Impermeable
Fig. 2.7 Flow net for seepage beneath a dam.
Impermeable
maximum along a path normal to the equipotentials and in isotropic soil the flow follows the paths of the steepest gradients, so that flow lines cross equipotential lines at right angles.
Figure 2.7 shows a typical flow net representing seepage through a soil beneath a dam. The flow is assumed to be two dimensional, a condition that covers a large number of seepage problems encountered in practice.
From Darcy's law q = Aki, so if we consider unit width of soil and if Aq = the unit flow through a flow channel (the space between adjacent flow lines), then:
Aq = b x 1 x k x i = bki where b = distance between the two flow lines.
In Fig. 2.7 the figure ABCD is bounded by the same flow lines as figure AiBiCiD, and by the same equipotentials as figure A2B2C2D2. For any figure in the net Aq = kib = kAhb/1, where
Ah = head loss between the two equipotentials 1 = distance between the equipotentials (see Fig. 2.8),
Referring to Fig. 2.7:
Row lines
Fig. 2.8 Section of a flow net.
li b2
Flow through ABCD = Aq = kAhy
If we assume that the soil is homogeneous and isotropic then k is the same for all figures and it is possible to draw the flow net so that bi = li, b2 = 12, b = 1. When we have this arrangement the figures are termed 'squares' and the flow net is a square flow net. With this condition:
Since square ABCD has the same flow lines as A1B1C1D1, Aq = Aqi
Since square ABCD has the same equipotentials as A2B2C2D2,
Hence, in a flow net, where all the figures are square, there is the same quantity of unit flow through each figure and there is the same head drop across each figure.
No figure in a flow net can be truly square, but the vast majority of the figures do approximate to squares in that the four corners of the figure are at right angles and the distance between the flow lines, b, equals the distance between the equipotentials, 1. As will be seen, a little imagination is sometimes needed when asserting that a certain figure is a square and some figures are definitely triangular in shape, but provided the flow net is drawn with a sensible number of flow channels (generally five or six) the results obtained will be within the range of accuracy possible. The more flow channels that are drawn the more the figures will approximate to true squares, but the apparent increase in accuracy is misleading and the extra work involved in drawing perhaps twelve channels is not worthwhile.
Note: Several computer programs for generating flow nets are widely available and these greatly speed up the task. Nevertheless, the method for drawing a flow net by hand is given in Section 2.13 for readers interested in learning the techniques involved.
2.12 Calculation of seepage quantities from a flow net
Nd = number of potential drops Nf = number of flow channels h = total head loss q = total quantity of unit flow.
Then
Nd Nf
.'. Total unit flow per unit length (q) — kh —-
2.13 Drawing a flow net
A soft pencil, a rubber and a pair of dividers or compasses are necessary. The first step is to draw in one flow line, upon the accuracy of which the final correctness of the flow net depends. There are various boundary conditions that help to position this first flow line, including:
(i) Buried surfaces (e.g. the base of the dam, sheet piling), which are flow lines as water cannot penetrate into such surfaces.
(ii) The junction between a permeable and an impermeable material, which is also a flow line; for flow net purposes a soil that has a permeability of one-tenth or less the permeability of the other may be regarded as impermeable.
(iii) The horizontal ground surfaces on each side of the dam, which are equipotential lines.
The procedure is as follows
(a) Draw the first flow line and hence establish the first flow channel.
(b) Divide the first flow channel into squares. At first the use of compasses is necessary to check that in each figure b = 1, but after some practice this sketching procedure can be done by eye.
(c) Project the equipotentials beyond the first flow channel, which gives an indication of the size of the squares in the next flow channel.
(d) With compasses determine the position of the next flow line; draw this line as a smooth curve and complete the squares in the flow channel formed.
(e) Project the equipotentials and repeat the procedure until the flow net is completed.
As an example, suppose that it is necessary to draw the flow net for the conditions shown in Fig. 2.9a. The boundary conditions for this problem are shown in Fig. 2.9b, and the sketching procedure for the flow net is illustrated in Figs c, d, e and f of Fig. 2.9.
If the flow net is correct the following conditions will apply.
(i) Equipotentials will be at right angles to buried surfaces and the surface of the impermeable layer.
(ii) Beneath the dam the outermost flow line will be parallel to the surface of the impermeable layer.
After completing part of a flow net it is usually possible to tell whether or not the final diagram will be correct. The curvature of the flow lines and the direction of the equipotentials indicate if there is any distortion, which tends to be magnified as more of the flow net is drawn and gives a good indication of what was wrong with the first flow line. This line must now be redrawn in its corrected position and the procedure repeated again, amending the first flow line if necessary, until a satisfactory net is obtained.
Generally the number of flow channels, Nf will not be a whole number, and in these cases an estimate is made as to where the next flow line would be if the impermeable layer was lower. The width of the lowest channel can then be found (in Fig. 2.9f, Nr = 3.3).
Note In flow net problems we assume that the permeability of the soil is uniform throughout the soil's thickness. This is a considerable assumption and we see therefore that refinement in the construction of a flow net is unnecessary, since the difference between a roughly sketched net and an accurate one is small compared with the actual flow pattern in the soil and the theoretical pattern assumed.
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