leachate flow in leakage collection layers due to defects in ...

GIROUD et al. D Leachate Flow in Leakage Collection Layers Due to Geomembrane Defects
Y
= y
(25)
Combining Equations 23, 24 and 25 gives the equation of the parabola in axes Ox and
Oy as follows:
y
2
2to
x to
= +
2
sinb
sin b
2
(26)
Equation 26 can also be written:
y
2
2
F
HG
to
x
= 1+
2
2
sin b
sin b
t
o
I
KJ
(27)
4.1.2 Comment on the Development of the Equations
It should be noted that the equation of the parabola was obtained with a minimum
amount of calculations because it was recognized in Section 2.2, using geometric considerations,
that the curve had to be a parabola. Thus, it was straightforward to establish
the equation of a parabola passing by three known points, P, Pi and V (Figure 6). If it
had not been recognized that the curve was a parabola, it would have been necessary
to use the analytical method to determine the equation of an unknown curve, which consists,
in this particular case, of determining in polar coordinates the intersection of a
cone and an inclined plane, and converting the obtained equation into cartesian coordinates.
The senior author has checked that this lengthy method yields Equation 27.
4.1.3 Equations for the Case Where the Leakage Collection Layer is not Full
Combining Equations 10, 23 and 26 gives the following equations for the parabola
that delimitates the wetted zone in the case where the leakage collection layer is not full:
2 Q 2 X
Y =
k sin b
(28)
hence:
y
y
2
2
2x Q/
k Q
= +
2
sin b k sin b
F
HG
Q x
1 2 sinb
= +
2
k sin b Q/
k
I
KJ
(29)
(30)
230 GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NOS. 3-4