Waterway level change benchmark (Water Module): Difference between revisions

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===Formulas===
===Formulas===
When the water level in the water way changes, the water table level will also change over time. The relation between water level in the waterway and the ground water table at distance x from the water way is described by the following formulas{{ref|Marsily86}}:
When the water level in the waterway changes, the water table level will also change over time. The relation between water level in the waterway and the ground water table at distance x from the waterway is described by the following formulas{{ref|Marsily86}}:


In case the water level is raised, the following formula should hold:
In case the water level is raised, the following formula should hold:


<math> \frac{h(x)}{h_0} = erfc (x\sqrt{\frac{S}{4kDt}})</math>
<math> \frac{h(x)}{h_0} = erfc (x\sqrt{\frac{S}{4kDt}})</math>
The formula erfc is the complementary error function.


In case the water level is lowered, the following formula should hold:
In case the water level is lowered, the following formula should hold:


<math> \frac{h(x)}{h_0} = erf (x\sqrt{\frac{S}{4kDt}})</math>
<math> \frac{h(x)}{h_0} = erf (x\sqrt{\frac{S}{4kDt}})</math>
The formula erf is the complementary error function.


<math> erf(z) = \frac{2}{\sqrt{\pi}}\int\limits_{0}^{z}e^{-t^2}dt</math>
<math> erf(z) = \frac{2}{\sqrt{\pi}}\int\limits_{0}^{z}e^{-t^2}dt</math>
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where:
where:
: <math>h(x)</math>: change in freatic groundwater table (m)
: <math>h(x)</math>: change in phreatic groundwater table (m)
: <math>h0</math>: change in water level in the waterway(m)
: <math>h0</math>: change in water level in the waterway(m)
: <math>x</math>: distance to the water way (m)
: <math>x</math>: distance to the waterway (m)
: <math>t</math>: time since the water level change
: <math>t</math>: time since the water level change
: <math>kD</math>: transmissivity of the aquifer (m2 / day)
: <math>kD</math>: transmissivity of the aquifer (m2 / day)
: <math>S</math>: freatic water storage fraction.
: <math>S</math>: phreatic water storage fraction.


[[File:graph_waterlevel_change_percentual_over_distance.png|frame|left| De grafiek laat de relatieve verandering zien van de grondwaterstand na 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 en 121 dagen, berekend met bovenstaande formule.]]
[[File:graph_waterlevel_change_percentual_over_distance.png|frame|left| De grafiek laat de relatieve verandering zien van de grondwaterstand na 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 en 121 dagen, berekend met bovenstaande formule.]]
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The terrain height is set up as followed:
The terrain height is set up as follows:
: 0.0 m for x <= 2
: 0.0 m for x <= 2
: 3.0 m for x == 3
: 3.0 m for x == 3
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The water will be raised to 4 meters, thereby simulating an <math>h0</math> of 2.
The water will be raised to 4 meters, thereby simulating an <math>h0</math> of 2.


One inlet is placed on the cells x = 1 and y = 1 to 3, with the following setup to create the sudden stable water level increment:
An inlet at x=1, y=1-3 creates a sudden stable water level increase:
:[[Inlet_upper_threshold_(Water_Overlay)|UPPER_THRESHOLD]] set to 4.
:[[Inlet_upper_threshold_(Water_Overlay)|UPPER_THRESHOLD]] set to 4.
:[[Inlet_lower_threshold_(Water_Overlay)|LOWER_THRESHOLD]] set to 4.
:[[Inlet_lower_threshold_(Water_Overlay)|LOWER_THRESHOLD]] set to 4.

Revision as of 14:32, 15 July 2026

In this benchmark, the ground water table is initially the same as the water level in the waterway. Then, the level in the waterway suddenly rises with h0 meters. This causes water to infiltrate into the ground, slowly raising the ground water table as well.

Situation of the waterway level change benchmark for groundwater

Formulas

When the water level in the waterway changes, the water table level will also change over time. The relation between water level in the waterway and the ground water table at distance x from the waterway is described by the following formulas[1]:

In case the water level is raised, the following formula should hold:

h(x)h0=erfc(xS4kDt)

The formula erfc is the complementary error function.

In case the water level is lowered, the following formula should hold:

h(x)h0=erf(xS4kDt)

erf(z)=2π∫0ze−t2dt

erfc(z)=1−erf(z)

where:

h(x): change in phreatic groundwater table (m)
h0: change in water level in the waterway(m)
x: distance to the waterway (m)
t: time since the water level change
kD: transmissivity of the aquifer (m2 / day)
S: phreatic water storage fraction.
De grafiek laat de relatieve verandering zien van de grondwaterstand na 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 en 121 dagen, berekend met bovenstaande formule.


Setup

The test setup uses the same parameters used in the research results published in literature[2]. The grid chosen is 112 by 5. The ground water table comparisons will be conducted in the x-direction. The grid cell size is set to 25 meters.


The terrain height is set up as follows:

0.0 m for x <= 2
3.0 m for x == 3
10.0 m for x > 3

The initial water level is set to 2, for both surface and the groundwater table, by using a water area set to 2 meters.

The water will be raised to 4 meters, thereby simulating an h0 of 2.

An inlet at x=1, y=1-3 creates a sudden stable water level increase:

UPPER_THRESHOLD set to 4.
LOWER_THRESHOLD set to 4.
Inlet Q set to 0, such that is unlimited.

This ensures that the water level is instantly raised to level 4 and that it remains at 4 at all times.

The ground is setup with: Water storage fraction set to 0.25.

An aquifer is added for the region as well, initialized with the Aquifer KD set to 700 m2/day.

For the test different simulation times T are tested, each resulting in a different relation graph as shown in the described case above.

Use a ground water overlay and set the Weather's rain M parameter to [T, 0.0] for a rainless simulation of T seconds. Note that T has to be converted to days to be used in the formula's described above.

The surface water simulation mode is set to surface averaging.

Results

The results are generated for four simulation periods: 16 days, 64 days, 81 days and 121 days. The value h is the groundwater level increase (m), where the waterway rose from 2 to 4 meters. The dx is the distance in meters to the waterway. As shown below, the lines align well.

16 Days

64 Days

81 Days

121 Days


References

  1. ↑ Quantitative Hydrogeology. Groundwater Hydrology for Engineers. Academic Press ∙ Marsily, Ghislain de, 1986. ∙ Page(s): 199 (originally in french)
  2. ↑ Verhoging van grondwaterstanden en vermindering van afvoer door opstuwing van beken. Overdruk uit Verslagen Technische Bijeenkomsten 11-12. Versl. Meded. Comm. Hydrol. Onderz. T.N.O. No. 3 (1958) ∙ Ernst, L.F. 1958
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