Motivation

For the stabilized flow the wellbore pressure profile is constant and wellbore temperature profile is changing very slowly.

This allows solving the pressure-temperature problem iteratively:

Inputs

$\begin{array}{l}T_s\end{array}$	Intake temperature	$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss
$\begin{array}{l}p_s\end{array}$	Intake pressure	$\begin{array}{l}\theta (l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$
$\begin{array}{l}q_s\end{array}$	Intake flowrate	$\begin{array}{l}d\end{array}$	Flow pipe diameter (tubing or casing depending on where flow occurs)
$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density	$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness
$\begin{array}{l}\mu(T, p)\end{array}$	Fluid viscosity

Outputs

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

Temperature distribution along the wellbore trajectory

Stationary fluid flow		Isothermal or Quasi-isothermal conditions	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole
Incompressible fluid $\begin{array}{l}\rho(T, p)=\rho_s = \rm const\end{array}$	Isoviscous $\begin{array}{l}\mu(T, p) = \mu_s = \rm const\end{array}$

The water injection wellbore temperature profile can be split into the following components:

Upward vertical heat conduction from Earth's Centre towards Earth's surface leading to a static geothermal profile
Upward & Downward vertical heat conduction from reservoir with non-geothermal temperature (invaded by injection water)
Heat exchange between wellbore fluid and surrounding rocks above and below the invaded reservoir
The temperature in water invaded reservoir stays constant from top to bottom

(1)	$\begin{array}{l}\displaystyle T(l) = Ts + ...\end{array}$

where