View source for Radial well aquifer benchmark (Water Module)

This [[Water_Module_benchmarks|benchmark]] demonstrates a situation where a well is extracting ground water. A characteristic ground water level curve will form over time.

[[File:drainage_freatic_benchmark.gif]]
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===Formulas=== 
Stationary lowering of the ground water table in a closed water transmissive layer can be described by the following formula <ref name="verruijt70"/> 

<math>\phi(r) - \phi_0 = \frac {Q_0}{2\pi\cdot kD} \ln{\left ( \frac{r}{R} \right )}</math>

where:
: <math>phi_0</math>: stable water level at the considered stable water table edge
: <math>phi(r)</math>: water level between the considered stable water table edge and the well
: <math>kD</math>: transmissivity of the aquifer in m² / day
: <math>r</math>: distance to the well
: <math>R</math>: distance of the considered stable water table edge to the well
: <math>Q_0</math>: amount of water pumped out in m³ / day

===Setup===
We use the following setup in our tests. The grid size used is 51 by 51, with a configurable cell size of <math>dx</math> in meters. There is one underground outlet, which pumps water away continuously with a default amount per second.

The terrain height is set to 0 meters (datum).

The outlet is placed on the cells x = 25 and y = 25 as an [[Inlet (Water Overlay)|inlet]] with a negative inlet.

:[[Inlet q (Water Overlay)|INLET Q]] is set to <math>\frac{-Q_0}{3600*24}</math>
:[[Inlet underground (Water Overlay)|UNDERGROUND]] is set to true (1.0) to place the outlet below the surface.

To stabilized the water levels on the edges of the test case , an additional underground inlet is used. It is located on all cells equal to or further away than the chosen R.
This inlet is configured as followed:
:[[Inlet q (Water Overlay)|Inlet Q]] set to 0, such that is unlimited.
:[[Inlet underground (Water Overlay)|UNDERGROUND]] is set to true (1.0) to place the outlet below the surface.
:[[Inlet_upper_threshold_(Water_Overlay)|UPPER_THRESHOLD]] set to -2 m.
:[[Inlet_lower_threshold_(Water_Overlay)|LOWER_THRESHOLD]] set to -2 m.

The [[Ground bottom distance m (Water Overlay)|ground bottom distance]] is configured as 10 meters, which places the bottom at -10 meters (datum). 

The [[Terrain water storage percentage (Water Overlay)|water storage fraction]] is set to 0.25.

An [[Aquifer (Water Overlay)|aquifer]] can be added to configure the [[Aquifer kd (Water Overlay)|kD]] value used in the test.

The simulation is run for 64 days with 0 rainfall, which is configured in the weather's [[Weather rain m (Water Overlay)|rain attribute]] as: <math>[64 \cdot 24 \cdot 3600, 0]</math>

===Test results===
Important in all these tests is the chosen R, which is the distance to the well that is considered a stable water level. Additionally, the measurements are done relative to the [[Bottom distance prequel (Water Overlay)|bottom boundary]], which is situated 10 meters below the surface.

====Test case 1====
:cell size: 5 m;
:<math>kD</math>: 88 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 50;
:Simulation days <math>n</math>: 64 days;
[[File:5m_50m3_88kd.png]]

====Test case 2====
:cell size: 5 m;
:<math>kD</math>: 44 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 50;
:Simulation days <math>n</math>: 64 days;
[[File:5m_50m3_44kd.png]]

====Test case 3====
:cell size: 5 m;
:<math>kD</math>: 22 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 50;
:Simulation days <math>n</math>: 64 days;
[[File:5m_50m3_22kd.png]]

====Test case 4====
:cell size: 5 m;
:<math>kD</math>: 11 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 50;
:Simulation days <math>n</math>: 64 days;
[[File:5m_50m3_11kd.png]]

====Test case 5====
:cell size: 5 m;
:<math>kD</math>: 22 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 25;
:Simulation days <math>n</math>: 64 days;
[[File:5m_25m3_22kd.png]]

====Test case 6====
:cell size: 5 m;
:<math>kD</math>: 22 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 100;
:Simulation days <math>n</math>: 64 days;
[[File:5m_100m3_22kd.png]]

====Test case 7====
:cell size: 2 m;
:<math>kD</math>: 22 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 1;
:Simulation days <math>n</math>: 64 days;
[[File:2m_1m3_22kd.png]]

====Test case 8====
:cell size: 2 m;
:<math>kD</math>: 22 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 16;
:Simulation days <math>n</math>: 64 days;
[[File:2m_16m3_22kd.png]]

====Test case 9====
:cell size: 2 m;
:<math>kD</math>: 22 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 4;
:Simulation days <math>n</math>: 64 days;
[[File:2m_4m3_22kd.png]]

====Test case 10====
:cell size: 2 m;
:<math>kD</math>: 44 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 4;
:Simulation days <math>n</math>: 64 days;
[[File:2m_4m3_44kd.png]]

====Test case 11====
:cell size: 2 m;
:<math>kD</math>: 88 m²/day;
:<math>R</math>: 110;
:<math>Q_0</math>: 4;
:Simulation days <math>n</math>: 64 days;
[[File:2m_4m3_88kd.png]]

===References===
<references>
<ref name="verruijt70>Verruijt, A. (1970). Theory of Groundwater Flow. Macmillan, London.</ref>
</references>

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