Geotechnical Engineering

DHARM

SEEPAGE AND FLOW NETS 185

One of the assumptions in the method is that length GH is equal to its projection on the

vertical, that is, to the z-co-ordinate of point G. The inaccuracy introduced is insignificant for

dams with flat slopes. The other assumption is with regard to the expression used for the

gradient. Along the top flow line the gradient is (– dz/ds), since the only head is the elevation

head. Since the variation in the size of the square in the vicinity of the equipotential line GH

is small, the gradient must be approximately constant. It is assumed that the gradient at the

top flow line is the average gradient for all points of the equipotential line.

With these assumptions, we have:

q = kF−

HG

I

KJ

dz

ds

. z ...(Eq. 6.17)

At point E the gradient equals sin α, and z equals a sin α. Thus, for the equipotential
EJ, we have:
q = k. a sin^2 α ...(Eq. 6.18)
Equating the expressions for q given by Eqs. 6.17 and 6.18, and rearranging and inte-
grating between appropriate limits,

a sin^2 α

0

() sin

.

Sa

h

a

ds z dz

i

−

zz=−

α

s is the distance along the flow line.

Starting at A, the value of a is zero at A (S – a) at E. The value of z at A is ht and that at

E is a sin α.

Solving, we get

a = S – Sh−()tcosecα^2 ...(Eq. 6.19)
The value of S differs only slightly from the straight distance AF. Using this approxi-
mation,

S = hdt^22 + ...(Eq. 6.20)

Substituting this in Eq. 6.19, we have:

a = hd dhtt^22 +− −2 22cot α ...(Eq. 6.21)
A graphical solution for the distance a, based on Eq. 6.21, was developed by L. Casagrande
and is given in Fig. 6.20, which is self-explanatory.

G. Gilboy developed a solution for the distance a sin α, which avoids the approximation
of Eq. 6.20; but the equation developed is too complicated for practical use. Instead, use of a
chart developed by him is made.

A graphical method of sketching of the top flow line after the determination of the
breakout point, which is based on L. Casagrande’s differential equation, is given in Fig. 6.21.

Combining Eqs. 6.17 and 6.18, we have:

∆S = z ∆z

asin^2 α

F

HG

I

KJ

...(Eq. 6.22)

If C is plotted such that CF = a sin α, the height of C above F is obviously a sin^2 α.