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 Introduction
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This example shows a homogeneous, isotropic and confined aquifer with a steady state flow and a groundwater withdrawal at a well. We are looking for the lowering of the groundwater surface caused by the withdrawal. For this problem analytical solutions can be found in literature.
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 Model and Parameters
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The following figure represents the principle of a lowered confined aquifer caused by a withdrawal at a well.
Model principle
The example should have the following characteristics:
| Permeability coefficient |
kf = 5*10-4 m/s |
KWER |
| Thickness |
m = 10 m |
MAEC = 10 |
| Area |
upper edge = 45 m
lower edge = 0,01m |
GELA = 45
UNTE = 0,01 |
| Reach |
R = 500 m |
Expansion of the model |
| Well radius |
r0 = 0,3 m |
- |
| Initial potential heads |
H0 = 40,00 m |
POTE = 40
(Model boundary) |
| Withdrawal rate |
Q = -600000 m3/s |
KNOT = -600000 |
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Theoretical Background
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A partial Laplace differential equation is derived from the potential function and the flow function to get the analytical solution of a 2D flow. The derivable function of a complex variable for this problem is:
With the potential- and flow function
circle potential lines and linear streamlines result. With the boundary conditions h(R) = H0 and h(r0) = HB the withdrawal and the potential heads can be determined. The well efficiency is determined with the given boundary conditions
to
 .
The potential head is determined with
 .
This is the well formulation for steady state flow by Thiem. If the withdrawal is preset and the adjusting altitude in the well should be calculated this is done by converting the formulation
to
With the information of the chapter "Model and parameters" concerning the characteristics of the aquifer the altitude in the well is determined
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Discretisation
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In the file Brunnen_ge_s.zip includes the Brunnen_gesp_stat.net. With this net file the validation was done. A horizontal model with the time unit "year" was created. The initial parameters are described in the table in the chapter "Model and parameters". The generated mesh is shown in the following figure.
FE-Model mesh in SPRING
The following figure represents the calculated potential heads (red lines) and the nodes to which the attribute POTE was assigned (blue points).
With SPRING calculated potential heads
Visualization of the calculated steady-state velocitiy field with the help of streamlets (blue).
STRING flow visualization
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Comparison of the results
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The following figure shows the results of SPRING arranged face to face to the analytical solution. The analytical solution is computed with the file Brunnen_gesp_stat.xls. This file is also saved in the file brunnen_ge_s.zip. You can see clearly that the solution of SPRING does not vary much. The maximum difference between the two calculated potential heads is 0,09m. If the mesh is refined around the well the difference will be smaller.
Comparison of the analytical solution with the results of SPRING |
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Literature
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[David] Ioan David; Grundwasserhydraulik Strömungs- und Transportvorgänge, Vieweg, 1997
[Kinzelbach] W. Kinzelbach; Numerische Methoden zur Modellierung des Transports von Schadstoffen im Grundwasser, Oldenbourg, 1987
[Kinzelbach] W. Kinzelbach und Rausch, R., Grundwassermodellierung Eine Einführung mit Übungen, Gebrüder Bornträger, 1995
[SPRING] SPRING; Simulation of Processes in Groundwater, Programm- beschreibung,Version 3.2
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