SOIL PERMEABILITY AND SEEPAGE
4.20 FLO W NET CONSISTING O F CONJUGATE CONFOCAL PARABOLAS As a prelude to the study of an ideal flow net comprising of parabolas a s flow and equipotential
lines, it is necessary to understand the properties of a single parabola. The parabola ACV illustrated in Fig. 4.21, is defined as the curve whose every point is equidistant from a point F called the focus and a line DG called the directrix. If we consider any point, say, A, on the curve, we can write FA = AG, wher e th e lin e A G i s norma l t o th e directrix . I f F i s th e origi n o f coordinates , an d th e coordinates of point A are (jc , y), we can write
AF =
-^- (4-54 )
where, yQ = FD
Eq. (4.54) is the equation of the basic parabola. If the parabola intersects the y-axis at C, we can write
FC=CE = y0
Similarly for the vertex point V, the focal distance aQ is
FV = VD = a0 = y0/2 (4.55 )
Figure 4.2 1 illustrate s th e idea l flo w ne t consistin g o f conjugat e confoca l parabolas . Al l th e parabolas have a common focus F.
The boundary lines of such an ideal flow net are:
1 . Th e upstream face AB, an equipotential line, is a parabola.
2. Th e downstream discharge face FV, an equipotential line, is horizontal.
3. ACV, the phreatic line, is a parabola.
4. BF, the bottom flow line, is horizontal.
The known boundary conditions are only three in number. They are , the two equipotential lines AB and FV, and the bottom flow line BF. The top flow line ACV is the one that is unknown. The theoretical investigatio n of Kozeny (1931) revealed tha t the flow net for suc h an ideal condition mentioned above with a horizontal discharge face FV consists of two families of confocal parabolas with a common focus F. Since the conjugate confocal parabolas should intersect at right angles to each other, all the parabolas crossing the vertical line FC should have their intersection points lie on this line.
Since the seepage lin e is a line of atmospheric pressur e only, the only type of head that can exist along it is the elevation head . Therefore, ther e must be constant drops in elevation between the points at which successive equipotentials meet the top flow line , as shown in Fig. 4.21.
In al l seepage problem s connecte d wit h flow throug h earth dams , th e focus F o f the basi c parabola is assumed to lie at the intersection of the downstream discharge face FV and the bottom flow lin e B F a s show n i n Fig. 4.21. Th e poin t F i s therefor e known . The poin t A, which i s th e intersection point of the top flo w lin e of the basic parabola and the upstream water level, is also supposed to be known. When the point A is known, its coordinates (d, K) wit h respect t o the origin F ca n b e determined . Wit h thes e tw o know n points , th e basi c parabol a ca n b e constructe d a s explained below. We may write
(4.56)
Seepage Los s Throug h the Da m
The seepage flo w q across any section can be expressed accordin g to Darcy's law as
q = kiA (4.57)
Considering th e sectio n F C in Fig. 4.21, wher e th e sectiona l are a A i s equal t o yQ, th e hydraulic gradient / can be determined analytically as follows:
From Eq. (4.54), the equation of the parabola can be expressed a s
'o+^o2 (4.58)
Directrix
Figure 4.21 Idea l flownet consisting of conjugate confocal parabola s
Soil Permeabilit y an d Seepag e 1 29 The hydraulic gradient i at any point on the seepage lin e in Fig. 4.21 can be expressed a s
dy yo
For the point C which has coordinates (0 , yQ), th e hydraulic gradient from Eq. (4.59) is
Therefore, th e seepage quantit y across sectio n F C is
dy (4.60)
Seepage Throug h Homogeneou s an d Isotropi c Eart h Dam s Types of Entr y an d Exi t of Seepag e lines
The flow net consisting of conjugate confocal parabolas is an ideal case which is not generally met in practice. Though the top flow line resembles a parabola for most of its length, the departure fro m the basic parabola takes place a t the faces of entry and discharge o f the flow line . The departure from th e basi c parabol a depend s upo n th e condition s prevailin g a t th e point s o f entranc e an d discharge of the flow lin e as illustrated in Fig. 4.22 from (a ) to (e).
The seepage line should be normal to the equipotential line at the point of entry as shown in Fig. 4.22(a). However , thi s conditio n i s violate d i n Fig. 4.22(b), wher e th e angl e mad e b y th e upstream fac e AB wit h the horizontal i s less tha n 90°. It can be assumed i n this case th e coars e material used to support the face AB is highly permeable an d does not offer an y resistance fo r flow.
In suc h cases AB taken as the upstream equipotential line . The top flow lin e cannot therefore be
Seepage
line Coarse /••'•''/*?
material St.*'-^''.' -
Seepage line
(a) (b)
Discharge face XN | P
/3<90°
(c) (d ) (e )
Figure 4.22 Type s o f entry an d exit of seepage lines
normal to the equipotential line. However, this line possesses zero gradient and velocity at the point of entry . Thi s zer o conditio n relieve s th e apparen t inconsistenc y of deviatio n fro m a norma l intersection.
The conditions prevailing at the downstream toe of the dam affect th e type of exit of the flow line at the discharge face . In Fig. 4.22(c) the material at the toe is the same a s in the other parts of the dam whereas in (d) and (e) rock toe drains are provided. This variation in the soil condition at the to e affect s th e exi t patter n of th e flo w line . The flo w lin e wil l mee t th e discharg e fac e F E tangentially i n 4.22(c). This ha s t o be so because th e particles of water a s they emerge fro m the pores a t the discharge fac e have to conform as nearly as possible t o the direction of gravity. But in cases wher e roc k toe drains are provided, the top flow line becomes tangential t o the vertical lin e drawn at the point of exit on the discharge face as shown in (d) and (e) of Fig. 4.22.
Method o f Locatin g Seepag e Lin e
The genera l metho d o f locatin g th e seepag e lin e i n an y homogeneou s da m restin g o n a n impervious foundation may be explained wit h reference t o Fig. 4.23(a) . As explained earlier, the focus F of the basic parabola i s taken as the intersection point of the bottom flow line BF and the discharge face EF . In this case the focus coincides with the toe of the dam. On e more point i s required t o construc t th e basi c parabola . Analysi s o f th e locatio n o f seepag e line s b y A. Casagrande ha s revealed tha t th e basic parabol a wit h focus F intersects th e upstream wate r surface at A such that AA'= 0.3 m, where m is the projected lengt h of the upstream equipotentia l line A'B on the water surface . Point A is called th e corrected entranc e point . The parabola APSV may no w b e constructe d a s pe r Eq . (4.54). Th e divergenc e o f th e seepag e lin e fro m th e basi c parabola i s shown as AT1 and SD in Fig. 4.23(a). For dams wit h flat slopes , th e divergences ma y be sketched b y eye keeping i n view the boundary requirements. The error involve d in sketching by eye , th e divergence o n the downstream side , migh t be considerable i f the slopes are steeper.
B' T
(a)
Basic parabol a
u.t1 0.3
< 0.2+a
a 0.1
n
--.----,
i— ^^'^ -^_^^
^
^^\
(b)
30° 60 ° 90 ° 120 ° 150 ° 180 °
/5-Slope of discharge face
Figure 4.23 Constructio n of seepag e line
Soil Permeability an d Seepage 13 1
Procedures hav e therefor e bee n develope d t o sketc h th e downstrea m divergence a s explained below. A s show n i n Fig . 4.23(a), E i s th e poin t a t whic h th e basi c parabol a intersect s th e discharge face. Let the distance ED be designated as Aa and the distance DF as a. The values of Aa and a + Aa vary with the angle, j3, made by the discharge fac e with the horizontal measured clockwise. The angl e may vary from 30 ° to 180° . The discharge face is horizontal as shown in Fig. 4.22(e). Casagrande (1937) determined th e ratios of Aa / (a + Aa) for a number of discharge slopes varyin g from 30° to 180 ° an d the relationship is shown in a graphical form Fig. 4.23(b).
The distanc e ( a + Aa) can be determined by constructing the basic parabol a wit h F as the focus. Wit h th e know n ( a + Aa ) an d th e discharg e fac e angl e j3 , A a ca n b e determine d fro m Fig. 4.23(b). The point D may therefore be marked out at a distance of Aa from E. With the point D known, the divergence DS may be sketched by eye.
It should be noted that the discharge length a, is neither an equipotential nor a flow line, since it is at atmospheri c pressure . I t is a boundary along which the head a t any point i s equal to the elevation.
Analytical Solutions fo r Determinin g a an d q
Casagrande (1937) proposed th e following equation for determining a for j8 < 30°
(4.61) cos/? ^jcos 2/? sin 2/?
L. Casagrande (1932 ) gave the following equation for a when {$ lies between 30° and 90°.
(4.62) The discharg e q per unit length through any cross-section o f the dam may be expressed a s follows:
For/?<30°, a = fcasin/?tan/ ? (4.63 )
For30°</?<90°, a = fca sin2/? (4.64) .