Motivation

One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary fluid transport.

In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.

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

Pressure profile

(1)	$\begin{array}{l}\displaystyle F(p, l)=\left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right) + \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right) \cdot \frac{l}{ 2 \, d} - (2/j_m^2) \, \int_p^{p_0} \frac{dp}{\rho} - (2/j_m^2) \, g \, \Delta z(l) = 0\end{array}$

(2)	$\begin{array}{l}\displaystyle \dot m = A \cdot \sqrt{ 2 \cdot \frac { g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho} } { \left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right) + \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right) \cdot \frac{l}{ 2 \, d} } }\end{array}$

(3)	$\begin{array}{l}\displaystyle q_0 = \frac{A}{\rho_s} \cdot \sqrt{ 2 \cdot \frac { g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho} } { \left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right) + \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right) \cdot \frac{l}{ 2 \, d} } }\end{array}$

(4)	$\begin{array}{l}\displaystyle j_m = \sqrt{ 2 \cdot \frac { g \, \Delta z + \int_p^{p_0} \frac{dp}{\rho} } { \left( \frac{1}{\rho^2} - \frac{1}{\rho_0^2} \right) + \left( \frac{f}{\rho^2} + \frac{f_0}{\rho_0^2} \right) \cdot \frac{l}{ 2 \, d} } }\end{array}$

where

$\begin{array}{l}\rho_0 = \rho(T_0, p_0)\end{array}$	Intake fluid density
$\begin{array}{l}j_m = \dot m / A\end{array}$	Intake mass flux
$\begin{array}{l}\displaystyle \dot m = \frac{dm }{ dt}\end{array}$	mass flowrate
$\begin{array}{l}\displaystyle q_0 = \frac{dV_0}{dt} = \dot m / \rho_0\end{array}$	Intake flowrate
$\begin{array}{l}u_0 = u(l=0) = q_0 / A = j_m / \rho_0\end{array}$	Intake Fluid velocity
$\begin{array}{l}\Delta z(l) = z(l)-z(0)\end{array}$	elevation drop along pipe trajectory
$\begin{array}{l}f_s = f({\rm Re}_s, \, \epsilon)\end{array}$	Darcy friction factor at intake point
$\begin{array}{l}\displaystyle {\rm Re} = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_0 q_0}{\pi d} \frac{1}{\mu}\end{array}$	Reynolds number at intake point
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)

Derivation

The first term in the right side of

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defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:

In most practical applications in water producing or water injecting wells, water can be considered as incompressible and friction factor can be assumed constant $\begin{array}{l}f(l) = f_s = \rm const\end{array}$ along-hole ( see Darcy friction factor in water producing/injecting wells ).