Motivation

Assess how Darcy friction factor is varying along the hole of water producing/injecting wells

Conclusion

For water flow Darcy friction factor along-hole variation is usually not exceeding 10 %  and in many practical applications can be considered as constant along hole $//<![CDATA[ \begin{array}{l}f = f_s = \rm const\end{array} //]]>$. 

The absolute value is staying between  $//<![CDATA[ \begin{array}{l}f = 0.04\end{array} //]]>$ for the very small flow rates (< 100 cmd) and  $//<![CDATA[ \begin{array}{l}f = 0.015\end{array} //]]>$ for the very high flow rates  (> 1,000 cmd). 

Derivation

Consider a ration between friction-based pressure gradient  $//<![CDATA[ \begin{array}{l}\displaystyle \left[ \frac{dp}{dl} \right]_f =\frac{\rho_s \, q_s^2 }{2 A^2 d} \, f_s\end{array} //]]>$ and gravity-based pressure gradient in vertical well  $//<![CDATA[ \begin{array}{l}\displaystyle \left[ \frac{dp}{dl} \right]_g= \rho_s \, g\end{array} //]]>$:

Furthermore, Darcy friction factor  $//<![CDATA[ \begin{array}{l}f\end{array} //]]>$ for wellbore flow can be written as:

The along-hole variation of Darcy friction factor  $//<![CDATA[ \begin{array}{l}f\end{array} //]]>$ is due to the influence of pressure  $//<![CDATA[ \begin{array}{l}p(l)\end{array} //]]>$ and temperature  $//<![CDATA[ \begin{array}{l}T(l)\end{array} //]]>$ variations on the fluid viscosity  $//<![CDATA[ \begin{array}{l}\mu(T, p)\end{array} //]]>$.

See also

Physics / Fluid Dynamics / Pipe Flow Dynamics / Darcy–Weisbach equation / Darcy friction factor