2. When flow occures through soil it exerts thrust on the soil particle along the
direction of flow. As the direction of any flow line changes from point to point,
which is defined by the tangent at any point, it is necessary to obtain pressure
per unit volume of the soil mass. Which is explained below.
SEEPAGE PRESSURE:
Seepage Pressure at any point, Seepage through Earthen Dam
etc.
A.
3. Flow nets are useful in the determination of the seepage pressure a
t any point along the flow path.
Consider the cubical element with all the sides equal t
o a. Let h, be the piezometric
The total force on = P, = ay,h,
The total force on = P,= y,h,
The differential force acting on the element is
P , - P , = P , = @ y , h , - h , )
Since (h, - h,) is the head drop Ah, we can write
2
� tJ,. I, 3
P, = y A h = a - y _ = a 'i y
3 w
' w w
where @' is the volume of the element. The force per unit volume o
f the element is, therefore,
This force exerts a drag o
n the element known a
s the seepage pressure. It has the dimension
of unit weight, and at any point its line o
f action is tangent to the flow line. The seepage pressure is
a very important factor in the stability analysis of earth slopes. If the line o
f action o
f the seepage
force acts in the vertical direction upward a
s o
n a
n element adjacent t
o , the
f
o
r
c
e that is acting downward t
o keep the element stable is the buoyant unit weight o
f the element.
When these two forces balance, the soil will just b
e a
t the point o
f being lifted
in the soil mass
head acting on the left face and h2 is the the head at the right face,
Left
Right
= net force acting on the element is,
downstream
up causing failure.
This force is acting along the gradient i.e. slope of the Stream Line
4. SEEPAGE FLOW THROUGH HOMOGENEOUS EARTH DAMS
In almost all problems concerning seepage beneath a sheet pile wall or through the foundation of a
concrete dam all boundary conditions are known. However, in the case o
f seepage through an earth
dam the upper boundary or the uppermost flow line is nor known. This upper boundary is a free
water surface and will b
e referred t
o a
s the line o
f seepage or phreatic line. The seepage line may
therefore be defined as the line above which there is no hydrostatic pressure and below which there
is hydrostatic pressure. In the design of all earth dams, the following factors are very important.
I. The seepage line should not cut th
e downstream slope.
2. The seepage loss through the dam should b
e the minimum possible.
The two important problems that are required to be studied in the design of earth dams are:
I . The prediction of the position o
f the line of seepage in the cross-section.
2. The computation o
f the seepage loss.
If the line of seepage is allowed t
o intersect the downstream f
a
c
e much above the toe, more o
r
less serious sloughing may take place and ultimate failure may result, This mishap can b
e prevented
b
y providing suitable drainage arrangements on the downstream side of the dam.
The section of an earth dam may b
e homogeneous or non-homogeneous. A homogeneous
dam contains the same material over the whole section and only one coefficient of permeability
may b
e assumed t
o hold f
o
r the entire section. In the non homogeneous or the composite section.
two or more permeability coefficients may have t
o be used according to the materials used in the
section. Following is the skemetic diagram of a homogeneous dam with seepage line.
B.
5. k Phreatit line (seepage line)
[Basie parabola
'
'
'
Discharge f
a
c
e
Figure Basic parabola and the phreatic line for a homogeneous earth dam
1
B
DIRECTRIX
F
(CASAGREDE METHOD)
A
A'
C D
FD=y0
O O'
P'
P
X
Y
6. 4.20 FLOW NET CONSISTING OF CONJUGATE CONFOCAL PARABOLAS
As a prelude t
o the study of a
n ideal flow net comprising o
f parabolas a
s flow and equipotential
lines, i
t i
s necessary t
o understand th
e properties o
f a single parabola. The parabola ACV illustrated
in , is defined as the curve whose every point is equidistant fr
o
m a point F called the focus
and a line DG called the directrix. If w
e consider a
n
y point, say, A, o
n the curve, w
e can write F
A =
AG, where the line AG is nonnaJ to the directrix. If F is th
e origin o
f coordinates. and the
coordinates of point A are (x, y), we can write
AF =he+y = A G = x + y
y - }
o
r x =
2
%
where. y
,, = F
D
Eq. ) i
s the equation of the basic parabola. If th
e parabola intersects the y-axis at C
, w
e
can write
( )
FC= C
E =y,
Similarly f
o
r th
e vertex point V
, the fo
c
a
l distance a
, i
s
FV= VD=a,=y02 ( )
Figure illustrates the ideal flow net consisting of conjugate confocal parabolas. All the
parabolas have a common focus F.
The boundary lines of such a
n ideal flow net are:
Figure
1
(1
2
2
7. ingbyV.Murthy x [] Untitled1
E
Discharge
fa
c
e
Directrix
- - - - E
V D
i
'
•
f.
i
s
{
8
-
,
; r e ,
G w , ' , t i s
x
h
B
Figure Ideal flownet consisting of conjugate confocal parabolas
2.
8. I . Th
e upstream face AB, a
n equipotential line, i
s a parabola.
2. T
h
e downstream discharge fa
c
e F
V, a
n equipotential line, is horizontal.
3. ACV, the phreatic line, is a parabola.
4. BF, the bottom flow line, i
s horizontal.
T
h
e known boundary conditions a
r
e only three i
n number. They are, the two equipotential
lines AB and F
V, and th
e bottom fl
o
w line B
F
. Th
e t
o
p fl
o
w line ACV i
s t
h
e one that is unknown. Th
e
theoretical investigation of Kozeny ( 1 9 3 1 ) revealed that the flow net f
o
r such an ideal condition
mentioned above with a horizontal discharge fa
c
e FV consists of two families of confocal parabolas
with a common focus F. Since the conjugate confocal parabolas should intersect a
t right angles t
o
each other, all the parabolas crossing the vertical line F
C should have their intersection points lie o
n
this line.
which is unknown and
9. Method of Locating Seepage Line
The general method of locating the seepage line in an
y homogeneous dam resting on an
impervious foundation may be explained with reference to Fig 3(a). As explained earlier, the
focus F of the basic parabola is taken as the intersection point o
f the bottom flow line B
F and the
discharge face E
F. In this case the focus coincides with the toe of the dam. One more point is
required t
o construct the basic parabola. Analysis of the location of seepage lines by
A. Casagrande has revealed that the basic parabola with focus F intersects the upstream water
surface at A such that AA ' - 0 . 3 m, where m is the projected length of the upstream equipotential
line AB on the water surface. Point A is called the corrected entrance point. The parabola APSV
may now be constructed a
s per Eq. ). The divergence of the seepage line from the basic
parabol a is shown as A P and S
D in Fig. 3(a). For dams with flat slopes, the divergences may
be sketched by eye keeping in view the boundary requirements. The error involved in sketching
b
y eye, the divergence on the downstream side, might be considerable if the slopes are steeper.
ure -
( 1
10. B'
h
8
A A'
I
,
f
' . '
, o ,
p
<
' " �
1
'y- 83.�ic parabola
'
F V
(a)
0.4
0.3
? o »
a
0.I
0
30°
(b)
60
° 90
° 120° 150°
B-Slope of discharge face
180°
�
�
........_,
•
i
----•
Figure Construction of seepage line
3.
T
11. Seepage 335
Example
The cross section of an earth dam is shown in Figure 7.35. Calculate
the rate of seepage through the dam [q in m3/(min ∙ m)] by (a) Dupuit’s
method; (b) Schaffernak’s method; (c) L. Casagrande’s method; and
(d) Pavlovsky’s method.
25 m 2
2
Impermeable layer
b
c
30 m
k =3 ×10–4
m/min
a
a΄
0.3 ×50
=15 m
1 1
5 m
60 m
5 m
10 m
50 m
Figure 7.35
Seepage through an earth dam.
Ref: Adv. Soil Mechanics - B.M.Das
x
y
x
y
'
'
Yo
Directrix
Focus
