Motivation
Assess how Darcy friction factor is varying along the hole of water producing/injecting wells
Conclusion
For the stationary water flow in a constant diameter pipe the Darcy friction factor can be considered as constant along hole: f(l)= f_s = \rm const and depends on the value of the flowrate q.
The along-hole variation Darcy friction factor is usually not exceeding 10 % but the contribution of the friction-based pressure loss to the gravity-based pressure build up in vertical and slanted wells is very minor which makes constant friction factor assumption become relevant.
The absolute value is staying between f = 0.04 for the very small flow rates (< 100 cmd) and f = 0.015 for the very high flow rates (> 1,000 cmd) and this should be taken into account in calculations.
For complex well designs with varying pipe flow diameters and water source/stocks which may lead to substantial variation of flowrate the model can be split in segments each having a constant friction factor.
Derivation
Consider a ration between friction-based pressure gradient
\displaystyle \left[ \frac{dp}{dl} \right]_f =\frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s and gravity-based pressure gradient in vertical well
\displaystyle \left[ \frac{dp}{dl} \right]_g= \rho_s \, g :
| (1) | \frac{\left[dp/dl\right]_f }{ \left[ dp/dl \right]_g } = \frac{q_s^2 }{2 A^2 \cdot d \cdot g} \, f_s = \frac{f_s \, u_s^2}{2 \cdot d \cdot g} |
In 3" tubing with high flowrate (500 m3/d) the flow velocity is going to be around 1.3 m/s and the ratio (1) is going to be \frac{\left[dp/dl\right]_f }{ \left[ dp/dl \right]_g } \sim 3.3 \%.
Furthermore, Darcy friction factor f for wellbore flow can be written as:
| (2) | {\rm Re}(l) = \frac{u(l) \cdot d}{\nu(l)} = \frac{4 \rho_s q_s}{\pi d} \frac{1}{\mu(T, p)} |
The along-hole variation of Darcy friction factor
f is due to the influence of pressure
p(l) and temperature
T(l) variations on the fluid viscosity
\mu(T, p) .
Both temperature and pressure are growing with depth.
The decrease in water viscosity with growing temperature is partially compensated by decrease in response to growing pressure thus making viscosity staying within 10% along-hole in most practical cases (usually slightly decreasing with depth).
Providing that friction losses are only 3.3 % of the hydrostatic column the further variation of Darcy friction factor by 10% provides only 0.33 % error against pressure modelling with constant Darcy friction factor.
See also
Physics / Fluid Dynamics / Pipe Flow Dynamics / Darcy–Weisbach equation / Darcy friction factor