Darcy's law with neat figure

Darcy's law: the discharge velocity of water through a saturated soil is directly proportional to the hydraulic gradient (laminar flow):

$$v = k i \quad \Rightarrow \quad Q = k\,i\,A = k\,\frac{h}{L}\,A$$

Figure (inclined soil specimen)

  ▽ ---------------------------- H1 = total head at entry
  |                  ^
  |                  | h = H1 - H2 (head loss)
  |                  v
  |              ▽ ------------- H2 = total head at exit
  |              |
 (1)             |
   \\            |
     \\  soil    |
       \\ L, A   |
         \\_____(2)
  z1 = height of (1)   z2 = height of (2)
 ================================== datum

The total head at entry (1) and exit (2), measured from a common datum, each equals elevation head + pressure head: $H_1 = z_1 + h_{p1}$, $H_2 = z_2 + h_{p2}$. The water levels in the piezometers at (1) and (2) show these total heads.

Terms

• $Q$ = discharge per unit time (m³/s)
• $k$ = coefficient of permeability (m/s), a property of the soil and the water
• $h = H_1 - H_2$ = head loss (difference in total head between entry and exit) (m)
• $L$ = length of the flow path measured along the soil (m)
• $i = h/L$ = hydraulic gradient
• $A$ = cross-sectional area of the soil normal to the flow (m²)
• $v = Q/A$ = discharge velocity

The inclination does not appear separately, because the hydraulic gradient uses the total head (which already includes the elevation head), and $L$ is the length along the inclined sample.

$$Q = k\,\frac{h}{L}\,A$$