Darcy's law and seepage

groundwater flowpermeabilityhydraulic conductivityseepage velocity

Groundwater through soil: hydraulic gradient, Darcy discharge, the faster real velocity in the pores, and layered permeability.

Hydraulic Gradient

i=ΔhLi = \frac{\Delta h}{L}

Hydraulic gradient as the loss of total head divided by the length of the flow path, the dimensionless driving force behind all seepage.

Darcy's Law for Groundwater Flow

Q=kiAQ = k\,i\,A

Darcy's law for laminar flow through soil: discharge equals hydraulic conductivity times hydraulic gradient times gross cross-sectional area.

Seepage Velocity from Discharge Velocity

vs=vnv_s = \frac{v}{n}

Actual seepage velocity through the pores, obtained by dividing Darcy's fictitious discharge velocity by the porosity of the soil.

Equivalent Horizontal Permeability of Layered Soil

keq=k1H1+k2H2H1+H2k_{eq} = \frac{k_1 H_1 + k_2 H_2}{H_1 + H_2}

Thickness-weighted equivalent permeability for flow parallel to the bedding of two soil layers, the parallel-resistance case of stratified seepage.

How they fit together

Darcy found in 1856, while designing Dijon's water supply, that flow through sand is proportional to the hydraulic gradient — head lost per unit distance travelled. The constant of proportionality, hydraulic conductivity, spans about ten orders of magnitude from clean gravel to intact clay, a wider range than almost any other engineering property.

The distinction that matters on site is discharge velocity versus seepage velocity. Darcy's v assumes flow across the whole cross-section, but water only moves through the pores, so the actual particle speed is v divided by porosity — typically two to three times faster. Use discharge velocity to size a dewatering pump and seepage velocity to predict when a contaminant arrives; swapping them makes a plume look slower than it is.