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.

Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.

Outputs

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

Assumptions

Equations

\left( 1 -  \frac{c(p) \, \rho_s^2 \, q_s^2}{A^2}   \right)  \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl}  - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}

q(l) = \frac{\rho_s \cdot q_0}{\rho}

u(l) = \frac{\rho_s \cdot q_s}{\rho \cdot A}

p(l=0) = p_s

q(l=0) = q_s

\rho(T_s, p_s) = \rho_s

where

See Derivation of Pressure Profile in Stationary Isothermal Homogenous Pipe Flow @model.

Alternative forms

-\frac{1}{c} \frac{d}{dl} \left( \frac{1}{\rho} \right) + \frac{\rho_s^2 q_s^2}{2A^2} \frac{d}{dl} \left( \frac{1}{\rho^2} \right) + \frac{\rho_s^2 q_s^2}{2A^2} \frac{f}{d} \left( \frac{1}{\rho^2} \right) - g \frac{dz}{dl} = 0

See derivation at .

It does not give much benefit in computations comparing to  but it makes a starting point for derivation of some popular proxy models (like Slightly Compressible Fluid and  Ideal Gas). 

Approximations

Incompressible pipe flow  with constant viscosity 

Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model

p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l

\frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s

q(l) =q_s = \rm const

u(l) = u_s = \frac{q_s}{A} = \rm const

where

\frac{dp}{dl} = \rho_s \, g \, \frac{dz}{dl}  - \frac{\rho_s \, q_s^2 }{2 A^2 d} f_s

The first term in the right side of  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  along-hole ( see  Darcy friction factor in water producing/injecting wells ).

References