4 - Cedergren (1989). Unconfined flow

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

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

> first guess is toe lines to be moves r lowered, a new net, examined. If ' flow channels & ntil a satisfactorsy flow nets can be doubt exists, the N engineer’s scale in the section, as y made, accuracy me true squares. eloping this king d the position of the downstream the slope and =2 ' number of flow ould be includeg .2 flow channels )W net is cut off. of two differems d on the ratio of 1an the first (&, he flow net cam several hundreg s should be useg NKNOWN s for composite ples of shortcu ¢ procedure for eatic lines. The ‘meability mate- youndary into 2 st elongate or fy Eq. 3.11: (3.11) 4.6 COMPOSITE SECTIONS WITH PHREATIC LINE UNKNOWN 137 ko =Sk "Head” lines Water surface /—r\[ 4 ! / Zone 1 u;\_g =dfc=05 (%) FIG. 4.5 Method for constructing flow nets for composite sections (Example 4). (a) First trial flow net (not correct). (b) Completed flow net, In Eq. 3.11 c is the length and d, the width of the figures in the second soil; the permeability of the first soil is ki, that of the second k. Figure 4.5 shows a common type of composite cross section with an indeter- minate phreatic line. It is for a zoned earth dam with a soil of slightly lower permeability in the upstream part than in the downstream part. This section is a type 2b (Sec. 4.3) because it is for unconfined flow in a composite section. In this example the downstream zone is assumed to be five times as permeable as the upstream zone. To develop a flow net for this type of section these steps should be followed. 1. Locate the reservoir level and the 1ail water level, noting the difference in head as 4 and dividing 4 into a convenient number of equal parts or increments Ah. Draw a series of light horizontal guide lines (head lines) at intervals of Ah across the downstream part of the section (Fig. 4.5q). 2. Guess a trial position for the phreatic line in both zones and draw a preliminary flow net as shown in Figure 4.5a, making squares in zone 1 and rectangles in zone 2. Make the length-to-width ratios of all of the rectangles in zone 2 approximately equal by adjusting the shape of the saturation line, using an engineer’s scale to measure the lengths and widths of the figures as in Example 3, Figure 4.4. When this step is com- pleted, the trial flow net should be reasonably well drawn. It should satisfy the basic shape requirements of a flow net, but the length-to- width ratio of the shapes in zone 2 probably will not satisfy Eq. 3.11.

ction, the ratio lly assumed for 1st constructed. ) adjacent equi- e trial flow net ) adjacent equi- “igure 4.5a n,_; one 2, d/c and e 4.5a can now 4.1) ine in zone 2 is ) is too low, the vered. Raise or s indicated and red accuracy i : (Fig. 4.54a) k; le was assumed he general leve For the secong ‘ulated ratio of y of seepage = (3.18 4.6 COMPOSITE SECTIONS WITH PHREATIC LINE UNKNOWN 139 Bzone |, g = kih(ns—,/ng), and in zone 2 = k>h(ns-,/n,). For a given head g~ k.(nf_./nd) -~ kz(nf..z/nd) and Q/I‘ld"" .k,n,-. ~ kznf..z. Therefore k]flf_|_ = Wz and k; = ki(ns-\/n;_,). This expression (Eq. 4.1) can be used for deter- mming the permeability ratios k:/ky, ki/k,, and so on, for any composite flow »= being examined for accuracy. It is an essential criterion to be used in con- mucting accurate flow nets for composite sections. The fundamental relationships represented in Egs. 3.11 and 4.1 should be =2t in mind when flow nets for composite sections are being studied or con- Iacted. More than one system of lines can be selected for a flow net. This point is “astrated by Figure 4.6 which gives three sets of lines for the flow net devel- ped in Figure 4.5, Although the three nets appear to be different, they are w=ually identical. The total head 4 has been divided into eight parts equal to “%. As drawn in Figure 4.6a, the portion in zone 1 has been constructed with wid lines that form Squares, the portion in zone 2 with elongated rectangles Water surface —_—= Ah=h/8 nyg=8 ng.; =35 nfeg =bfa =07 h kg = 5k ‘a - ky Ib 4 (b) Ah = h/40; ng = 40 = ni.g =35 ne_y =53.5)=175 hy = 51 (makes squares < s 2 1 in zone 2) h ~ k i I1 s (c) FIG. 4.6 Three forms of one flow net.

140 FLOW-NET CONSTRUCTION (solid lines) with length-to-width ratios of 5. The dashed lines in zone 2 divide the elongated rectangles into squares. For this flow net n,_, = 3.5, n;_, = 0.7, and n, = 8. The seepage quantity g can be computed by using &, or k,. Using ki, g = kih(ne_1/ng) = k\h(3.5/8) = 0.44k,h. Using k,, g = kh(ns-o/ng) = k2h(0.7/8) = 0.088k,h, but k, = 5k, and g = 0.088(5k\)h = 0.44k,h. The line system in Figure 4.6q¢ is typical of that frequently used for this type of flow net. The dashed lines may be omitted unless they are needed in stability studies or to bring out detail. Sometimes all intermediate lines in one or more zones may be omitted, as in zone 2, Figure 4.6b, In this flow net, as in Fig. 4.6a the head 4 is divided into eight parts equal to Ak and the number of equipotential drops n; = 8. In zone 1 the number of flow channels n,_, = 3.5; in zone 2 the number of flow channels n,_, = 0.7 (as in Fig. 4.6a). Seepage quantities determined by using ki and k; are identical to those made from the flow net in Figure 4.6a. A third system of lines for this flow net shown in Figure 4.6¢ was obtained by dividing the total head 4 into 40 parts, each one-fifth as large as Ah in Figures 4.6a and 4.6b. The flow net in Figure 4.6¢ has been constructed with squares in zone 2 and shortened rectangles in zone 1. Because k; = 5k, the figures in zone 1 must foreshorten to lengths equal to one-fifth of their widths. This relationship can be verified readily by Eq. 3.11. With the flow net drawn as shown in Figure 4.6c, n,_; = 3.5 and n,_, = 3.5(5) = 17.5, for it is deter- mined by the number of squares between equipotentials. The number of equi- potential drops n;, = 40. Computing the seepage quantity in zone 1, using k,, qg = kih(n;_\/ng) = kh(17.5/40) = 0.44k,h, which was obtained from the flow net drawn in Fig. 4.6a. Computing the seepage quantity in zone 2, using ki q = kxh(ng-,/ng) = k,h(3.5/40) = 0.088k,h, which was also obtained for the flow net drawn in Figure 4.6a. Recalling that k, = 5k,, ¢ = 0.088(5k))h = 0.44kh, as obtained before. By inspection of the three forms of the flow net in Figure 4.6 we see that the flow patterns and hydraulic gradients are identical in all three. This should be true because this is but one flow net. If a flow net is being constructed for a section in which k,/k, is large, it may not be practical to subdivide the elongated rectangles in the manner de- scribed above. In such cases (Sec. 3.3) the measurement of the lengths and widths of flow-net figures with an engineer’s scale can establish the accuracy of basic shapes. Example 5 Earth Dam on Low Permeability Foundation. Figure 4.7 shows an earth dam (or levee) on a foundation that is relatively low in permeability in relation to the embankment. In this problem it is assumed that the perme- ability of the foundation &, is one-tenth of the permeability of the embank- ment k.. When seepage occurs through two dissimilar but more or less parallel zones, as in Figure 4.7, flow through the more permeable zone dominates the combined seepage pattern. Time can often be saved by constructing a flow net for the more permeable part, the dam in this case, assuming temporarily that 46 COMPOSITE SE ke Foundation 5% FIG. 4.7 Development of flow net fo 5). (@) Construct flow net for dam, a (b) Extend equipotential lines into fouw (¢) Adjust equipotentials and flow line: the other part is completely impervi extended down into the less permea 2 balanced flow net is developed (F nets of this kind, fundamental rel kept in mind. In the completed flow foundation conducts only one-tent the more permeable embankment. | tion are shown as dashed lines beca The embankment in Figure 4.7 act nowever, the first full flow channe son. Seepage beneath this flow lins 22l, thus raising the total value of

142 FLOW-NET CONSTRUCTION drops in this flow net is 7 and the shape factor is 1.32/7 or 0.19. Through the embankment alone (Fig. 4.7a) the shape factor would be approximately 0.9/ 7, or 0.13; hence the relatively deep foundation increases the water losses only about 50%. . In constructing a flow net for a section of the type re;?resented p.y.thls exarp- ple, it is helpful to think in terms of the water-conducting capab111t1e§ ‘of soils of different permeabilities. Thus, if a foundation has lower permeability th{m an embankment, as in this example, a greater thickness of the lower permeabil- ity soil is needed to conduct the same amount of water. .Becaus.e mpst ?f the flow is through the embankment, the position of the equipotential lines in the embankment is influenced only slightly by the flow through the foundation. Example 6 Earth Dam on a Highly Permeable Foundation. Figure 4.8 illus- trates the same physical cross section used in Figure 4.7, but hefe the founda- tion is 10 times more permeable than the embankment. In this example the general shape of the net is controlled more by the foundation than by the dam. k. (Assumed zero for Water surface / this flow net) ) :_——_ Wfi' y’\ kg = 10k, (a) — |ncompatible condition — e that must be corrected M‘jf (‘ ' -(v L L == L o pr e At e (b) 1.0 20 2.8 (c) FIG. 4.8 Development of flow net for dam on more germeal?le foundation (Exarppk 6). (@) Construct flow net assuming dam is completely.lmpfarvx_()us. b) Extenq equipe- tentials up into dam locating initial position of satutatlon l_me in dam. (¢) Adjust equi- potentials and flow lines until a balanced flow net is obtained. A good starting procedure is to ¢ temporarily that the embankmes 4.8a) is for a confined flow sys phreatic line) is known. This trial this trial net has been drawn (Fi up into the embankment (Fig. 4 are then adjusted until a flow ¢ differences between the permeabi extending the flow net into the ¢ vided into the correct number of the number of equipotential dro in Figure 4.4a, horizontal kead i the dam and each equipotential I phreatic line. (See also Figs. 3.6 In developing flow nets of the position of the phreatic line must ment of the flow net. This type ¢ At the start of the construction, the flow pattern can save much e trated in Figures 4.7 and 4.8 hel; work. Initial approximations of 1 never be permitted to overshados 47 EXAMPLES OF COMPLE When the principles illustrated i lowed, interesting and useful flow cross sections. Two examples of Figures 4.9 and 4.10. When large zones as in Figure 4.9 considerab only a few lines may be needed i out properly. The flow net in Fig assumed to have horizontal perm: net is constructed on a transforme dimensions have been reduced by sorizontal permeabilities, or V1, redrawn on the natural section, a Figure 4.10 gives two flow nets = 25 k,) in a study of seepage ben ing a reservoir on the left from a I thin layer of low permeability soil zravel to a depth of about 50 ft. = a 50- to 60- ft thick silty form permeable gravelly materials. Far

10.19. Through the \pproximately 0.9 e water losses only 2nted by this exam- apabilities of soils - permeability thas e lower permeabil- scause most of the tential lines in the h the foundation . Figure 4.8 illus- t here the founda- 1 this example the 1 than by the dam undation (Example (b) Extend equipe- m. (¢) Adjust equs 4.7 EXAMPLES OF COMPLEX FLOW NETS 143 % 2ood starting procedure is to draw a flow net for the foundation, assuming =mporarily that the embankment is completely impervious. Such a net (Fig. “%a) is for a confined flow system in which the upper saturation line (the shreatic line) is known. This trial net is the type illustrated in Figure 4.3. After s trial net has been drawn (Fig. 4.8a) the equipotential lines are extended i into the embankment (Fig. 4.8b). The lines in both dam and foundation we then adjusted until a flow net (Fig. 4.8c) compatible with the assumed &ifferences between the permeabilities of the two soil units is obtained. Before =uiending the flow net into the embankment, the total head 4 should be di- wded into the correct number of parts (eight in this example) to conform to = number of equipotential drops in the trial flow net in the foundation. As = Figure 4.4a, horizontal kead lines are drawn across the downstream part of e dam and each equipotential line must intersect the correct head line at the pareatic line. (See also Figs. 3.6 and 4.4c.) In developing flow nets of the type illustrated in Figures 4.7 and 4.8, the position of the phreatic line must be adjusted simultaneously with the refine- ment of the flow net. This type of flow net is the most difficult to construct. At the start of the construction, time taken to appraise the broad nature of the flow pattern can save much effort. Frequently shortcuts of the kind illus- trated in Figures 4.7 and 4.8 help to obtain correct solutions with minimum work. Initial approximations of the type made in these two examples should mever be permitted to overshadow the basic rules governing flow nets. 47 EXAMPLES OF COMPLEX FLOW NETS When the principles illustrated in the preceding examples are carefully fol- lowed, interesting and useful flow nets can be developed for a wide variety of cross sections. Two examples of fairly complex flow nets are given here in Figures 4.9 and 4.10. When large differences in permeabilities exist in various zones as in Figure 4.9 considerable detail may be needed in one material but only a few lines may be needed in another when the basic checks are carried out properly. The flow net in Figure 4.9 is for an earth dam and foundation assumed to have horizontal permeabilities nine times vertical; hence the flow net is constructed on a transformed section (Fig. 4.9b) in which the horizontal dimensions have been reduced by the square root of the ratio of vertical-to- horizontal permeabilities, or V1/9 = 1/3 (see sec. 3.3). the flow net is then redrawn on the natural section, as in Figure 4.9q. Figure 4.10 gives two flow nets (constructed on transformed sections for Kk = 25 k) in a study of seepage beneath a dam built in a natural saddle separat- ing a reservoir on the left from a lower valley on the right. Beneath a relatively thin layer of low permeability soil is a stratum of highly permeable sand and gravel to a depth of about 50 ft. Beneath the pervious sand and gravel layer is a 50- to 60- ft thick silty formation, which in turn is underlain by fairly permeable gravelly materials. Far to the right is a fairly steep slope (about 1