Motivation

One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the steady-state 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

$\begin{array}{l}T_0\end{array}$	Fluid temperature at inlet point ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}T(l)\end{array}$	Along-pipe temperature profile
$\begin{array}{l}p_0\end{array}$	Fluid pressure at inlet point ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}\rho(T, p)\end{array}$	Fluid density
$\begin{array}{l}q_0\end{array}$	Fluid flowrate at inlet point ( $\begin{array}{l}l=0\end{array}$ )	$\begin{array}{l}\mu(T, p)\end{array}$	Dynamic 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)	$\begin{array}{l}\displaystyle \left(\rho(p) - j_m^2 \cdot c(p) \right) \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f({\rm Re}, \, \epsilon)\end{array}$

(2)	$\begin{array}{l}\displaystyle p(l=0) = p_0\end{array}$

(3)	$\begin{array}{l}\displaystyle u(l) = \frac{j_m}{\rho(l)}\end{array}$

(4)	$\begin{array}{l}\displaystyle q(l) =A \cdot u(l)\end{array}$

where

$\begin{array}{l}\displaystyle j_m =\frac{ \rho_0 \, q_0}{A}\end{array}$	mass flux
$\begin{array}{l}q_0 = q(l=0)\end{array}$	Fluid flowrate at inlet point ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}\rho_0 = \rho(T_0, p_0)\end{array}$	Fluid density at inlet point ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}\rho(l) = \rho(T(l), p(l))\end{array}$	Fluid density at any point $\begin{array}{l}l\end{array}$
$\begin{array}{l}с(p)\end{array}$	Fluid Compressibility
$\begin{array}{l}f({\rm Re}, \, \epsilon)\end{array}$	Darcy friction factor
$\begin{array}{l}\displaystyle {\rm Re} = \frac{j_m \cdot d}{\mu(T, p)}\end{array}$	Reynolds number in Pipe Flow
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$	Characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)

Derivation

See Derivation of Pressure Profile in Steady-State Homogeneous Pipe Flow @model.

Approximations

Incompressible pipe flow $\begin{array}{l}\rho(T, p) = \rho_0\end{array}$ with constant viscosity $\begin{array}{l}\mu(T, p) = \mu_0\end{array}$

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

Pressure profile

Pressure gradient profile

Fluid velocity

Fluid rate

(5)	$\begin{array}{l}\displaystyle p(l) = p_0 + \rho_0 \, g \, z(l) - \frac{j_m^2 f_0}{2 \, \rho_0 \, d} \cdot l\end{array}$

(6)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{j_m^2 f_0}{2 \, \rho_0 \, d}\end{array}$

(7)	$\begin{array}{l}\displaystyle q(l) =q_0 = \rm const\end{array}$

(8)	$\begin{array}{l}\displaystyle u(l) = u_0 = \frac{q_0}{A} = \rm const\end{array}$

where

$\begin{array}{l}f_0 = f({\rm Re}_0, \, \epsilon)\end{array}$	Darcy friction factor at inlet point ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}\displaystyle {\rm Re}_0= \frac{j_m \, d}{\mu_0}\end{array}$	Reynolds number at inlet point ( $\begin{array}{l}l=0\end{array}$ )
$\begin{array}{l}\mu_0 = \mu(T_0, p_0)\end{array}$	Dynamic Fluid Viscosity at inlet point ( $\begin{array}{l}l=0\end{array}$ )

Derivation

Incompressible fluid $\begin{array}{l}\rho(T, p) = \rho_0 = \rm const\end{array}$ means that compressibility vanishes $\begin{array}{l}c(p) = 0\end{array}$ and fluid velocity is going to be constant along the pipeline trajectory $\begin{array}{l}u(l) = u_0 = \frac{q_0}{A} = \rm const\end{array}$ .

For the constant viscosity $\begin{array}{l}\mu(T, p) = \mu_0 = \rm const\end{array}$ along the pipeline trajectory the Reynolds number $\begin{array}{l}\displaystyle {\rm Re}(l) = \frac{j_m^2 \, d}{\mu_0} = \rm const\end{array}$ and Darcy friction factor $\begin{array}{l}f(l) = f({\rm Re}, \, \epsilon) = f_0 = \rm const\end{array}$ are going to be constant along the pipeline trajectory.

Equation (1) becomes:

(9)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl} - \frac{j_m^2 f_0}{2 \, \rho_0 \, d}\end{array}$

which leads to (6) after substituting $\begin{array}{l}\displaystyle \cos \theta(l) = \frac{dz(l)}{dl}\end{array}$ and can be explicitly integrated leading to (5).

The first term in the right side of (6) defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:

In many practical applications the water in water producing wells or water injecting wells can be considered as incompressible and friction factor can be assumed constant $\begin{array}{l}f(l) = f_0 = \rm const\end{array}$ along-hole ( see Darcy friction factor in water producing/injecting wells ).

Pressure Profile in GF-Proxy Pipe Flow @model

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Pressure Profile in Homogeneous Steady-State Pipe Flow @model