Bottom flow pankow benchmark (Water Module): Difference between revisions

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This testcase demonstrates a situation where a parcel of land is situated between two waterways with a stable water level. In combination with seepage, a characteristic curve will form over time, as shown in the image below. It is described in a water balance research in 1968 of Pankow and Rijtema <ref name="Pankow68"/>, and also contain the accompanying formulas that describe the curve.  
This [[Water Module benchmarks|benchmark]] demonstrates a situation where a parcel of land is situated between two waterways with a stable water level. In combination with [[Ground bottom flow formula (Water_Overlay)|ground water flowing from deeper ground layers through the bottom boundary]], a characteristic curve will form over time, as shown in the image below. It is described in a water balance research in 1968 of Pankow and Rijtema <ref name="Pankow68"/>, and also contains the accompanying formulas that describe the curve.  
In this case, we use the case which does not take into account the additional water flow resistance of the waterway. Secondly, rain could also be taken into account, but we set that to 0 to exclusively benchmark the seepage mechanics. The continuous rainfall case is already tested in [[Freatic_groundwater_levels_benchmark_(Water_Module)|freatic groundwater levels benchmark]].
In this case, we use the case which does not take into account the additional water flow resistance of the waterway. Secondly, rain could also be taken into account, but we set that to 0 to exclusively benchmark the bottom flow mechanics. The continuous rainfall case is already tested in [[Freatic_groundwater_levels_benchmark_(Water_Module)|freatic groundwater levels benchmark]].


[[File:pankow_situation_benchmark.gif|A parcel of land situated between two waterways with seepage and optional continuous rainfall]]
[[File:pankow_situation_benchmark.gif|frame|left|A parcel of land situated between two waterways with ground bottom flow and optional continuous rainfall<ref name="pankow"/>]]
<br style="clear: both;">


===Formulas===
===Formulas===
Due to seepage and two stable water levels left and right, a specific ground water table curve will form. Part of the water seeped in will flow left and part of the it will flow right.
Due to ground bottom flow and two stable water levels left and right, a specific ground water table curve will form. Part of the water flowed in from the bottom will flow left and part of the it will flow right.
Note that the time at which this balance occurs is dependent on the starting situation.
Note that the time at which this balance occurs is dependent on the starting situation.


Line 20: Line 21:
Simplified without rain N:
Simplified without rain N:


<math>h_d-h_1 = \frac{(h_1-h_2) cosh \frac{x_1}{\sqrt{KD \cdot c}}}{( cosh (\frac{x_2}{\sqrt{KD \cdot c}
<math>h_1 = h_d - \frac{(h_1-h_2) cosh \frac{x_1}{\sqrt{KD \cdot c}}}{( cosh (\frac{x_2}{\sqrt{KD \cdot c}
}) -1) - ( cosh (\frac{x_1}{\sqrt{KD \cdot c}}) -1) } </math>
}) -1) - ( cosh (\frac{x_1}{\sqrt{KD \cdot c}}) -1) } </math>


To test the correctness of the ground bottom flow, the formula can test the following condition with the accepted error margin <math>\epsilon</math>:
<math>\left | h_d - \frac{(h_1-h_2) cosh \frac{x_1}{\sqrt{KD \cdot c}}}{( cosh (\frac{x_2}{\sqrt{KD \cdot c}
}) -1) - ( cosh (\frac{x_1}{\sqrt{KD \cdot c}}) -1) } -  h_1 \right |< \epsilon</math>


where:
where:
: <math>x_1</math>: distance (m) of the point of measurement compared to the middle of the parcel.
: <math>x_1</math>: distance (m) of the point of measurement compared to the middle of the parcel.
: <math>x_2</math>: distance (m) of the second point of measurement, which is always situated 3 meters from the edge of the waterway.
: <math>x_2</math>: distance (m) of the second point of measurement, which is always situated 3 meters from the edge of the waterway.
: <math>h_1</math>: measured ground water level (m) at x_1
: <math>h_1</math>: measured ground water level (m) at <math>x_1</math>
: <math>h_2</math>: measured ground water level (m) at x_2
: <math>h_2</math>: measured ground water level (m) at <math>x_2</math>
: <math>h_d</math>: seepage head
: <math>h_d</math>: bottom head pressure
: <math>KD</math>: distance between both waterways (m)
: <math>KD</math>: measured horizontal transmissivity of the ground layer (m2/day)
: <math>k</math>: horizontal infiltration speed (m / day)
: <math>N</math>: additional ground water due to rainfall (m/day).
: <math>N</math>: additional ground water (m/day).
: <math>\epsilon</math>: accepted error margin


===Setup===
===Setup===
The following setup has been taken from the use case ''Peilverschil over een strook grond (freatisch)'' at [http://grondwaterformules.nl/index.php/formules/peilverschil/rechte-strook-freatisch grondwaterformules.nl].
We use the following setup. The grid size used is 53 by 5, with a configurable cell size of <math>dx</math> in meters. There are two waterways, left and right, both with a stable water level of 3 meters.
 
One inlet is placed on the cells x = 1 and y = 1 to 3 and an other is placed on the cells x = 52 and y - 1 to 3, with the following setup to ensure a stable water level:
:[[Inlet_upper_threshold_(Water_Overlay)|UPPER_THRESHOLD]] set to 3 m.
:[[Inlet_lower_threshold_(Water_Overlay)|LOWER_THRESHOLD]] set to 3 m.
:[[Inlet q (Water Overlay)|Inlet Q]] set to 0, such that is unlimited.
 
The terrain height is set to:
<math>\begin {cases}
0, & \text{if }x < 4 \text{ or } x > 48\\
1.5, & \text{if }x = 4 \text{ or } x = 48\\
3.5, & \text{if }x = 5 \text{ or } x = 47\\
5, & \text{otherwise}
\end{cases}</math>


In the situation described there, the chosen Length L is set to 500 m. This is achieved with a grid size of 28 by 5, with a cell size of 20 m. There are two stable ground water levels, one on left of 11 meters (datum) and one on the right at 10 meters (datum). The rainfall is 0.8 mm per day, the horizontal infiltration speed is 3 m per day.
Depending on the specific test case, the [[Bottom pressure prequel (Water Overlay)|Bottom head pressure]] <math>h_p</math> is set to:


<math>\begin {cases}
h_p, & \text{if }x > 5 \text{ and } x < 47\\
none, & \text{otherwise}
\end{cases}</math>


The terrain height is set to 12 for all cells and the [[Ground bottom distance m (Water Overlay)|ground bottom distance]] is also set to 12.
Similarly, the [[Bottom resistance prequel (Water Overlay)|Bottom resistance]] (in days) is set to:


The terrain type's [[Ground infiltration md (Water Overlay)|infiltration speed]] can be configured to 3 m / day.
<math>\begin {cases}
c, & \text{if }x > 5 \text{ and } x < 47\\
none, & \text{otherwise}
\end{cases}</math>


For x = 1 the initial ground water level is set to 11 and for x = 26 the ground water level is set to 10. The initial ground water levels between this points is linearly interpolated.
An aquifer can be added to configure the [[Aquifer kd (Water Overlay)|horizontal infiltration speed]]. An important thing to note here, is that the KD value used in the report <ref name="Pankow68"/> is derived from changes in water head over time, and does not mention the storage capacity of the soil. Therefor, in order to configure the Aquifer KD value in the {{software}}, the KD value has to be multiplied with the storage percentage.


Additionally, two [[Inlet (Water Overlay)|underground inlets]] are placed, one on x=1 and one on x=26, as an area over y = 1 to 3.
The [[Water level (Water Overlay)|water level]] is configured to 3 m for all cells, resulting in an initial water level above and below ground of 3 meters.


The simulation time is set to n days, with a rainfall of 0.8 mm per day. To configure this, the [[Weather rain m (Water Overlay)|rain]] set is set to <math>[3600 \cdot 24 \cdot n, \frac{0.8}{1000 \cdot 3600 \cdot 24}]</math>.
The simulation time is set to n days, with rain set to 0 mm. To configure this, the [[Weather rain m (Water Overlay)|rain]] set is set to <math>[3600 \cdot 24 \cdot n,0]</math>.


===Results===
===Results===
====365 days====
====Test case 1====
The first result is generated for n = 365:<br>
:cell size: 10 m;
[[File:benchmark_freatic_365.png]]
:<math>KD</math>: 17 m²/day;
:Water storage fraction: 0.10;
:Aquifer KD configured to 1.7 m²/day in the {{software}};
:Bottom resistance <math>c</math>: 20 days
:Bottom head pressure <math>h_d</math>: 4.1
:Simulation days <math>n</math>: 128 days
[[File:128days_10m_kd17_c20_ws10_h4_1.png]]
 
====Test case 2====
:cell size: 10 m;
:<math>KD</math>: 22 m²/day;
:Water storage fraction: 0.25;
:Aquifer KD configured to 5.5 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 40 days
:Bottom head pressure <math>h_d</math>: 4.1
:Simulation days <math>n</math>: 128 days
[[File:128days_10m_kd22_c40_ws25_h4_1.png]]
 
====Test case 3====
:cell size: 10 m;
:<math>KD</math>: 22 m²/day;
:Water storage fraction: 0.25;
:Aquifer KD configured to 5.5 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 40 days
:Bottom head pressure <math>h_d</math>: 3.8
:Simulation days <math>n</math>: 128 days
[[File:128days_10m_kd22_c40_ws25_h3_8.png]]
 
====Test case 4====
:cell size: 10 m;
:<math>KD</math>: 22 m²/day;
:Water storage fraction: 0.25;
:Aquifer KD configured to 5.5 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 10 days
:Bottom head pressure <math>h_d</math>: 4.8
:Simulation days <math>n</math>: 128 days
[[File:128days_10m_kd22_c10_ws25_h4_8.png]]
 
====Test case 5====
:cell size: 10 m;
:<math>KD</math>: 17 m²/day;
:Water storage fraction: 0.10;
:Aquifer KD configured to 1.7 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 20 days
:Bottom head pressure <math>h_d</math>: 4.1 (but leads to <math>h_d=4.077</math> for the comparisons with the middle value)
:Simulation days <math>n</math>: 64 days
[[File:64days_10m_kd17_c20_ws10_h4_1.png]]
 
====Test case 6====
:cell size: 10 m;
:<math>KD</math>: 22 m²/day;
:Water storage fraction: 0.33;
:Aquifer KD configured to 7.26 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 10 days
:Bottom head pressure <math>h_d</math>: 4.5 (but leads to <math>h_d=4.499</math> for the comparisons with the middle value)
:Simulation days <math>n</math>: 64 days
[[File:64days_10m_kd22_c10_ws33_h4_5.png]]
 
====Test case 7====
:cell size: 10 m;
:<math>KD</math>: 220 m²/day;
:Water storage fraction: 0.25;
:Aquifer KD configured to 55 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 20 days
:Bottom head pressure <math>h_d</math>: 4.1
:Simulation days <math>n</math>: 64 days
[[File:64days_10m_kd220_c20_ws25_h4_1.png]]
 
====Test case 8====
:cell size: 10 m;
:<math>KD</math>: 220 m²/day;
:Water storage fraction: 0.10;
:Aquifer KD configured to 55 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 20 days
:Bottom head pressure <math>h_d</math>: 4.1
:Simulation days <math>n</math>: 64 days
[[File:64days_10m_kd220_c20_ws10_h4_1.png]]
 
====Test case 9====
:cell size: 5 m;
:<math>KD</math>: 22 m²/day;
:Water storage fraction: 0.10;
:Aquifer KD configured to 2.2 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 20 days
:Bottom head pressure <math>h_d</math>: 4.1
:Simulation days <math>n</math>: 64 days
[[File:64days_5m_kd22_c20_ws10_h4_1.png]]


====730 days====
====Test case 10====
The second result is generated using n = 730:<br>
:cell size: 2 m;
[[File:benchmark_freatic_730.png]]
:<math>KD</math>: 22 m²/day;
:Water storage fraction: 0.10;
:Aquifer KD configured to 2.2 m²/day in the {{software}};
:Bottom resistance  <math>c</math>: 20 days
:Bottom head pressure <math>h_d</math>: 4.1
:Simulation days <math>n</math>: 64 days
[[File:64days_2m_kd22_c20_ws10_h4_1.png]]


===Notes===
===Notes===
* The amount of days it takes to reach the stable solution is highly dependent on the starting situation.
* A higher KD value can lead to a lower stable ground water level than the [[Bottom pressure prequel (Water Overlay)|head pressure]].


===References===
===References===
<references>
<references>
<ref name="Bear79"> Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill. (formula on p.180, equation 5-212)</ref>
<ref name="Pankow68">J. Pankow en P.E. Rijtema, 1970. De resultaten van het waterbalansonderzoek in 1968 voor de objecten met een constant slootpeil in Hoenkoop. Nota 567. Instituut voor Cultuurtechniek en Waterhuishouding, Wageningen.</ref>
<ref name="pankow">Schatten van kwel (Pankow), Grondwaterformules.nl/index.php/formules/ontwatering/kwel-berekenen-pankow</ref>
</references>
</references>


{{Water Module buttons}}
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Latest revision as of 14:50, 5 March 2024

This benchmark demonstrates a situation where a parcel of land is situated between two waterways with a stable water level. In combination with ground water flowing from deeper ground layers through the bottom boundary, a characteristic curve will form over time, as shown in the image below. It is described in a water balance research in 1968 of Pankow and Rijtema [1], and also contains the accompanying formulas that describe the curve. In this case, we use the case which does not take into account the additional water flow resistance of the waterway. Secondly, rain could also be taken into account, but we set that to 0 to exclusively benchmark the bottom flow mechanics. The continuous rainfall case is already tested in freatic groundwater levels benchmark.

A parcel of land situated between two waterways with ground bottom flow and optional continuous rainfall[2]


Formulas

Due to ground bottom flow and two stable water levels left and right, a specific ground water table curve will form. Part of the water flowed in from the bottom will flow left and part of the it will flow right. Note that the time at which this balance occurs is dependent on the starting situation.

The following formula, taken from [1], describes the curve of ground water levels when the ground water flow to the left and right have become stable:

Simplified without rain N:

To test the correctness of the ground bottom flow, the formula can test the following condition with the accepted error margin :

where:

: distance (m) of the point of measurement compared to the middle of the parcel.
: distance (m) of the second point of measurement, which is always situated 3 meters from the edge of the waterway.
: measured ground water level (m) at
: measured ground water level (m) at
: bottom head pressure
: measured horizontal transmissivity of the ground layer (m2/day)
: additional ground water due to rainfall (m/day).
: accepted error margin

Setup

We use the following setup. The grid size used is 53 by 5, with a configurable cell size of in meters. There are two waterways, left and right, both with a stable water level of 3 meters.

One inlet is placed on the cells x = 1 and y = 1 to 3 and an other is placed on the cells x = 52 and y - 1 to 3, with the following setup to ensure a stable water level:

UPPER_THRESHOLD set to 3 m.
LOWER_THRESHOLD set to 3 m.
Inlet Q set to 0, such that is unlimited.

The terrain height is set to:

Depending on the specific test case, the Bottom head pressure is set to:

Similarly, the Bottom resistance (in days) is set to:

An aquifer can be added to configure the horizontal infiltration speed. An important thing to note here, is that the KD value used in the report [1] is derived from changes in water head over time, and does not mention the storage capacity of the soil. Therefor, in order to configure the Aquifer KD value in the Tygron Platform, the KD value has to be multiplied with the storage percentage.

The water level is configured to 3 m for all cells, resulting in an initial water level above and below ground of 3 meters.

The simulation time is set to n days, with rain set to 0 mm. To configure this, the rain set is set to .

Results

Test case 1

cell size: 10 m;
: 17 m²/day;
Water storage fraction: 0.10;
Aquifer KD configured to 1.7 m²/day in the Tygron Platform;
Bottom resistance : 20 days
Bottom head pressure : 4.1
Simulation days : 128 days

128days 10m kd17 c20 ws10 h4 1.png

Test case 2

cell size: 10 m;
: 22 m²/day;
Water storage fraction: 0.25;
Aquifer KD configured to 5.5 m²/day in the Tygron Platform;
Bottom resistance : 40 days
Bottom head pressure : 4.1
Simulation days : 128 days

128days 10m kd22 c40 ws25 h4 1.png

Test case 3

cell size: 10 m;
: 22 m²/day;
Water storage fraction: 0.25;
Aquifer KD configured to 5.5 m²/day in the Tygron Platform;
Bottom resistance : 40 days
Bottom head pressure : 3.8
Simulation days : 128 days

128days 10m kd22 c40 ws25 h3 8.png

Test case 4

cell size: 10 m;
: 22 m²/day;
Water storage fraction: 0.25;
Aquifer KD configured to 5.5 m²/day in the Tygron Platform;
Bottom resistance : 10 days
Bottom head pressure : 4.8
Simulation days : 128 days

128days 10m kd22 c10 ws25 h4 8.png

Test case 5

cell size: 10 m;
: 17 m²/day;
Water storage fraction: 0.10;
Aquifer KD configured to 1.7 m²/day in the Tygron Platform;
Bottom resistance : 20 days
Bottom head pressure : 4.1 (but leads to for the comparisons with the middle value)
Simulation days : 64 days

64days 10m kd17 c20 ws10 h4 1.png

Test case 6

cell size: 10 m;
: 22 m²/day;
Water storage fraction: 0.33;
Aquifer KD configured to 7.26 m²/day in the Tygron Platform;
Bottom resistance : 10 days
Bottom head pressure : 4.5 (but leads to for the comparisons with the middle value)
Simulation days : 64 days

64days 10m kd22 c10 ws33 h4 5.png

Test case 7

cell size: 10 m;
: 220 m²/day;
Water storage fraction: 0.25;
Aquifer KD configured to 55 m²/day in the Tygron Platform;
Bottom resistance : 20 days
Bottom head pressure : 4.1
Simulation days : 64 days

64days 10m kd220 c20 ws25 h4 1.png

Test case 8

cell size: 10 m;
: 220 m²/day;
Water storage fraction: 0.10;
Aquifer KD configured to 55 m²/day in the Tygron Platform;
Bottom resistance : 20 days
Bottom head pressure : 4.1
Simulation days : 64 days

64days 10m kd220 c20 ws10 h4 1.png

Test case 9

cell size: 5 m;
: 22 m²/day;
Water storage fraction: 0.10;
Aquifer KD configured to 2.2 m²/day in the Tygron Platform;
Bottom resistance : 20 days
Bottom head pressure : 4.1
Simulation days : 64 days

64days 5m kd22 c20 ws10 h4 1.png

Test case 10

cell size: 2 m;
: 22 m²/day;
Water storage fraction: 0.10;
Aquifer KD configured to 2.2 m²/day in the Tygron Platform;
Bottom resistance : 20 days
Bottom head pressure : 4.1
Simulation days : 64 days

64days 2m kd22 c20 ws10 h4 1.png

Notes

  • A higher KD value can lead to a lower stable ground water level than the head pressure.

References

  1. 1.0 1.1 1.2 J. Pankow en P.E. Rijtema, 1970. De resultaten van het waterbalansonderzoek in 1968 voor de objecten met een constant slootpeil in Hoenkoop. Nota 567. Instituut voor Cultuurtechniek en Waterhuishouding, Wageningen.
  2. Schatten van kwel (Pankow), Grondwaterformules.nl/index.php/formules/ontwatering/kwel-berekenen-pankow