Drain Spacing Equations — Hooghoudt & Ernst

The design problem

For a given soil, drain depth, drain radius and drainage requirement, the designer must find the spacing $L$ between parallel drains so that, under the design recharge, the water table midway between the drains does not rise above a safe level (usually 0.5–1.0 m below the surface).

Inputs:

• $K$ — hydraulic conductivity of the soil (m/day), measured by the auger-hole or piezometer method.
• $q$ — the drainage coefficient (discharge per unit area) (m/day) = the recharge to be removed.
• $h$ — hydraulic head midway between drains, measured above the drain level (m).
• $D$ — depth of the impermeable (barrier) layer below the drain (m).
• $d$ — the equivalent depth (m).
• $\mu$ — drainable porosity (specific yield) for non-steady flow.
• $r_0$ — radius of the drain; $u$ — wetted perimeter.

Assumptions (Dupuit–Forchheimer)

1. Flow is horizontal and the velocity is uniform at any vertical section.
2. The hydraulic gradient equals the slope of the water table (Dupuit assumption).
3. The soil is homogeneous and isotropic (or in layers with simple values).
4. A steady recharge exists, equal to the discharge (steady state).

With these, the water table between two parallel drains is an ellipse, and the discharge per unit area is:

Steady-state — Hooghoudt's equation

Hooghoudt (1940) extended the formula to drains above the barrier by introducing the equivalent depth $d$, which replaces the actual depth $D$ to the barrier to account for the convergence (radial flow) near the drain:

• $K_1$ = hydraulic conductivity of the soil above the drain level; $K_2$ = conductivity below the drain level.
• For a homogeneous soil ($K_1=K_2=K$): $L^2=\frac{8Kdh+4K{h}^2}{q}$.
• The first term in the numerator is the flow below the drain level (through the equivalent layer $d$); the second is the flow above the drain.
• Equivalent depth $d$ depends on the spacing $L$, the depth $D$ and the drain radius $r_0$ — it is obtained from Hooghoudt's tables or from Moody's formulas; since $d$ depends on $L$, the equation is solved by trial (assume $L$, get $d$, compute $L$, repeat).
• For $D/L$ large, $d$ is limited by the radial flow; for small $D$, $d\approx D$.

Where $d$ is limited: $d\le D$; and when $D$ is greater than about $L/4$, the extra depth has little effect.