gen-define-darcy-s-law-and-give-one-short-worked-example-d4bb

Define Darcy's law and give one short worked example.

# Darcy's Law ## Definition Darcy's law describes the flow of fluid through a porous medium. It states that the volumetric flow rate is proportional to the pressure gradient and inversely proportional to the fluid viscosity. **Mathematical form:** $$Q = -\frac{kA}{\mu}\frac{dP}{dL}$$ Where: - Q = volumetric flow rate (m³/s) - k = permeability of the porous medium (m²) - A = cross-sectional area (m²) - μ = dynamic viscosity of the fluid (Pa·s) - dP/dL = pressure gradient (Pa/m) **Alternative form (using hydraulic head):** $$Q = -KA\frac{dh}{dL}$$ Where K = hydraulic conductivity (m/s) and h = hydraulic head (m) ## Worked Example **Problem:** Water flows through a sand column with cross-sectional area 0.01 m². The hydraulic conductivity is 1×10⁻⁴ m/s. If the hydraulic head drops from 5 m to 2 m over a 10 m length, find the flow rate. **Solution:** $$Q = -KA\frac{dh}{dL} = -(1 \times 10^{-4})(0.01)\frac{(2-5)}{10}$$ $$Q = -(1 \times 10^{-6})\frac{-3}{10} = 3 \times 10^{-7} \text{ m}^3/\text{s}$$ $$Q = 0.3 \text{ mL/s}$$ The positive result indicates flow in the direction of decreasing head (downward).