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 Introduction
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This example shows a homogeneous, isotropic and confined aquifer with a unsteady 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 an analytical solution by Theis 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 = 725 m |
Expansion of the model |
| Well radius |
r0 = 0,3 m |
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| Initial potential head |
H0 = 40,00 m |
POTE = 40 (Rand)
EICH = 40 |
| Withdrawal rate |
Q = -600000 m3/s |
KNOT = -600000 |
| Storage coefficient |
S = 0,2 |
SPEI = 0,2 |
The period of time is 100 days. The computation is done for 200 time steps with a time step width of 0,5 days.
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Theoretical Background
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The analytical solution by Theis makes it possible to calculate unsteady flow conditions. The derivation based on the following conditions. The aquifer has to be confined, homogeneous and isotrop. The expansion of the
aquifer is assumed to be unlimited. Also the aquifer has a constant thickness and horizontal potential heads.
The withdrawal rate must be constant and the storage coefficient of the well is neglected. The following equation must be solved:
The initial conditions h(0,r) = h0 are set for all r. For the boundary conditions is valid:
 for t > 0
 for all t.
From this the analytical solution can be derivated to:
which means:
So you get:
In practice the following approximation is used:
Here an error ε is tolerated from a point in time t. The error comes to:
 ε < 5%
 ε < 2%
 ε < 1%
Thus the course of the function for the approximation is:
A comparison of the two results shows that the maximum difference is 0.16 m in a distance of 725 m from the well.
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Discretisation
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The file Brunnen_ge_is.zip includes the Brunnen_gesp_instat.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
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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_instat_e.xls. This file is also saved in the brunnen_ge_is.zip . You can see clearly that the solution of SPRING does not vary much.
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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