Motivation

Outputs

$\begin{array}{l}T(t, l)\end{array}$

Along-hole Temperature Profile

$\begin{array}{l}t\end{array}$	Flowing duration	$\begin{array}{l}T_b(l)\end{array}$	Background temperature
$\begin{array}{l}q_s\end{array}$	Intake flowrate	$\begin{array}{l}a_b(l)\end{array}$	Thermal diffusivity of the surroundings
$\begin{array}{l}r_f\end{array}$	Flowing pipe radius	$\begin{array}{l}\lambda_b(l)\end{array}$	Thermal Conductivity of the surroundings
$\begin{array}{l}r_w\end{array}$	Wellbore radius	$\begin{array}{l}U(l)\end{array}$	Heat Transfer Coefficient (HTC) between pipe fluid and surroundings

where

$\begin{array}{l}l\end{array}$

wellbore trajectory Measured Depth with reference to Earth's surface ( $\begin{array}{l}l=0\end{array}$ )

Constant rate	Constant intake temperature
$\begin{array}{l}q_s(t) = q_s = {\rm const}\end{array}$	$\begin{array}{l}T_s(t) = T_s = \rm const\end{array}$

Equations