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

Stationary flow	Homogenous flow	Isothermal or Quasi-isothermal conditions	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole
$\begin{array}{l}\theta (l) = \theta = \rm const\end{array}$	$\begin{array}{l}f(l) = f = \rm const\end{array}$	$\begin{array}{l}\rho = \rho_0 \cdot ( 1 + c^* \cdot p/p_0)\end{array}$

Pressure profile along the pipe

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

(2)	$\begin{array}{l}\displaystyle \cos \theta \leq 0\end{array}$

(3)	$\begin{array}{l}\displaystyle L = \frac{\rho_0}{j_m^2 \cdot f/(2d)} \left[ (p_0-p) + \frac{c^*}{2 p_0} \left( p_0^2 - p^2 \right) \right] - j_m^2 \, \ln \frac{\rho_0}{\rho}\end{array}$

(4)	$\begin{array}{l}\displaystyle \cos \theta = 0\end{array}$

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 $\begin{array}{l}p\end{array}$
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
$\begin{array}{l}G = g \, \cos \theta\end{array}$
$\begin{array}{l}F = j_m^2 \cdot f/(2d)\end{array}$

Derivation