Motivation

Outputs

	Pressure distribution along the pipe
	Flowrate distribution along the pipe
	Flow velocity distribution along the pipe

Inputs

	Intake temperature		Along-pipe temperature profile
	Intake pressure		Fluid density
	Intake flowrate		Fluid viscosity
	Pipeline trajectory TVDss		Pipe cross-section area
	Pipeline trajectory inclination,		Inner pipe wall roughness

Stationary flow	Homogenous flow	Isothermal or Quasi-isothermal conditions	Constant cross-section pipe area along hole

Pressure profile along the pipe

L = \frac{1}{2 \, G \, c^*  \rho^*}  \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}

 \cos \theta \neq 0

L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
 - \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho}

 \cos \theta = 0

where

	mass flux
	mass flowrate
	Intake volumetric flowrate
	Intake fluid density
	elevation drop along pipe trajectory
	Darcy friction factor
	Reynolds number in Pipe Flow
	dynamic viscosity as function of fluid temperature and density
	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
	gravity acceleration along pipe

The equation for horizontal pipelines can be re-written explicitly in terms of pressure:

\frac{fL}{2d} = (\rho^*/j_m^2)  \cdot (p_0-p) \cdot (1+ 0.5 \, c^* \cdot (p+p_0)) -  \ln \frac{1+c^* \cdot p_0}{1+c^* \cdot p}