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Inputs
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| Pressure profile | Pressure gradient profile |
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| LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| p(l) = p_s0 + \rho_s \, g \, \Delta z(l) - \frac{\rho_s0 \, q_s^20^2 }{2 A^2 d} \, f_s0 \, l |
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| LaTeX Math Block |
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| \frac{dp}{dl} = \rho_s0 \, g \cos \theta(l) - \frac{\rho_s0 \, q_s^20^2 }{2 A^2 d} \, f_s0 |
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| Mass Flux | Mass Flowrate |
| LaTeX Math Block |
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| anchor | MassFlux |
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| alignment | left |
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| j_m = \rho_s0 \cdot \sqrt{\frac{2 \, d}{f_s0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_s0 - p)/ \rho_s0}
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| LaTeX Math Block |
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| anchor | MassFlowrate |
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| alignment | left |
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| \dot m = j_m \cdot A = \rho_s0 \cdot A \cdot \sqrt{\frac{2 \, d}{f_s0 \, l }} \cdot \sqrt{g \, \Delta z(l) + (p_s0 - p)/ \rho_s}
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Volumetric Flowrate | Intake Fluid velocity |
| LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| q_s0 = \dot m / \rho_s0 = A \cdot \sqrt{\frac{2 \, d }{ f_s0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_s0 - p)/ \rho_s }
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| LaTeX Math Block |
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| anchor | PPconst |
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| alignment | left |
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| u_s0 = j_m/ \rho_s0 =q_s0 / A = \sqrt{\frac{2 \, d }{ f_s0 \, l }} \cdot \sqrt{ g \, \Delta z(l) + (p_s0 - p)/ \rho_s }
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|
where
| Intake mass flux |
| mass flowrate |
| Intake Fluid velocity |
| LaTeX Math Inline |
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| body | \Delta z(l) = z(l)-z(0) |
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| elevation drop along pipe trajectory |
| LaTeX Math Inline |
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| body | --uriencoded--f_s 0 = f(%7B\rm Re%7D_s0, \, \epsilon) |
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|
| Darcy friction factor at intake point |
| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle %7B\rm Re%7D_s 0 = \frac%7Bu(l) \cdot d%7D%7B\nu(l)%7D = \frac%7B4 \rho_s 0 q_s%7D%7B0%7D%7B\pi d%7D \frac%7B1%7D%7B\mu_s%7D0%7D |
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| Reynolds number at intake point |
| LaTeX Math Inline |
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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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In most practical applications in water producing or water injecting wells, water can be considered as incompressible and friction factor can be assumed constant
| LaTeX Math Inline |
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| body | f(l) = f_s 0 = \rm const |
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|
along-hole ( see
Darcy friction factor in water producing/injecting wells ).
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