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Comparing steady and non-steady state subsurface drainage using
calculations with relevant models
R.J. Oosterbaan, 20-10-2019
Abstract
The results of model developed for steady state subsurface drainage calculating the level of
the water table using the traditional Darcy equation as well as the energy balance of
groundwater flow are compared with those of a model developed for non-steady state
subsurface drainage, based on the non-linear reservoir concept, calculating the fluctuations of
the water-table, while the characteristics of the drainage system and the soil conditions are the
same.
The average water level in the non-steady state using the non-linear reservoir approach
corresponds, after an initial period, well with the steady state energy balance approach.
Contents
1. Introduction
2. Experimentation
3. Conclusions
4. References
1. Introduction
The RainOff model [Ref. 1] uses the non-linear reservoir concept to simulate rainfall-runoff
relations in watersheds [Ref. 2] as well as the recharge – water level - discharge relations for
subsurface drainage systems.
The EnDrain model [Ref. 3] uses both the classical Darcy equation as well as the energy
balance of groundwater flow to obtain the steady state shape of of the water table between
two parallel subsurface drains (ditches or pipes, [Ref. 4] ). Input data are the recharge, the
drain depth, the drain spacing, the drain dimensions, the hydraulic conductivity of the soil
above and below drain level, and the depth of the impermeable layer, see the following figure.
Figure 1.
The symbols used in the above figure are clarified with a sketch of the properties of the
drainage system in the next figure.
Figure 2.
The RainOff model uses the equations: Q= A.H + B and dH/dT = d {(Q-B)/A}/ dT =
(R ‒ Q) / P, where Q is the runoff or discharge, H is the water storage or hydraulic head, A
and B are parameters depending on the dimensions of the drainage system, R is the recharge
(rainfall minus change in water storage), and P is the drainable porosity of the soil. The
calculation method for A and B is shown in the next figure. To compare the results of both
models, the RainOff model needs regularly fluctuating recharges so that in the long run the
fluctuations reach an equilibrium.
2. Experimentation
A calculator of RainOff to find the values of A and B is pictured in the next figure. The data
used here are the same as those used in the EnDrain program (figure 1).
Figure 3.
The resulting A and B values are transferred to the input tab-sheet, as depicted in the next
figure, where it is also shown how the calculator is activated.
Figure 4.
In the above illustration it can be seen that the rainfall is 0 and 10 mm every other day, giving
a recharge of 5 mm/day on average. Subtracting the escape rate (representing in this case the
evaporation), being 3 mm/day, results in a net average recharge of 5 – 3 = 2 mm/day, the
same as used in the EnDrain program (figure 1).
The results of the EnDrain software are demonstrated in the following picture. It presents the
shape of the water-table over the distance from the drain to midway between the drains, using
the classical Darcy equation and the energy balance of the groundwater flow respectively.
Figure 5.
The elevation of the water table midway between the drains above drain level is 0.38 and 0.52
m for the Darcy and energy balance respectively. The energy balance takes into account the
energy supplied by the downward percolating water to the water table, whereas the Darcy
equation does not. Reason why the water table is deeper in the case of the energy balance.
The results of the RainOff model are shown in the next illustration. It depicts the fluctuations
of the water-table midway between the drains in the course of the time. The green line
corresponds with the average water level in time towards the end of the calculation period.
Figure 6.
The level of the green line is at 0.41 m, which corresponds well with the level found with
EnDrain in the case of the energy balance (0.38), while the Darcy option gives a much higher
value (0.52). The reason is that the non-steady state model adds the percolation water to the
water-table, so that it rises attaining a higher energy level, thus taking the energy balance also
into account. The model based on the Darcy equation does not do that and therefor misses an
energy component so that the water-level gets higher.
3. Conclusions
The EnDrain software for steady state drainage gives good results when the full energy
balance of groundwater flow is used. It shows the shape of the steady state water-table in the
region from the drain to midway between the drains and represents the average shape of the
fluctuations in time.
The RainOff model gives for non-steady state drainage gives good results as it automatically
includes the proper energy balance. It shows the fluctuation of the water-table in time at the
point midway between the drains. When the rainfall-recharge pattern is not too irregular, the
model shows a stabilization in the long run. The stabilized fluctuations give, on average, the
same value as the one calculated with EnDrain with the full energy balance.
4.References
[Ref. 1] RainOff, free software for the calculation of rainfall-runoff relations in watersheds
and non steady groundwater flow to subsurface drains. Download from:
https://www.waterlog.info/rainoff.htm
[Ref. 2] RAINFALL-RUNOFF RELATIONS OF A SMALL VALLEY ASSESSED WITH
A NON-LINEAR RESERVOIR MODEL. In: International Journal of Environmental
Science, 2018. On Line: https://www.iaras.org/iaras/filedownloads/ijes/2019/008-
0002(2019).pdf
[Ref. 3] EnDrain, free software for the calculations of subsurface drainage (hydraulic
conductivity, hydraulic head, drain spacing, level of the water table). Download from:
https://www.waterlog.info/endrain.htm
[Ref. 4] THE ENERGY BALANCE OF GROUNDWATER FLOW APPLIED TO
SUBSURFACE DRAINAGE IN ANISOTROPIC SOILS BY PIPES OR DITCHES WITH
ENTRANCE RESISTANCE. Download from: https://www.waterlog.info/pdf/enerart0.pdf