GGU-3D-SSFLOW: Differential equation

The application solves the differential equation:

Where

k_x, k_y, k_z = permeability in m/s, for example, for the x, y and z directions

h = piezometric head in m, for example

k_r = coefficient for determining the permeability in the unsaturated zone
(dimensionless)

Q = discharge in m³/s, for example

x, y, z = coordinates in m, for example

The value k_r records the change in permeability in the unsaturated zone r, above groundwater, and can take on values between 0.0 and 1.0. In the saturated area of the system kr is 1.0. The value k_r is a function of the pore water pressure u. The pore water pressure u is calculated from the piezometric head h, the elevation head z and the unit weight of water .

u = (h - z) ·

In Figure 4, the curves are presented for three typical soils. Using this value in the differential equation has the distinct advantage, for the following finite-element calculation, that phreatic lines can be very easily calculated.

Figure 5 serves as a further explanation. For the case of a horizontal groundwater table, all four variables h, y, u and kr are given as functions.

If unsaturated zones are present in the system, and a phreatic line analysis is to be carried out, an iteration process is required, as the position of the phreatic line is not known from the outset. In the first iteration step, the program assumes that all areas of the system are saturated:

kr = 1.0

After the initial calculation of the piezometric heads at the system nodes, the pore water pressures u are calculated and kr determined. With the altered permeabilities, a new calculation of the system follows, with new piezometric heads and correspondingly new pore water pressures and kr values. The iteration process is carried on until the difference in piezometric head in iteration step (i) and (i - 1) falls below the user-defined limit value (iteration divergence).

Note on boundary conditions:
The case of an impermeable boundary is automatically taken in to account by the finite-element-method. Valid is, that all system boundaries or partial system boundaries which have no water level or source boundary conditions, are automatically impermeable.