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

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

$\begin{array}{l}T_s\end{array}$	Intake temperature	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_s\end{array}$	Intake pressure	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_s\end{array}$	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

Assumptions

Stationary flow

Homogenous flow

Isothermal or Quasi-isothermal conditions

Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole

Equations

(1)	$\begin{array}{l}\displaystyle \bigg( 1 - \frac{c(p) \, \rho_s^2 \, q_s^2}{A^2} \bigg ) \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl} - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho}\end{array}$

(2)	$\begin{array}{l}\displaystyle q(l) = \frac{\rho_s \cdot q_0}{\rho}\end{array}$

(3)	$\begin{array}{l}\displaystyle u(l) = \frac{\rho_s \cdot q_s}{\rho \cdot A}\end{array}$

(4)	$\begin{array}{l}\displaystyle p(l=0) = p_s\end{array}$

(5)	$\begin{array}{l}\displaystyle q(l=0) = q_s\end{array}$

(6)	$\begin{array}{l}\displaystyle \rho(T_s, p_s) = \rho_s\end{array}$

where

$\begin{array}{l}f({\rm Re}, \, \epsilon)\end{array}$

Darcy friction factor

$\begin{array}{l}\displaystyle {\rm Re} = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_s q_s}{\pi d} \frac{1}{\mu(T, p)}\end{array}$

Reynolds number

$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$

characteristic linear dimension of the pipe

(or exactly a pipe diameter in case of a circular pipe)

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

Approximations

Incompressible pipe flow $\begin{array}{l}\rho(T, p) = \rho_s\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_s\end{array}$

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

Pressure profile

Pressure gradient profile

Fluid velocity

Fluid rate

(7)	$\begin{array}{l}\displaystyle p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l\end{array}$

(8)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s\end{array}$

(9)	$\begin{array}{l}\displaystyle q(l) =q_s = \rm const\end{array}$

(10)	$\begin{array}{l}\displaystyle u(l) = u_s = \frac{q_s}{A} = \rm const\end{array}$

Derivation

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

The first term in the right side of (8) 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 ).

Page tree

Motivation

Outputs

Inputs

Assumptions

Equations

Approximations

Incompressible pipe flow $\begin{array}{l}\rho(T, p) = \rho_s\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_s\end{array}$

See also

References

Page tree

Pressure Profile in Stationary Quasi-Isothermal Homogenous Pipe Flow @model

Motivation

Outputs

Inputs

Assumptions

Equations

Approximations

Incompressible pipe flow \rho(T, p) = \rho_s//<![CDATA[ \begin{array}{l}\rho(T, p) = \rho_s\end{array} //]]> with constant viscosity \mu(T, p) = \mu_s//<![CDATA[ \begin{array}{l}\mu(T, p) = \mu_s\end{array} //]]>

See also

References

Incompressible pipe flow $\begin{array}{l}\rho(T, p) = \rho_s\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_s\end{array}$