Calculation of temperatures of deep tunnels as well as ducts/pipes (e.g. directional drilling) involves special considerations regarding thermal inertia.

• Depth value can easily reach hundreds of meters (tunnel between two valleys in a mountainous area)
• The initial temperature of rocky soils at the core of mountains can reach unexpected temperature values (more than 50 C)
• The heating source can be the combination of cable losses, other circulating fluids thermal effects (in multipurpose structures), but also water-cooling and forced ventilation with cold air renewal.
The installation of cables in shared tunnels (e.g., railway tunnels) can lead to large diameters, because some tunnel-boring machines are capable of excavating a diameter greater than eight meters. Therefore, some of the assumptions regarding the depth and the diameter may no longer be valid and a more accurate, although somewhat more complex, solution may be required as shown below.

The calculation of the equivalent depth of a deep tunnel is based on the IEEE paper 'Ampacity calculation for deeply installed cable' by E. Dorison, dated 2010. This method is considered valid also for deeply buried ducts/pipes.

Symbol
$L_{\mathrm{deep}}$
Unit
m
Formulae
 $\frac{Do_{\mathrm{d}}}{2000} \cosh{\left (\ln{\left (e^{- \frac{1}{2} \operatorname{expi}{\left (- \frac{Do_{\mathrm{d}}^{2}}{16000000 \delta _{\mathrm{soil}} \tau_{t}} \right )} + \frac{1}{2} \operatorname{expi}{\left (- \frac{L_{\mathrm{cm}}^{2}}{\delta _{\mathrm{soil}} \tau_{t}} \right )}} \right )} \right )}$ cables $\frac{D_{\mathrm{hs}}}{2000} \cosh{\left (\ln{\left (e^{- \frac{1}{2} \operatorname{expi}{\left (- \frac{D_{\mathrm{hs}}^{2}}{16000000 \delta _{\mathrm{soil}} \tau_{t}} \right )} + \frac{1}{2} \operatorname{expi}{\left (- \frac{L_{\mathrm{cm}}^{2}}{\delta _{\mathrm{soil}} \tau_{t}} \right )}} \right )} \right )}$ heat source $\frac{Do_{\mathrm{t}}}{2} \cosh{\left (\ln{\left (e^{- \frac{1}{2} \operatorname{expi}{\left (- \frac{Do_{\mathrm{t}}^{2}}{16 \delta _{\mathrm{soil}} \tau_{t}} \right )} + \frac{1}{2} \operatorname{expi}{\left (- \frac{L_{\mathrm{cm}}^{2}}{\delta _{\mathrm{soil}} \tau_{t}} \right )}} \right )} \right )}$ tunnel
Related
$D_{\mathrm{hs}}$
$\delta _{\mathrm{soil}}$
$Do_{\mathrm{d}}$
$Do_{\mathrm{t}}$
$L_{\mathrm{cm}}$