Radial well freatic benchmark (Water Module)

This testcase demonstrates a situation where a well is extracting ground water in a confined freatic layer. There is no aquifer present. A characteristic ground water level curve will form over time.

Formulas

The lowering of the ground water table in a closed freatic layer, without additional rainfall, can be described by the following formula

${h(r)}^{2}={h_{0}}^{2}+{\frac {Q_{0}}{\pi \cdot k}}\ln {\left({\frac {r}{R}}\right)}$

where:

$h_{0}$: stable water level at the considered stable water table edge

$h(r)$: water level between the considered stable water table edge and the well

$k$: hydraulic conductivity of the freatic layer in m / day

$r$: distance to the well

$R$: distance of the considered stable water table edge to the well

$Q_{0}$: 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 $dx$ 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 with a negative inlet.

INLET Q is set to ${\frac {-Q_{0}}{3600*24}}$

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 set to 0, such that is unlimited.

UNDERGROUND is set to true (1.0) to place the outlet below the surface.

UPPER_THRESHOLD set to -2 m.

LOWER_THRESHOLD set to -2 m.

The ground bottom distance is configured as 10 meters, which places the bottom at -10 meters (datum).

The water storage fraction is set to 0.25.
The vertical infiltration is set to 0.05 m / day.
The hydraulic conductivity is set to k m / day.

The simulation is run for 64 days with 0 rainfall, which is configured in the weather's rain attribute as: $[64\cdot 24\cdot 3600,0]$

Test results

Important in all these tests in 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 impenetrable soil, which is situated 10 meters below the surface.

Test case 1

cell size: 5 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 50;

Simulation days $n$: 64 days;

Test case 2

cell size: 5 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 50;

Simulation days $n$: 64 days;

Test case 3

cell size: 5 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 50;

Simulation days $n$: 64 days;

Test case 4

cell size: 5 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 50;

Simulation days $n$: 64 days;

Test case 5

cell size: 5 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 25;

Simulation days $n$: 64 days;

Test case 6

cell size: 5 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 100;

Simulation days $n$: 64 days;

Test case 7

cell size: 2 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 1;

Simulation days $n$: 64 days;

Test case 8

cell size: 2 m;

$k$: m/day;

$R$: 110;

$Q_{0}$: 16;

Simulation days $n$: 64 days;

Test case 9

cell size: 2 m;

$kD$: m/day;

$R$: 110;

$Q_{0}$: 4;

Simulation days $n$: 64 days;

Test case 10

cell size: 2 m;

$k$: m/day;

$R$: 110;

$Q_{0}$: 4;

Simulation days $n$: 64 days;

Test case 11

cell size: 2 m;

$k$: m/day;

$R$: 110;

$Q_{0}$: 4;

Simulation days $n$: 64 days;

References

Water Module

Tutorials

Overview

Theory

Benchmarks