Permeability and Flow of Water in Soils - Elements of Soil Mechanics - page 61

Fig. 2.22 The parabola.
Fig. 2.23 Determination of upper flow line.
upstream end, actual dams do not differ substantially from this imaginary example, so that the flow net
for the middle and downstream portions of the dam are similar to the theoretical parabolas (a parabola
is a curve, such that any point along it is equidistant from both a fixed point, called the focus, and a fixed
straight line, called the directrix). In Fig. 2.22, FC = CB.)
The graphical method for determining the phreatic surface in an earth dam was evolved by Casagrande
( 1937) and involves the drawing of an actual parabola and then the correction of the upstream end.
Casagrande showed that this parabola should start at the point C of Fig. 2.23 (which depicts a cross-
section of a typical earth dam) where AC ≈ 0.3AB (the focus, F, is the upstream edge of the filter). To
determine the directrix, draw, with compasses, the arc of the circle as shown, using centre C and radius
CF; the vertical tangent to this arc is the directrix, DE. The parabola passing through C, with focus F and
directrix DE, can now be constructed. Two points that are easy to establish are G and H, as FG = GD and
FH = FD; other points can quickly be obtained using compasses. Having completed the parabola a cor-
rection is made as shown to its upstream end so that the flow line actually starts from A.
This graphical solution is only applicable to a dam resting on a permeable material. When the dam is
sitting on impermeable soil the phreatic surface cuts the downstream slope at a distance (a) up the
slope from the toe (Fig. 2.19a ). The focus, F, is the toe of the dam, and the procedure is now to establish
point C as before and draw the theoretical parabola (Fig. 2.24a ). This theoretical parabola will actually
cut the downstream face at a distance Δ a above the actual phreatic surface; Casagrande established a