A drainage equation is an equation describing the relation between depth and spacing of parallel subsurface drains, depth of the watertable, depth and hydraulic conductivity of the soils. It is used in drainage design.
Parameters in Hooghoudt's drainage equation
A well known steady-state drainage equation is the Hooghoudt drain spacing equation. Its original publication is in Dutch. The equation was introduced in the USA by van Schilfgaarde.
Hooghoudt's equation can be written as:
Q L2 = 8 Kb d (Di - Dd) (Dd - Dw) + 4 Ka (Dd - Dw)2
where:
Q = steady state drainage discharge rate (m/day)
Ka = hydraulic conductivity of the soil above drain level (m/day)
Kb = hydraulic conductivity of the soil below drain level (m/day)
Di = depth of the impermeable layer below drain level (m)
Dd = depth of the drains (m)
Dw = steady state depth of the watertable midway between the drains (m)
L = spacing between the drains (m)
d = equivalent depth, a function of L, (Di-Dd), and r
r = drain radius (m)
Steady (equilibrium) state condition
In steady state, the level of the water table remains constant and the discharge rate (Q) equals the rate of groundwater recharge (R), i.e. the amount of water entering the groundwater through the watertable per unit of time. By considering a long-term (e.g. seasonal) average depth of the water table (Dw) in combination with the long-term average recharge rate (R), the net storage of water in that period of time is negligibly small and the steady state condition is satisfied: one obtains a dynamic equilibrium.
Derivation of the equation
For the derivation of the equation Hooghoudt used the law of Darcy, the summation of circular potential functions and, for the determination of the influence of the impermeable layer, de method of mirror images and superposition.
Hooghoudt published tables for the determination of the equivalent depth (d), because the function (F) in d = F (L,Di-Dd,r) consists of long series of terms.
Crop yield and seasonal average depth of the water table
Determining:
the discharge rate (Q) from the recharge rate (R) in a water balance as detailed in the article: hydrology (agriculture)
the permissible long term average depth of the water table (Dw) on the basis of agricultural drainage criteria
the soil's hydraulic conductivity (Ka and Kb) by measurements
the depth of the bottom of the aquifer (Di)
the design drain spacing (L) can be found from the equation in dependence of the drain depth (Dd) and drain radius (r).
Drainage criteria
One would not want the water table to be too shallow to avoid crop yield depression nor too deep to avoid drought conditions. This is a subject of drainage research. The figure shows that a seasonal average depth of the water table shallower than 70 cm causes a yield depression [4]
The figure was made with the SegReg program for segmented regression.
