not understand how from the formula No. 3, substituting in it the formula No. 4, it is possible to receive the formula No. 5 (may be K depends on x, may be in formula No. 3 instead of K there should be D):

$$\frac{\partial }{\partial x}(K\frac{\partial \psi}{\partial x})=\frac{\partial \theta_u}{\partial t}+\frac{\rho_i}{\rho_w}\frac{\partial \theta_i}{\partial t}(3)$$

$$D = K\frac{\partial \psi}{\partial \theta_u}(4)$$

where D - The soil water diffusivity [m^2/s]

$$\frac{\partial }{\partial x}(D\frac{\partial \theta_u}{\partial x})=\frac{\partial \theta_u}{\partial t}+\frac{\rho_i}{\rho_w}\frac{\partial \theta_i}{\partial t} - \frac{\partial K}{\partial x} (5)$$

The second question: Do you know the quality and intuitive (with detailed explanations) articles, books, theses (on English language) on the subject: modeling of heat and mass transfer processes in frozen and thawed soils by the control (finite) volume method. This subject is very interesting. I am searching of one, two and three-dimensional problems, as well as software environments where you can realize the solution of these problems by “my” formulas (instead of formulas 'built into' these systems).

Here is the article:

https://www.researchgate.net/public..._in_freezing_soils_using_finite_volume_method