Motivation

Outputs

$\begin{array}{l}T(l)\end{array}$	Temperature distribution along the flow pipe	$\begin{array}{l}q(l)\end{array}$	Flowrate distribution along the flow pipe
$\begin{array}{l}p(l)\end{array}$	Pressure distribution along the flow pipe	$\begin{array}{l}u(l)\end{array}$	Flow velocity distribution along the flow pipe

where

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

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

In stabilised wellbore flow the wellbore pressure profile is constant and wellbore temperature profile is changing very slowly.

This allows solving the pressure-temperature problem iteratively: