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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


p(l)

Pressure distribution along the pipe

q(l)

Flowrate distribution along the pipe

u(l)

Flow velocity distribution along the pipe

Inputs


T_0

Intake temperature 

\rho_0

 Fluid density

p_0

Intake pressure 

\mu_0

q_0

Intake flowrate 

A

z(l)

Pipeline trajectory TVDss


\epsilon
 Inner pipe wall roughness
\theta (l)


Pipeline trajectory inclination, \displaystyle \cos \theta (l) = \frac{dz}{dl}



Assumptions


Steady-State flowIsothermal

\displaystyle \frac{\partial p}{\partial t} = 0

T(t, l)=T_0 = \rm const

Homogenous flow

Constant cross-section pipe area A along hole

\displaystyle \frac{\partial p}{\partial \tau_x} =\frac{\partial p}{\partial \tau_y} =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)

A(l) = A = \rm const

Incompressible fluid 

\rho(T, p)=\rho_0 = \rm const → \mu(T, \rho) =\mu_0 = \rm const


Equations


Pressure profilePressure gradient profile
(1) p(l) = p_0 + \rho_0 \, g \, \Delta z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l
(2) \frac{dp}{dl} = \rho_0 \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0
Mass FluxMass Flowrate
(3) j_m = \rho_0 \cdot \sqrt{\frac{2 \, d}{f_0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_0 - p)/ \rho_0}
(4) \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}

 Volumetric Flowrate

Intake Fluid velocity

(5) 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 }
(6) 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 }

where

j_m

Intake mass flux

\dot m

mass flowrate

u_0 = u(l=0)

Intake Fluid velocity

\Delta z(l) = z(l)-z(0)

elevation drop along pipe trajectory

f_0 = f({\rm Re}_0, \, \epsilon)

Darcy friction factor at intake point

\displaystyle {\rm Re}_0 = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_0 q_0}{\pi d} \frac{1}{\mu_0}

Reynolds number at intake point

\displaystyle d = \sqrt{ \frac{4 A}{\pi}}

characteristic linear dimension of the pipe

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



 Derivation

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

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

Equation (7) becomes:

(7) \frac{dp}{dl} = \rho_0 \, g \, \frac{dz}{dl} - \frac{j_m^2 f_0}{2 \, \rho_0 \, d}

which leads to (2) after substituting \displaystyle \cos \theta(l) = \frac{dz(l)}{dl}  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  f(l) = f_0 = \rm const along-hole ( see  Darcy friction factor in water producing/injecting wells ).


See also

References




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