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| \bigg( 1 - \frac{c(p) \, \rho_0^2s^2 \, q_0^2s^2}{A^2} \bigg ) \frac{dp}{dl} = \rho \, g \, \frac{dz}{dl} - \frac{\rho_0^2s^2 \, q_0^2s^2 }{2 A^2 d} \frac{f({\rm Re}, \, \epsilon)}{\rho} |
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| q(l) = \frac{\rho_0s \cdot q_0}{\rho} |
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| u(l) = \frac{\rho_0s \cdot q_0s}{\rho \cdot A} |
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| p(l=0) = p_0s |
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| q(l=0) = q_0s |
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| \rho(T_0s, p_0s) = \rho_0s |
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where
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| body | --uriencoded--f(%7B\rm Re%7D, \, \epsilon) |
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| Darcy friction factor |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_0 s q_0%7D%7Bs%7D%7B\pi d%7D \frac%7B1%7D%7B\mu(T, p)%7D |
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| Reynolds number |
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| body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D |
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| characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
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See Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model.
Approximations
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Incompressible pipe flow
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with constant viscosity ...
| Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
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| p(l) = p_0s + \rho_0s \, g \, z(l) - \frac{\rho_0s \, q_0^2s^2 }{2 A^2 d} \, f_0s \, l |
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| \frac{dp}{dl} = \rho_0s \, g \cos \theta(l) - \frac{\rho_0s \, q_0^2s^2 }{2 A^2 d} \, f_0 s |
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| q(l) =q_0s = \rm const |
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| u(l) = u_0s = \frac{q_0s}{A} = \rm const |
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where
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| Incompressible fluid | LaTeX Math Inline |
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| body | \rho(T, p) = \rho_0 s = \rm const |
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means that compressibility vanishes and fluid velocity is going to be constant along the pipeline trajectory | LaTeX Math Inline |
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| body | --uriencoded--u(l) = u_0 s = \frac%7Bq_0%7D%7BA%7D s%7D%7BA%7D = \rm const |
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.For the constant viscosity | LaTeX Math Inline |
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| body | \mu(T, p) = \mu_0 s = \rm const |
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along the pipeline trajectory the Reynolds number | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D = \frac%7B4 \rho_0 s q_0%7D%7Bs%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_0%7D s%7D = \rm const |
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and Darcy friction factor | LaTeX Math Inline |
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| body | --uriencoded--f(%7B\rm Re%7D, \, \epsilon) = f_0 s = \rm const |
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are going to be constant along the pipeline trajectory.Equation | LaTeX Math Block Reference |
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becomes:| LaTeX Math Block |
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| \frac{dp}{dl} = \rho_0s \, g \, \frac{dz}{dl} - \frac{\rho_0s \, q_0^2s^2 }{2 A^2 d} f_0s |
which leads to | LaTeX Math Block Reference |
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after substituting | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \cos \theta(l) = \frac%7Bdz(l)%7D%7Bdl%7D |
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and can be explicitly integrated leading to | LaTeX Math Block Reference |
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. |
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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_0 s = \rm const |
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along-hole ( see
Darcy friction factor in water producing/injecting wells ).
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