Motivation
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One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary fluid transport.
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Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
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Assumptions
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Equations
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| \bigg( 1 - \frac{c(p) \, \rho_s^2 \, q_s^2}{A^2} \bigg ) \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl} - \frac{\rho_s^2 \, q_s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho} |
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| q(l) = \frac{\rho_s \cdot q_0}{\rho} |
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| u(l) = \frac{\rho_s \cdot q_s}{\rho \cdot A} |
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| p(l=0) = p_s |
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| q(l=0) = q_s |
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| \rho(T_s, p_s) = \rho_s |
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See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.
Approximations
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Incompressible pipe flow
with constant viscosity
| Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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| anchor | PPconst |
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| alignment | left |
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| p(l) = p_s + \rho_s \, g \, z(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s \, l |
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| \frac{dp}{dl} = \rho_s \, g \cos \theta(l) - \frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s |
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| q(l) =q_s = \rm const |
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| u(l) = u_s = \frac{q_s}{A} = \rm const |
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In most practical applications in water producing or water injecting wells, water can be considered as incompressible and friction factor can be assumed constant
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| body | f(l) = f_s = \rm const |
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along-hole ( see
Darcy friction factor in water producing/injecting wells ).
References
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| title | ARAX |
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