Geotechnical Engineering

DHARM

180 GEOTECHNICAL ENGINEERING

Two simple cases will be dealt with in regard to the determination of the top flow line:

1. Discharge takes place into a horizontal filter inside the downstream toe.


2. Downstream slope of the dam forms in itself a medium for discharge and a horizontal
   filter is outside the downstream toe.

6.6.1 Top Flow Line for an Earth Dam with a Horizontal Filter

We have seen that the flow lines and equipotentials, analytically speaking, are based on conju-
gate functions; one of the simplest examples of conjugate functions is given by nests of confocal
parabolas, shown in Fig. 6.15 (b). A simple parabola is shown in Fig. 6.15(a). It is defined as the
curve, every point on which is equidistant from a point called the ‘focus’ and a line called the
‘directrix’. If a cross-section of an earth dam can be conceived of satisfy the boundary condi-
tions so that the flow lines and equipotentials conform to this parabolic shape, Fig. 6.15 (b)
gives a flow net for this dam (Kozeny, 1931).

Figure 6.15(c) shows an earth dam cross-section for which a flow net consisting of confocal
parabolas holds rigorously. In this case, BC and DF are flow lines, and BD and FC are
equipotentials. The upstream equipotential is the only unusual feature of this flow net.
Fig. 6.15(d) shows the common case of an earth dam with underdrainage and the correspond-
ing top flow line. The flow net for this will resemble parabolas but there will be departures at
the upstream side. There will be reverse curvature near B for a short distance. A.Casagrande
(1937) suggests that BA is approximately equal to 0.3 times BE where B is the starting point
of the Kozeny parabola at the upstream water level and E is on the upstream water level
vertically above the heel D of the dam.

Thus, the top flow line may be obtained by constructing the parabola with focus at F,
the starting point of the filter, and passing through A, as per Casagrande’s suggestion. The
short section of reversed curvature can be easily sketched by visual judgement.

The following are the steps in the graphical determination of the top flow line:
(i) Locate the point A, using BA = 0.3 (BE). A will be the starting point of the Kozeny
parabola.

S

x F

Focus

z

C G

Directrix

B E

F(permeable)

(a) Parabola (b) Conjugate confocal parabolas

(Kozeny, 1931)

A D

Fig. 6.15(Contd.)