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
T_0 | Intake temperature | T(l) | Along-pipe temperature profile |
p_0 | Intake pressure | \rho(T, p) | Fluid density |
q_0 | Intake flowrate | \mu(T, p) | |
z(l) | Pipeline trajectory TVDss | A | Pipe cross-section area |
\theta (l) | Pipeline trajectory inclination, \displaystyle \cos \theta (l) = \frac{dz}{dl} | \epsilon | Inner pipe wall roughness |
Assumptions
Stationary flow | Homogenous flow | Isothermal or Quasi-isothermal conditions | Constant cross-section pipe area A along hole |
Equations
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where
q_0 = q(l=0) | fluid flow rate at pipe intake |
\rho_0 = \rho(T_0, p_0) | fluid density at intake temperature and pressure |
с(p) | Fluid Compressibility |
f({\rm Re}, \, \epsilon) | Darcy friction factor |
\displaystyle {\rm Re} = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_0 q_0}{\pi d} \frac{1}{\mu(T, p)} | Reynolds number in Pipe Flow |
\displaystyle d = \sqrt{ \frac{4 A}{\pi}} | characteristic linear dimension of the pipe |
See Derivation of Pressure Profile in Stationary Isothermal Homogenous Pipe Flow @model.
Approximations
Incompressible pipe flow
\rho(T, p) = \rho_s with constant viscosity
\mu(T, p) = \mu_s
Pressure Profile in Incompressible Isoviscous Stationary Quasi-Isothermal Pipe Flow @model
Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate | ||||||||
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where
f_s = f({\rm Re}_s, \, \epsilon) | Darcy friction factor at intake point |
\displaystyle {\rm Re}_s = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_s q_s}{\pi d} \frac{1}{\mu_s} | Reynolds number at intake point |
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
f(l) = f_s = \rm const along-hole ( see Darcy friction factor in water producing/injecting wells ).
See also
Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Pressure Profile in Quasi-Isothermal Stationary Pipe Flow @model
[ Darcy friction factor ] [ Darcy friction factor @model ] [ Reynolds number in Pipe Flow ]
[ Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model ]
[ Temperature Profile in Homogenous Pipe Flow @model ]
[ Fluid Compressibility ] [ Fluid Compressibility @model ]
References