4 - Cedergren (1989)

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2 _ J>e 90° ow lines wider at ces from et piling ures that t square ntersections angles e line all boundaries nes. (b) Look or equipoten- ), sketch first (e) Erase and and accuracy. 4.5 UNCONFINED FLOW SYSTEMS 133 noting that flow radiates around the bottoms of the cutoff walls and that the lines tend to be somewhat equally spaced under the centers of wide structures. In Figure 4.34 a trial family of equipotentials has been added to obtain a trial flow net, and several nonsquare figures have been crosshatched. The flow net obtained by progressive correction of errors is given in Figure 4.3e. 4.5 UNCONFINED FLOW SYSTEMS (Phreatic Line Unknown) In the examples given in Sec. 4.4 the line of saturation, or Phreatic line, is known in advance, which makes the procedure for obtaining the flow nets simpler than it is for cross sections in which the upper saturation line is not known. Cross sections with an unknown phreatic line are considered the most challenging because the phreatic line must be located simultaneously with the drawing of the flow net. The procedures described in this section permit flow nets of this kind to be constructed with minimum time and effort. Example 3 Seepage Through Homogeneous Earth Dam. The cross section and known conditions for a type 2a flow net are given in Figure 4.4a. Line AB, the face of the dam, is the maximum equipotential line; line AC, the base of the dam, is a flow line. The exact position of the phreatic line is unknown, but it can reasonably be expected to lie somewhere within the shaded zone BDE. The general condition at the free surface is known (Fig. 3.6) and it is known that the net must be composed of squares. Before starting to construct a flow net with an unknown phreatic line, the total head 4 should be divided into a convenient number of equal parts Ah, and light guidelines should be drawn across the region in which the phreatic line is expected to lie. In Figure 4.4a four intermediate guidelines (for conve- nience, called head lines in this section) are drawn at a vertical spacing Al = g/, The conditions that establish the position of the phreatic line in Fig. 4.4 are the following;: 1. Equal amounts of head must be consumed between adjacent pairs of equipotential lines. 2. Equipotential lines must intersect the phreatic line at the correct eleva- tion. 3. To satisfy requirements 1 and 2, each equipotential line must intersect the phreatic line at the appropriate head line. this key requirement must be satisfied by all flow nets having an unknown phreatic line. After drawing the horizontal guidelines or head lines across the region in which the phreatic line is expected to lie a trial saturation line (line ab, Fig. 4.4)) should be drawn; make a reasonable guess about its probable location

134 FLOW-NET CONSTRUCTION Water surface B = 907 E ,—"Head" line Ak Maximum o 3 equipotential 90° >Flow lines ‘L_Ah h 90° Equipotentials ‘ ,P-Ah A 90° ~\90° 90° D1 AR Flow line c (a) 5o Arrows show directions of needed corrections Number of full_/y 15 1.3 1.0 0.7 0.4 flow channels 1.2 1.2 1.2 1.2 1.2 (c) FIG. 4.4 Flow net for homogeneous dam on impervious foundation (Example 3) (@) Known conditions. (b) Trial saturation line and flow net. (¢) Final flow net. and shape. Next a family of equipotential lines should be drawn, making at intersections with flow lines right angles. Then draw one or more intermediats flow lines to establish a trial flow net as shown in Figure 4.4b. If all intersec- tions are right angles, the first trial net will be composed of rectangular figures Some of the figures may happen to be squares, but most will probably be elongated rectangles, as in this example. Usually it is possible by inspection alone to note the kinds of correctios that must be made and to develop accurate flow nets with an indeterminats phreatic line (e.g., Fig. 4.4c). At this point it is well to review a systematis checking method that takes the guesswork out of flow nets for unconfines seepage. This check is based on the rule: In any flow net the number of flow channels must remain the same throughout the net. This rule simply states tha all the water entering a cross section must flow through the section and emergs at the low potential side. It ag sources feeding water into them If a flow net has been corre squares, the rule given here is ot net in Figure 4.3e, which has tw ate equipotential lines. To satisf lines must encompass the same 1 figure that meets the requireme usually a true square, for it is fundamental requirement if its a ure 4.3e each pair of equipotent: basic rule is met. A flow net tha in Fig. 4.4b; another that does f Because the last two flow nets a possible to detect inaccuracies a done in two ways. 1. By inspection note the figus shift lines in directions that apy Figure 4.4b show initial adjustm be necessary. If this procedure is will become progressively more sguares and has the same numbse you are not confident enough to use the second method. 2. Use an engineer’s scale to : a trial flow net and calculate th squipotentials. A figure that ha flow channel; one that has a wi channel, and so on. Thus in Fign of ef to cd is 1.0; hence one full fractional space below this squa therefore, the number of flow ct + 0.3, or 1.3. This number is wr similar procedure of measuring fi channels is carried out for the bs 4.4p. If the numbers written dow of balance and must be redraws irial flow net (Fig 4.4b) these nu adjustments indicated by the an raise the phreatic line above the t “squares’’ above a given point a: too low above that point; if the | above the point. To correct a trial flow net tha

e 1Ak T TR Y h -—.fi!_Ah A F_Ah D Fax C E. R 0 C 1.2 L2 ation (Example 3 nal flow net. awn, making o ore intermedia b. If all interses tangular figurss vill probably & ds of correctis n indeterminas EW a systemans for unconfins number of flow mply states tha ion and emergs 4.5 UNCONFINED FLOW SYSTEMS 135 % e low potential side. It applies to all sections that have no secondary Wweces feeding water into them or drains removing water from them. " 2 flow net has been correctly constructed and is composed entirely of W= squipotential lines. To satisfy this rule each adjacent pair of equipotential W& must encompass the same number of ““square” figures, a square being a ‘wure that meets the requirement described in Sec. 3.3. A “‘square’’ is not wually a true square, for it is enclosed in curved lines, but jr satisfies the wadamental requirement if its average width equals its average length. In Fig- Secause the last two flow nets are somewhat similar in appearance, how is it possible to detect inaccuracies and make the necessary corrections? It can be L. By inspection note the figures in a trial flow net that are not squares and Wit lines in directions that appear reasonable; for example, the arrows in 2. Use an engineer’s scale to measure the widths and lengths of figures in 2 trial flow net and calculate the number of flow channels between pairs of fractional space below this square has a width-to-length ratio of about 0.3; therefore, the number of flow channels between equipotentials 3 and 4 is 1.0 + 0.3, or 1.3. This number js written below the base of the dam as shown. A similar procedure of measuring figures and recording the total number of flow adjustments indicated by the arrows is now apparent. This adjustment will raise the phreatic line above the trial position in Figure 4.4p. Generally, if the “squares’’ above a given point are elongated horizontally, the phreatic line is foo low above that point; if the figures are elongated vertically, it is too high above the point. To correct a trial flow net that has an indeterminate phreatic line the line

136 FLOW-NET CONSTRUCTION should be raised if the first guess is too low or lowered if the first guess is too high. This correction automatically forces the equipotential lines to be moved in the correct direction. After the phreatic line is raised or lowered, a new family of equipotentials should be drawn and the new flow net, examined. If the figures are not all squares and the calculated number of flow channels is not the same throughout, the procedure should be repeated until a satisfactory flow net is obtained (Fig. 4.4c). With experience accurate flow nets can be developed with the help of visual inspection alone. If any doubt exists, the lengths and widths of the figures should be measured with an engineer’s scale and the number of flow channels calculated at several places in the section, as described in the preceding paragraph. If this check is properly made, accuracy is guaranteed. As a flow net is improved and all figures become true squares, the phreatic line will be forced into its correct position. In developing this kind of flow net we are solving simultaneously for the flow net and the position of the phreatic line. When seepage emerges along a sloping surface, such as the downstream slope of a dam (Fig. 4.4¢), some of the water flows down the slope and a portion of the flow net is thereby cut off. When counting the number of flow channels in the extremities of flow nets, cut-off portions should be included in the totals. Thus in Figure 4.4c¢ the finished flow net has 1.2 flow channels throughout, even though a portion of the right end of the flow net is cut off. If line CF in Figure 4.4c were a boundary between zones of two different permeabilities &, to k», the shape of the flow net would depend on the ratio of k; to k,. If the second zone is considerably more permeable than the first (k,/ ki = 1000 or more), the influence of the second zone on the flow net can usually be ignored. If the ratio k,/k, is in a moderate range, several hundred or less, the method described in Sec. 4.6 for composite sections should be used in developing the flow net. 4.6 COMPOSITE SECTIONS WITH PHREATIC LINE UNKNOWN In this section a method of checking the accuracy of flow nets for composite sections with unknown phreatic lines is described and examples of shortcut methods are given. Example 4 Zoned Earth Dam. Figure 4.5 illustrates a basic procedure for drawing composite flow nets for sections with unknown phreatic lines. The transfer conditions at boundaries between soils of different permeability mate- rials were described in Sec. 3.3. When water flows across a boundary into a soil of different permeability, the figures in the second soil must elongate or shorten to make the length-to-width ratios of the figures satisfy Eq. 3.11: = — (3.11) 4.6 COMPOSITE SE Water surface —— S — — Zone 1 FIG. 4.5 Method for constructing flow First trial flow net (not correct). (b) Com In Eq. 3.11 c is the length and d, th the permeability of the first soil is &, Figure 4.5 shows a common type o minate phreatic line. It is for a zone permeability in the upstream part th is a type 2b (Sec. 4.3) because it is for In this example the downstream zone as the upstream zone. To develop a flow net for this type - 1. Locate the reservoir level and t in head as /# and dividing /4 ints increments Ak, Draw a series o at intervals of Ak across the do 2. Guess a trial position for the ; preliminary flow net as shown i and rectangles in zone 2. Maks rectangles in zone 2 approxima saturation line, using an engin widths of the figures as in Exam pleted, the trial flow net shoul satisfy the basic shape requires width ratio of the shapes in zo

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