Smith's Elements of Soil Mechanics. Ian Smith

      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, k_x, than that in the vertical direction (the anisotropic condition). The value of k_z is usually 1/5 to 1/10 the value of k_x.

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)

      Unless k_x is equal to k_z the equation is not a true Laplacian and cannot therefore be solved by a flow net. To obtain a graphical solution, the equation must be written in the form (i.e. divide through by k_z):

      or

      (2.29)

      where

      or

      i.e.

      (2.30)

This equation is Laplacian and involves the two coordinate variables x_t and z. It can be solved by a flow net provided that the net is drawn to a vertical scale of z and a horizontal scale of:

      (2.31)

      2.15.2 Calculation of seepage quantities in an anisotropic soil

      This is exactly as before:

      (2.32)

      and the only problem is what value to use for k.

Using the transformed scale, a square flow net is drawn, and N_f and N_d are obtained. If we consider a ‘square’ in the transformed flow net, it will appear as shown in Fig. 2.26a. The same figure, drawn to natural scales (i.e. scale x = scale z), will appear as shown in Fig. 2.26b.

      Let k′ be the effective permeability for the anisotropic condition. Then k′ is the operative permeability in Fig. 2.26a.

      Hence, in Fig. 2.26a:

Fig.