Firstly, we may declare the thermal conductivity of the soil layers as constant value (2nd assumption). Indeed, if heat transfer is plane and steady, and if is not large (in our case ), one can make a reasonably accurate approximation using a constant average value of (Lienhard &Lienhard 2003, p. 51).
Then, we may declare the density and the specific heat of the soil layers as constants and (3rd assumption). It is necessary to note that such assumption is somewhat groundless, especially for the cases of strictly inhomogeneous soils. However, only after this we may introduce a constant diffusion coefficient () of the soil (Bird et al. 2002, p. 268).
Finally, we may declare that the rate of internal energy conversion is negligibly small (4th assumption). In fact, such assumption is declaration of absence of the heat generation or consumption within the soil. Again we note that this assumption can be groundless for the inhomogeneous soils with stone inclusions, fluid- or air-filled interstices with internal convective flows. Moreover, the heat transfer in such porous and composite media is very difficult to analyze (Bird et al. 2002, p. 281-283).
for the steady boundary conditions and ; is the thickness of the soil upon the rock background. ...