Motivation

Outputs

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

Pressure 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
$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss	$\begin{array}{l}\mu(T, p)\end{array}$	Fluid viscosity
$\begin{array}{l}\theta(l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$	$\begin{array}{l}A\end{array}$	Pipe cross-section area

Steady-State flow	Quasi-isothermal flow
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = 0\end{array}$	$\begin{array}{l}\displaystyle \frac{\partial T}{\partial t} =0 \rightarrow T(t,l) = T(l)\end{array}$
Homogenous flow	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)\end{array}$	$\begin{array}{l}A(l) = A = \rm const\end{array}$
Constant inclination	Linear density
$\begin{array}{l}\displaystyle \theta(l) = \theta = {\rm const} \rightarrow \cos \theta = \frac{dz}{dl} = {\rm const}\end{array}$	$\begin{array}{l}\rho = \rho^* \cdot ( 1 + c^* \cdot p)\end{array}$

Pressure profile in static fluid column, no flow:

$\begin{array}{l}\dot m = 0, \, q_0 = 0\end{array}$

(1)	$\begin{array}{l}\displaystyle p(L) = \frac{1}{c^} \cdot \left[ -1 + (1+c^ \, p_0) \cdot \exp(c^* \rho^* G \, L) \right]\end{array}$

(2)	$\begin{array}{l}\displaystyle p(L) = p_0 + \frac{1}{c_0} \cdot \left[ -1 + \exp(c_0 \, \rho_0 \, G \, L) \right]\end{array}$

where

$\begin{array}{l}G = g \cdot \cos \theta\end{array}$

gravity acceleration along pipe

Derivation

See ...