Derive expression for tile drain spacing

The spacing of tile drains is obtained by applying Darcy's law to the steady flow of recharge water towards parallel drains laid over an impervious layer (Dupuit–Forchheimer assumptions: flow is horizontal, and the hydraulic gradient equals the slope of the water table).

Assumptions

• Soil homogeneous and isotropic, permeability $K$
• Drains parallel, equal spacing $S$, laid at height $a$ above the impervious layer
• Uniform recharge $q$ (m/s, per unit area) from rainfall/irrigation
• Steady state; the water table has height $b$ above the impervious layer midway between two drains, and $a$ at the drain

          ground level
 ------------------------------------------
        .--------. water table (b at middle)
     _.'           '._
    y|      q        |   <- recharge q
  ---o-----------------o---  drain level (height a)
 a   drain   S/2    drain
 //////////// impervious layer ///////////
        origin x = 0 at the drain

Derivation

Take $x$ from the drain, and $y$ the height of the water table above the impervious layer at $x$. Flow per unit length of drain through a vertical section at $x$ comes from the area between $x$ and the mid-point, a width $(S/2-x)$:

$$q_x = q\left(\frac{S}{2}-x\right)$$

By Darcy's law (area = $y\times1$, gradient $= dy/dx$):

$$q\left(\frac{S}{2}-x\right) = K\,y\,\frac{dy}{dx}$$

Integrating:

$$\frac{K y^2}{2} = q\left(\frac{S x}{2}-\frac{x^2}{2}\right) + C$$

At $x=0$, $y=a$, so $C = \dfrac{K a^2}{2}$. At the mid-point $x=S/2$, $y=b$:

$$\frac{K b^2}{2} = q\left(\frac{S^2}{4}-\frac{S^2}{8}\right)+\frac{K a^2}{2} = \frac{q S^2}{8}+\frac{K a^2}{2}$$ $$q S^2 = 4K\left(b^2-a^2\right)$$ $$\boxed{\,S = \sqrt{\frac{4K\,(b^2-a^2)}{q}}\,}$$

This is Donnan's formula. With $h=b-a$ (height of the water table above the drain at the mid-point) and $d=a$, it can be written as Hooghoudt's form:

$$q = \frac{8Kdh + 4Kh^2}{S^2}$$

Use

• $q$ = (fraction of rainfall to be drained) / (time), for example 1.5 % of annual rainfall in 24 h
• $b$ = depth of impervious layer − depth of highest water table below ground
• $a$ = depth of impervious layer − depth of drain below ground

Then $S$ is calculated. If the actual drain is on the impervious layer ($a=0$), the formula reduces to $S=\sqrt{4Kb^2/q}$.