Equivalent depth for deep burial

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

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. Note: We consider a time period of 10 years for $\tau_L$

For soils with a thermal diffusivity varying from 0.35E-6 to 1.2E-6 m⊃2/s a soil layer with a depth of about 100 m results in a time constant of 30 to 100 years (refer to the book Sustainable Energy Technologies, chapter 9.3.1, by E. Rincon-Mejia, 2008). One year has more than 31.54 Mio seconds during Based on this We approximate $\tau_L$ with

Symbol
$L_{deep}$
Unit
m
Formulae
$\frac{D_{o} \cosh{\left(\ln{\left(e^{- \frac{\operatorname{expi}{\left(- \frac{D_{o}^{2}}{16 \delta _{soil} \tau_{L}} \right)}}{2} + \frac{\operatorname{expi}{\left(- \frac{L_{cm}^{2}}{\delta _{soil} \tau_{L}} \right)}}{2}} \right)} \right)}}{2}$exact
$0.374655935106899 e^{0.5 \operatorname{expi}{\left(- \frac{L_{cm}^{2}}{\delta _{soil} \tau_{L}} \right)} + 0.5 \ln{\left(4 \delta _{soil} \tau_{L} \right)}}$approximative
Related
$D_{o}$
$\delta _{soil}$
$L_{cm}$