Motivation
One of the key problems in designing the pipelines and wells and controlling the fluid transport along is to predict the pressure along-hole pressure distribution during the stationary fluid transport.
In many cases the flow can be considered as Isothermal or Quasi-isothermal.
Pipeline flow simulator is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
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Pipeline trajectory LaTeX Math Inline |
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body | {\bf r} = {\bf r}(l) = \{ x(l), \, y(l), \, z(l) \} |
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| along-pipe distribution of stabilised pressure |
| along-pipe distribution of stabilised flow rate |
| along-pipe distribution of stabilised average flow velocity |
Inner pipe wall roughness |
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Assumptions
Equations
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| \bigg( 1 - \frac{c(p) \, \rho_0^2 \, q_0^2}{A^2} \bigg ) \frac{dp}{dl} = \rho(p) \, g \, \frac{dz}{dl} - \frac{\rho_0^2 \, q_0^2 }{2 A^2 d} \frac{f(p)}{\rho(p)} |
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| u(l) = \frac{\rho_0 \cdot q_0}{\rho(p) \cdot A(l)} |
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| q(l) = \frac{\rho_0 \cdot q_0}{\rho(p)} |
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(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
Approximations
Incompressible fluid with constant friction
Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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| p(l) = p_0 + \rho \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l |
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| \frac{dp}{dl} = \rho \, g \cos \theta(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 |
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| u(l) = \frac{q_0}{A(l)} |
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| q(l) =q_0 = \rm const |
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where
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body | \displaystyle \cos \theta(l) = \frac{dz(l)}{dl} |
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| correction factor for trajectory deviation |
The first term in
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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 the water can be considered as incompressible and friction factor an 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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bgColor | papayawhip |
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title | ARAX |
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