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
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
\displaystyle j_m =\frac{ \rho_0 \, q_0}{A} | |
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 |
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