Darcy's Law for Groundwater Flow

Henry Darcy was the municipal engineer of Dijon, and in 1856 — as an appendix to a report on the town's water supply — he described pouring water through columns of sand and finding the flow strictly proportional to the head loss. That appendix founded groundwater hydraulics. A sand with k = 0.01 cm/s under a gradient of 0.05 across 2 m² passes Q = 1×10⁻⁴ m/s × 0.05 × 2 m² = 1×10⁻⁵ m³/s, or 0.6 litres per minute. Typical k values span eleven orders of magnitude: clean gravel 1–100 cm/s, sand 10⁻³–10⁻¹, silt 10⁻⁶–10⁻⁴, intact clay below 10⁻⁷ cm/s.

The trap is the area. A is the gross cross-section, grains included, so the velocity v = ki that comes out of Darcy's law is a fictitious discharge velocity that no water molecule ever travels at — the real pore velocity is higher by 1/n. The second trap is the laminar assumption: in coarse gravel or under steep gradients the flow turns turbulent and the linear law overpredicts discharge. And remember k is a property of soil and fluid; the same sand carrying hot water or a hydrocarbon has a different k, which is why contaminant work uses intrinsic permeability instead.