Geotextiles and Geomembranes I I (1992) i-28
Rate of Leakage through a Composite Liner due to
Geomembrane Defects
J.P. Giroud, K. Badu-Tweneboah
GeoSyntec Consultants (formerly GeoServices Inc. Consulting Engineers) 1200 South
Federal Highway. Suite 204. Boynton Beach, Florida 33435. USA
&
R. Bonaparte
GeoSyntec Consultants (formerly GeoServices Inc. Consulting Engineers) 5950 Live Oak
Parkway. Suite 330, Norcross. Georgia 30093. USA
(Received 31 August 1990: accepted 9 October 1990)
ABSTRACT
This paper presents the development of a method to evaluate the rate of
leakage through geomembrane defects and. in particular, defective seams, in
cases where the geomembrane is placed on a layer of low-permeability soil to
form a composite liner. The method is validfor a wide range of liquid heads on
top of the composite liner. The method is presented in the form of equations.
tables, and charts that can be usedfor defects rangingfrom small holes to long
cracks. The use of the method is illustrated by examples, and a limited
parametric study is presented.
1 INTRODUCTION
1.1 Purpose
The purpose of this paper is to provide a comprehensive calculation
method to evaluate leakage through geomembrane defects in composite
liners. The method is applicable to defects ranging from small holes to
1
Geotextiles and Geomembranes 0266-1144/91/$03.50 © 1991 Elsevier Science Publishers
Ltd. England. Printed in Great Britain.
2
J.P Giroud, K. Badu-Tweneboah, R. Bonaparte
long cracks, as well as to a wide range of liquid heads on top of the
composite liner.
1.2 Composite liners
A composite liner is a liner composed ofa geomembrane and a layer of
low-permeability soil placed in close contact with each other (Fig. 1).
Composite liners are increasingly used in liquid storage and wastecontainment facilities.
Leakage through a composite liner can result from flow through
geomembrane defects or permeation through the geomembrane. In the
case of a geomembrane defect, the rate of leakage through a composite
liner is significantly less than the rate of leakage through a similar defect
in a geomembrane placed on a high-permeability soil, as shown in
several papers (Brown et al.. 1987; Jayawickrama et al.. 1988: Bonaparte et
al., 1989; Giroud & Bonaparte, 1989b; Giroud et ai.. 1989). On the other
hand, as discussed by Giroud and Bonaparte (1989a), the rate of leakage
due to permeation through an intact geomembrane is not significantly
different whether the geomembrane is placed on a low-permeability soil
(to form a composite liner) or a high-permeability soil. In this paper, only
leakage due to geomembrane defects is considered.
1.3 Flow through a composite liner with a defect in the geomembrane
In this paper, the contained liquid is assumed to be on the geomembrane
side of the composite liner. If there is a defect in the geomembrane, the
liquid flows first through the geomembrane defect, then laterally some
distance between the geomembrane and the low-permeability soil, and,
~.....~
L LOW-PERMEABISOIL
LITY
Fig.1. Compositeliner.
Rate of leakage through geomembrane defects
hi
~.---
(~--~~~---~
I~
3
GEOMEMBRANE
SPACE
/~ 'I'
"h Y, X Y Y
~ t / ( V, J,"~/
/
111111
R
Fig. 2. Flow of liquid through a composite liner. The space between the geomembrane
and the low-permeability soil is exaggerated to show interface flow. The flow in the soil is
assumed to be vertical and R is the radius of the wetted area.
finally, into and through the low-permeability soil layer (Fig. 2). Flow in
the space between the geomembrane and the soil is called interface flow,
and the area covered by the interface flow is called the wetted area.
The quality of the contact between the two components of a composite
liner (i.e. the geomembrane and the low'permeability soil) is one of the
key factors governing the rate of flow through the composite liner,
because it governs the radius of the wetted area (Fig. 2), Good and poor
contact conditions have been defined by Bonaparte et al. (1989) as
follows.
• Good contact conditions correspond to a geomembrane installed,
with as few wrinkles as possible, on top of a low-permeability soil
layer that has been adequately compacted and has a smooth
surface.
• Poor contact conditions correspond to a geomembrane that has
been installed with a certain number of wrinkles, and/or placed on a
low-permeability soil that has not been well compacted and does
not appear smooth.
Other factors affecting the rate of flow through a composite liner are
the size of the defect, the hydraulic conductivity of the low-permeability
soil underlying the geomembrane, and the head of liquid on top of the
geomembrane. If hydrostatic conditions prevail, the head of liquid is
equal to the depth of liquid (Fig. 3a) and, if the liquid is unconfined and
flowing along a slope (Fig. 3b), the head of liquid is given by the
following equation:
h = Dcos2fl
where D = depth of liquid; and fl = slope angle (in degrees).
(1)
4
J.P. Giroud. K. Badu-Tweneboah. R. Bonaparte
h
=
D
=
T =
HEAD
DEPTH
THICKNESS
(b)
Fig. 3. Head ofliquid: (a) if hydrostatic conditions prevail, the head is equal to the depth
of liquid: (b) if the liquid is unconfined and flowing, the head is less than the depth of
liquid and is given by eqn (I).
1.4 Organization of the paper
Section 2 presents leakage-rate equations for the case of small
geomembrane defects that can be modeled by circular or square holes.
These equations are derived from work previously published by the
authors (Bonaparte et al.. 1989; Giroud & Bonaparte, 1989b; Giroud et aL.
1989). The equations given in Section 2 are only valid for rather small
liquid heads.
Section 3 presents an extension of the equations given in Section 2 for
cases where the liquid head on top of the geomembrane is large.
Section 4 presents the development of the method to evaluate leakage
rates for the case of long geomembrane defects.
Section 5 presents the application of the method.
Finally, Section 6 presents conclusions.
1.5 Limitations of the method
Geomembranes are not absolutely impermeable and, even if the
geomembrane has no defects, there is always leakage due to permeation.
Rates of leakage due to permeation are usually very small. The method
presented in this paper does not consider the rate of leakage due to
permeation, which must therefore be added for a complete evaluation of
the rate of leakage through a geomembrane liner.
Rate of leakage through geomerabrane defects
5
The liquid considered in this paper is water. Chemical solutions are
not considered because, in certain cases, they could attack the
geomembrane and/or the low-permeability soil, which could lead to
relatively large leakage rates. Also, some chemicals, either pure or in
solution, have large rates of permeation through geomembranes in
comparison to the rate for water, as reported in other studies (e.g. August
& Tatzky, 1984: Haxo et al.. 1984).
The method presented in this paper has been developed with the
assumption that steady-state flow conditions are established and that the
low-permeability soil layer is saturated in a zone adjacent to the
geomembrane defect: this assumption requires that the head of liquid on
top of the geomembrane be constant for a long period of time. Another
assumption used to develop the method is that the head of liquid on top
of the geomembrane is not affected by the leakage: in other words, it is
assumed that there is sufficient liquid supply on top of the geomembrane
to keep the head constant.
The equations giving the rate of leakage through small defects were
established on the basis of a combination of theoretical analyses and
large-scale model tests, as discussed by Giroud and Bonaparte (1989b).
The equations giving the rate of leakage through long defects were
established as an extension of the method for small defects, using
assumptions that have not yet been verified experimentally.
2 LEAKAGE THROUGH SMALL DEFECTS DUE TO SMALL
LIQUID HEADS
2.1 Rate of leakage through a circular hole
Analytical studies and model tests have led to the following two
empirical equations proposed by Giroud et ai. (1989) for the rate of
leakage through a hole in the geomembrane component of a composite
liner:
• for good contact conditions,
Q = 0-21 a °'l h~ 9 kC2"74
(2)
• for poor contact conditions,
Q = l'15a°l h~w9k0'74
(3)
where Q = rate of leakage through a hole in the geomembrane
component of the composite liner; a = geomembrane hole area;
J P Giroud, K. Badu-Tweneboah. R. Bonaparte
hw = head of liquid on top of the geomembrane; and k~ = hydraulic
conductivity of the low-permeability soil component of the composite
liner. These empirical equations are not dimensionally homogeneous
and can only be used with the following units: Q(m3/s), a(m2), hw(m), and
k~(m/s). The good and poor contact conditions were defined in Section
1.3. Fundamental studies leading to the above equations were conducted
by Giroud and Bonaparte and first published by USEPA (1987); they
were then refined and published by Giroud and Bonaparte (1989a, b).
These studies were derived from analyses and/or laboratory tests by
Faure (1979, 1984), Fukuoka (1985, 1986), Sherard (1985), Brown et aL
(1987), and Jayawickrama et al. (1988).
2.2 Radius of the wetted area
Equations (2) and (3) were developed for circular holes using the
assumption that the flow in the low-permeability soil layer is perpendicular to the plane of the geomembrane (i.e. the flow is vertical if the
composite liner is horizontal, as in Fig. 2). It was also assumed that the
hydraulic gradient in the low-permeability soil layer is equal to one,
which restricts the use ofeqns (2) and (3) to cases where the liquid head
on top of the geomembrane (hw) is small (theoretically zero, and, as a
reasonable approximation, less than the thickness of the low-permeability
soil layer). The assumption of a hydraulic gradient equal to one explains
why the leakage rates given by eqns (2) and (3) do not depend on the
thickness of the low-permeability soil layer.
As a result of the above assumptions, the rate of leakage through a
composite liner due to a circular hole can be expressed as follows:
Q = rrR 2 k~
(4)
whereR = radius of the wetted area (i.e. the area of the low-permeability
soil layer where flow takes place) (Fig. 2); and k~ = hydraulic conductivity
of the low-permeability soil.
Equation (4) can be equated with eqn (2) forgood contact conditions,
and with eqn (3) forpoor contact conditions, to yield the following radii
of the wetted area:
• for good contact conditions,
R = 0"26a °5 h~45ks °'13
(5)
• for poor contact conditions,
R = 0.61 a °'°5 h~w45 k~-°'13
(6)
Rate of leakage through geomembrane defects
Table I
Typical Values of Radii of Wetted Areas
Liquid head
on geomembrane
Hydraulic conductivity
of low-permeability soil
~hw)
Radius of wetted area (R)
m (~)
~kO
m
(ft)
0.0003
(0.001)
0.003
(O.Ol)
0-03
(0.1)
0"3
(I)
3
(10)
30
(100)
mZ~
(crn/s)
10-9
i0_ s
IO-9
lO_ s
10-9
10_ s
10-9
10-s
10-9
i0_ s
IO-9
i0_ ~
(10 -7 )
(10_6)
(lO -7)
00_6)
(10 -7)
(10 -6)
(10 -7)
(10 -6)
(10 -7)
(10_6)
(iO -7)
(10_6)
Good contact
0.06
0.05
0.18
0-13
0-50
0.37
1"4
1"05
4
2.9
II
8.3
(0.21)
(0.15)
(0.58)
(0.43)
(i.6)
(I.2)
(4'6)
(3"4)
(13)
(9.7)
(37)
(27)
Poor contact
0.15
0.11
0.42
0.31
I-2
0.87
3"3
2"5
9"3
6.9
26
19
(0.50)
(0.36)
(I.4)
(1.O)
(3.9)
(2-9)
(11)
(8"0)
(31)
(23)
(86)
(64)
The table is related to a geomembrane hole area of I cm 2 (•. 16 in-') and togood and
poor contact conditions between the geomembrane and the low-permeability soil.
Equations (5) and (6) were used to calculate values of R for the good and poor
contact conditions, respectively. These two equations are valid if the thickness of
the low-permeability soil layer is greater than the liquid head on the geomembrane.
These empirical equations are not dimensionally h o m o g e n e o u s a n d can
only be used with the following units: R(m), a(m2), hw(m), a n d k~(m/s).
Typical values of radii of wetted areas are presented in Table 1.
2.3 Rate of leakage through a square hole
As shown in Table 1, the wetted area is large c o m p a r e d to the
g e o m e m b r a n e hole area. Therefore, in the case of a square hole, the
wetted area can be considered approximately circular. As a result, eqns
(2), (3), (5), a n d (6) can be rewritten approximately as follows in the case
o f a square hole with a side length b:
Q = 0-21 b °'2 h~ 9 k °'74
(7)
Q = 1-15 b°2h~w9k~ 74
(8)
R = 0-26 b °'t h~w45 k s °'13
(9)
R = 0-61 b ~t h~ 45 k~-°'13
(10)
8
J.P Giroud, K. Badu-Tweneboah, R. Bonaparte
where eqns (7) and (9) are related togood contact conditions, and eqns (8)
and (10) to poor contact conditions. These empirical equations are not
dimensionally homogeneous and can only be used with the following
units: Q(m3/s), R(m), b(m), hw(m), and k~(m/s).
3 EXTENSION TO LARGE LIQUID HEADS
3.1 Head of liquid on top of the low-permeability soil
Equations (2) to (10) were established assuming that the hydraulic
gradient in the low-permeability soil layer is one. However, the average
hydraulic gradient in the soil is always greater than one. Assuming that
the flow is vertical, the average hydraulic gradient is given by
(11)
i = 1 + h/H~
where h = liquid head on top of the low-permeability soil layer (i.e. in the
space between the geomembrane and the low-permeability soil layer):
and H~ = thickness of the low-permeability soil layer.
The head of liquid on top of the low-permeability soil layer decreases
progressively from a maximum value (hw) in the geomembrane hole area
to zero at the edge of the wetted area (Fig. 4). As a result, the hydraulic
gradient in the low-permeability soil varies from a maximum value
under the geomembrane hole area to one at the edge of the wetted area.
According to eqn (11), the maximum value is given by
/max
=
1 + hw/Hs
(12)
1
r~
Fig. 4. Distribution of liquid head on top of the low-permeability soil. The geomembrane
hole is assumed to be circular with a radius R~. and R is the radius of the wetted area. The
liquid head on top of the geomembrane (hw) is directly applied to the soil through the
geomembrane hole.
Rate of leakage through geomembranedefects
9
where hw = head of liquid on top of the geomembrane; and H.~ = thickness
of the low-permeability soil layer.
In the case of a circular hole in the geomembrane, a relationship
between the head of liquid on top of the soil (h) and the radial distance
(r) from the center of the hole is given by the following equation (Giroud
& Bonaparte, 1989b):
R2k~[21n R
where R -- radius of the wetted area: k~ -- hydraulic conductivity of the
low-permeability soil: and 0--hydraulic transmissivity of the space
between the geomembrane and the low-permeability soil.
Equation (13) was established assuming that the liquid flows through
the low-permeability soil with a gradient ofone. In reality, the gradient is
greater than one since h is greater than zero (see eqn (11)). Consequently,
for a given head of liquid (hw) on the geomembrane and, therefore, for a
given leakage rate, the actual radius of the wetted area is less than the
radius calculated using eqn (5), (6), (9), or(10). As a result, the actual head
of liquid on top of the low-permeability soil is less than the value
calculated with eqn (13) for radii greater than the geomembrane hole
radius. For the sake of simplicity, the following approximate equation
can conservatively be used to calculate the head of liquid:
h - R2k~ ln(R/r)
(14)
2O
This equation is simpler than eqn (13) and gives a greater value of h, as
shown in Fig. 5. Since h = hw (the head of liquid on top of the
geomembrane) for r = R0 (the radius of the hole in the geomembrane),
eqn (14) gives
h = hwlln(R/r)]/ln(R/R0)
(15)
In conclusion, the head of liquid on top of the low-permeability
soil is
h = hw
if0<r<Ro
h given by eqn (15)
if R0 < r < R.
3.2 Influence of head on leakage rate
The rate of leakage through a circular hole in the geomembrane
component of a composite liner can be calculated using the following
equation:
J.P Giroud, K. Badu-Tweneboah, R. Bonaparte
10
h
o
r~
Fig. 5. Curves ofliquid head on top of the low-permeability soil. Curve (I) was obtained
with eqn (13) and Curve (2) with eqn (14).
r=R
Q = k~(l + hw/HOnRo + f,=n, dQ
(16)
The first term of eqn (16) is the rate of leakage through the soil located
directly beneath the geomembrane hole and was obtained using the
following form of Darcy's equation:
Q = k~imaxa
(17)
where Q = leakage rate; k~ = hydraulic conductivity of the lowpermeability soil;/max = hydraulic gradient in the soil located directly
beneath the geomembrane hole and given by eqn (12); and a = nR~ =
geomembrane hole area.
In the second term o f e q n 16, dQ is the rate of change in leakage rate as
a function of radius, which can be written using Darcy's equation as
follows:
dQ = k, idA
(18)
with i given by eqn (II), and
dA = 2nrdr
(19)
Hence,
dQ = k~ (1 + h/H~) 2nrdr
(20)
Combining eqns (15), (16), and (20), and solving for the integral, yields
Q = k,
1+
(1 - ( R o / R ) 2 ) ] n n 2
?.H~]-n~
J
(21)
Rate of leakage through geomembrane defects
11
It should be noted that, as R tends to R0, the bracket approaches
1 + h,,,/Hs, which is correct. As shown in Table 1, R is always much
greater than R0 and, therefore, a close approximation of eqn (21) is
Q = k~ {1 + hw/[2 H~ ln(R/R0)]} rrR 2
(22)
Equation (22) is identical to Darcy's equation, which can be written as
follows:
Q = k~ iavgAw
(23)
where Q = leakage rate; k, = hydraulic conductivity of the lowpermeability soil; iav~ = average hydraulic gradient in the low-permeability
soil located under the wetted area; and Aw = wetted area (which is equal
to nR'- in eqn (22)).
From eqns (22) and (23), the average hydraulic gradient in the case of a
circular hole of radius R0 is given by
iavg = 1 + hw/[2 H~ ln(R/Ro)]
(24)
Similarly, the average hydraulic gradient in the case of a square hole of
side length b would be given by
iav~ = 1 + hw/[2 H~ ln(2R/b)]
(25)
3.3 Rate of leakage through holes due to large liquid heads
The only unknown in eqns (22), (24), and (25) is the radius of the wetted
area (R). The values of R given by eqns (5), (6), (9), and (10) were obtained
with the assumption of a hydraulic gradient of one. These values of R are
larger than the value of R for a hydraulic gradient larger than one, as
indicated in Section 3.1. Therefore, the flow extends further laterally for
the case of a hydraulic gradient of one than it does for the case of a
hydraulic gradient larger than one.
It follows from the preceding discussion that, if eqn (22) is used with
the value of R given by eqns (5), (6), (9), or (10), the value of the leakage
rate (Q) thus calculated will be greater than if more rigorously calculated.
This conclusion is valid because the increase in the term R 2 in eqn (22)
(resulting from the use of eqns (5), (6), (9), or (10) rather than a more
rigorous equation) has a more significant effect on the calculated value
of Q than does the slight decrease in iavg(resulting from the use ofeqns (5),
(6), (9), or (10)). Therefore, a conservative theoretical value of the leakage
rate can be obtained by combining eqns (22) and (24) (or 25) with eqn (4),
and with eqns (2), (3), (7), or (8), depending on the case considered. The
12
J.P Giroud. K. Badu-Tweneboah. R. Bonaparte
resulting empirical equations for a small hole, approximately square or
circular, are
• for good contact conditions,
Qo = 0.21 iavga °' h~ 9 k°74
(26)
Qo = 0-21 i~v~b °'-' h~w~ kl~"74
(27)
• for poor contact conditions,
Qo = 1-15 i,vg a °1 h~ ~ k12'74
(28)
Q0 = 1.15 i,,vgb°2 --w
h0"9 "~sk0"74
(29)
where Q0 = rate of leakage through a small hole; iavg = average hydraulic
gradient given by eqn (24) or eqn (25); a = geomembrane hole area;
b -- side length of a square hole or, as a good approximation, diameter of
a circular hole (i.e. b = 2R0) in the geomembrane; hw = head of liquid on
top of the geomembrane; and k~ = hydraulic conductivity of the lowpermeability soil. These empirical equations are not dimensionally
homogeneous and can only be used with the following units: Qo(m3/s).
a (m-'), b (m), hw (m), and k~ (m/s): i~vg is dimensionless.
4 EXTENSION TO L O N G D E F E C T S
4.1 Derivation of leakage rate equation for long defects
Long defects can be modeled by a rectangular hole of length B and width
b. The method presented in this paper to evaluate the rate of leakage
through a rectangular hole is based on the assumption that the wetted
area in the case of a rectangular hole has the shape shown in Fig. 6, where
the radius is the same as the radius of the wetted area in the case of a
square hole with a side length (b) equal to the width of the rectangular
hole. This assumption has not been verified experimentally or through a
more rigorous analysis.
Using eqn (23) for the wetted area shown in Fig. 6 gives
Q = k~iavgrtR 2 + k~i*v~2R (B - b)
(30)
where Q -- rate of leakage through a rectangular hole; ks = hydraulic
conductivity of the low-permeability soil; R = radius of the wetted area;
B = length of the rectangular opening; b = width of the rectangular
opening; iavg = average hydraulic gradient in the soil beneath the
circular portion of the wetted area; and ia*vg = average hydraulic gradient
Rate of leakage throughgeomembranedefects
13
--LIMIT OF THE WETTED AREA
--OPENING IN THE GEOMEMBRANE
F
I
I
I
R
I------'I
B-b
Fig. 6. Assumed wetted area in the case of a rectangular defect in the geomembrane.
in the soil beneath the rectangular portion of the wetted area.
When B is much greater than R, eqn (30) becomes
Q = 2RBk~i*vg
(31)
In this case, the end effects on Fig. 6 become negligible, i.e. the circular
portion of the wetted area becomes negligible compared to the
rectangular portion. The flow in the soil is then planar and ia*vgcan be
obtained by a bi-dimensional analysis, as discussed below.
4.2 Average hydraulic gradient in the case of planar flow
In the case of a defect of infinite length, the flow is planar and the rate of
leakage per unit length (Q*) is given by the following equation derived
from eqn (31):
Q* = Q/B = 2 R ks i*vg
(32)
The value of the average hydraulic gradient in the case of planar flow
can be obtained by calculations similar to those presented in Section 3.2
for a circular hole (i.e. axisymmetric case).
Equation (16) becomes
A" =
R
Q* = k~ (1 + hw/Hs) b + 2 f,
dQ*
(33)
= b/2
where
dO* = ks (1 + h/Hs) dx
(34)
Equation (15) becomes
h = hw Iln(R/x)/ln(2R/b)l
(35)
14
J.P Giroud. K. Badu-Tweneboah.R. Bonaparte
Combining eqns (33), (34), and (35), and solving for the integral,
yields
Q* = 2R k~ 1 +
hw (1 - (b/2R))]
],_/~l n - ( 2 ~ ) ]
(36)
Since R is much greater than b, a good approximation of eqn (36) is
given by
Q* = 2R k~ {1 + hw/[H~ ln(2R/b)]}
(37)
Comparing eqns (32) and (37) shows that the average hydraulic
gradient in the case of planar flow is given by
i*vg = 1 + hw/lH~ ln(2R/b)]
(38)
4.3 Rate of leakage through an infinitely long defect
Combining eqn (32) and eqn (9) or (10) leads to the following empirical
equations for the rate of leakage through an infinitely long defect:
• for good contact conditions,
Q* = 0.52 l,vg
"* b °'l -w
h °45 .,~
k °'87
(39)
• for poor contact conditions,
Q* = 1.22 t.v~
"* b °1 h-w°'45k-.~°'s7
(40)
where Q* is the notation used for rate of leakage per unit length, and i*~g
is given by eqn (38). The empirical eqns (39) and (40) are not
dimensionally homogeneous and can only be used with the following
units: Q* (m2/s), b (m), hw (m), and k~ (m/s).
4.4 Rate of leakage through a long defect
Equation (30) can be written as follows:
Q = Q0+(B-b)Q*
(41)
Combining eqn (41) with eqns (27) and (39) (in the case ofgood contact
conditions), and eqns (28) and (40) (in the case of poor contact
conditions), leads to the following empirical equations for the rate of
leakage through a long defect:
• for good contact conditions,
Q = 0.52 i*vg(B - b) b °~ h~ 45k °'g7 + 0.21 iavgb °2 h~ 9 k'~74
(42)
Rate of leakage throughgeomembranedefects
15
• for poor contact conditions,
Q = 1.22i*vg(B - b)b °1 h~w45k°~7 + 1.15 iavgb°'2h~wgk°'74
(43)
where iavgis given by eqn (24) and ia*vgby eqn (38). The empirical eqns (42)
and (43) are not dimensionally homogeneous and can only be used with
the following units: Q (m3/s),B (m), b (m), hw(m), and k~(m/s): ia*v~and iavg
are dimensionless.
5 APPLICATION OF THE M E T H O D
5.1 Practical use of the equations
The use of the equations in the preceding sections can be simplified by
introducing the parameter ~. given by:
= 1/[2 ln(2R/b) l
(44)
whereR = radius of the wetted area (given by eqn (5), (6), (9), or (10)): and
b = width of a rectangular hole or diameter of a circular hole.
For a small defect modeled by a circular or a square hole, the average
hydraulic gradient (iavg) given by eqn (24) can be rewritten as
iavg =
1 + ¢hw/Hs
(45)
For a long defect modeled by a rectangular hole, the average hydraulic
gradient (i*vg) given by eqn (38) can be rewritten as
i*vg =
1 + 2¢hw/H~
(46)
Tables and charts can be used to obtain ~ as follows:
• The value of ~ is given in Table 2 for four hole widths and a full
range of values ofhw and ks. The hole widths considered are 1 m m
(0.04 in), 3.16 m m (0.12 in), 10 m m (0.4 in), and 31.6 m m (1.25 in).
These include a 10-mm 2 (0.1-cm 2 = 0.016-in 2) hole sometimes used
in the performance analysis o f g e o m e m b r a n e liners that have been
installed using rigorous construction quality assurance procedures,
and a 100-mm 2 (1-cm 2= 0-16-in 2) hole sometimes used to size
leakage collection systems.
• Values of ~ are also given by charts presented in Figs 7 and 8.
The values of~thus obtained are used in eqns (45) and (46) to calculate
iavg and t'*avg,which in turn are used in eqns (42) or (43) to calculate the
leakage rate.
-
-
-
-
good
poor
good
poor
good
poor
0.1
0.01
0.001
0.01
0.001
0.001
--
good
poor
I
0.01
0.001
good
poor
I0
0.1
0.01
O. I
I
I0
good
poor
1
0.1
I00
good
poor
i00
I0
good
poor
I
00316 m
(1.25 in)
good
poor
001 m
(0 4 in)
!00
000316 m
(0 125 in)
Contact
conditions
I0
100
~001 m
(0 04 in)
Head (hw) (m) at various hole widths (b)
O-189
O-143
O-136
0. I I0
0.106
0.090
0.087
0.076
0.074
0.066
0.064
0.058
0-057
0.052
0.051
0.047
0.046
0.042
10 -II m/s"
(!0 -9 cm/~)
TABLE 2
Values of the Parameter
0-214
0.157
O. 148
0.118
0.113
0.095
0-092
0.079
0.077
0.068
0.066
0.060
0.058
0.053
0.052
0.048
0.047
0.044
10 -m m/~
(!0 -~ cm/~)
0-245
0.173
O. 163
O. 127
0.122
O. I01
0.097
0.083
0-081
0.07 !
0.069
0.062
0.061
0.055
0.054
0.049
0.048
0.045
!0 -v m/s"
(10 -7 cm/s)
0.287
0.193
O. 180
O-138
0.131
O. 107
O. 103
0.088
0.085
0-074
0-072
0.064
0-063
0.057
0.056
0.051
0-050
0.046
!0 -~ m/s
(!0 -~ cmZ~)
at various soil hydraulic conductivities (k s)
0.347
0.218
0.202
O. 150
0-142
0.115
O-! I0
0.093
0-090
0.078
0.076
0.067
0.065
0.059
0.058
0.052
0-051
0.047
10 -7 mZs
(10 -~ cm/~)
.a.~
~"
~-
.~
"~
¢~
~'
Rate of leakage throughgeomembranedefects
h.
17
GOOD CONTACT
(m)
10.35
- -
k = = l O "7 m / s
P.30
- 10
k , = l O "a m / s
).25
- -
1()1
k = = l O "g r n / s
L15
~.10
,,
\L
Lo
Fig. 7. Values of ff in the case of good contact conditions. The lines with arrows are
related to the example calculation presented in Section 5.2.
The leakage rate can also be obtained directly using the charts given in
Figs 9 and 10 for the following special cases: b = 10 m m (0.4 in) and
3.16mm (0-12 in), ks = 1 X 10-9 m/s (1 X 10-7cm/s), H~= 1 m (3.3 ft),
and good contact conditions.
5.2 Example calculation
5.2.1 Description of example calculation
A liquid i m p o u n d m e n t is lined with a composite liner. The depth of
liquid (hw) is 3 m (10 ft), the thickness of the layer of low-permeability
soil (Hs) is 0.9 m (3 ft) and its hydraulic conductivity (ks) is 1 X 10-9 m/s
(1 X 10 -7 cm/s). Good geomembrane/soil contact conditions are assumed.
A crack has developed along a geomembrane seam; the crack length (B)
18
J.P Giroud, K. Badu-Tweneboah, R.
h
POOR
Bonaparte
CONTACT
(~
=10.7 m/s
=10a m/s
=10 -9 m / s
=lo-'°m/~
=10-I1 m/s
Fig. 8. Values offf in the case of poor contact conditions. (See Fig, 7 for use of this chart.)
is 2 m ( 7 9 i n ) a n d its width (b) is 3 m m (0.12in). The o w n e r o f the
i m p o u n d m e n t requires a n estimate o f the steady-state leakage from the
i m p o u n d m e n t due to the crack.
For simplicity, the e x a m p l e calculation is presented in SI units only,
since most o f the equations presented in this p a p e r c a n o n l y be used with
these units.
5.2.2. Direct evaluation using charts
T h e above values o f the p a r a m e t e r s are close to those used to develop the
chart given in Fig. 9: hr., = 0.9 m instead o f 1 m a n d b = 0.003 m instead
o f 0-00316 m. Therefore, Fig. 9 can be used to obtain a n a p p r o x i m a t e
value o f the leakage rate. T h e following values are read on the chart:
Q = 2.6 x l0 -8 ma/s
for hw =
Q = 2-5 x 10-7 ma/s
for hw = 10 m.
1m
Rate of leakage through geomembrane defects
Q
19
CHART FOR:
b=0.00316 m. Hs =1 m, ks =lxlO4m/s
GOOD CONTACTCONDITIONS
(~3/,)
10-3 .~
r
h,,=lOOm (R=17.2m)
S
S
io' I
l
10-5 I
S
S
h,=iOm
(R=6.1m)
h,,---1.0m
(R=2.16m)
¢,
J
/
/s
lO- [,,
.e S
#
,j
h,,=O.tOm (R=O.77m)
h ---O.01rn (R=O.27m)
hw==O.OOlm (R=O.tm)
10-7 .~
I0-8
~
,"
~
¢.
f Pf / S
$
•
i w
P jS
•
IO-9:
h=0.0001m (R=O.O3m)
P
~ ""
S
S
/
16~°
1611
I~)12
j
/
," / ,
--
/
._..~
= ,IHI,
| ,,,||i
i ,IH.i
, , , , , ,
0.001
0.01
0.10
1.0
|
,,|,,,,
1o
,
1oo
,,
i'aoo
Fig. 9. Leakage rate through a composite liner due to a rectangular defect in the
geomembrane for good contact conditions. The width ofthe defect is 0.00316 m (0.125 in)
and the low-permeability soil has a thickness H~ = I m (3 ft) and a hydraulic conductivity
of I x l0 -~ m/s (! × l0 -7 cm/s). The dashed portion of each curve is a straight line that
corresponds to planar flow. The lines with arrows are related to the example calculation
presented in Section 5.2. The charts are valid only if the radius of the wetted area is less
than half the distance between adjacent defects. Notations: B -- length of rectangular
defect in the geomembrane: b = width of the rectangular defect: h , = head of liquid on
top of the geomembrane: K, = hydraulic conductivity of the low permeability soil
underlying the geomembrane: and R = half-width of the wetted area (see Fig. 6).
A l i n e a r i n t e r p o l a t i o n for hw = 3 m g i v e s
Q = 6.7 x l 0 -s m3/s.
A l s o , u s i n g Fig. 9, t h e f o l l o w i n g v a l u e s c a n b e r e a d o n t h e c h a r t for t h e
r a d i u s o f w e t t e d area:
J.P Giroud. K. Badu-Tweneboah. R. Bonaparte
20
Q
I
(m3/=) /
,.3
1u
CHART FOR:
b=O.01 m, H s = l r e , k s = l x l 0 " ~ m / s ,
GOOD CONTACT CONDITIONS
i
/
164
~'
P
/ ~
4
105 ,
/
J
r36
1U
~
1 (~7
i
-"
(R=6.Sm)
j h==lm
(R=2.4,..,~1"t)
/ - ,,
/
168
i h =lOre
./:
h =OJOin
~-0.86m)
%=o.oomm(
s s~,s/"
./
f
~
f
I
/ ~J/J"
I
/
i~ I
1(~
I
2
,
0.01
,,11111
,
tll,ii
0.1
i iiii1,~
I
i ,,111,,
10
100
,
lll,lt
1000
(m)
Fig. 10. Leakage rate through a composite liner due to a rectangular defect in the
geomembrane for good contact conditions. The width of the defect is 0-01 m (0-4 in) and
the low-permeability soil has a thickness H, = I m (3 ft) and a hydraulic conductivity of
I X l0 -q m/s (1 X 10-7 cm/s). The dashed portion of each curve is a straight line that
corresponds to p l a n a r flow. The charts are valid only if the radius of the wetted area is less
than half the distance between adjacent defects. Notations: B = length of rectangular
defect in the geomembrane: b = width of the rectangular defect: h,, = head of liquid on
top of the geomembrane: K~ = hydraulic conductivity of the low permeability soil
underlying the geomembrane: and R = half-width of the wetted area (see Fig. 6).
R = 2.16m
forhw =
R = 6.1
forh.
m
=
1m
10m
A l i n e a r i n t e r p o l a t i o n f o r hw = 3 m gives
R=3m.
Rate of leakage through geomembranedefects
21
5.2.3. Evaluation using ~ value
Leakage rate calculation c a n also be p e r f o r m e d with e q n s (42) or (43),
using the value o f ~ given by Fig. 7 or 8. T h e following value o f ~: is
o b t a i n e d f r o m Fig. 7:
-- 0.06.
T h e value o f ~: c o u l d also be o b t a i n e d f r o m Table 2:
-- 0.069
for b = 0-00316m a n d hw --
= 0.061
forb = 0.00316mandhw
1m
-- 1 0 m .
A linear i n t e r p o l a t i o n gives ~ = 0-067 for hw = 3 m. For the e x a m p l e
calculation below, the value ~ -- 0.063 (which is between 0.06 a n d 0.067)
is a s s u m e d ; ia~g a n d i*~g can t h e n be calculated u s i n g eqns (45) a n d (46),
respectively:
iavg = 1 + 0"063 (3/0"9) = 1"21
i*vg = 1 + 2 × 0"063 (3/0"9) = 1"42.
T h e leakage rate Q is t h e n calculated using e q n (42):
Q = 0.52 x 1.42 x 1.997 x 0-003 ~l x 3 ~r45 X 10 -gx°'s7
+ 0.21 X 1.21 X 0-003 tr2 X 3 °9 X 10 -9x074
Q = 6.7 x IO-X m3/s.
5.2.4 Calculation using equations only
In all cases, especially those b e y o n d the range o f the tables a n d charts, it
is possible to calculate the leakage rate u s i n g e q u a t i o n s only.
First the radius o f the wetted area is calculated u s i n g e q n (9):
R = 0.26 X 0-003 °l x 3 o.45 X 10 -9x1-0"13)
= 3.53m
T h e n , t~ is calculated using e q n (44):
= 1/I2 In(2 X 3.53/0-003)1 = 0.064
T h e n , i.~v~a n d i*vgare calculated using eqns (45) a n d (46), respectively:
ia~g = 1 + 0"064(3/0-9)--
1"21
ia*g = 1 + 2 X 0"064 (3/0"9) = 1"43
T h e leakage rate Q is t h e n calculated u s i n g e q n (42):
22
J.P Giroud, K. Badu-Tweneboah. R. Bonaparte
Q = 0-52 X 1"43 X 1.997 X 0.003 °l X
+ 0.21 X 1.21 X 0.OO30.2 X 30.9 X
3 ~45 X 10 -9x(~87
l 0 -t) xff74
Q = 6"7 x l0 -8 m3/s
The leakage rate values obtained using the equations, tables, a n d figures
are in good agreement.
5.2.5 Comment on the wetted area
In cases where the g e o m e m b r a n e has several defects, it is useful to
calculate R to determine whether the wetted areas related to different
defects overlap. The wetted areas do not overlap if the distance between
two defects is greater t h a n 2R. If they do overlap, the rate of leakage
through each defect is less than calculated using the equations in this
paper, which were established for an isolated defect. In the above
example, the radius of the wetted area is 3-53 m. Therefore, the rate of
leakage through one defect calculated above is valid only if the distance
between defects is greater t h a n 7.06 m.
5.3 Parametric study
5.3.1 Influence of liquid head and defect length
Equation (42) was used to c o n d u c t a limited parametric study for the case
of good contact conditions. Leakage rates were calculated for the
following values of parameters:
eH~--lm
• k~ = l X 10 - g m / s
• b = 0.OO316 m a n d 0.01 m
• B = 0.oo316m, 0.01 m, 0.1 m, l m , 10m, 100m, a n d 10oom
• hw = 0.OO01 m, 0.OO1 m, 0.01 m, 0-10 m, 1 m, l0 m, a n d 1OO m.
The results of the parametric study are presented in Figs 9 a n d 10. This
study illustrates the influence of the head of liquid a n d the defect length
on the calculated leakage rates. It appears in Figs 9 a n d l0 that the
leakage rate ratio between a very long defect a n d a small defect depends
on the head of liquid on top of the g e o m e m b r a n e :
• for small liquid heads typically encountered in landfills, a long
defect will cause m u c h more leakage t h a n a small defect: whereas
• for large liquid heads, such as those encountered in liquid
i m p o u n d m e n t s , a long defect may only cause a few times more
leakage than a small defect.
Rate of leakage through geomembrane defects
o
o
23
1
2
b2
/
J
Fig. i 1. Comparison between leakage rates due to small and long defects. For a given
liquid head, the leakage rate is a function of the wetted area. Case (a) is related to a small
liquid head. in this case, a large number of small defects (al) is required to generate a total
wetted area approximately equal to the wetted area due to the long defect (a2). Case (b) is
related to a large liquid head. In this case, a small n u m b e r of small defects (bl) is required
to generate a total wetted area approximately equal to the wetted area due to the long
defect (b_,).
The reason for this difference between small and large liquid heads is
illustrated by Fig. 11. From the figure, it can be seen that if the liquid
head is large, the radius of the wetted area is so large that a large defect
length is required to generate significantly greater leakage than that
generated by one defect.
5.3.2 Influence of liquid head and soil-layer thickness
Equations (45) and (46) were used to conduct a parametric study giving a
range of values for the average hydraulic gradients, iav~ and i*vr The
results of this study are shown in Fig. 12, which makes it possible to
compare the efficiency of composite liners constructed with lowpermeability soil layers of different thicknesses, such as
J.P Giroud, K. Badu-Tweneboah. R. Bonaparte
24
I°'F
C~
103 t
102!
I0
C~
>
lay9 ~
I
-
E
i
L
1 0 -1
1
I0 -e
I I 1111
I0 -l
I
I l IIIIII
1
l
II Illll
I0
I
I[Illll
I
10 2
IIIIII
I
II l[lll
~/H~
10 3
10 4
Fig. 12. Average hydraulic gradient. The two curves encompass the range of values ofi..g
and the dotted area represents the range of values for i*vg. Each of these two ranges
encompasses good and poor contact conditions and the following values of the
parameters: hole width (b} from 0-001 to 0.0316 m (0-04 to 1-25 in): liquid head on top of
the geomembrane (h~,) from 0-001 to 100 m (0.04 in to 330 ft): thickness of the soil
underlying the geomembrane (H,) from 0.001 to 100 m (0.04 in to 330 ft}: and hydraulic
conductivity ofthe soil underlying the geomembrane (k~) from I × l0 -7 to I x l0 -t° m/s
(l X l0 -5 to i × 10-~ cm/s).
• a clay layer of 0.9 m (3 ft), which is the m i n i m u m thickness
permitted by the US Environmental Protection Agency for
composite bottom liners in double-lined hazardous waste disposal
landfills;
• a clay layer of 0.45 m (1.5 ft), which is the thinnest clay layer for
which proper placement of at least one 0.15 m (6 in) thick lift can be
ensured, based on the authors' experience: and
• a fabric-contained clay panel of 0.01 m (0.4 in).
The following assumptions were made:
• the three low-permeability soil layers have the same hydraulic
conductivity, e.g. 1 x 10-9 m/s (1 X 10 -7 c m / s ) ; and
• the contact conditions between the three low-permeability soil
layers and the adjacent geomembranes are the same (i.e. in the case
of the fabric-contained clay panel, the influence of the fabric on the
contact conditions was neglected).
Rate of leakage throughgeomembranedefects
25
Based upon the above assumptions, the following results are derived
from Fig. 12.
• When hw < H~, iavgand i*vgare close to one. Therefore, the fabriccontained clay panel of 0.01 m (0-4 in) is as efficient (with respect to
steady-state leakage) as thicker clay layers if the liquid head on top
of the geomembrane is less than the thickness of the fabriccontained clay panel.
• When hw/H, = 0.3/0.9 or 0.3/0.45, iavg and i*vgare close to one, and,
when hw/H~ 0.3/0.01 = 30, 1,~g and ~.,v~
"* are in the order of 5.
Therefore, with a liquid head on top of the geomembrane of 0.3 m
(1 ft), which is often considered as a maximum design head in
landfills, clay layers of 0.45 m (1.5 ft) and 0-9 m (3 ft) are equivalent
(with respect to steady-state leakage) and are approximately 5 times
more efficient than the fabric-contained clay panel of 0-01 m
(0.4 in) when associated with the same geomembrane to form a
composite liner.
• When hw/H~ = 10/0.9 = 11. i ~ and t~,,~
"* are approximately 1.5 times
less than for hw/H~ = 10/0.45 = 22. and 30 times less than for hw/
H, = 10/0.01 = 1000. Therefore. with a liquid head on top of the
geomembrane of 10m (33 ft) as in a liquid impoundment.
composite liners with a clay layer of 0.9 m (3 ft) are approximately
1.5 times more efficient (with respect to steady-state leakage) than
composite liners constructed with a clay layer of 0.45 m (1.5 ft). and
approximately 30 times more efficient than composite liners
constructed with a fabric-contained clay panel of 0.01 m (0.4 in).
•
=
"
The above conclusions are only valid if the fabric-contained clay
panel has the same hydraulic conductivity as a layer of compacted clay.
However. fabric-contained panels are usually made with bentonite.
which has a hydraulic conductivity that is significantly lower than the
hydraulic conductivity of compacted clay. The case of fabric-contained
bentonite panels with a hydraulic conductivity less than that of
compacted clay is discussed below.
5.3.3 lnfluence ofliquid head, soil-layerthickness, and hydraulic conductivity
This case is identical to the case discussed above, with the fabriccontained clay panels replaced by fabric-contained bentonite panels.
The bentonite hydraulic conductivity is assumed to be 1 X 10-" m/s
(1 X 10-gcm/s). Using eqn (42), the following conclusions can be
drawn.
• When hw < Hs, composite liners constructed with fabric-contained
26
J.P Giroud, K. Badu-Tweneboah, R. Bonaparte
bentonite panels are approximately 30 times more efficient (with
respect to steady-state leakage) than the composite liners constructed
with layers of compacted clay having thicknesses ofO.90 m (3 fi) or
0.45 m (1.5 It).
When hw = 0-3 m (1 fi), composite liners constructed with fabriccontained bentonite panels are approximately 10 times more
efficient (with respect to steady-state leakage) than composite liners
constructed with layers of compacted clay having thicknesses of
0.90 m (3 fi) or 0.45 m (1-5 fi).
When hw = 10 m (30 ft), composite liners constructed with fabriccontained bentonite panels are as efficient (with respect to steadystate leakage) as composite liners constructed with layers of
compacted clay having thicknesses of 0.90 m (3 ft) or 0.45 m
(1-5 ft).
5.3.4 Influence of the hydraulic gradient
As indicated in Section 1.4, equations that were previously available to
evaluate the rate of leakage through composite liners were established
with the assumption that the hydraulic gradient in the soil was equal to
one. The method presented in this paper takes into account the average
hydraulic gradient in the soil obtained by integration of a differential
form of Darcy's equation. This average hydraulic gradient is always
greater than one.
Figure 12 provides an opportunity to evaluate the influence of the
hydraulic gradient. This figure shows the following.
• For values of liquid head on top of the geomembrane less than the
thickness of the low-permeability soil (i.e. hw < H~), leakage rates
calculated assuming a gradient of one in the soil are a good
approximation of leakage rates calculated with the gradient
obtained by integration of a differential form of Darcy's equation.
Therefore, in this case, eqns (2), (3), (7), and (8) can be used.
• For hw > 3 H~ and, in particular, hw > 10 H~, the influence of the
hydraulic gradient on the rate of leakage is very significant, and
equations which were established with a gradient of one are no
longer valid. Equations given in Section 4 of this paper should then
be used.
6 CONCLUSIONS
This paper extends the earlier work of the authors (Bonaparte et al.. 1989;
Giroud et al.. 1989: and Giroud & Bonaparte, 1989a, b) on the steady-state
Rate of leakage through geomembrane defects
27
rate of leakage through a defect in the geomembrane component of a
composite liner. The paper extends the earlier work to include leakage
through a defect subjected to a large hydraulic head (which allows the
method to be applied to liquid impoundments and dams), and leakage
through a long defect (such as might be associated with a defective seam).
The paper represents the culmination of a calculation method for
leakage through composite liners that was initiated by Giroud and
Bonaparte for the US Environmental Protection Agency in 1987
(USEPA, 1987).
The method available in this paper allows the calculation of leakage
rates for conditions beyond those for which experimental verification
exists. The reader is therefore urged to use the equations carefully and to
apply judgement in interpreting calculation results. At the same time,
researchers are encouraged to conduct experiments to quantify leakage
rates through composite liners and thereby contribute to the body of
available data.
ACKNOWLEDGEMENTS
The authors are indebted to J. Lebredonchel, S.L. Berdy, A.C. Mozzar,
R. Rodriguez, and D. Pallanck for assistance during the preparation of
this paper, and are grateful to B.A. Gross for her extensive review of the
manuscript.
REFERENCES
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polymeric liners for hazardous landfill leachate organic constituents. In
Proceedings of the International Conference on Geomembranes (Vol. l) IFAI, St.
Paul, Minnesota, pp. 163-8.
Bonaparte, R., Giroud, J.P. & Gross, B.A. (1989). Rates of Leakage through
Landfill Liners. In Proceedings of Geosynthetics '89 (Vol. l) IFAL St. Paul,
Minnesota, pp. 18-29.
Brown, K.W., Thomas, J.C., Lytton, R.L., Jayawickrama, P. & Bahrt, S.C. (1987).
Quantification of leak rates through holes in landfill liners. USEPA Report
CR 810940, Cincinnati, Ohio, USA, 147 p.
Faure, Y.H. (1979).Nappes Etanches: Debit et Forme de I'Ecoulement en Cas de
Fuite. Thesis, University of Grenoble, France. (In French.)
Faure, Y.H. (1984). Design of drain beneath geomembranes: discharge
estimation and flow patterns in case of leak. In Proceedings of the International
Conference on Geomembranes (Vol. 2) IFAI, St. Paul, Minnesota, pp. 463-8.
Fukuoka, M. (1985).Outline of large scale model test on waterproof membrane.
Unpublished report, Japan, May 1985, 24 p.
28
J.P Giroud K. Badu-Tweneboah. R. Bonaparte
Fukuoka, M. (1986). Large scale permeability tests for geomembrane-subgrade
system. In Proceedings of 3rd International Conference on Geotextiles (Vol. 3)
Balkema Publishers, Rotterdam, The Netherlands, pp. 917-22.
Giroud, J.P. & Bonaparte, R. (1989a). Leakage through liners constructed with
geomembranes. Part I. Geomembrane liners. Geotextiles and Geomembranes.
8(1), 27-67.
Giroud, J.P. & Bonaparte, R. (1989b). Leakage through liners constructed with
geomembranes. Part I1. Composite liners. Geotextiles and Geomembranes,
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