Darcy's law

Darcy's law states that the rate of flow of water through a porous medium is directly proportional to the hydraulic gradient and to the cross-sectional area normal to the flow.

$$Q = K\,i\,A = K\,\frac{\Delta h}{L}\,A$$

where $Q$ is the discharge (m$^3$/s), $K$ is the coefficient of permeability (hydraulic conductivity, m/s), $i = \Delta h / L$ is the hydraulic gradient, and $A$ is the total cross-section area (m$^2$). The Darcy velocity is $v = Q/A = K i$. The actual seepage velocity is $v_s = v/n$, where $n$ is the porosity.

   h1 ->  _________ L _________
         |   soil sample (A)   |-> Q
   h2 ->  ---------------------
   head loss  dh = h1 - h2

How water flows underground

Rain infiltrates, moves down through the unsaturated zone, and reaches the water table. In the saturated zone it moves slowly (cm/day to m/day) through pores and fractures, from high to low hydraulic head, until it comes out at springs, rivers or wells.

Applicability

• Valid for laminar flow in saturated, homogeneous and isotropic media, with Reynolds number $R_e$ less than about 1 (up to 10).
• Not valid for very coarse gravel or fractured rock with turbulent flow, very high gradients, or very fine clay at very low gradient.

Example: $K = 1\times10^{-4}$ m/s, $i = 0.02$, $A = 50$ m$^2$ gives $Q = 1\times10^{-4}\times0.02\times50 = 1\times10^{-4}$ m$^3$/s.

Answer: $Q = 1\times10^{-4}$ m$^3$/s, i.e. 0.1 litre/s.