Motivation

Outputs

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

Inputs

$\begin{array}{l}T_0\end{array}$	Intake temperature	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_0\end{array}$	Intake pressure	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_0\end{array}$	Intake flowrate	$\begin{array}{l}\mu(T, p)\end{array}$	Fluid viscosity
$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss	$\begin{array}{l}A\end{array}$	Pipe cross-section area
$\begin{array}{l}\theta (l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$	$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness

Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole

Pressure profile along the pipe

(1)	$\begin{array}{l}\displaystyle L =\int_{p_0}^{p} \frac{ \rho - j_m^2 \, c(p) }{G \, \rho^2 - F(\rho)} \, dp = \int_{p_0}^{p} \frac{ \rho }{G \, \rho^2 - F(\rho)} \, dp -\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}\end{array}$

where

$\begin{array}{l}\displaystyle j_m = \frac{ \dot m }{ A}\end{array}$	mass flux
$\begin{array}{l}\displaystyle \dot m = \frac{dm }{ dt}\end{array}$	mass flowrate
$\begin{array}{l}\displaystyle q_0 = \frac{dV_0}{dt} = \frac{ \dot m }{ \rho_0}\end{array}$	Intake volumetric flowrate
$\begin{array}{l}\rho_0 = \rho(T_0, p_0)\end{array}$	Intake fluid density
$\begin{array}{l}\Delta z(l) = z(l)-z(0)\end{array}$	elevation drop along pipe trajectory
$\begin{array}{l}f(T,p) = f({\rm Re}(T,p), \, \epsilon)\end{array}$	Darcy friction factor
$\begin{array}{l}\displaystyle {\rm Re}(T,p) = \frac{u(l) \cdot d}{\nu(l)} = \frac{j_m \cdot d}{\mu(T,p)}\end{array}$	Reynolds number in Pipe Flow
$\begin{array}{l}\mu(T,p)\end{array}$	dynamic viscosity as function of fluid temperature $\begin{array}{l}T\end{array}$ and pressure $\begin{array}{l}p\end{array}$
$\begin{array}{l}\displaystyle c(T,p) = \frac{1}{\rho} \left( \frac{\partial \rho}{\partial p} \right)_T\end{array}$	fluid compressibility
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
$\begin{array}{l}G = g \, \cos \theta\end{array}$	gravity acceleration along pipe
$\begin{array}{l}F = j_m^2 \cdot f/(2d)\end{array}$

Derivation