Hello, dear colleague. Now I'm dealing with issues of modeling heat and mass transfer in frozen and thawed soils. I am solving this problem numerically using the finite volume method. Below I give an equation from Harlan for mass transfer,

∂/∂x(K(∂ψ/∂x)) = (∂θ_{u}/∂t) + (ρ_{i}/ρ_{w})(∂θ_{i}/∂t),

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 [m]; 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}].

Give me please the most thorough explanation for these four questions:

What is the physical meaning of this equation?

What is the physical meaning of its left and right side?

In the right side of this equation, why has the second term a factor (ρ_{i}/ρ_{w})?

In which program can I put this equation to see its physical meaning?

P.S. Do you know quality and intuitive (with detailed explanations) articles, books, theses (in English language) on the subject of 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 looking for the treatment 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).