The thermal diffusivity can change as a function of depth and time depending on the soil

layer structure. To simplify calculations, the vertical node increments, ∆zi, are held fixed.

The time increment, ∆t, is allowed to vary according to

⎡ 0.45 ( ∆*z*i )2 ⎤

∆*t*

< 0.5 → ∆*t *= min ⎢

⎥.

(4.22)

( ∆*z*i )

2

⎢

⎥

⎣

⎦

Ensuring that the conditions set forth in Equation (4.21) are met, a lower value of 0.45 is

used to calculate the time step. Determination of the soil thermal diffusivity, *k*th, and other

physical properties needed to solve Equation (4.16) are discussed in Section 4.3. At each

calculation time step, ∆t is updated as necessary depending on the current soil physical

properties.

To determine the vertical temperature profile in the ground as governed by Equation

(4.16), several physical characteristics of the material are needed. They are the surface

diffusivity, *k*. The latter two parameters are, among other things, dependent on the

temperature and moisture content of the soil.

Following the recommendations of Farouki (1981), Johanson's method (1975) is used to

calculate the soil thermal conductivity as a function of soil type, porosity (*n*), dry density

volume (*ww*), and the Kersten number (*K*e). The governing equation is

(4.23)

where

⎧ *provided by user*

⎪ 135γ + 64.7

⎪

⎪ 2700 - 947γ 20%

κ dry = ⎨

(4.24)

⎪

0.05

⎪

0.55

⎪

⎩

⎧ 0.57n κ s(1-*n *)

⎪ n (1-*n *)

⎪2.2 κ s 0.269

κ sat = ⎨

(4.25)

0.55

⎪

⎪

1.80

⎩

⎧0.7 log *S*r + 1.0 *T * j,*i *> 273.15*K coarse*

⎪

(4.26)

⎪

⎩

35