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

$\begin{array}{l}T_0\end{array}$	Intake temperature	$\begin{array}{l}\rho_0\end{array}$	Fluid density
$\begin{array}{l}p_0\end{array}$	Intake pressure	$\begin{array}{l}\mu_0\end{array}$	Fluid viscosity
$\begin{array}{l}q_0\end{array}$	Intake flowrate	$\begin{array}{l}A\end{array}$	Pipe cross-section area
$\begin{array}{l}z(l)\end{array}$	Pipeline trajectory TVDss	$\begin{array}{l}\epsilon\end{array}$	Inner pipe wall roughness
$\begin{array}{l}\theta (l)\end{array}$	Pipeline trajectory inclination, $\begin{array}{l}\displaystyle \cos \theta (l) = \frac{dz}{dl}\end{array}$

Assumptions

Steady-State flow	Isothermal
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial t} = 0\end{array}$	$\begin{array}{l}T(t, l)=T_0 = \rm const\end{array}$
Homogenous flow	Constant cross-section pipe area $\begin{array}{l}A\end{array}$ along hole
$\begin{array}{l}\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)\end{array}$	$\begin{array}{l}A(l) = A = \rm const\end{array}$
Incompressible fluid
$\begin{array}{l}\rho(T, p)=\rho_0 = \rm const\end{array}$ → $\begin{array}{l}\mu(T, \rho) =\mu_0 = \rm const\end{array}$

Equations

Pressure profile

Pressure gradient profile

(1)	$\begin{array}{l}\displaystyle p(l) = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{8}{\pi^2} \frac{f_0 \, l}{d^5} \rho_0 \, q_0^2\end{array}$

(2)	$\begin{array}{l}\displaystyle \frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0\end{array}$

Mass Flux

Mass Flowrate

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

(4)	$\begin{array}{l}\displaystyle \dot m = j_m \cdot A = \rho_0 \cdot A \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_s}\end{array}$

Volumetric Flowrate

Intake Fluid velocity

(5)	$\begin{array}{l}\displaystyle q_0 = \dot m / \rho_0 = A \cdot \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_0 - p)/ \rho_0 }\end{array}$

(6)	$\begin{array}{l}\displaystyle u_0 = j_m/ \rho_0 =q_0 / A = \sqrt{\frac{2 \, d }{ f_0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_0 - p)/ \rho_s }\end{array}$

where

$\begin{array}{l}j_m\end{array}$	Intake mass flux
$\begin{array}{l}\dot m\end{array}$	mass flowrate
$\begin{array}{l}u_0 = u(l=0)\end{array}$	Intake Fluid velocity
$\begin{array}{l}\Delta z(l) = z(l)-z(0)\end{array}$	elevation drop along pipe trajectory
$\begin{array}{l}f_0 = f({\rm Re}_0, \, \epsilon)\end{array}$	Darcy friction factor at intake point
$\begin{array}{l}\displaystyle {\rm Re}_0 = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_0 q_0}{\pi d} \frac{1}{\mu_0}\end{array}$	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

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 (7) becomes:

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

which leads to (2) after substituting $\begin{array}{l}\displaystyle \cos \theta(l) = \frac{dz(l)}{dl}\end{array}$ and can be explicitly integrated leading to (1).

The first term in the right side of (2) 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 $\begin{array}{l}f(l) = f_0 = \rm const\end{array}$ along-hole ( see Darcy friction factor in water producing/injecting wells ).

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Motivation

Inputs

Assumptions

Equations

References

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Pressure Profile in Incompressible Isothermal Proxy Pipe Flow @model

Motivation

Inputs

Assumptions

Equations

References