2.20 Conditions at the downstream end of an upper flow line. (a) Flow line tangential to downstream slope. (b) Flow line vertical at exit.
The second condition is that, as the upstream face of the dam is an equipotential, the flow line must start at right angles to it (see Fig. 2.19a), but an exception to this rule is illustrated in Fig. 2.19b where the coarse material is so permeable that the resistance to flow is negligible and the upstream equipotential is, in effect, the downstream face of the coarse material. The top flow line cannot be normal to this surface as water with elevation head only cannot flow upwards, so that in this case the flow line starts horizontally.
The third condition concerns the downstream end of the flow line where the water tends to follow the direction of gravity and the flow line either exits at a tangent to the downstream face of the dam (Fig. 2.20a) or, if a filter of coarse material is inserted, takes up a vertical direction in its exit into the filter (Fig. 2.20b).
2.14.2 Types of flow occurring in an earth dam
From Fig. 2.20, it is seen that an earth dam may be subjected to two types of seepage: when the dam rests on an impermeable base, the discharge must occur on the surface of the downstream slope (the upper flow line for this case is shown in Fig. 2.21a), whereas when the dam sits on a base that is permeable at its downstream end, the discharge will occur within the dam (Fig. 2.21b). This is known as the underdrainage case. From a stability point of view, underdrainage is more satisfactory since there is less chance of erosion at the downstream face and the slope can therefore be steeper but, on the other hand, seepage loss is smaller in dams resting on impermeable bases.
Fig. 2.21 Types of seepage through an earth dam. (a) Impermeable base. (b) Base permeable at down‐stream end.
Fig. 2.22 Flow net for a theoretical earth dam.
2.14.3 Parabolic solutions for seepage through an earth dam
In Fig. 2.22 is shown the cross‐section of a theoretical earth dam, the flow net of which consists of two sets of parabolas. The flow lines all have the same focus, F, as do the equipotential lines. Apart from the 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. 2.23, 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. 2.24 (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 correction 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. 2.20a). 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. 2.25a). This theoretical parabola will actually cut the downstream face at a distance Δa above the actual phreatic surface; Casagrande established a relationship between a and Δa in terms of α, the angle of the downstream slope (Fig. 2.25b). In Fig. 2.25, the point J can thus be established, and the corrected flow line sketched in as shown.
Fig. 2.23 The parabola.
Fig. 2.24 Determination of upper flow line.
Fig. 2.25 Dam resting on an impermeable soil. (a) Construction for upper flow line. (b) Relationship between a and ∆a (after Casagrande).
2.15 Seepage through non‐uniform soil deposits
2.15.1 Stratification in compacted soils
Most loosely tipped deposits are probably isotropic, i.e. the value of permeability in the horizontal direction is the same as in the vertical direction. However, in the construction of embankments, spoil heaps, and dams, soil is placed and spread in loose layers which are then compacted. This construction technique results in a greater value of permeability in the horizontal direction, kx, than that in the vertical direction (the anisotropic condition). The value of kz is usually 1/5 to 1/10 the value of kx.
The general differential equation for flow was derived earlier in this chapter (Equation 2.16):
For the two‐dimensional, i.e. anisotropic case, the equation becomes:
(2.28)
