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 Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...

of the soils. It is used in

drainage Drainage is the natural or artificial removal of a surface's water and sub-surface water from an area with excess of water. The internal drainage of most agricultural soils is good enough to prevent severe waterlogging (anaerobic conditio ...

design.

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

Hooghoudt's equation can be written as:. :Q L² = 8 Kb d (Dd - Dw) + 4 Ka (Dd - Dw)² where: * Q =

steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ' ...

drainage discharge rate (m/day) * Ka =

of the soil above drain level (m/day) * Kb =

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

, the level of the water table remains constant and the discharge rate (Q) equals the rate of

groundwater recharge Groundwater recharge or deep drainage or deep percolation is a hydrologic process, where water moves downward from surface water to groundwater. Recharge is the primary method through which water enters an aquifer. This process usually occurs ...

(R), i.e. the amount of water entering the

groundwater Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidat ...

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 In chemistry, a dynamic equilibrium exists once a reversible reaction occurs. Substances transition between the reactants and products at equal rates, meaning there is no net change. Reactants and products are formed at such a rate that the ...

. ''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 image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...

s 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. R-3VAR1

''Determining:'' *the discharge rate (Q) from the recharge rate (R) in a

water balance The law of water balance states that the inflows to any water system or area is equal to its outflows plus change in storage during a time interval. In hydrology, a water balance equation can be used to describe the flow of water in and out of ...

as detailed in the article:

hydrology (agriculture) Agricultural hydrology is the study of water balance components intervening in agricultural water management, especially in irrigation and drainage. Water balance components The water balance components can be grouped into components correspond ...

*the permissible long term average depth of the water table (Dw) on the basis of ''agricultural drainage criteria'' *the soil's

(Ka and Kb) by measurements *the depth of the bottom of the

aquifer An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials ( gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their characteri ...

(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
The figure was made with the SegReg program for

segmented regression Segmented regression, also known as piecewise regression or broken-stick regression, is a method in regression analysis in which the independent variable is partitioned into intervals and a separate line segment is fit to each interval. Segmented r ...

Equivalent depth

In 1991 a

closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...

was developed for the equivalent depth (d) that can replace the Hooghoudt tables: :d = πL / 8 where: *x = 2π (Di − Dd) / L *F(x) = Σ 4e^−2nx / n (1 − e^−2nx), with n = 1, 3, 5, . . .

Extended use

Theoretically, Hooghoudt's equation can also be used for ''sloping land''. The theory on drainage of sloping land is corroborated by the results of sand tank experiments. In addition, the ''entrance resistance'' encountered by the water upon entering the drains can be accounted for. Image:SlopingLandDrains.jpg, Geometry of drains in sloping land used in the theory Image:SandTankDrains.jpg, Corroboration with drainage experiments in a sloping sand tank Image:Entrance resistance.jpg, Definition of entrance resistance

Amplification

The drainage formula can be amplified to account for (see figure on the right): *the additional energy associated with the incoming percolation water ( recharge), see

groundwater energy balance The groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of f ...

*multiple soil layers * anisotropric hydraulic conductivity, the vertical conductivity (Kv) being different from the horizontal (Kh) *drains of different dimensions with any width (W)

Computer program

The amplified drainage equation uses an hydraulic equivalent of Joule's law in electricity. It is in the form of a

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

that cannot be solved analytically (i.e. in a closed form) but the solution requires a

numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...

for which a

computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer progra ...

is indispensable. The availability of a computer program also helps in quickly assessing various alternatives and performing a

sensitivity analysis Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty ana ...

. The blue figure shows an example of results of a computer aided calculation with the amplified drainage equation using the EnDrain program.The free computer program EnDrain can be downloaded from web page

, or from

/ref> It shows that incorporation of the incoming energy associated with the recharge leads to a somewhat deeper watertable, water table.

References

External links

Drainage Hooghoudt's equation calculator
{{DEFAULTSORT:Drainage Equation Drainage Hydrology Hydraulic engineering Soil Soil science Soil physics Agricultural soil science