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

(1)	$\begin{array}{l}\displaystyle T(t, l) = T_b - R(t) \, G_b + \left( T_s - T_b(0) + R(t) \, G_b \right) \, e^{ - l/R(t)}\end{array}$

(2)	$\begin{array}{l}\displaystyle G_b = \frac{dT_b}{dl}\end{array}$

(3)	$\begin{array}{l}\displaystyle R(t) = \frac{q_s}{2 \pi \, a_b} \, \left( T_D(t) + \frac{\lambda_b}{r_f \, U} \right)\end{array}$

(4)	$\begin{array}{l}\displaystyle T_D(t) = \ln \left[ e^{-0.2 \, t_D} + (1.5 - 0.3719 \, e^{-t_D}) \, \sqrt{t_D} \right]\end{array}$

(5)	$\begin{array}{l}\displaystyle t_D(t) = \frac{a_b \, t}{r_w^2}\end{array}$