Contents preface (VII) introduction 1—37

Fig. 15.8 Comparison of flownet figures in the (I) transformed and (II) original section



500 IRRIGATION AND WATER RESOURCES ENGINEERING

∆q =

∆h

b = Kx

∆h

b

K

b K /K

b

y

x

∴

= K x Ky

K

(15.12)

One can also determine the seepage quantity using Eq. (9.46) which is rewritten as

q = Kh	^Nf	(15.13)
Nd

Equation (15.13) is valid for the isotropic condition but can also be used for the anisotropic
condition by replacing K with effective permeability K [Eq. (15.12)]. It should be noted that the shape factor (i.e., N_f /N_d) remains the same for both the original and transformed sections.

The major difficulty in the seepage analysis of an embankment dam is that the topmost streamline, i.e., the seepage line or the phreatic line, is not known. The seepage line is defined as the line above which there is no hydrostatic pressure and below which there is hydrostatic pressure (6). If the embankment is composed of coarse material, the capillary effects are negligible and the seepage line is practically the line of saturation. But, in case of an embankment of fine-grained soil, there is saturation without hydrostatic pressure and also a negligible flow occurs in the capillary fringe above the seepage line. The prediction of the seepage line helps in drawing the flownent and determines the piping potential. In case of a homogeneous earth dam founded on impervious foundation, the seepage line cuts the downstream face above the base of the dam unless, of course, special drainage measures are adopted.

The equipotential lines must intersect the seepage line at equal vertical intervals (Fig. 15.9). This requirement permits graphical determination of the seepage line simultaneously while a flownet is being drawn. Alternatively, one can determine the seepage line using Kozeny’s solution according to which the equation of seepage line for an embankment with parabolic upstream face and downstream horizontal drain, shown in Fig. 15.10, is expressed as

	y^2 – y0^2 + 2xy0 = 0	(15.14)
where,	y0 = 2a0 = d ^2 + h ^2 − d	(15.15)
	Free water surface
	(uppermost line of seepage)
	∆		h
Flow	∆		h
lines
					h
	∆		h
	∆		h
	∆		h
	Equipotential
	lines

Fig. 15.9 General conditions for line of seepage

EMBANKMENT DAMS
Further, the seepage discharge per unit length of embankemnt, embankment, as per Kozeny’s solution, is given as

q = Ky₀ = K ^L_MN d ² + h ² − d^O_PQ


501
q, through the

(15.16)

where, K is the coefficient of permeability, and other symbols are as explained in Fig. 15.10.

	B	Y
		h
			y0	X
A		F C		D
		d	a0

Fig. 15.10 Embankment dam with Kozeny’s conditions
It should be noted that the location of seepage line and the point at which it cuts the downstream face (the downstream drain in Fig. 15.10) are dependent only on the cross-section of the dam and are not affected by the coefficient of permeability of the embankment material. Further, actual embankment dams do not have parabolic upstream faces though many of them have downstream horizontal drain. The seepage line follows a parabola, Eq. (15.14), with some departures at the entry and the exit of common types of embankment dams.
15.3.2. Casagrande’s Solution for Common Embankment Dams
Casagrande (9) has described important conditions which must be met by flownets at the points of entry, discharge, and transfer across boundaries between dissimilar soils. These conditions have been given in Fig. 15.11. Casagrande (9) has also extended Kozeny’s solution for common embankment dams with usual upstream face and the downstream drain other than the horizontal drain.
Casagrande obtained accurate solutions for different embankment sections by drawing flownet and compared the seepage lines with Kozeny’s parabolic seepage line (also called ‘base parabola’). He, thus, found that in the central portion, the seepage line coincided with the base parabola. Further, the base parabola meets the water surface at the corrected entry point B which lies on the water surface, and is at a distance of 0.3 times the horizontal projection of the water-covered upstream face, from the junction of the water surface with the upstream face (Fig. 15.12). This means that EB = 0.3 EG. The actual seepage line, however, starts from E and is at right angles to the upstream face which is an equipotential surface. It, then, takes a reverse curvature and meets the base parabola tangentially.
When the downstream drain is other than a horizontal drain (in which case angle of the discharge face, α = 180°), the actual seepage line departs from the base parabola in the exit region also. For α less than 90°, the seepage line meets the discharge face, i.e., the downstream slope, tangentially at C. For 90° < α < 180°, the seepage line becomes vertical at the discharge face. Comparing his graphical solutions with the corresponding base parabola, Casagrande obtained a graphical relation between the ratio ∆a/(a + ∆a) and the angle of discharge face α (Fig. 15.13). Here, a is the distance from the focus F, along the discharge face, to the point where the seepage line meets the discharge face while (a + ∆a) is the corresponding distance

502 IRRIGATION AND WATER RESOURCES ENGINEERING
for the base parabola. Using this graph, ∆ a and, hence, the corrected exit point C can be determined. The base parabola in the exit region is, therefore, suitably modified so that it meets the discharge face at C .

< 90°	= 90°	> 90°
		Horizontal	Horizontal
	r	r = 90°	90°
Open
	water
			(a) Entrance conditions

For £90°lineofseepage tangent todischargeface

For 90° £ £ 180° line of seepage tangent to vertical at point of discharge

Vertical
Vertical

Parabola

h

a

= 90°

yo

o

(b) Exit (or discharge) conditions

k1 > k2

k1 < k2

k1 < k2

Vertical

k1

^k1

k2

^k1^k2

^k2

k

1

^k2

= =

= = 0

= 270° – –

Discontinuity,

Exceptional case

usual case

= 180°
a ₀

k ₁ > k ₂


k ₁ k ₂
= 270° – –

(c) Transfer conditions at boundaries between dissimilar soils

Fig. 15.11 Different conditions for seepage line

G B E

K Ñ a

EB = 0.3 EG

Top flow line
C Base parabola

a
+	Ñ
	a
a

a
F

Fig. 15.12 A Casagrande’s solution

The seepage quantity can be determined by using Kozeny’s equation [Eq.(15.16)] with d being replaced by the horizontal distance between the corrected entry point B and focus F of the base parabola. Alternatively, one can draw the flownet and use Eq.(15.13).

EMBANKMENT DAMS


503

Discharge face

h

d

For 60°< < 90°

	0.4
	0.3
a
aa+	0.2
c=
	0.1
	0

30°

Discharge

Discharge

face

face

x

a

x

h

y

a

h

y

a

y

0

y

0

a

3

d

3

a0

d

a =

y0 =

2

2

( h + d – d)

4

4

For = 90°

For 90°< < 180°

(a) Diagrams for determining a + a

face Vertical

Overhanging slope 60° 90° 120° 150°

(a) Diagram for determining a and a

x

h

y

Discharge

y

a

o

face

d

a0

a = a0 =

1

2

2

( h + d – d)

2

For = 180°

180°