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

GIROUD et al. D Leachate Flow in Leakage Collection Layers Due to Geomembrane Defects
A
wrand
1 L
=
Lz
A x wd
o
(116)
Combining Equations 24 and 116 gives:
A
wrand
1
=
L
z
L + to
/( 2 sin b )
to
/( 2 sin b )
A
w
dX
(117)
Equations for A w , and therefore integrations, are simpler with X than with x. Therefore,
Equation 117 will be used instead of Equation 116.
Two cases must be considered for integration of Equation 117. The first case is defined
by t o /(2 sinβ) ≤ L (i.e. Equation 51). Combining Equations 51 and 109 gives the
following condition for the first case of integration:
m£2
(118)
In this case, there are two different expressions for A w depending on X: Equation 49
(complete parabola) if X meets the condition expressed by Equation 52; and Equation
54 (truncated parabola) if X meets the condition defined by Equation 57. Therefore, if
μ ≤ 2, Equation 117 can be written as follows:
A
wrand
L + to
/( 2 sin b )
z z L
L
= 1 Aw1
dX
+
1
L to
/( 2 sin b ) L
A
w2
dX
(119)
where A w1 is the value of A w expressed by Equation 49 and A w2 is the value of A w expressed
by Equation 54.
Integration of Equation 119 gives:
A
w rand
2
15 L
=
F H G
to
sinb
I LF
+
KJ 1
NM
HG
2 L sin b
t
o
/
I
-
KJ
3 5 2
2
O
QP
(120)
Combining Equations 115 and 120 gives the following expression for the average
wetted fraction in the random scenario:
R
w rand
2 F
15 L
=
F H G
to
sinb
I LF
+
KJ 1
NM
HG
2 L sin b
t
o
/
I
-
KJ
3 5 2
2
O
QP
(121)
For practical calculations, it is convenient to use the following dimensionless expression:
R
w rand
=lrand
F L
where λ rand is a dimensionless factor derived from Equation 121 and defined by:
2
(122)
248 GEOSYNTHETICS INTERNATIONAL S 1997, VOL. 4, NOS. 3-4