Introduction

This example shows a homogeneous and isotropic 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.

Model and parameters

The following figure represents the principle of a lowered aquifer caused by a withdrawal at a well.

The example should have the following characteristics:

Theoretical background

A partial Laplace differential equation is derived from the potential function and the flow function to get the analytic 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. The boundary conditions are given with h(R) = H₀ and h(r₀) = H_B. The productiveness of the well is calculated with the formulation

and with one of the boundary conditions to

If the withdrawal is preset and the adjusting altitude in the well should be calculated this is done by converting the function for H_B given above

The potential head is determined with

and the boundary condition h(r₀) = H_B. For this the equation given above is converted to h so that

results. In this equation the boundary condition h(r₀) = H_B is entered and the constant c can be determined.

Entering the constant c into h(r) results in this:

This equation can be summarized by transformation:

With the boundary condition of the model area mentioned above and the equation for the reach of the well by Sichardt the altitude in the well can be calculated.

From this

(see description concerning the model parameters mentioned above)
The productiveness of the well is determined to:

Discretisation

The file Brunnen_f_s_e.zip includes the file "Brunnen_frei_stat.net". With this mesh 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.

The next figure shows the calculated potential heads (red lines) and the nodes to which the attribute POTE was assigned(blue points).

Comparison of the calculation results

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_frei_stat_e.xls. This file is also saved in the file brunnen_f_s_e.zip. You can see clearly that the solution of SPRING does not vary much. The maximum difference between the two calculated potential heads is 0,05m. The productiveness which is calculated in SPRING varies about -0,00241 m³/s. This is an amount which can be neglected. If the grid is refined around the well the difference will be smaller.

Literature

[David] Ioan David; Grundwasserhydraulik Strömungs- und Transportvorgänge, Vieweg, 1997
[Kinzelbach] W. Kinzelbach; Numerische Methoden zur Modellierung des Transports von Schadstoffen im Grundwasser, Oldenbourg, 1987
[SPRING] SPRING; Simulation of Processes in Groundwater, Programm- beschreibung,Version 3.2