Which of the following statements regarding the physical validity and formulation of Darcy's Law ($v = -K \frac{dh}{dl}$) in porous media groundwater hydraulics are correct?
Options:
(A) Darcy's law is valid strictly for laminar flow where the pore Reynolds number is small ($Re = \frac{\rho v d_{10}}{\mu} < 1 \text{ to } 10$)
(B) The Darcy velocity $v = Q/A$ is an apparent macroscopic discharge velocity through the entire gross cross-sectional area
(C) The actual microscopic seepage pore velocity is given by $v_s = \frac{v}{n}$, where $n$ is effective soil porosity ($v_s > v$)
(D) The velocity head term $\frac{v^2}{2 g}$ in total hydraulic head $h = z + \frac{p}{\gamma_w} + \frac{v^2}{2 g}$ is routinely neglected because flow velocities in aquifers are very small
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
All four statements are foundational principles in groundwater hydraulics (Todd):
- A: Viscous forces dominate over inertia, ensuring linear laminar resistance.
- B: Darcy velocity ignores solid particle volume, treating soil as open area.
- C: Since pore area $A_p = n A < A$, conservation of mass dictates $v_s = v/n > v$.
- D: Flow velocities are on the order of $\text{m/day}$ ($10^{-5}\text{ m/s}$), making velocity head $\sim 10^{-11}\text{ m}$, which is physically negligible compared to elevation and pressure heads.