Motivation

Outputs

$\begin{array}{l}T_G(t, {\bf r})\end{array}$	Along-hole Geothermal Temperature Profile
$\begin{array}{l}G_T({\bf r})\end{array}$	Geothermal Temperature Gradient
$\begin{array}{l}H_n\end{array}$	Neutral Temperature Layer (NTL)

Inputs

$\begin{array}{l}t\end{array}$	Local Calendar Time	$\begin{array}{l}\delta T_A\end{array}$	Annual average surface temperature variation based on weather reports
$\begin{array}{l}z(l)\end{array}$	Wellbore trajectory True Vertical Depth Sub-Sea (TVDss)	$\begin{array}{l}A_T\end{array}$	Period of annual temperature variation cycle: $\begin{array}{l}A_T = 1 \, {\rm year}\end{array}$
$\begin{array}{l}j_z\end{array}$	True vertical component of regional Earth's Heat Flux	$\begin{array}{l}\delta t_A\end{array}$	Time shift of annual highest temperature with respect to January 1
$\begin{array}{l}T_s\end{array}$	Local annual average surface temperature based on weather reports	$\begin{array}{l}\delta T_D\end{array}$	Daily average surface temperature variation based on weather reports
$\begin{array}{l}a_{en}\end{array}$	Local average Thermal diffusivity of the soil between Earth's surface and NTL	$\begin{array}{l}D_T\end{array}$	Period of daily temperature variation cycle: $\begin{array}{l}A_D = 1 \, {\rm day}\end{array}$
$\begin{array}{l}\lambda_e({\bf r})\end{array}$	Subsurface Thermal Conductivity profile as function of position vector	$\begin{array}{l}\delta t_D\end{array}$	Time shift of daily highest temperature with respect to Midnight 00:00
		$\begin{array}{l}\delta T_{\rm cut}\end{array}$	Temperature measurement threshold (usually $\begin{array}{l}\delta T_{\rm cut} = 0.01 \, {\rm °C}\end{array}$ )

where

$\begin{array}{l}l\end{array}$	Measured Depth of wellbore trajectory with reference to Earth's surface ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}z_s = z(l=0)\end{array}$	TVDss of the Earth's surface in a given location. In case the Earth's surface is at sea level then $\begin{array}{l}z_s = 0\end{array}$

(1)	$\begin{array}{l}\displaystyle T_G(t, {\bf r}) = T_{GS}({\bf r}) + T_Y(t, z) + T_D(t, z)\end{array}$

(2)	$\begin{array}{l}\displaystyle G_T({\bf r}) = \frac{j_z}{\lambda_r({\bf r})}\end{array}$

(3)	$\begin{array}{l}\displaystyle \nabla T_{GS} = \lambda^{-1}({\bf r}) \cdot {\bf j}\end{array}$

(4)	$\begin{array}{l}\displaystyle T_{GS}(x, y, z = z_s) = T_s\end{array}$

(5)	$\begin{array}{l}\displaystyle T_Y(t,z) = \delta T_A \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_{en} \, A_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_A}{A_T} + (z_s -z) \sqrt {\frac{\pi}{a_{en} \, A_T}} \, \right]\end{array}$

(6)	$\begin{array}{l}\displaystyle T_D(t,z) = \delta T_D \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_{en} \, D_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_D}{D_T} + (z_s -z) \sqrt {\frac{\pi}{a_{en} \, D_T}} \, \right]\end{array}$

Neutral Layer

(7)	$\begin{array}{l}\displaystyle z_n = z_s + H_n\end{array}$

(8)	$\begin{array}{l}\displaystyle H_n = \sqrt{\frac{a_{en} \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} }\end{array}$