Tile Drainage Field Lab (3D)

Darcy flow in a thin aquifer (Dupuit-Forchheimer)

Water moves through saturated soil by Darcy's law, q = -K dh/dx. For a shallow water table the flow is nearly horizontal, so the flux through a vertical column of height (d_e + h) is Q(x) = -K (d_e + h) dh/dx. Mass conservation with rain recharge R gives the Boussinesq equation used by the simulation:

n_e dh/dt = d/dx [ K (d_e + h) dh/dx ] + R(t) - ET, with h = 0 at both drains (h is measured above drain level, so the soil below the drain is always saturated).

Hooghoudt's steady-state drain-spacing equation

For steady recharge the equation integrates exactly to q = (8 K d_e h_m + 4 K h_m^2) / L^2: the first term is flow below drain level, the second above it. h_m is the water-table height midway between drains, L the drain spacing. The inverse, h_m = -d_e + sqrt(d_e^2 + q L^2 / 4K), is drawn as the dashed line in the profile graph; the design-spacing readout solves the same equation for L by bisection.

Equivalent depth d_e

Close to a drain the flow lines converge radially and the real depth D to the impermeable layer overstates the flow area. Hooghoudt replaced D by a smaller d_e (here the van Beers closed form): d_e = D / [1 + (D/L)((8/pi) ln(D/r0) - 3.4)] for D/L <= 0.3, and d_e = pi L / (8 (ln(L/r0) - 1.15)) for deeper barriers. With a very shallow barrier (D tiny) d_e tends to D and the drains must sit close together.

Why it matters: SEW30 and yield

Roots need air. The classic indicator SEW30 = sum over days of max(0, 30 cm - depth to water table) (cm days) accumulates the time the water table spends inside the top 30 cm of soil. Relative yield here is an illustrative linear loss, Y = max(0.2, 1 - 0.0018 SEW30), evaluated at every position between the drains and averaged. Real crop-response functions depend on crop, growth stage and temperature.

Checks run in the page

• Steady state: the transient code is run to equilibrium and compared with the closed-form Hooghoudt midpoint head and with the requirement that drain outflow equals recharge.
• No-barrier limit: as d_e tends to 0 the equation has the exact Dupuit ellipse h(x)^2 = (R/K) x (L - x); the nonlinear solver must reproduce it.
• Recession: for small heads and fixed d_e the equation is linear and the lowest mode decays at alpha = pi^2 K d_e / (n_e L^2); the measured decay rate must match.
• Mass balance: rain minus ET, drain outflow, surface runoff and change in stored water is monitored live (readout at the bottom).

Honest simplifications

One-dimensional cross-section between two identical parallel drains; homogeneous isotropic soil; drainable porosity constant (no unsaturated-zone dynamics, no capillary fringe, no hysteresis); infiltration is instant and rain above the surface capacity runs off; ET is a simple sink that fades as the water table drops; the barrier is perfectly impermeable and drains are always flowing full with no clogging or siltation. The tracer dots and the vertical stretch of the scene (the field is compressed horizontally so the cut face is readable) are schematic.

What to try

• Choose Wet fortnight and watch the water table climb into the root zone; compare drain spacings of 10 m and 30 m.
• Press Heavy clay: tiles must be 8 m apart to do the work a 30 m spacing does in sand.
• Set the barrier to 0.5 m: d_e collapses and the same spacing leaves the field waterlogged.
• Raise the drain depth: deeper drains lower the water table but cost more; see the design chart curve shift.
• Start the guided tour for the section view, the radial flow near the pipe, the observation well and the weir.