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Technical Paper by J.P. Giroud and K.L. Soderman 

CRITERION FOR ACCEPTABLE BENTONITE 

LOSS FROM A GCL INCORPORATED 

INTO A LINER SYSTEM 

ABSTRACT: The following case is considered in the present paper: a geosynthetic 

clay liner (GCL) used in the composite primary liner of a double-lined landfill is overlain 

by a geomembrane and underlain by the secondary leachate collection layer, which 

consists of a geonet. If bentonite particles migrate downward out of the GCL, through 

the geotextile(s) that separate(s) the bentonite layer from the geonet, they penetrate into 

the secondary leachate collection layer. Depending on the amount of bentonite particles 

migrating from the GCL, these particles may decrease the hydraulic transmissivity of 

the secondary leachate collection layer. Also, the loss of bentonite particles from the 

GCL may decrease the performance of the composite liner. The present paper provides 

theoretical analyses of these detrimental effects of bentonite particle loss, which lead 

to an acceptance criterion that can be used to evaluate the results of tests performed to 

determine if the geotextile(s) that separate(s) the bentonite from the geonet is (are) suitable 

or if another (or additional) geotextile is necessary. 

KEYWORDS: Geosynthetic clay liner (GCL), Bentonite loss, Landfill liner, 

Composite liner, Leachate collection layer, Liquid collection layer, Acceptance 

criterion, Laboratory test, Geotextile, Filter. 

AUTHORS: J.P. Giroud, Chairman Emeritus, and K.L. Soderman, Project Engineer, 

GeoSyntec Consultants, 621 N.W. 53rd Street, Suite 650, Boca Raton, Florida 33487, 

USA, Telephone: 1/561-995-0900, Telefax: 1/561-995-0925, E-mail: 

jpgiroud@geosyntec.com and kriss@geosyntec.com, respectively. 

PUBLICATION: Geosynthetics International is published by the Industrial Fabrics 

Association International, 1801 County Road B West, Roseville, Minnesota 

55113-4061, USA, Telephone: 1/651-222-2508, Telefax: 1/651-631-9334. 

Geosynthetics International is registered under ISSN 1072-6349. 

DATES: Original manuscript received 6 July 1999, revised version received 3 August 

2000, and accepted 26 September 2000. Discussion open until 1 June 2001. 

REFERENCE: Giroud, J.P. and Soderman, K.L., 2000, “Criterion for Acceptable 

Bentonite Loss From a GCL Incorporated Into a Liner System”, Geosynthetics 

International, Special Issue on Liquid Collection Systems, Vol. 7, Nos. 4-6, pp. 

529-581. 

GEOSYNTHETICS INTERNATIONAL S 2000, VOL. 7, NOS. 4-6 

529

GIROUD AND SODERMAN D Criterion for Acceptable Bentonite Loss From a GCL 

1 INTRODUCTION 

1.1 Description of the Considered Liner System 

A landfill double liner system is considered (Figure 1). This double liner system 

comprises the following layers, from top to bottom: 

S the primary leachate collection layer is a geonet or a granular layer; 

S the primary liner is a composite liner that consists of a geomembrane and a geosynthetic 

clay liner (GCL); 

S the secondary leachate collection layer consists of a geonet; and 

S the secondary liner is a geomembrane (which may or may not be underlain by a GCL 

or a compacted clay layer). 

The secondary leachate collection layer collects the leachate that leaks through the 

primary liner and conveys it by gravity to a collector swale. The collector swale typically 

contains multiple layers of geonet or a granular material (e.g. gravel) and may contain 

a pipe. The collector swale conveys the leachate by gravity to a sump. In the sump, 

the leachate is detected (leakage detection), measured (leakage rate evaluation), and 

pumped (leachate removal). The secondary leachate collection layer, collector swale, 

and sump constitute the secondary leachate collection system. 

1.2 Purpose, Approach, and Presentation 

The purpose of the present paper is: (i) to review and evaluate the detrimental effects 

of a loss of bentonite particles from the GCL component of the composite primary liner 

Soil protective layer 

Geonet primary leachate collection layer 

Geomembrane - GCL composite 

primary liner 

Geonet secondary leachate collection layer 

Geomembrane - GCL composite 

secondary liner 

Permeable subgrade 

Figure 1. Cross section of a double liner system with two composite liners. 

Note: The double liner system depicted in this figure includes a geomembrane-GCL secondary composite 

liner. As discussed in Section 1.1, other options for the secondary liner include a secondary composite liner 

consisting of a geomembrane placed on a compacted clay layer or simply a geomembrane secondary liner. 

530 GEOSYNTHETICS INTERNATIONAL S 2000, VOL. 7, NOS. 4-6


of the considered double liner; and (ii) to propose a methodology to evaluate if the loss 

measured in a laboratory test is acceptable. The approach used is purely theoretical, but 

it leads to an acceptance criterion that can be used to evaluate the results of tests performed 

to determine if the geotextile(s) that separate(s) the bentonite from the geonet 

is (are) suitable or if another (or additional) geotextile is necessary. Therefore, the practical 

implications of the theoretical study described in the present paper are important, 

and an essential part of the present paper is the discussion of the application of the results 

of the theoretical study to the evaluation of laboratory test results. 

Some of the analyses are complex and a number of assumptions were made regarding 

the mechanisms involved. The authors are aware that some of these assumptions 

can be criticized. To allow the readers to evaluate the validity of the approach, the assumptions 

and the mathematical derivations are presented in detail. However, to facilitate 

reading, several mathematical developments are presented in appendices. 

As a result of the detailed presentation of the assumptions and derivations, the present 

paper can be a guide for design engineers facing a similar regulatory situation and 

a reference document for future research work on the subject. Also, the authors hope 

that the publication of the present paper could trigger interesting discussions on the 

functioning of filters and liquid collection layers. 

2 MECHANISMS AND CONSEQUENCES OF BENTONITE LOSS 

2.1 Mechanisms of Bentonite Loss 

There are two types of GCLs: (i) the GCLs that consist of a layer of bentonite encapsulated 

between two layers of geotextiles; and (ii) the GCLs that consist of a layer of 

bentonite glued to a geomembrane. The study presented herein is related to the case 

where the bentonite layer is separated from the geonet by one or two layers of geotextile, 

but not by a geomembrane. Since the most typical case is that of a GCL that consists 

of a layer of bentonite encapsulated between two geotextiles and that is placed directly 

on top of the geonet, the geotextile separating the bentonite layer from the geonet is 

herein referred to as the “lower geotextile of the GCL”. Bentonite particles are very 

small, and a certain amount of these particles may migrate through geotextiles. (Only 

bentonite migration through openings between fibers or yarns of the geotextiles is considered. 

Bentonite migration through holes in the geotextiles that might result from tear 

or puncture is not considered.) Migration of bentonite particles may be due to two 

mechanisms: 

S When bentonite is hydrated, it has a very low shear strength and it is conceivable that 

the overburden stress (i.e. load applied on the GCL due to the weight of the overlying 

layers) may cause a certain amount of hydrated bentonite to be extruded through 

geotextile openings. Evidence of this mechanism is provided by Fox et al. (1998). 

S If there is a hole in the geomembrane overlying the GCL, the resulting flow of leachate 

through the hole in the geomembrane, then through the GCL and toward the 

geonet, may dislodge some bentonite particles from the GCL and may carry some 

of these particles through geotextile openings. 


531


Both mechanisms are involved in the migration of bentonite particles through the 

lower geotextile of the GCL, whereas only the first mechanism, extrusion, is involved 

in the migration of bentonite particles through the upper geotextile of the GCL. Migration 

of bentonite particles through the upper geotextile of the GCL may improve the 

quality of the contact between the GCL and the overlying geomembrane and, as a result, 

may improve the performance of the composite liner. (It should be noted that the phrase 

“quality of contact” refers to the intimate contact between the geomembrane and the 

GCL required for the geomembrane-GCL composite liner to be effective; it does not 

refer to the interface shear strength, which in fact is likely to decrease as a result of bentonite 

extrusion.) Since the migration of bentonite particles through the upper geotextile 

is not expected to adversely impact the hydraulic performance of either the 

composite liner or the secondary leachate collection layer, only the migration of bentonite 

particles through the lower geotextile of the GCL is considered in the remainder 

of the present paper. 

Migration of bentonite particles due to the overburden stress, if it occurs, may occur 

over the entire area covered by the primary liner GCL; however, the amount of particles 

migrating per unit area may be greater where the overburden stress is greater (i.e. where 

the landfilled waste is higher) and/or where the rate of loading is greater (i.e. where rate 

of waste placement is greater). In contrast, migration of bentonite particles due to leachate 

flow through the GCL occurs only where there is a hole in the geomembrane overlying 

the GCL. Giroud et al. (1997b) have shown that, in the case of geomembrane holes 

ranging between 1 and 10 mm in diameter: (i) the diameter of the GCL area where flow 

due to a geomembrane hole takes place is of the order of 1 and 5 m for liquid heads above 

the geomembrane of 5 and 300 mm, respectively; and (ii) for typical numbers of geomembrane 

holes and typical liquid heads, the ratio between the GCL surface area where 

flow takes place and the total GCL surface area is of the order of 0.01% (head of 5 mm) 

to 1% (head of 300 mm) for 5 holes per hectare, and 0.04% (head of 5 mm) to 4% (head 

of 300 mm) for 20 holes per hectare. Clearly, the risk of migration of bentonite particles 

from the primary liner GCL due to leachate leaking through the liner exists only on a 

small fraction of the surface area of the liner system. Hence, it would be unreasonable 

to assume that the migration of bentonite particles due to leachate flow through the GCL 

occurs over any more than a small fraction of the surface area of the liner system. 

The two mechanisms (i.e. extrusion of hydrated bentonite due to overburden stress 

and migration of bentonite particles due to leachate flow) are governed by a number 

of parameters, including two parameters they have in common, the cohesion of the hydrated 

bentonite and the opening size of the lower geotextile of the GCL. These two 

parameters are briefly discussed below: 

S The cohesion of the hydrated bentonite is the mechanical property that measures the 

magnitude of the bonds that keep the bentonite particles together. The higher the cohesion, 

the less likely is the hydrated bentonite to be extruded through the geotextile 

openings and the less likely are the bentonite particles to be dislodged and carried 

by the flow of liquid. 

S The opening size of the lower geotextile of the GCL depends on the type of geotextile. 

Typical needle-punched nonwoven geotextiles have opening sizes of the order 

of 50 to 250 μm (Giroud 1996). Such opening sizes are much greater than the dimensions 

of bentonite particles (less than 1 μm). Therefore, bentonite particles carried 



individually, or in clusters, by liquid flow can pass through the geotextile(s) located 

between the bentonite and the geonet. However, it is known that, in the case of cohesive 

soils such as bentonite (Giroud 1996), only a few particles typically pass 

through a geotextile filter having openings smaller than approximately 100 to 200 

µm, because such soils form, over geotextile openings, bridges that can withstand 

the normal stress due to the weight of overlying materials and the drag forces exerted 

by the flow of leachate. 

2.2 Consequences of Bentonite Loss 

Migration of bentonite particles through the lower geotextile of the GCL has potentially 

detrimental effects on two important components of the liner system, the primary 

liner and the secondary leachate collection layer: (i) the loss of particles from the GCL 

may decrease the effectiveness of the geomembrane-GCL composite primary liner; and 

(ii) the accumulation of bentonite particles at certain locations may reduce the hydraulic 

capacity of the secondary leachate collection system. These two effects are addressed 

in Sections 3 and 4, respectively. 

In evaluating the potential detrimental effects of the migration of bentonite particles, 

one should remember that the two functions of the secondary leachate collection system 

are: (i) to detect leakage through the primary liner; and (ii), more importantly, to collect 

and remove the leachate to ensure that the liquid head on top of the secondary liner is 

small. Clogging of the secondary leachate collection layer, collector swale, or sump by 

bentonite particles is detrimental with respect to both functions because it impairs leak 

detection and increases the liquid head on top of the secondary liner. 

3 EFFECT OF BENTONITE LOSS ON THE PRIMARY LINER 

3.1 Role of GCL in Composite Liner 

In a composite liner, the function of the low-permeability soil component that underlies 

the geomembrane (i.e. the GCL in the case of a geomembrane-GCL composite liner) 

is to control the liquid flow that results from defects in the geomembrane. The 

effectiveness of the low-permeability soil component depends on its hydraulic conductivity 

and thickness. In the case of a GCL, a loss of bentonite particles causes a decrease 

of thickness, or a decrease in hydraulic conductivity, or both. Therefore, to evaluate the 

influence of a loss of bentonite particles on the effectiveness of a geomembrane-GCL 

composite liner, one should evaluate the influence of a decrease of GCL thickness and 

the influence of a decrease of GCL hydraulic conductivity on the rate of leakage 

through the composite liner. The influence of a decrease in GCL thickness will be evaluated 

in Section 3.2 and the influence of a decrease of GCL hydraulic conductivity will 

be evaluated in Section 3.3. 

The rate of leakage through a circular hole in a geomembrane underlain by a GCL 

can be calculated using the following semi-empirical equation (Giroud 1997): 

b 

095 . 02 . 09 . 074 . 

qo GCL GCL 

Q= 0. 976C 1+ 

01 . h t d h k 

g 

(1) 


533


where: d = hole diameter; h = liquid head on top of the geomembrane; k GCL = hydraulic 

conductivity of the GCL; t GCL = thickness of the bentonite layer in the GCL; and C qo 

= contact quality factor for a circular hole. It should be noted that Equation 1 can only 

be used with the following units: Q (m 3 /s), h (m), t GCL (m), d (m), and k GCL (m/s). The 

contact quality factor, C qo , is dimensionless and is such that: 

Cqo good 

£ Cqo £ Cqo poor 

(2) 

where: C qo good = value of C qo in the case of good contact conditions; and C qo poor =value 

of C qo in the case of poor contact conditions. “Good” and “poor” contact conditions refer 

to the contact between the geomembrane and the GCL and are defined by Giroud 

(1997). In the case of a GCL, the contact conditions can be assumed to be good. As indicated 

by Giroud (1997): 

Combining Equations 1 and 3 gives: 

C qo good 

= 021 . 

b 

Q= 0. 205 1+ 

01 . h t d h k 

GCL 

g 

095 . 02 . 09 . 074 . 

GCL 

(3) 

(4) 

3.2 Influence of GCL Thickness on Leakage Rate Through Composite Liner 

It is assumed that, as a result of the loss of bentonite particles, the thickness of the 

bentonite layer decreases while its porosity (and, therefore, its hydraulic conductivity) 

remains constant. Based on Equation 4, the relative derivative of Q with t GCL as the sole 

variable is: 

hence: 

b 

b 

dQ 

d 1+ 

01 . ht 

= 

Q 1+ 

01 . ht 

dQ 

0095 . ht 

=- 

Q 1 + 01 . ht 

From Equation 6, it appears that: 

b 

b 

GCL 

GCL 

GCL 

GCL 

g 

g 

g 

g 

095 . 

095 . 

095 . 

095 . 

dt 

t 

GCL 

GCL 

(5) 

(6) 

dQQ 

0 < < 095 . 

dt 

t 

GCL 

GCL 

(7) 

In other words, the relative variation of Q is always less than the relative variation 

of t GCL . Therefore, it is conservative to use the following approximate equation: 



dQ 

Q 

≈− 

dt 

t 

GCL 

GCL 

(8) 

For example, if the bentonite layer thickness decreases by 10%, Equation 8 shows 

that the flow rate through the GCL increases by approximately 10%. 

As indicated by Giroud et al. (1997b), the porosity of the bentonite in a GCL is: 

n 

GCL 

µ 

d 

= 1 − 

ρ t 

where: μ d = mass per unit area of dry bentonite; and Ã B = density of bentonite particles. 

Hence: 

t 

GCL 

= 

ρ 

B 

µ 

B 

d 

GCL 

( 1− 

n ) 

GCL 

(9) 

(10) 

Based on Equation 10 the relative derivative of t GCL with respect to μ d is: 

dt 

t 

GCL 

GCL 

dµ 

d 

= 

µ 

d 

(11) 

Equation 11 provides a relationship between the decrease in mass per unit area of 

the bentonite layer due to bentonite loss and the resulting decrease in bentonite layer 

thickness in the case where the bentonite porosity remains constant while particles are 

being lost. Combining Equations 8 and 11 gives: 

dQ 

Q 

dµ 

d 

≈− 

µ 

d 

(12) 

3.3 Influence of GCL Hydraulic Conductivity on Leakage Rate Through 

Composite Liner 

It is assumed that, as a result of the loss of bentonite particles, the thickness of the 

bentonite layer remains constant, while its porosity (and, therefore, its hydraulic conductivity 

) increases. Based on Equation 4, the relative derivative of Q with respect to 

k GCL is: 

dQ 

Q 

d kGCL 

= 0.74 

k 

Based on the classical Kozeny-Carman’s equation (Carman 1937): 

GCL 

(13) 

k 

GCL 

= 

λ 

n 

3 

GCL 

( 1− 

n ) 

GCL 

2 

d 

2 

B 

(14) 


535


where: λ = constant depending on the acceleration due to gravity, density of liquid, and 

viscosity of liquid; and d B = diameter of bentonite particle. 


k 

GCL 

2 3 

⎛ ρ 

Bt 

⎞ ⎛ 

GCL 

µ ⎞ 

d 

= λ ⎜ ⎟ ⎜1− 

⎟ d 

⎝ µ 

d ⎠ ⎝ ρBtGCL 

⎠ 

2 

B 

(15) 

Based on Equation 15, the relative derivative of k GCL with μ d as the sole variable is: 

dk 

k 

GCL 

GCL 

⎛ µ 

d 

d⎜1 

− 

dµ 

ρ 

d 

B 

t 

=− 2 + 3 

⎝ 

µ 

µ 

d 

d 

1 − 

ρ t 

B 

GCL 

GCL 

⎞ 

⎟ 

⎠ 

(16) 

hence: 


dkGCL dµ d 

dµ 

d 

=−2 − 3 

k 

µ ρ t − µ 

GCL d B GCL d 

dk 

k 

GCL 

⎡ 3 1 

=− ⎢2 

+ 

⎣ n 

( − n ) 

GCL 

⎤ dµ 

d 

⎥ 

⎦ µ 

GCL GCL d 

(17) 

(18) 

Equation 18 provides a relationship between the decrease in mass per unit area of 

the bentonite layer due to bentonite loss and the resulting decrease in GCL hydraulic 

conductivity in the case where the thickness of the bentonite layer remains constant 

while particles are being lost. It should be noted that n GCL is a variable; however, it can 

be eliminated by using a typical value. As indicated by Giroud et al. (1997b), a typical 

value for the porosity of bentonite hydrated under confined conditions is 0.75, hence: 

dkGCL 

dµ 

d 

≈−3 

(19) 

k 

µ 

GCL 

d 


d Q dµ 

d 

≈−2.22 Q 

µ 

d 

(20) 

3.4 Development of a Criterion 

As indicated in Section 3.1, the two mechanisms through which the loss of bentonite 

particles from a GCL can affect the rate of flow through a composite liner are a decrease 

of GCL thickness and a decrease of GCL hydraulic conductivity. The two mechanisms 

can occur simultaneously. However, since thickness and hydraulic conductivity are 

linked as shown by Equation 15, if the two mechanisms occur simultaneously, the effect 



on flow rate will be intermediate between the effects of each of the two mechanisms 

considered separately. 

Comparing Equations 12 and 20 shows that the effect of the second mechanism is 

greater than that of the first mechanism. Therefore, to be conservative, Equation 20 will 

be used for the development of a criterion for acceptable bentonite loss. 

For the sake of simplicity, the following notation is used: 

dm = −dµ 

L 

d 

(21) 

where dm L is an increment of bentonite mass loss per unit area. 

The mass per unit area of dry bentonite, μ d , is generally not reported. It can be derived 

from the value that is generally reported, which is the mass per unit area of bentonite 

with the initial water content (i.e. the bentonite in the GCL as delivered to the site), 

using the following equation (Giroud et al. 1997b): 

b 

m = m 1 +w 

d o o 

where: μ o = mass per unit area of bentonite with the initial water content; and w o = initial 

water content. 

Combining Equations 20 to 22 gives: 

dQ 

1+ 

wo 

= 2.2 dmL 

(23) 

Q µ 

The mass per unit area of bentonite at the initial water content in a GCL is typically 

μ o = 5 kg/m 2 . The initial water content is typically between 10 and 20%. Assuming a 

conservative value of 20%, Equation 23 becomes: 

o 

g 

(22) 

dQ 

Q 

≈ 

0.5 dm 

L 

(24) 

For example, Equation 24 gives dm L = 0.2 kg/m 2 = 200 g/m 2 for a leakage rate increase 

of 10%, and dm L = 0.02 kg/m 2 =20g/m 2 for a leakage rate increase of 1%. 

3.5 Bentonite Particles Entrapped in the Lower Geotextile of the GCL 

The “bentonite loss” discussed in Section 2 and in Sections 3.1 to 3.4 is the amount 

of bentonite that has been lost by the bentonite layer of the GCL, whereas the amount 

of migrating bentonite measured in the laboratory test described in Section 5 is the 

amount of bentonite that passes through the lower geotextile of the GCL. The difference 

is the amount of bentonite entrapped in the lower geotextile of the GCL. This can be 

summarized as follows: 

mL = mGT + mGN + mFL 

mMIGR = mGN + mFL = mL -mGT 

(25) 

(26) 


537


where: m L = mass per unit area of bentonite lost from the GCL; m GT = mass per unit area 

of bentonite entrapped in the lower geotextile of the GCL; m GN = mass per unit area of 

bentonite accumulated in the geonet secondary leachate collection layer; m FL =mass 

per unit area of bentonite flowing in suspension in the secondary leachate collection 

layer; and m MIGR = mass per unit area of bentonite migrating from the GCL into the secondary 

leachate collection layer. 

It is important to properly evaluate the amount of bentonite entrapped in the lower 

geotextile of the GCL to develop a criterion to interpret the laboratory test results. As 

shown in Appendix A, the amount of bentonite particles entrapped in the lower geotextile 

of the GCL may be very large, e.g. up to a value of the order of 100 g/m 2 for a needlepunched 

nonwoven geotextile with a mass per unit area of 200 g/m 2 , i.e. a typical 

geotextile used in a GCL. In Section 5.4, it will be seen that the mass of migrating bentonite 

particles per unit area may be of the order of 10 g/m 2 . Therefore, according to 

Equation 26, if the lower geotextile of the GCL contains a large amount of bentonite 

particles (e.g. 100 g/m 2 ), there is a large difference between the amount of migrating 

bentonite particles (m MIGR ) measured in the laboratory test described in Section 5 and 

the amount of bentonite particles lost from the GCL bentonite layer. It should, however, 

be noted that if the lower geotextile of the GCL contains a large amount of bentonite 

particles, these particles contribute to some degree to the leakage control function of 

the GCL. Therefore, the fact that these particles are not detected in the laboratory test 

described in Section 5 may not be critical. 

4 EFFECT OF BENTONITE LOSS ON THE SECONDARY LEACHATE 

COLLECTION SYSTEM 

4.1 Fate of Bentonite Particles in the Secondary Leachate Collection System 

A bentonite particle that passes through the lower geotextile of the GCL penetrates 

into the secondary leachate collection layer at a certain location. It may stay approximately 

at that location or it may be carried downslope in the secondary leachate collection 

layer by leachate flow. 

In a well designed and constructed liner system, there should not be much leakage 

through the primary liner and, therefore, there should not be much leachate flow in the 

secondary leachate collection system. Furthermore, if there is leachate flow, it is localized 

in “wetted zones” (Giroud et al. 1997a). Conservative calculations (Appendix B) 

based on equations developed by Giroud et al. (1997a) show that the combined surface 

area of the wetted zones is less than 1% of the surface area of the secondary leachate 

collection layer, because the rate of leakage through a composite primary liner incorporating 

a GCL is very small. Migration of bentonite particles due to extrusion may occur 

over the entire area covered by the GCL. Since leachate flow in the secondary leachate 

collection layer is localized in the wetted zones, only a small fraction of the bentonite 

that passes through the lower geotextile of the GCL due to extrusion is likely to be carried 

by leachate flow. On the other hand, migration of bentonite particles due to leachate 

leaking through the liner occurs within the wetted zones, and therefore virtually all of 

the bentonite that passes through the lower geotextile of the geotextile due to leachate 

leaking may be carried by leachate flow. 



Bentonite particles are very small. Therefore, in the zones of the secondary leachate 

collection system where there is leachate flow (“wetted zones”), the bentonite particles 

will be initially in suspension in leachate. However, as shown in Appendix C, the depth 

of leachate in the wetted zones is extremely small (a fraction of a millimeter) if the geomembrane-GCL 

composite primary liner functions as designed. Consequently, as 

shown in Appendix D, bentonite particles are more likely to settle in the secondary leachate 

collection layer than to remain in suspension and travel to the sump. As shown 

in Appendix D, it would require a large leak through the primary liner (e.g. a failure of 

the GCL at a location where there is a defect in the geomembrane) to generate sufficient 

leachate flow in the secondary leachate collection layer to carry bentonite particles in 

suspension to the sump. 

The bentonite particles that reach the sump may settle if there are relatively long periods 

without pumping leachate from the sump, which may happen since the rate of leachate 

flow into the sump is low. Therefore, the secondary leachate collection layer sump 

should be designed to accommodate the settling of bentonite particles. For example, 

it might be possible to remove settled bentonite particles from the sump by periodically 

flushing the sump with clean water and removing water containing bentonite particles 

in suspension using the pumps normally used to remove leachate from the secondary 

leachate collection layer. 

From the foregoing discussion it is clear that bentonite particles that pass through 

the lower geotextile of the GCL are likely to remain in the leachate collection layer for 

the following two reasons: (i) the “wetted zones”, i.e. the zones where leachate flows 

and could carry bentonite particles in suspension, generally occupy only a very small 

fraction of the secondary leachate collection layer; and (ii) in the wetted zones, generally, 

there is not sufficient flow to carry bentonite particles to the sump. Accordingly, in 

the remainder of the present paper, it will be assumed that bentonite particles stay in 

the secondary leachate collection layer approximately at the location where they penetrate 

into the secondary leachate collection layer after they pass through the lower geotextile 

of the GCL. Assuming that the particles move from this location would be less 

conservative since the particles would then be distributed over a larger area of geonet 

and the impact on the performance of the geonet would be less. 

Based on the foregoing discussion, m FL = 0 in Equation 26, which becomes: 

mMIGR = mGN = mL -mGT 

(27) 

Herein, only geonet secondary leachate collection layers are considered. Bentonite 

particles that accumulate in a geonet can either adhere to the geonet strands or form a 

layer on top of the secondary liner geomembrane. Therefore: 

mGN = mGNs + mGNa 

(28) 

where: m GNs = mass per unit area of bentonite particles adhering to geonet strands; and 

m GNa = mass per unit area of particles accumulating in the geonet on the secondary liner. 

These two cases are addressed in Sections 4.2 and 4.3, respectively. 


539


4.2 Effect of Bentonite Particles Adhering to the Geonet Strands 

4.2.1 Effect of Bentonite Particles on Geonet Hydraulic Conductivity 

Bentonite particles that adhere on geonet strands can affect the hydraulic conductivity 

of the geonet in two ways: (i) by increasing the diameter of the geonet strands; and 

(ii) by decreasing the geonet porosity. These two effects, which exist simultaneously, 

are accounted for by the following equation (Giroud 1996), which was developed for 

nonwoven geotextiles but can also be used for geonets: 

k 

GN 

3 

n 

= l 

1- 

n 

b 

where: k GN = geonet hydraulic conductivity; n GN = geonet porosity; and d GN = diameter 

of geonet strands. Equation 29 is based on the classical Kozeny-Carman’s equation 

(Carman 1937) and experimental evidence on the applicability of this equation to geonets 

is provided by Giroud et al. (2000). 

The relative derivative of k GN with respect to the two variables, n GN and d GN ,is: 

dk 

dn 

d n 

GN 

3 2 

GN 

b1- 

GNg 

2d 

dGN 

= - 

+ 

(30) 

k n 1- 

n d 

hence: 

GN 

GN 

GN 

GN 

g 

2 

GN 

d 

2 

GN 

dkGN 

3 - nGN 

dnGN 

2dd 

= 

k 1- 

n n d 

GN 

F 

HG 

I 

+ 

GN KJ 

GN 

Both the porosity variation, dn GN , and the strand diameter variation, dd GN , are related 

to the amount of bentonite adhering to geonet strands. Therefore, dn GN and dd GN 

can be expressed as a function of an incremental amount of bentonite, dm GNs .Toestablish 

the relationships between dn GN ,dd GN ,anddm GNs , it is necessary to consider the 

area of geonet strands per unit area of geonet. This is the specific surface area per unit 

area of geonet given by the following equation (Giroud 1996): 

S 

a 

b 

41- n 

= 

d 

where t GN is the thickness of the geonet. 

An incremental mass of bentonite per unit area of geonet, dm GNs , corresponds to an 

incremental volume (per unit area of geonet) of bentonite coating on the geonet strands, 

V, of: 

GN 

GN 

g 

t 

GN 

GN 

GN 

GN 

(29) 

(31) 

(32) 

dV 

= 

dmGNs 

r 1- n 

where n c is the porosity of the bentonite coating on the geonet strands. 

B 

b 

c 

g 

(33) 



The incremental thickness of bentonite coating on the geonet strands, dt c , is equal 

to the volume of bentonite coating (per unit area of geonet) divided by the surface area 

(per unit area of geonet) of geonet strands, hence from Equation 33: 

dt 

c 

dm 

= 

r 1- n 


dt 

c 

b 

GNs 

g 

S 

B c a 

dmGNs 

dGN 

= 

4 1-n 1-n t 

r b gb g 

B c GN GN 

The increase in geonet strand diameter is twice dt c , hence: 

dd 

GN 

dGN 

dmGNs 

= 

2 1-n 1-n t 

r b gb g 

B c GN GN 

(34) 

(35) 

(36) 

The decrease in geonet porosity is equal to the volume occupied by the bentonite 

coating on the geonet strands per unit volume of geonet, hence: 

dn 

GN 


Sadt 

=- 

t 

GN 

c 

(37) 

dn 

GN 

dm 

=- 

r 1- n 

Combining Equations 31, 36, and 38 gives: 

b 

GNs 

g 

t 

B c GN 

dk 

⎛ 3 ⎞ 

GN 

dmGNs 

= ⎜2 

− ⎟ 

k ⎝ n ⎠ ρ ( 1−n )( 1−n ) t 

GN GN B c GN GN 

(38) 

(39) 

Knowing that n GN is less than 1, Equation 39 shows that dk GN /dm GNs is negative, 

which indicates that a migration of bentonite particles resulting in an increase of the 

amount of bentonite particles adhering to the geonet strands (dm GNs > 0) causes a decrease 

of geonet hydraulic conductivity (dk GN


t 

max 

= 

Q 

k 

GN 

(40) 

The relative derivative of t max with respect to k GN is: 

dt 

t 

max 

max 

dk 

=- 1 2 k 

GN 

GN 

(41) 


dt 

F 

max 

3 I dmGNs 

= -1 

tmax 

HG 

2 nGN 

KJ 

1-n 1-n t 

r b gb g 

B c GN GN 

Equation 42 shows that dt max /dm GNs is positive. Thus, an increase of the amount of 

bentonite particles adhering to geonet strands (dm GNs > 0) causes an increase of leachate 

thickness (dt max ) in the secondary leachate collection layer. With n GN =0.8,Ã B = 2,700 

kg/m 3 , n c = 0.93 (typical porosity of bentonite hydrated under unconfined conditions 

as shown in Appendix A, Equation A-7), and t GN =4.5mm=4.5× 10 -3 m, Equation 

42 gives: 

d tmax 

= 5144 . dm 

with d m in kg m 2 

GNs d GNs i 

(43) 

t 

max 

For example, Equation 43 gives dm GNs = 0.0194 kg/m 2 = 19.4 g/m 2 for a leachate 

thickness (or head) increase of 10% (i.e. dt max /t max =0.1),anddm GNs = 0.0019 kg/m 2 = 

1.9 g/m 2 for a leachate thickness (or head) increase of 1% (i.e. dt max /t max = 0.01). 

It is important to note that equations based on derivatives, such as Equations 42 and 

43, while they provide short and elegant demonstrations, are accurate only for small 

values of the considered increments (i.e. dt max and dm GNs ). The degree of approximation 

provided by such equations depend on the considered function and can only be evaluated 

by performing numerical calculations using the function itself. This requires lengthy 

calculations presented in Appendix E. These calculations show that Equations 42 and 

43 provide excellent approximations for dt max /t max up to 10%, which is satisfactory. 

4.3 Effect of Bentonite Particles Accumulating on the Secondary Liner 

Geomembrane 

If bentonite particles accumulate on the secondary liner geomembrane, they do not 

decrease the geonet hydraulic conductivity, but they decrease the geonet thickness 

available for leachate flow. The thickness of the layer formed by bentonite particles accumulated 

on the secondary liner geomembrane is inversely proportional to the geonet 

porosity, n GN , and to the bentonite dry density, Ã B (1 -- n B ), hence the following equation: 

t 

B 

= 

n 

mGNa 

1- n gr 

b 

GN B B 

(42) 

(44) 



where n B is the porosity of the bentonite in the layer formed by bentonite particles accumulated 

at the bottom of the geonet on the secondary liner geomembrane. Hereafter, 

t B is called the “effective thickness decrease” of the geonet. 

With n GN =0.8,n B = 0.93 (typical porosity of bentonite hydrated under unconfined 

conditions, as shown in Appendix A, Equation A-7), and Ã B = 2,700 kg/m 3 , Equation 

44 gives: 

d 

t = 0. 0066 m with t in m and m in kg m 2 

B GNa B GNa 

For example, if m GNa is 50 g/m 2 (0.05 kg/m 2 ), then t B = 0.00033 m = 0.33 mm, which 

is approximately 15 times less than a geonet thickness. 

Equation 44 can be written as follows: 

t 

B 

mGNa 

= 

(46) 

t n r 1- n t 

GN 

b 

g 

GN B B GN 

It should be noted that no derivation was necessary to develop Equation 46 (in contrast 

with the development of Equation 42) because Equation 44 is linear. However, for 

the sake of consistency with the equations related to other mechanisms, the differential 

notation is used below, with --dt GN instead of t B : 

dtGN 

dmGNa 

=- 

(47) 

t n r 1- n t 

GN 

b 

g 

GN B B GN 

With n GN =0.8,Ã B = 2,700 kg/m 3 , n B =0.93andt GN = 4.5 mm (4.5 × 10 -3 m), Equation 

47 becomes: 

dt 

t 

GN 

GN 

=-1470 . 

dm 


GNa 

For example, Equation 48 gives dm GNa = 0.068 kg/m 2 =68g/m 2 for an effective 

thickness decrease of 10%, and dm GNa = 0.0068 kg/m 2 =6.8g/m 2 for an effective thickness 

decrease of 1%. 

GNa 

i 

(45) 

(48) 

5 APPLICATION OF THE THEORETICAL ANALYSIS TO 

LABORATORY TESTING 

5.1 Need for Laboratory Testing 

While it is possible to theoretically quantify the detrimental consequences of bentonite 

particle migration, as shown in Sections 3 and 4, it is not possible to theoretically 

quantify the amount of bentonite particles likely to migrate through the geotextile, because 

there is no generally accepted theory to quantify the amount of cohesive soil particles 

passing through a filter. Therefore, laboratory testing is necessary. The theoretical 

analyses presented in Sections 3 and 4 provide a rational approach to establish a criterion 

necessary to interpret the laboratory tests, which is the purpose of the present paper. 

Therefore, it is important to discuss laboratory testing at this point. 


543


5.2 Bentonite Hydration 

5.2.1 Bentonite Hydration in the Field 

The behavior of bentonite and, consequently, the ability of particles to migrate depend 

to a large extent on the degree of hydration of the bentonite. In a GCL as delivered 

to a site, the bentonite is in a relatively dry state: its water content is typically 10 to 20%. 

If a GCL is hydrated, its water content raises to a value that depends on the amount of 

swelling undergone by the GCL during the hydration process. Swelling is limited by 

the overburden stress applied at the time of hydration. Under typical overburden 

stresses, the water content of hydrated bentonite is of the order of 150% (Giroud et al. 

1997b) and its porosity is then 0.8 (Appendix A). The bentonite is then saturated and 

its shear strength is less than at the initial water content. Therefore, to be conservative, 

testing should be conducted with saturated bentonite. 

For bentonite to be hydrated, there must be a water supply. This supply exists in the 

field at the location of holes in the geomembrane overlying the GCL: the leachate that 

leaks through the geomembrane holes and flows through the GCL hydrates the bentonite 

of the GCL. As indicated in Section 2.1, only a small fraction of a GCL is thus hydrated 

(e.g. 0.01 to 4% of the liner system surface area). The remainder of the GCL 

either remains non-hydrated or becomes progressively hydrated by absorbing water vapor 

from the humid air contained in the secondary leachate collection layer. To the best 

of the authors’ knowledge, there are no published data indicating that bentonite can become 

saturated by absorbing water vapor from the humid air and, if it becomes saturated, 

how much time is needed to achieve saturation. The important topic of absorption 

of water vapor from air by bentonite is discussed in Appendix F, where it is concluded 

that the bentonite in a GCL overlying a secondary leakage collection layer (Figure 1) 

is unlikely to become saturated by absorbing water vapor from the air in the secondary 

leachate collection layer, unless the air is saturated due to extensive leachate ponding 

in the secondary leachate collection layer, which normally should not happen. 

From the foregoing discussion, it appears that: (i) it is conservative to consider that 

the bentonite is saturated; and (ii) the bentonite of the GCL may not be saturated, except 

in relatively small areas associated with leakage through the geomembrane overlying 

the GCL. Therefore, to consider that the GCL is saturated over its entire surface area 

(which is implied in test interpretation) is conservative. 

5.2.2 Representative Conditions in a Laboratory Test 

A laboratory test that conservatively simulates the field conditions consists of measuring 

the amount of particles migrating through the lower geotextile of saturated GCL 

in contact with the underlying geonet under a normal stress that simulates the overburden 

load in the field, with or without liquid flow through the GCL. The case without 

flow corresponds to the larger portions of GCL in the field that are away from leaks 

through the geomembrane overlying the GCL, and the case with flow corresponds to 

the small portions of GCL in the field through which leachate that has passed through 

geomembrane holes is flowing. The fact that saturated bentonite is used in the test is 

conservative for the case where there is no leachate flow, as indicated in Section 5.2.1. 



The GCL should be hydrated and permeated with leachate equivalent to that anticipated 

in the field. It is important that the GCL be hydrated under a compressive stress 

equal to the overburden stress at the landfill construction or operation stage where the 

bentonite is likely to become hydrated. If not, the bentonite would swell freely and its 

water content could increase to values such as 500% or more, which would not be representative 

of field conditions. It should be noted that, by conducting the test at the 

maximum stress to be applied by the overburden materials, and with the GCL in contact 

with the underlying geonet, the effect of this stress on the porometry of the lower 

geotextile of the GCL, and the associated effect on bentonite migration, are accounted 

for in the test. 

5.3 Calculation of Representative Testing Time 

5.3.1 Testing Time 

The influence of time on the amount of bentonite particles likely to pass through the 

geotextile is not the same for the two mechanisms presented in Section 2.1: 

S In the case of the extrusion of saturated bentonite through the geotextile openings, 

the bentonite will migrate more easily if the load is applied more rapidly, because 

the bentonite will have less time to consolidate. Therefore, a laboratory test will be 

conservative if the load is applied more rapidly than in the field, which is usually 

the case since, in the field, load application (i.e. waste placement) takes months. 

S In the case of the migration of bentonite particles due to leachate flow, the effect of 

time is linked to the effect of hydraulic gradient. In order to decrease testing time 

to an acceptable value (i.e. a time much shorter than the time in the field, typically 

in the order of years, during which leachate may flow through a GCL), the flow rate 

through the GCL is typically much greater in the laboratory than in the field. This 

is achieved by using, in the laboratory test, a hydraulic gradient that is much greater 

than the hydraulic gradient in the field, since flow velocity is proportional to hydraulic 

gradient. It is well known from the theory of porous media that the drag forces 

applied by a liquid to the porous medium through which it is flowing are proportional 

to the hydraulic gradient. Therefore, the drag forces are greater in the laboratory test 

than in the field and, as a result, the bentonite particles are more likely to be dislodged 

by flow in the laboratory test than in the field. For a given flow velocity, the 

number of particles dislodged is likely to increase with time, i.e. with the flow volume, 

since many of the particles likely to move must wait for other particles to move 

first. It may, therefore, be assumed that, if two identical GCLs are exposed to the 

same total volume of flow, but different flow velocities, the flow that has the highest 

velocity is likely to dislodge more particles than the other. Therefore, if the laboratory 

test is conducted with the same flow volume as in the field, it is likely more conservative 

than the field situation (i.e. more bentonite particles are likely to move in 

the laboratory test than in the field) because the flow velocity is greater in the laboratory 

test than in the field. 


545


5.3.2 Direct Calculation of Representative Testing Time 

The amount of leachate that passes per unit of time and per unit area of GCL in the 

field can be calculated by dividing the leakage rate expressed using Equation 4 by the 

surface area where the flow takes place in the GCL, which may be calculated using the 

following equation (Giroud et al. 1997b): 

A = 0. 

205d h k 

02 . 09 . -026 

. 

w FIELD GCL 

(49) 

where h FIELD is the leachate head on top of the geomembrane overlying the GCL in the 

field. 

Dividing Equation 4 by Equation 49 gives: 

QFIELD 

095 . 

= kGCL 1+ 

01 . bhFIELD tGCLg 

(50) 

A 

w 

where Q FIELD is the rate of leakage through a defect in the composite liner in the field. 

Equation 50 could have been derived from the classical Darcy’s equation since the 

term in brackets in Equations 4 and 50 is the average hydraulic gradient in the GCL as 

pointed out by Giroud (1997). 

The total amount (i.e. volume V FIELD ) of leachate that passes through the area A w of 

the GCL in the field is obtained by multiplying the leakage rate, Q FIELD , by the total 

duration of the flow in the field, t FIELD 

: 

VFIELD = QFIELD tFIELD 

(51) 

Therefore, the total amount of leachate passing through a unit area of GCL (associated 

with a geomembrane leak) in the field is: 

VFIELD 

095 . 

= kGCL 1+ 

01 . bhFIELD t t 

GCLg 

FIELD 

(52) 

A 

w 

The right and left sides of Equation 52 have the dimension of a length and can be 

designated as “the height of the column of leachate that passes through the portion of 

GCL associated with a geomembrane leak”, hence: 

H 

FIELD 

VFIELD 

095 . 

= = k 1+ 

01 . h t t 

A 

w 

GCL FIELD GCL FIELD 

where H FIELD is the height of the leachate column that passes through the portion of GCL 

associated with a geomembrane leak over the time t FIELD 

. 

It is interesting to note that this column height depends only on the hydraulic conductivity 

of the GCL, its thickness, the head of liquid on top of the geomembrane, and the 

duration of the flow in the field. It does not depend on the size of the geomembrane hole 

(which has the same influence on leakage rate and on the size of the area of GCL where 

flow takes place, according to Equations 4 and 49). 

If the leachate head on top of the geomembrane in the field, h FIELD , is not a constant, 

Equation 53 can be used with an average value of h FIELD ; alternatively, a more complex 

b 

g 

(53) 



equation given in Appendix G can be used. Appendix G provides a set of equations for 

cases where the leachate head on top of the geomembrane in the field, h FIELD , and/or the 

leachate head on top of the geomembrane in the laboratory test, h LAB , are not constant. 

As indicated above, the total volume of flow per unit area (i.e. the height of the “leachate 

column”) in the laboratory test should be the same as in the field. The height of 

the leachate column, H LAB , or total volume of flow per unit area in the laboratory test 

is given by Darcy’s equation as follows: 

H 

LAB 

VLAB 

= = k i t 

A 

GCL 

GCL LAB LAB 

(54) 

where: V LAB = total volume of flow in the laboratory test; A GCL = surface area of the GCL 

specimen in the laboratory test; and t LAB 

= duration of the laboratory test. The hydraulic 

gradient in the laboratory test is: 

1 b g 

i = + h t 

LAB LAB GCL 

(55) 


H 

LAB 

VLAB 

= = k + h t t 

A 

GCL 

1 b g 

GCL LAB GCL LAB 

(56) 

The requirement that the flow volume per unit area (or leachate column) be the same 

in the laboratory test and in the field is expressed as follows: 

H 

LAB 

VLAB 

V 

= = HFIELD 

= 

A 

A 

GCL 

FIELD 

w 

(57) 

Combining Equations 52, 57, and 54 or 56 gives: 

t 

LAB 

( ) ( ) 

i 1+ 

( h t ) 

⎡1 + 0.1 h t ⎤t ⎡1 + 0.1 h t ⎤t 

= 

⎣ ⎦ 

= 

⎣ ⎦ 

0.95 0.95 

FIELD GCL FIELD FIELD GCL FIELD 

LAB LAB GCL 

(58) 

An example of use of Equation 58 is given in Section 5.3.4. 

5.3.3 Calculation of Representative Testing Time Using the Pore-Volume Approach 

Traditionally, this column height (Equation 53) is expressed in terms of pore volumes, 

or more accurately, pore volumes per unit area. The pore volume per unit area 

of GCL (i.e. the “pore height” of the GCL) is equal to the thickness of the bentonite layer 

in the GCL multiplied by the porosity of the bentonite in the GCL: 

H 

P 

VP 

= = n 

A 

w 

GCL 

t 

GCL 

(59) 

where V P is the pore volume in the portion of surface area A w of the GCL. 

Dividing Equation 53 by Equation 59 gives: 


547


N 

FIELD 

b 

095 . 

GCL FIELD GCL FIELD 

k 1+ 

01 . h t t 

= 

n t 

GCL 

GCL 

g 

(60) 

where N FIELD is the number of pore volumes corresponding to the liquid flow in the field 

through the area of GCL associated with a geomembrane hole during time t FIELD 

,the 

time during which leachate flow is expected to occur in the field. For example, 

t FIELD 

may be the time elapsing between the beginning of waste landfilling and the 

completion of the landfill final cover. 

For example, considering a flow duration of 10 years in the field, a thickness of the 

bentonite layer in the GCL of 7 mm, a bentonite porosity in the GCL of 0.8, a hydraulic 

head on top of the geomembrane of 5 or 300 mm, and a GCL hydraulic conductivity 

of 1 × 10 -11 or 5 × 10 -11 m/s, the values of pore volumes presented in Table 1 are obtained 

using Equation 60. 

The condition expressed by Equation 59 can also be expressed by saying that the 

leachate flow in the laboratory test must correspond to the same number of pore volumes 

as the leachate flow in the field: 

N 

LAB 

= N 

FIELD 

(61) 

The “leachate column” in the laboratory, H LAB , can be expressed in terms of pore 

volume by dividing the expressions of H LAB by n GCL t GCL (Equation 54), hence: 

N 

LAB 

k i t k + h t t 

GCL LAB LAB 

= = 

n t 

n t 

GCL 

GCL 

1 b g 

GCL LAB GCL LAB 

It should be noted that the use of pore volumes is not practical because it requires 

two more parameters, n GCL and k GCL (Equation 62), than the equation that gives directly 

the required duration of the laboratory test (i.e. Equation 58). This is illustrated by the 

following example. 

GCL 

GCL 

(62) 

Table 1. 

Numbers of pore volumes corresponding to 10 years of flow in the field. 

Liquid head on top of the geomembrane 

overlying the GCL, h (mm) 

Hydraulic conductivity of theGCL,k k GCL 

1 × 10 -11 m/s 

(1 × 10 -9 cm/s) 

5 × 10 -11 m/s 

(5 × 10 -9 cm/s) 

5 0.6 3.0 

300 2.6 12.8 

Note: The tabulated numbers of pore volumes were calculated using Equation 60 with n GCL =0.8,t GCL = 

7 mm, and t FIELD = 10 years. 



5.3.4 Example of Calculation of Representative Testing Time 

Example 1. The duration of leachate flow in the field is 10 years (3.154 × 10 8 s), the 

average leachate head on top of the geomembrane in the field is 3 mm during this time, 

and the thickness of the bentonite layer in the GCL is 7 mm. Calculate the minimum 

testing time to ensure that the test is representative. 

Equation 58 provides a relationship between the duration of leachate flow in the 

field and the required duration of the laboratory test, as a function of the hydraulic gradient 

in the test. Assuming a hydraulic gradient of 1000, Equation 58 gives: 

t LAB 

= 

095 . 

8 

b g d i 

1+ 01 . 3 7 3154 . ¥ 10 

1000 

5 

= 33 . ¥ 10 s= 

38 . days 

If one wants to use the pore volume, it is necessary to assume a value of k GCL (e.g. 

1 × 10 -11 m/s) and a value of n GCL (e.g. 0.8) to use Equation 60: 

−11 095 . 

8 

d1× 10 i 1+ 01 . b3 7g d3154 . × 10i 

N FIELD 

= 

= 059 . 

−3 

08 . 7× 

10 

b gd 

Then the required duration of the laboratory test can be calculated using Equation 

62 with N LAB = N FIELD (Equation 61) as follows: 

−3 

b059 . gb08 . 7× 

10 

t LAB 

gd i 

5 

= 

= 33 . × 10 s = 3.8 days 

−11 

1× 

10 1000 

d ib g 

The above example confirms that using the pore-volume approach involves more 

calculations than calculating directly the time required for the laboratory test. However, 

using the pore-volume approach is appropriate if there is a difference in one or several 

of the GCL characteristics (t GCL , n GCL , k GCL ) between the field conditions and the 

laboratory conditions. 

The above example also shows that the laboratory test may take a long time: several 

days for a small number of pore volumes and several weeks for a larger number of pore 

volumes. 

i 

ENDOF EXAMPLE1 

5.4 Criterion of Acceptable Bentonite Loss 

5.4.1 Summary of Theoretical Analyses 

Three effects of bentonite loss from the GCL and the resulting migration of bentonite 

particles into the secondary leachate collection layer were analyzed: 

S the effect of bentonite loss on the primary liner (Section 3); and 


549


S two effects of bentonite migration into the geonet secondary leachate collection layer: 

(i) the effect of bentonite particles adhering to geonet strands (Section 4.2); and 

(ii) the effect of a layer of bentonite particles accumulating on the secondary liner 

(Section 4.3). 

Quantifications of these three effects based on the analyses presented in Sections 3 

and 4 can be summarized as follows: 

S Effect of bentonite loss on the primary liner (from Equation 24): 

dQ 

2 

0.5dmL 

( with d mL 

in kg m ) 

(63) 

Q 

≈ 

S Effect of bentonite migration into the geonet secondary leachate collection layer: 

S bentonite particles adhering on geonet strands (from Equation 43): 

dt 

t 

max 

max 

= 

2 

( mGNs 

) 

5.144dm 

with d in kg m 

GNs 

(64) 

S bentonite particles accumulating on the secondary liner (from Equation 48): 

dt 

t 

GN 

GN 

=-1 470dm 


. 

GNa d GNa i 

(65) 

5.4.2 Impact of Bentonite Particle Migration on Geonet Liquid Collection Layer 

Performance 

Bentonite particles that pass through the lower geotextile of the GCL have two 

choices (assuming conservatively that they do not migrate downslope in the secondary 

leachate collection layer): they may adhere to the geonet strands, or they may accumulate 

in the geonet on the secondary liner geomembrane, but they cannot do both. Hence: 

dm = P dm 

GNs s GN 

dm = P dm 

GNa a GN 

(66) 

(67) 

where: P s = probability that a bentonite particle migrating into the geonet secondary 

leachate collection layer will adhere to geonet strands; and P a = probability that a bentonite 

particle migrating into the geonet secondary leachate collection layer will accumulate 

on the secondary liner. If it is assumed that bentonite particles do not migrate 

downslope in the secondary leachate collection layer: 

P + P =1 

s 

a 

(68) 

A relative increase in leachate thickness in the secondary leachate collection layer 

(dt max /t max ) and a relative decrease in geonet secondary leachate collection layer thickness 

(--dt GN /t GN ) are equivalent with respect to the performance of the secondary lea- 



chate collection layer. Therefore, in subsequent calculations, these two quantities will 

be treated generically as dt/t. Accordingly, combining Equations 64 to 68 gives: 

dt 

= 5144 . b1 - Pag 

+ 1. 

470 Pa dmGN 

(69) 

t 

As shown in Appendix H: 

P 

a 

≥ R 

GN 

(70) 

where R GN is the geonet relative open area. 

Since 5.144 > 1.470, it is conservative to use: 

P 

a 

= R 


dt 

= b5144 . -3. 

674 R 

t 

GN 

GN 

g 

dm 

GN 

(71) 

(72) 

According to Appendix H, in the case of a typical geonet: 

R GN 

= 0. 

556 


dt 

= 31 dm 

with m in kg m 3 

GN 

GN 

t 

. d i 

(73) 

(74) 

Equation 74 was used to establish Table 2 that gives a relationship between the 

amount of bentonite particles migrating into the geonet and the decrease in geonet secondary 

leachate collection layer performance. Based on Table 2, it is recommended to 

use m GN =10g/m 2 as a criterion for the test. This will ensure that the decrease in geonet 

performance will be less than 3.1%, a small value that is likely to be acceptable. 

Table 2. Relationship between the mass of migrating bentonite particles per unit area and 

the impact of bentonite particles on the secondary leachate collection layer performance. 

Amount of bentonite particles 

migrating into the geonet, m GN 

(g/m 2 ) 

Impact on secondary leachate 

collection layer performance, dt/t 

(%) 

3.23 1.0% 

6.45 2.0% 

9.68 3.0% 

10.00 3.1% 

12.90 4.0% 

16.13 5.0% 

Note: The tabulated values of dt/t were calculated using Equation 74 after converting m GN into kg/m 2 . 


551


5.4.3 Impact of Loss of Bentonite Particles on Liner Performance 

The impact on primary liner performance of a loss of bentonite particles consistent 

with the criterion proposed in Section 5.4.2 can be evaluated using Equation 63. If there 

is no migration of bentonite particles downslope in the geonet secondary leachate 

collection layer and if there are no bentonite particles entrapped in the lower geotextile 

of the GCL, according to Equation 27: 

m 

L 

= m 

GN 

(75) 

and Equation 63 becomes: 

dQ 

Q 

≈ 

0.5 dm 

GN 

(76) 

With the proposed criterion of dm GN =10g/m 2 , Equation 76 gives: 

dQ 

Q 

≈ 

0.5% 

The impact on the primary liner performance (i.e. a variation in flow rate through 

the primary liner of 0.5%) is very small. However, if bentonite particles are entrapped 

in the lower geotextile of the GCL, Equation 76 becomes the following, in accordance 

with Equation 27: 

dQ 

0.5 d 

GN 

d 

Q ≈ + 

( m m ) 

GT 

(77) 

As discussed in Section 3.5, dm GT may be as high as 100 g/m 2 . With dm GN =10g/m 2 

and dm GT = 100 g/m 2 , Equation 77 gives: 

dQ 

Q 

≈ 

5.5% 

This indicates that even if a large amount of bentonite particles (100 g/m 2 ) migrates 

into the lower geotextile of the GCL, the impact of bentonite loss from the GCL (110 

g/m 2 including 100 g/m 2 into the geotextile and 10 g/m 2 into the geonet) on the primary 

liner performance is small (i.e. a variation in flow rate of 5.5%). Furthermore, the actual 

value of dQ/Q may be less than 5.5% because, as noted at the end of Section 3.5, the 

bentonite entrapped in the lower geotextile of the GCL is not entirely lost and contributes 

to the performance of the composite liner. This shows that even in the worst case 

where a large amount of bentonite particles are accumulated in the lower geotextile of 

the GCL, the impact on the primary liner performance of an amount of bentonite migration 

detected in a laboratory test equal to the proposed criterion is small. 



5.4.4 Proposed Criterion 

In conclusion, the proposed criterion is that the amount of bentonite particle loss detected 

in a representative test be less than 10 g/m 2 . Since conservative assumptions are 

made in all demonstrations, no factor of safety is required on the proposed criterion. 

5.4.5 Representative Testing 

The criterion presented in Section 5.4.4 is applicable to the test described in Section 

5. To be representative, the laboratory test should be conducted under the following 

conditions: 

S The GCL must be hydrated under a normal stress equal to the overburden stress at 

the landfill construction or operation stage where the bentonite is likely to become 

hydrated, as discussed in Section 5.2. 

S The test must be conducted under a normal stress equal to the maximum stress to 

be applied on the GCL in the field, as discussed in Section 5.2. 

S The duration of the test can be calculated directly using Equation 58 (Section 5.3.2). 

The same result is obtained using Equations 60 to 62 that follow the pore-volume 

approach. In the calculation to determine the test duration required to ensure that the 

test is representative of the field conditions, the entire duration of the expected leachate 

flow in the field should be considered. 

6 CONCLUSION 

The present paper provides a detailed analysis of the mechanisms and consequences 

of bentonite loss from the GCL component of a composite primary liner of a double 

liner system. Bentonite loss may affect the performance of the composite primary liner, 

and the resulting bentonite migration into the geonet may affect the performance of the 

secondary leachate collection layer. Based on the results of the analysis, a criterion for 

acceptable bentonite migration is proposed. This criterion can be used to evaluate the 

results of laboratory testing performed to estimate the amount of bentonite migration 

into the geonet likely to occur in the field. The criterion sets the limit for acceptable 

bentonite migration into the geonet secondary leachate collection layer beneath the 

GCL at 10 g/m 2 . The analysis demonstrates that, if this criterion is met, the expected 

performances of the composite primary liner and the geonet secondary leachate collection 

layer are not significantly affected by the bentonite migration. 

ACKNOWLEDGMENTS 

The present paper was inspired by an actual situation faced by the authors. The authors 

would like to acknowledge Waste Management for supporting the preparation of 

the report upon which this paper is based. The support of GeoSyntec Consultants for 

the preparation of the present paper is acknowledged. The authors are grateful to D.E. 


553


Daniel who suggested the approach and provided detailed information for Appendix F. 

In addition, the authors are grateful to B.A. Gross for her review of Appendix F and to 

M.A. Othman and L.G. Tisinger for providing useful information and K.E. Holcomb, 

S.L. Berdy, R. Ortiz, and J.A. Simons for assistance during the preparation of the present 

paper. 

REFERENCES 

Carman, P.C., 1937, “Fluid Flow Through Granular Beds”, Transactions of the Institution 

of Civil Engineers, Vol. 15, pp. 150-166. 

Daniel, D.E., Shan, H.Y., and Anderson, J.D., 1993, “Effects of Partial Wetting on the 

Performance of the Bentonite Component of a Geosynthetic Clay Liner”, Proceedings 

of Geosynthetics ’93, IFAI, Vol. 3, Vancouver, British Columbia, Canada, March 

1993, pp. 1483-1496. 

Daniel, D.E., 2000, Personal Communication on Water Absorption by Bentonite, May 

2000. 

Fox, P.J., Triplett, E.J., Kim, R.H., and Olsta, J.T., 1998, “Field Study of Installation 

Damage for Geosynthetic Clay Liners”, Geosynthetics International, Vol. 5, No. 5, 

pp. 492-520. 

Giroud, J.P., 1996, “Granular Filters and Geotextile Filters”, Proceedings of GeoFilters 

’96, Lafleur, J. and Rollin, A.L., Editors, Montréal, Quebec, Canada, May 1996, pp. 

565-680. 

Giroud, J.P., 1997, “Equations for Calculating the Rate of Liquid Migration Through 

Composite Liners Due to Geomembrane Defects”, Geosynthetics International, Vol. 

4, Nos. 3-4, pp. 335-348. 

Giroud, J.P. and Perfetti, J., 1977, “Classification des textiles et mesure de leurs proprietes 

en vue de leur utilisation en geotechnique”, Proceedings of the International 

Conference on the Use of Fabrics in Geotechnics, Session 8, Paris, April 1977, pp. 

345-352. (in French) 

Giroud, J.P., Gross, B.A., Bonaparte, R., and McKelvey, J.A., 1997a, “Leachate Flow 

in Leakage Collection Layers Due to Defects in Geomembrane Liners”, Geosynthetics 

International, Vol. 4, Nos. 3-4, pp. 215-292. 

Giroud, J.P., Rad, N.S., and McKelvey, J.A., 1997b, “Evaluation of the Surface Area 

of a GCL Hydrated by Leachate Migrating Through Geomembrane Defects”, Geosynthetics 

International, Vol. 4, Nos. 3-4, pp. 433-462. 

Giroud, J.P., Zhao, A., and Richardson, G.N., 2000, “Effect of Thickness Reduction on 

Geosynthetic Hydraulic Transmissivity”, Geosynthetics International, Special Issue 

on Liquid Collection Systems, Vol. 7, Nos. 4-6, pp. 433-452. 

Lange, A.R.G., 1967, “Osmotic Coefficients and Water Potentials of Sodium Chloride 

Solutions From 0 to 40 degrees C”, Australian Journal of Chemistry, Vol. 20, pp. 

2017-2023. 



Rawlings, S.L. and Campbell, G.S., 1986, “Water Potential: Thermocouple Psychrometry”, 

in Methods of Soil Analysis, Part 1, Physical and Mineralogical Methods, A. 

Lkute, Editor, American Society of Agronomy, Madison, Wisconsin, USA, pp. 

597-618. 

NOTATIONS 

This notation list includes symbols used in the Appendices. Basic SI units are given 

in parentheses. 

A GCL = surface area of GCL specimen in laboratory test (m 2 ) 

A GT = surface area of geotextile (m 2 ) 

A w = surface area where leakage flow takes place in a GCL (m 2 ) 

C qo = contact quality factor for circular hole (dimensionless) 

C qo good = value of C qo in the case of good contact conditions (dimensionless) 

C qo poor = value of C qo in the case of poor contact conditions (dimensionless) 

D = depth of leachate in secondary leachate collection layer (m) 

d = hole diameter (m) 

d B = diameter of bentonite particle (m) 

d GN = diameter of geonet strands (m) 

d′ GN 

= diameter of geonet strands including bentonite layer (m) 

g = acceleration due to gravity (m/s 2 ) 

H FIELD = height of leachate column that passes through the portion of GCL 

associated with a geomembrane leak (m) 

H LAB = height of leachate column or total volume of flow per unit area in 

laboratory test (m) 

H P = “pore height” (m) 

H r = relative humidity (dimensionless) 

h = liquid head on top of geomembrane (m) 

h FIELD = leachate head on top of geomembrane overlying GCL in the field (m) 

h FIELDi = leachate head on top of geomembrane overlying GCL during time t FIELD 

(m) 

h LAB = leachate head in laboratory test (m) 

h LABi = hydraulic head in laboratory test during time t LABi 

(m) 

i = hydraulic gradient (dimensionless) 

i L = hydraulic gradient in secondary leachate collection layer 

(dimensionless) 

i LAB = hydraulic gradient in laboratory test (dimensionless) 

i LABi = hydraulic gradient in laboratory test during time t LABi 



555


k GCL = GCL hydraulic conductivity (m/s) 

k GN = geonet hydraulic conductivity (m/s) 

k L = hydraulic conductivity of secondary leachate collection layer material 

(m/s) 

k′ GN 

= hydraulic conductivity of geonet with bentonite on geonet strands (m/s) 

L = length of secondary leachate collection layer slope (m) 

M = molecular mass (kg mol -1 ) 

m FL = mass per unit area of bentonite flowing in suspension in secondary 

leachate collection layer (kg/m 2 ) 

m GN = mass per unit area of bentonite accumulated in geonet secondary leachate 

collection layer (kg/m 2 ) 

m GNa = mass per unit area of particles accumulating in geonet on secondary liner 

(kg/m 2 ) 

m GNs = mass per unit area of bentonite particles adhering to geonet strands 

(kg/m 2 ) 

m GT = mass per unit area of bentonite entrapped in lower geotextile of GCL 

(kg/m 2 ) 

m L = mass per unit area of bentonite lost from GCL (kg/m 2 ) 

m MIGR = mass per unit area of bentonite migrating from GCL into secondary 

leachate collection layer (kg/m 2 ) 

N FIELD = number of pore volumes per unit area corresponding to liquid flow 

through area of GCL with geomembrane hole during time t FIELD 


N LAB = number of pore volumes per unit area corresponding to liquid flow 

through GCL specimen in laboratory test (dimensionless) 

n B = porosity of bentonite in layer formed by bentonite particles accumulated 

at bottom of geonet on secondary liner geomembrane (dimensionless) 

n GCL = porosity of bentonite in GCL (dimensionless) 

n GN = geonet porosity (dimensionless) 

n GT = geotextile porosity (dimensionless) 

n L = porosity of secondary leachate collection layer material (dimensionless) 

n c = porosity of bentonite coating on geonet strands (dimensionless) 

n h = porosity of hydrated bentonite (dimensionless) 

n′ GN 

= geonet porosity accounting for presence of bentonite layer on strands 


O GN = distance between geonet strands (m) 

P a = probability that a bentonite particle migrating into a geonet secondary 

leachate collection layer will accumulate on the secondary liner 


P s = probability that a bentonite particle migrating into a geonet secondary 

leachate collection layer will adhere on geonet strands (dimensionless) 



p s = saturated water vapor pressure (Pa) 

Q = rate of leakage through defect in composite primary liner, in particular, 

rate of leakage through circular hole in geomembrane underlain by GCL 

(m 3 /s) 

Q FIELD = value of Q in the field (m 3 /s) 

R = ideal gas constant (J_K -1 mol -1 ) 

R GN = geonet relative open area (dimensionless) 

R w rand = wetted fraction of surface area of secondary leachate collection layer 


S E = elementary surface area of geonet hole including one strand in each of 

two directions (m 2 ) 

S H = surface area of geonet hole (m 2 ) 

S a = specific surface area per unit area of geonet (dimensionless) 

s = “water potential”/suction (Pa) 

T = absolute temperature (_K) 

T C = temperature (_C) 

t = thickness of leachate in secondary leachate collection layer (m) 

t B = thickness of layer formed by bentonite particles accumulated on 

secondary liner (m) 

t GCL = thickness of bentonite layer in GCL (m) 

t GN = thickness of geonet (m) 

t GT = thickness of geotextile (m) 

t L = thickness of secondary leachate collection layer (m) 

t avg rand = average depth of leachate in secondary leachate collection layer (m) 

t c = thickness of bentonite coating on geonet strands (m) 

t max = maximum thickness of leachate in secondary leachate collection layer 

(m) 

t′ max 

= maximum thickness of leachate in secondary leachate collection layer 

when geonet hydraulic conductivity is k′ GN 

(m) 

t FIELD 

= time during which flow is expected to occur in the field (s) 

t FIELDi 

= time during which leachate head on top of geomembrane overlying GCL 

is h FIELDi (s) 

t L 

= time for leachate to flow along secondary leachate collection layer slope 

(s) 

t LAB 

= time required for flow in laboratory, i.e. duration of laboratory test (s) 

t LABi 

= time during which hydraulic gradient is i LABi and hydraulic head is h LABi 

in laboratory test (s) 

t v = time for vertical movement of bentonite particle in leachate (s) 

V = volume (per unit area of geonet) of bentonite coating on geonet strands (m) 


557


V FIELD = total volume of leachate that passes through area A w of GCL in the field 

(m 3 ) 

V LAB = total volume of flow in laboratory test (m 3 ) 

V P = pore volume in portion of surface area A w of GCL (m 3 ) 

V v = volume of voids in geotextile (m 3 ) 

v L = velocity of flow along slope (m/s) 

v V = vertical velocity of bentonite particle (m/s) 

w = bentonite water content (dimensionless) 

w h = water content of hydrated bentonite (dimensionless) 

w o = initial water content (dimensionless) 

x rand = horizontal distance between primary liner defect and low end of leakage 

collection layer slope in random scenario (m) 

z L = maximum height of landfilled material above considered GCL (m) 

β = slope of secondary leachate collection layer (°) 

η w = viscosity of leachate assumed to be equal to that of water (kg/(m⋅s)) 

λ = constant depending on acceleration due to gravity, density of liquid, and 

viscosity of liquid (Equations 14 and 29) (m -1 s -1 ) 

λ rand = coefficient used in Appendix C (dimensionless) 

μ = coefficient used in Appendix C (dimensionless) 

μ GT = mass per unit area of geotextile (kg/m 2 ) 

μ d = mass per unit area of dry bentonite (kg/m 2 ) 

μ o = mass per unit area of bentonite with initial water content (kg/m 2 ) 

Ã B = density of bentonite particles (kg/m 3 ) 

Ã F = density of geotextile fibers (kg/m 3 ) 

Ã L = average density of landfilled material (including waste and soil layers) 

(kg/m 3 ) 

Ã d = dry density of bentonite (mass of bentonite particles divided by volume 

occupied by bentonite) (kg/m 3 ) 

Ã w = density of water (also, density of leachate assumed to be equal to that of 

water) (kg/m 3 ) 

θ = angle between geonet strands of two different layers (°) 

σ = compressive stress applied to GCL (Pa) 



APPENDIX A 

AMOUNT OF BENTONITE PARTICLES CONTAINED IN A 

NEEDLE-PUNCHED NONWOVEN GEOTEXTILE 

The volume of voids per unit area of geotextile, V v /A GT , is equal to the geotextile 

thickness multiplied by its porosity: 

Vv AGT = tGT nGT 

(A-1) 

The following classical relationship exists between the thickness of a geotextile, 

t GT , its porosity, n GT , its mass per unit area, μ GT , and the density of fibers, Ã F (Giroud 

and Perfetti 1977): 

t 

GT 

m 

GT 

= 

r 1- 

n 

F 

b 

GT 

g 

(A-2) 

Combining Equations A-1 and A-2 gives the volume of voids per unit area of geotextile 

as follows: 

V 

v 

A 

GT 

m 

GT 

n 

= 

r 1- 

n 

F 

b 

GT 

GT 

g 

(A-3) 

The dry density of bentonite (i.e. the mass of bentonite particles divided by the volume 

occupied by the bentonite) is given by the following classical relationship: 

b 

r = r 1 -n 

d B B 

g 

(A-4) 

where: Ã B = density of bentonite particles; and n B = porosity of bentonite. 

Multiplying the volume of voids per unit area (Equation A-3) by the dry density of 

bentonite (Equation A-4) gives the mass per unit area of bentonite particles that can be 

entrapped in the geotextile: 

m 

GT 

m 

= 

r 1- 

n 

r 1- 

n 

n 

GT B B GT 

F 

b 

b 

GT 

g 

g 

(A-5) 

Numerical values calculated using Equation A-5 are very sensitive to the value of 

the geotextile porosity, n GT . Porosities of needle-punched nonwoven geotextiles range 

typically between 0.85 and 0.92 under zero compressive stress. However, the porosity 

of needle-punched nonwoven geotextiles significantly decreases with increasing compressive 

stresses. According to Giroud (1996), the porosity of a needle-punched nonwoven 

geotextile having a porosity of 0.90 under no stress is 0.74 under a compressive 

stress of 250 kPa and 0.69 under a compressive stress of 500 kPa. 

Numerical values calculated using Equation A-5 are also very sensitive to the value 

of bentonite porosity, n B . Since bentonite particles move individually into the geotextile 

where they are free to hydrate under negligible compressive stress, it is logical to 

assume that the water content of the bentonite inside the geotextile has a high value, 


559


such as 500%, that corresponds to hydration under zero stress. As indicated by Giroud 

et al. (1997b), the relationship between the water content and porosity of hydrated bentonite 

is: 

w 

h 

= 

nh 

r 

w 

b1 

- nh 

r 

B 

g 

(A-6) 

where: w h = water content of hydrated bentonite; n h = porosity of hydrated bentonite; 

and Ã w = density of water. 

Equation A-6 can be written as follows: 

n 

h 

= 

b 

r 

w 

r 

h 

+ w 

w B h 

(A-7) 

Equation A-7 gives n h =0.80forw h = 1.5 (150%), a value used in the main text of 

the present paper for confined conditions, and n h =0.93forw h = 5.0 (500%) (unconfined 

conditions, as mentioned above). 

Using n B = n h = 0.93, Ã B = 2700 kg/m 3 , Ã F = 910 kg/m 3 ,andμ GT = 0.2 kg/m 2 (200 

g/m 2 ), Equation A-5 gives m GT = 0.118 kg/m 2 (118 g/m 2 )forn GT = 0.74 (which corresponds 

to an overburden stress of 250 kPa) and m GT = 0.092 g/m 2 (92 g/m 2 )forn GT = 

0.69 (which corresponds to an overburden stress of 500 kPa). If the final compressive 

stress is 500 kPa, it is not known if the extrusion and migration of bentonite particles 

will take place at that stress level or at a lower stress level. Therefore, only an order of 

magnitude, such as 100 g/m 2 , should be considered instead of more precise values such 

as 92 or 118 g/m 2 . This amount of bentonite particles should be regarded as a maximum 

amount that can be entrapped in a 200 g/m 2 needle-punched nonwoven geotextile under 

the considered overburden stress, because it is unlikely that bentonite particles will be 

able to occupy the entire void space of the geotextile considering that this void space 

is tortuous and some portions of it are difficult to access. Nevertheless, 100 g/m 2 is a 

large value compared to the other values of mass per unit area of bentonite particles discussed 

in the present paper. 

g 



APPENDIX B 

FRACTION OF THE SECONDARY LEACHATE COLLECTION LAYER 

WETTED BY LEACHATE LEAKING THROUGH THE PRIMARY LINER 

The rate of leakage through a hole in a geomembrane underlain by a GCL can be 

calculated using Equation 4 in the main text of the present paper. The following conservative 

(i.e. large) values are considered: hole diameter, d = 10 mm; leachate head on 

top of the primary liner, h = 0.3 m; and GCL hydraulic conductivity, k GCL =1× 10 -11 

m/s. With a thickness of the bentonite layer in the GCL of t = 7 mm, Equation 4 gives: 

− 

Q = 0. 205 1+ 0. 1 0. 3 0. 007 10 × 10 0. 

3 1× 

10 

Q = 9. 

106 × 10 

−10 

b g d i b g d i 

m 

3 

s 

095 . 3 02 . 09 . −11 074 . 

Assuming the leachate collection layer has a length of 50 m and a slope of 2%, Equation 

111 from Giroud et al. (1997a) gives: 

µ = 

−10 

9106 . × 10 01 . 

= 954 . × 10 

50 0. 

02 

b gb 

Using Equation 123 from Giroud et al. (1997a): 

d 

L 

i 

F 

NM 

HG 

2 

− 

λ rand 

= × + 

15 954 10 1 2 

. 

954 . × 10 

g 

5 3 −5 

52 

I K J − 

−5 

O 

P 

Q 

2 

P = 7. 

4 × 10 

Assuming a geomembrane hole frequency of five per hectare, Equation 122 from 

Giroud et al. (1997a) gives the “wetted fraction” of the surface area of the secondary 

leachate collection layer as follows: 

d ib gb g 

−3 2 −3 

R w rand 

= 7. 4 × 10 5 10, 000 50 = 9. 3 × 10 = 0. 

93% 

It appears that less than 1% of the surface area of the secondary leachate collection 

layer is wetted by leachate leaking through the composite primary liner if there are five 

holes per hectare. 

−3 


561


APPENDIX C 

DEPTH OF LEACHATE FLOW IN THE SECONDARY LEACHATE 

COLLECTION LAYER 

Leachate flow is due to leachate that leaks through a hole in the primary liner geomembrane. 

Two cases are considered: (i) the case of the geomembrane-GCL composite 

primary liner; and (ii) the case of a geomembrane primary liner, which is a conservative 

case that occurs if, for some reason, the GCL is not present or all the bentonite of the 

GCL, at the location of the considered geomembrane hole, has migrated. 

In the case of the geomembrane-GCL composite primary liner, the rate of leakage 

has been calculated in Appendix B: Q =9.11× 10 -10 m 3 /s for a geomembrane hole diameter 

of 10 mm and a leachate head on top of the geomembrane of 0.3 m. The average 

depth of leachate in the wetted zone can then be calculated using the following equation 

(Giroud et al. 1997a): 

t 

avg rand 

( ) + ( ) 

⎡ 53 152 xrand 

sinβ 

kL 

Q⎤Lsinβ 

= 

⎣ 

⎦ 

1+ 2 sin −2 

( L β k ) 52 

L 

Q 

(C-1) 

where: L = length of the secondary leachate collection layer; β = slope of the secondary 

leachate collection layer; k L = hydraulic conductivity of the secondary leachate collection 

layer; x rand = horizontal distance between the primary liner defect and the low end 

of the leakage collection layer slope in the random scenario; and: 

L 

F 

NM 

HG 

53 

xrand m 

= 1+ 

23 

L 10 2 

e 

j 

2 

m 

52 

I 

- 

KJ 

2 

O 

QP 

23 

m 

- 

2 

(C-2) 

Using the value of μ =9.54× 10 -5 calculated in Appendix B, Equation C-2 gives: 

Assuming L =50m: 

xrand = 0538 . 

L 

x rand 

= 26. 90 m 

Assuming a 2% slope (sinβ = 0.02) and k L = 0.1 m/s, Equation C-1 gives: 

− 

t avg rand 

= 665 . × 10 7 m 

In this case, the calculated depth of leachate in the wetted zone is so small that there 

is virtually no flow. 

In the case of the geomembrane primary liner, the rate of leakage can be calculated 

as follows using Bernoulli’s equation, where g is the acceleration due to gravity: 



2 

πd 

Q= 06 . a 2gh 

= 06 . 2 gh 4 

(C-3) 

hence, for d =10mmandh = 0.3 m (a high head in a granular primary leachate collection 

layer): 

- 

Q = 114 . ¥ 10 4 3 

m s 

The average depth of leachate in the wetted zone can then be calculated using the 

following equation (Giroud et al. 1997a), where t L is the thickness of the secondary leachate 

collection layer: 

t 

avg rand 

= 

L 

b g 

N 

M 

L 

N 

M 

15 xrand 

sin b 

53+ 

L sin b 

F Q I 

t 

L 

1+ 

2 P 

k t 

1+ 

t 

L 

HG 

L 

4 L sin b 

F 

HG 

Q 

1+ 

k t 

L 

2 

L 

O 

P 

LKJ 

Q 

O 

I 

KJ 

Q 

P 

52 

- 2 

(C-4) 

To calculate x rand , it is necessary to calculate μ using the following equation (Giroud 

et al. 1997a): 

t F I 

L 

Q 

m = 1+ 

(C-5) 

2 

2 L sin b k t 

hence, for t L =5× 10 -3 m (5 mm): 

hence, using Equation C-2: 

HG 

m = 0117 . 

L 

x rand 

= 26. 89 m 

Then, the average leachate depth can be calculated using Equation C-4: 

t avg rand 

= 0026 . m 

L 

KJ 


563


APPENDIX D 

SETTLING OF BENTONITE PARTICLES IN THE SECONDARY 

LEACHATE COLLECTION LAYER 

A bentonite particle settles in the secondary leachate collection layer if the time required 

for the particle to reach the base of the secondary leachate collection layer (i.e. 

the secondary liner geomembrane) is less than the time required for leachate flow to 

reach the toe of the secondary leachate collection layer slope (Figure D-1). 

The time for vertical movement of a particle is: 

t 

V 

D 

= 

v 

v 

(D-1) 

where: D = depth of leachate in the secondary leachate collection layer; v v = vertical 

velocity of a bentonite particle; and: 

t 

D = cos b 

(D-2) 

where t is the thickness of leachate in the secondary leachate collection layer. 

The time for leachate to flow along the secondary leachate collection layer slope is: 

t 

L 

L 

= 

v 

L 

(D-3) 

where: L = length of secondary leachate collection layer slope; and v L = velocity of flow 

along the slope. 

v L 

v V 

• 

t 

L 

b 

Figure D-1. 

Velocity of bentonite particle carried by leachate flow. 



It is assumed that the velocity of a bentonite particle falling in leachate is given by 

Stokes equation as follows: 

v 

v 

= 

2 

b B wg B 

r - r gd 

18h 

w 

(D-4) 

where: Ã B = density of bentonite particle; Ã w = density of leachate assumed to be equal 

to that of water; g = acceleration due to gravity; d B = diameter of bentonite particle; and 

η w = viscosity of leachate (assumed to be equal to that of water). 

The velocity of flow along the slope is given by the following equation derived from 

Darcy’s equation: 

vL = kLiL nL = kLsin b nL 

(D-5) 

where: k L = hydraulic conductivity of the secondary leachate collection layer material; 

n L = porosity of the secondary leachate collection layer material; i L =sinβ = hydraulic 

gradient in the secondary leachate collection layer; and β = slope of the secondary leachate 

collection layer. 

A bentonite particle does not settle if t L 

< t V 

, hence from Equations D-1 to D-5: 

L < 

18hwtkL 

tan b 

2 

r - r gn d 

b 

g 

B w L B 

(D-6) 

With η w =10 -3 kg/(m⋅s), tanβ =0.02(2%),Ã B = 2,700 kg/m 3 , Ã w = 1,000 kg/m 3 ,and 

g =9.81m/s 2 , Equation D-6 becomes: 

- tkL 

L < 216 . ¥ 10 8 2 

(D-7) 

n d 

with L (m), t (m), k L (m/s), and d B (m). 

Considering a geonet secondary leachate collection layer with n L = n GN = 0.8, Equation 

D-7 becomes: 

L 

B 

- tk 

L < 27 . ¥ 10 8 2 

d 

B 

L 

(D-8) 

with L (m), t (m), k L (m/s), and d B (m). 

Considering a hydraulic conductivity of 0.1 m/s for the geonet, the values of L presented 

in Table D-1 are obtained using Equation D-8. It appears in Table D-1 that, for 

small values of the average depth of leachate, such as the value of 6.65 × 10 -7 m calculated 

in Appendix C, bentonite particles are likely to settle in the secondary leachate 

collection layer. In contrast, if the secondary leachate collection layer is full or almost 

full due to a major leak through the primary liner, then bentonite particles can travel 

a long distance and reach the sump. 


565


Table D-1. Maximum value of the length, L (m), of the secondary leachate collection layer 

to ensure that there will be no bentonite particle settling in the secondary leachate 

collection layer. 

Thickness of leachate, t 

Maximum value of length of secondary leachate collection layer, L (m) 

(mm) For d B =1μm For d B =0.1μm 

0.1 0.3 27 

1.0 2.7 270 

5.0 13 1,350 

Notes: d B = diameter of bentonite particle. The tabulated values of L (m) were calculated using Equation D-8 

with k L = 0.1 m/s. The tabulated values can also be defined as “the flow length beyond which bentonite particles 

with a diameter d B will settle”. 



APPENDIX E 

DECREASE IN GEONET HYDRAULIC CONDUCTIVITY AND INCREASE 

IN LEACHATE THICKNESS DUE TO BENTONITE PARTICLES ON 

GEONET STRANDS 

The presence of bentonite particles on geonet strands increases the diameter of the 

geonet strands and decreases the porosity of the geonet. Asa result,the hydraulic conductivity 

of the geonet decreases. The relationship between geonet hydraulic conductivity, 

porosity, and strand diameter is (see Equation 29 in the main text of the present paper): 

k 

GN 

3 

n 

= l 

1- 

n 

b 

GN 

GN 

g 

2 

d 

2 

GN 

(E-1) 

The thickness, t c , of the layer of bentonite on the geonet strands is linked to the mass 

per unit geonet area of bentonite particles as follows (see Equation 34 in the main text 

of the present paper): 

t 

c 

mGNs 

= 

r 1- n 

b 

g 

S 

B c a 

(E-2) 

The geonet specific surface area per unit area of geonet is (see Equation 32 in the 

main text of the present paper): 

S 

a 

b 

41- n 

= 

d 

GN 

GN 

g 

t 

GN 

(E-3) 

hence, from Equations E-2 and E-3: 

t 

c 

mGNs 

dGN 

= 

4 1-n 1-n t 

r b gb g 

B c GN GN 

(E-4) 

The diameter of the geonet strands, including the bentonite layer is: 

d¢ = d + 2 t 

GN GN c 

(E-5) 

hence from Equations E-4 and E-5: 

L 

NM 

mGNs 

dGN 

¢ = dGN 

1+ 

2 1 -n 1 -n t 

r b gb g 

B c GN GN 

O 

QP 

(E-6) 

The porosity of the geonet, accounting for the presence of a bentonite layer of thickness 

t c on the strands is: 

Sa 

tc 

nGN 

¢ = nGN 

- 

(E-7) 

t 

GN 


567


hence, from Equations E-2 and E-7: 

mGNs 

nGN 

¢ = nGN 

- 

r 1- 

n t 

b 

g 

B c GN 

(E-8) 

From Equation E-1, the hydraulic conductivity of the geonet having bentonite on the 

strands is: 

b g 

3 

nGN 

¢ 

kGN 

¢ = l dGN 

¢ 

b - nGN 

¢ g b g 2 

1 

Combining Equations E-6, E-8, and E-9 gives: 

k¢ = l 

GN 

L 

N 

M 

n 

GN 

- 

r 

m O 

- Q 

P 

L 

GNs 

mGNs 

+ 

N 

M1 b1 

ngt 

2 r b 1 - gb 1 - g 

2 

L 

m 

- + 

N 

M 

O 

GNs 

1 nGN 

- Q 

P 

r 

Bb1 

ncgtGN 

B c GN 

2 

n n t 

B c GN GN 

O 

Q 

P 

3 2 

d 

2 

GN 

(E-9) 

(E-10) 

The relationship between the maximum thickness of leachate in a secondary leachate 

collection layer, t max , and the geonet hydraulic conductivity is (see Equation 40 

in the main text of the present paper): 

t 

max 

= 

Q 

k 

GN 

(E-11) 

In the case where the geonet strands are covered with bentonite particles, Equation 

E-11 becomes: 

t¢ = 

max 

Q 

k¢ 

GN 

(E-12) 

where t′ max 

is the maximum thickness of liquid in the secondary leachate collection layer 

when the geonet hydraulic conductivity is k′ GN 

. 

Combining Equations E-1 and E-10 to E-12 gives, after simplifications: 

tmax 

¢ - t 

t 

max 

max 

= 

F 

HG 

32 

nGN 

1- 

n 

GN 

I 

KJ 

L 

N 

M 

n 

GN 

- 

r 

mGNs 

1- 

n 

P - 1 

1 2 1 n 1 n t 

1- nGN 

+ 

r 

Bb 

cgtGN 

32 

m O 

- Q 

P 

L 

GNs 

mGNs 

+ 

b N 

M 

1 ngt 

r b - gb - g 

B c GN 

B c GN GN 

O 

Q 

(E-13) 

Numerical calculations done with n GN =0.8,n c = 0.93, Ã B = 2,700 kg/m 3 ,andt GN = 

4.5 × 10 -3 m give the values presented in Table E-1. Also presented in Table E-1 are 

values of dt max / t max (from Equation 43 in the main text of the present paper) obtained 



by calculating relative derivatives. It appears in Table E-1 that, even for values of 

(t′ max − t max )∕t max as large as 0.1 (10%), the relative derivative provides an excellent 

approximation (error less than 1%). 

Table E-1. Influence of bentonite particles adhering to geonet strands on leachate 

thickness in the secondary leachate collection system. 

m GNs 

(kg/m 2 ) 

t′ max 

− t max 

t max 

dt max 

t max 

Approximation 

(error) 

0.0001 0.0005144 (0.1%) 0.0005144 (0.1%) 0.00% 

0.001 0.005146 (0.5%) 0.005144 (0.5%) 0.04% 

0.01 0.05165 (5.2%) 0.05144 (5.1%) 0.4% 

0.02 0.10384 (10.4%) 0.10288 (10.3%) 0.9% 

Note: The tabulated values were calculated using Equation 43 from the main text of the present paper and 

Equation E-13 from Appendix E. 


569


APPENDIX F 

HYDRATION OF BENTONITE EXPOSED TO HUMID AIR 

F.1 Purpose of This Appendix 

The purpose of this appendix is to try to evaluate whether or not a layer of bentonite 

exposed to humid air can become saturated by absorbing water vapor from air. To that 

end, the mechanisms involved in the absorption of water vapor by bentonite are discussed 

and quantified. 

F.2 Relative Humidity of Air 

The maximum amount of water vapor that air can contain at a certain temperature 

is quantified by the “saturated water vapor pressure”, p s , which is defined as the partial 

pressure of water vapor that would exist if the air was saturated with water vapor at the 

considered temperature. The “partial pressure” of the water vapor is defined as the pressure 

that the water vapor would have if it occupied by itself the entire volume actually 

occupied by air. The saturated water vapor pressure increases as temperature increases. 

For example, values of p s for typical temperatures are: 0.61 kPa for 0_C, 2.34 kPa for 

20_C, and 101.3 kPa for 100_C. This last value results from the definition of the Celsius 

scale of temperature: water boils at 100_C at the atmospheric pressure, which is 101.3 

kPa. The fact that the saturated water vapor pressure increases as temperature increases 

shows that hot air can contain more water vapor than cool air. 

At a given temperature, the air cannot contain more water vapor than is indicated 

by the saturated water vapor pressure, but it can contain less. The relative humidity of 

air is the ratio between the actual water vapor pressure and the saturated water vapor 

pressure. The relative humidity is zero if the air is absolutely dry, and is 1 (100%) if the 

air is saturated. In temperate climates, the relative humidity of air is typically above 

50%, and air is considered relatively dry if the relative humidity is less than 75%. 

F.3 Suction Required to Absorb Water Vapor From Air 

To absorb water vapor from air, a porous material must have a certain “water potential”, 

i.e. it must exert a suction that can be calculated using the following equation 

(Daniel 2000; Rawlings and Campbell 1986): 

s 

= − 

ρw 

RT 

M 

ln H 

r 

(F-1) 

where: Ã w = density of water; R = ideal gas constant; T = absolute temperature; M =molecular 

mass of water; and H r = relative humidity of air. With Ã w = 1000 kg/m 3 , R =8.31 

J _K -1 mol -1 ,andM = 0.018 kg mol -1 , Equation F-1 gives: 

( ) 

s = − 461,666.67 273 + T ln H 

where T C is the temperature expressed in _C (whereas T was expressed in _K). 

C 

r 

(F-2) 



Values of the suction, s, calculated using Equation F-2 are represented by a family of 

curves for different values of the temperature (Figure F-1). Inspection of Figure F-1 reveals 

that the magnitude of suction is not significantly affected by temperature. Figure 

F-1 also shows that the relationship between s and H r is almost linear: this is true only for 

large values of H r because lnH r is approximately equal to (1--H r )forH r close to unity. 

F.4 Suction Exerted by Bentonite 

The suction exerted by a porous material depends on its pore size and water content, 

among other parameters. The smaller the pore size and the dryer the porous material, 

the greater the suction exerted by the material. Bentonite pore size is extremely small 

and, therefore, bentonite can exert significant suction and absorb a large amount of water 

from adjacent media. Bentonite has such a high affinity for water that it will swell 

to absorb more water. A compressive stress exerted on bentonite limits the absorption 

of water by limiting the swelling. 

The magnitude of suction exerted by bentonite typically used in GCLs has been measured 

by Daniel et al. (1993). The approximate curves shown in Figure F-2 for compressive 

stresses of 0 and 15 kPa are based on data provided by Daniel et al. (1993). Also, 

elementary calculations presented by Giroud et al. (1997) show that a 5 mm-thick layer 

of bentonite with an initial water content of 17% (which is typical for a GCL as delivered) 

and an initial mass per unit area of 5 kg/m 2 would be saturated without swelling 

at a water content of 80%. This has allowed the authors of the present paper to draw the 

beginning of the curve that corresponds to the very large compressive stress that would 

prevent swelling (dashed line in Figure F-2). 

7 

Water potential (suction), s (MPa) 

6 

5 

4 

3 

2 

1 

0 o 

C 

20 o 

C 

40 o 

C 

0 

95 96 97 98 99 100 

Air relative humidity, H r (%) 

Figure F-1. Suction required to absorb water vapor from humid air. 

Note: The above curves were obtained using Equation F-1. 


571


7 

6 


5 

4 

3 

2 

1 

σ =0kPa 

σ =15kPa 

σ = ∞ 

0 

0 50 100 150 

Bentonite water content, w (%) 

Figure F-2. Suction exerted by bentonite typically used in GCLs at 20_C. 

Note: The curves for a compressive stress, σ, of 0 kPa and 15 kPa were drawn by the authors of the present 

paper based on experimental data provided by Daniel et al. (1993). The curve for 0 kPa reaches w = 500% 

at s = 0 (i.e. for H r = 100%). The dashed curve is based on volumetric calculations by Giroud et al. (1997). 

When the bentonite is saturated, the suction is zero; this occurs for a water content of approximately 500% 

under zero compressive stress, 150% under a compressive stress of 15 kPa, and 80% under a high 

compressive stress that is sufficient to prevent swelling of the bentonite. 

Inspection of Figure F-2 indicates that the curve, which represents the suction of 

bentonite as a function of its water content, is not greatly affected by the magnitude of 

the compressive stress applied on the bentonite layer, except for the parts of the curve 

that correspond to water contents greater than 60% (i.e. the lowermost parts of the 

curves in Figure F-2). 

F.5 Water Content of Bentonite in the Presence of Air 

There is always some humidity in air. Therefore, a specimen of bentonite that is initially 

dry absorbs some water vapor from air. The resulting water content of the bentonite 

can be obtained by comparing the suction required to absorb water vapor from air 

(Figure F-1) and the suction exerted by bentonite (Figure F-2). As shown in Figure F-3 

(which combines Figures F-1 and F-2), an equilibrium is reached when the water content 

of the bentonite is such that the suction exerted by the bentonite is equal to the suction 

required to absorb water vapor from the air. If the bentonite specimen is initially 

at a water content that is greater than the equilibrium water content, water will evaporate 

from the bentonite and, as a result, the water content of the bentonite specimen will 

decrease until the equilibrium water content is reached. The time for equilibrium to be 

reached depends on the amount of water transfer that is required. 




7 

6 

5 

4 

3 

2 

1 

0 

6 

20 o C 

5 

4 

3 

2 

σ =15kPa 

1 

0 

95 96 97 98 99 100 0 50 100 150 

Air relative humidity, H r 

(%) 

7 


Figure F-3. Equilibrium between bentonite water content and air relative humidity. 

Note: For example, a bentonite water content of 40% corresponds to an air relative humidity of 98.8%. The 

left-hand figure shows the curve for 20_C from Figure F-1, and the right-hand figure shows the curve for σ 

= 15 kPa from Figure F-2. 

Figure F-3 shows that, for a given temperature and a given compressive stress, there 

is a relationship between the relative humidity of air and the water content of bentonite. 

The curve presented in Figure F-4a was obtained using the curve for 20_C from Figure 

F-1 and the curve for 15 kPa from Figure F-2, as shown in Figure F-3. It should be noted 

that the relationship represented by the curve in Figure F-4a is valid only for the bentonite 

used in the tests by Daniel et al. (1993) that were used to generate the curves in Figure 

F-2. Figure F-4b presents the same curve as in Figure F-4a, completed by interpolation 

between H r = 0 and 95%. 

Figure F-4 shows that bentonite can become saturated in contact with air only if the 

air is completely saturated, i.e. if the relative humidity is 100%. As seen in Figure F-4, 

the water content of the considered bentonite at equilibrium is only approximately 50% 

if the relative humidity of the air is 99% and 33% if the relative humidity of the air is 

98%, whereas the water content of saturated bentonite is typically 80% under very high 

compressive stress, 150% under a typical stress, and 500% under zero stress. 

F.6 Humidity of Air 

Figure F-4 shows that the water content of a bentonite specimen can be derived from 

the relative humidity of the air in contact with the specimen provided that enough time 

is allotted for reaching equilibrium. Therefore, one needs to know the relative humidity 

of air to predict the water content of a bentonite specimen exposed to that air. In particu- 


573


(a) 

150 



(b) 

100 

50 

0 

150 

100 

50 

95 

96 

97 98 

Air relative humidity, (%) 

H r 

99 

100 

0 

0 

10 

20 

30 40 50 60 70 80 

Air relative humidity, (%) 

H r 

90 100 

Figure F-4. Relationship between bentonite water content at a compressive stress of 15 

kPa and air relative humidity at 20_C: (a) for 95% ≤ H r ≤ 100%; (b) for 0 ≤ H r ≤ 

100%. 

Note: The curve in Figure F-4a was derived from Figures F-1 and F-2, as shown in Figure F-3. Curves for 

other temperatures between 0 and 40_C and for other compressive stresses would not be very different. The 

curve in Figure F-4b was interpolated between H r = 0 and 95%. The curve in Figure F-4b between H r =95 

and 100% is identical to the curve in Figure F-4a. 



lar, to determine if the bentonite can be saturated, it is necessary to determine if the air 

can be saturated. 

A limited volume of air that is confined in the vicinity of a body of pure water can 

become saturated, i.e. its relative humidity can be 100%. This may happen in a drainage 

layer where water stagnates due to inadequate design or operation. If water contains 

salts in solution (which is the case of leachate), the relative humidity of air confined on 

top of such solution is less than 100% because, like bentonite, salty water has a suction 

potential. Values of the suction potential of a solution of NaCl in water and values of 

the corresponding relative humidity of air confined with a solution of NaCl are given 

in Table F-1. 

Salt concentration in municipal solid waste leachate is typically less than a few 

grams per liter. Inspection of Table F-1 reveals that, for such small values of the salt 

concentration, the air that is confined with the solution can become saturated. Therefore, 

if leachate stagnates in large areas of a leachate collection system, and if the air 

in the leachate collection system is not vented, a GCL in contact with that air can hydrate 

and become saturated. 

Salt concentration may be very high (e.g. 100 g/liter) in liquid or leachate from some 

industrial waste. Table F-1 shows that, for such high salt concentration, the relative humidity 

of air confined with such liquid or leachate is of the order of 90 to 95%. Figure 

F-4 shows that, for a relative humidity of air of 90 to 95%, the water content of bentonite 

is less than 20%. Therefore, bentonite cannot become saturated by absorption of water 

vapor from air in the presence of very salty liquid or leachate. 

Table F-1. Water potential (expressed as a suction) of a solution of NaCl at 20_C and 

corresponding relative humidity of air confined with this solution. 

Concentration of NaCl solution Water potential (suction) Relative humidity of air 

(MPa) (%) 

Moles/liter 

g/liter 

0.05 

0.1 

2.9 

5.9 

0.23 

0.45 

99.8 

99.7 

0.2 

0.5 

1.0 

2.0 

11.7 

29.3 

58.5 

117.0 

0.90 

2.24 

4.55 

9.57 

99.3 

98.4 

96.7 

93.2 

Note: The values of water potential (suction) are from Daniel (2000) and Lange (1967). The values of relative 

humidity of air were calculated from the suction values using Equation F-1. 


575


F.7 Amount of Water Available 

Consider a 5 mm-thick layer of bentonite, with an initial water content of 17% 

(which is typical for a GCL as delivered) and an initial mass per unit area of 5 kg/m 2 , 

and assume that this layer of bentonite would absorb water and swell to a thickness of 

8 mm. Elementary calculations presented by Giroud et al. (1997) show that the water 

content of this bentonite would be 150%. The 133% water content in excess of the initial 

17% corresponds to 5,684 g/m 2 of water, or a thickness of water of approximately 5.7 

mm. A geonet with a thickness of 4.5 mm and a porosity of 0.8 contains only 3.6 mm 

of water. Therefore, even if the geonet secondary leachate collection layer was completely 

filled with water or leachate, it would not contain a sufficient amount of water 

to saturate the bentonite layer. Furthermore, a properly designed and operated secondary 

leachate collection layer is not full of liquid. If there is liquid stagnating in the secondary 

leachate collection layer, the air above that liquid may have a high relative 

humidity, as indicated in Section F.6. However, the amount of water contained in air 

is small, even if the air has a relative humidity of 100%. Therefore, to saturate the bentonite 

layer in a typical GCL, water should not only be ponding in the geonet leachate 

collection layer but should also be resupplied as it evaporates and water vapor migrates 

into the bentonite. If, instead of being a geonet, the secondary leachate collection layer 

was made of gravel, it would be possible to have a sufficient amount of ponding water 

to saturate the bentonite layer of a GCL. 

F.8 Conclusion 

Hydration of bentonite by water vapor contained in humid air is possible only if the 

relative humidity of air is very high, e.g. a relative humidity greater than 95%, and bentonite 

can become saturated only if the relative humidity of air is extremely high, e.g. 

99.5% or more. In the case of a liner system such as the system considered in the present 

paper, such an extremely high relative humidity of air in contact with the GCL is possible 

only if leachate has a low salt concentration and stagnates for a long period of time 

in a large area of the secondary leachate collection layer, which may happen due to improper 

design or maintenance. Furthermore, if the secondary leachate collection layer 

is a geonet, there is not a sufficient amount of water in the geonet to saturate the bentonite 

of the GCL; therefore, a significant amount of water or leachate should be resupplied, 

as water vapor evaporates and migrates into the bentonite. Clearly, the situations 

where there is sufficient water in the secondary leachate collection layer to maintain 

a saturated air able to saturate the bentonite should be rare. From all of the foregoing 

discussions, it can be concluded that, in general, GCLs are not saturated by sole exposure 

to air because the air to which they are exposed is generally not saturated. 



APPENDIX G 

CALCULATION OF REQUIRED TESTING TIME WHEN LEACHATE 

HEAD IS NOT CONSTANT 

The purpose of this appendix is to present equations that should be used instead of 

equations given in Section 5.3 of the main text of the present paper if the leachate head 

on top of the geomembrane in the field, h FIELD , and/or in the laboratory, h LAB , is not a 

constant. 

The following equation should be used instead of Equation 53 of the main text of 

the present paper if the leachate head on top of the geomembrane in the field, h FIELD , 

is not a constant: 

⎡ 

{ 1 0.1( ) } 

0.95 ⎤ 

∑ ⎢ 

⎥ 

V 

H = = k + h t t 

FIELD 

FIELD GCL FIELD i GCL FIELD i 

A 

⎣ 

⎦ 

w 

(G-1) 

where t FIELDi 

is the time during which the leachate head on top of the geomembrane 

overlying the GCL is h FIELDi . 


the present paper if the leachate head on top of the geomembrane in the laboratory,h LAB , 


V 

H = = k ∑ i t 

( ) 

LAB 

LAB GCL LABi LABi 

AGCL 

(G-2) 

where t LABi 

is the time during which the hydraulic gradient is i LABi and the hydraulic head 

is h LABi in the laboratory test. 




∑{ ⎡1 

( ) ⎤ } 

V 

H = = k + h t t 

LAB 

LAB GCL LABi GCL LABi 

A 

⎣ 

⎦ 

GCL 

(G-3) 




t 

LAB 

∑{ 1+ 0.1( ) } ∑ 1+ 

0.1( ) 

i 1+ 

( h t ) 

{ } 

⎡ h t ⎤t ⎡ h t ⎤t 

⎣ ⎦ ⎣ ⎦ 

= = 

0.95 0.95 

FIELDi GCL FIELDi FIELDi GCL FIELDi 

LAB LAB GCL 

(G-4) 





577


{ 1 ⎤ } ⎡1 0.1( ) 0.95 

∑( ) ∑ ⎡ ( ) 

i t = h t t h t ⎤ 

⎣ + ⎦ = + 

t 

⎣ ⎦ 

LABi LABi LABi GCL LABi FIELD GCL FIELD 

(G-5) 


the present paper if the leachate head on top of the geomembrane in both the field and 

the laboratory is not a constant: 

{ 1 } ⎡1 0.1( ) 

∑( ) ∑ ⎡ ( ) ⎤ ∑ 

0.95 

{ 

⎤ 

} 

i t = ⎣ + h t ⎦t = + h t t 

⎣ ⎦ 

LABi LABi LABi GCL LABi FIELDi GCL FIELDi 

(G-6) 




N 

FIELD 

∑{ ⎡1+ 

0.1( ) } 

0.95 ⎤ 

k h t t 

⎣ 

⎦ 

= 

n t 

GCL FIELDi GCL FIELDi 

GCL 

GCL 

(G-7) 




k ( ) { 1 ( ) 

GCL ∑ iLABi t k 

LABi GCL ∑ ⎡⎣ 

+ hLABi tGCL ⎤⎦tLABi} 

N 

(G-8) 

LAB 

= = 

n t n t 

GCL GCL GCL GCL 



APPENDIX H 

PROBABILITY THAT A BENTONITE PARTICLE WILL MEET A GEONET 

STRAND 

For a bentonite particle, which has migrated through the lower geotextile of a GCL 

overlying a geonet, the probability that the particle will meet a geonet strand is one minus 

the probability that the particle will pass through the geonet. Assuming that bentonite 

particles that migrate from the GCL travel in a direction normal to the GCL, the 

probability that a bentonite particle will pass through a geonet is equal to the relative 

open area, R GN , of the geonet (known as “percent open area” when expressed as a percentage). 

Considering a simple model of a geonet shown in Figure H-1a, the surface 

area of a geonet hole is: 

S 

H 

2 

OGN 

= 

sinq 

(H-1) 

where: O GN = distance between geonet strands; and θ = angle between geonet strands 

of two different layers. 

The elementary surface area including one strand in each of the two directions (i.e. 

from strand center to strand center) is: 

bOGN 

+ dGNg 2 

S 

(H-2) 

E 

= 

sinq 

where d GN is the diameter of geonet strands. 

The relative open area is equal to S H /S E , hence: 

R 

GN 

= 

F 

HG 

OGN 

O + d 

GN 

GN 

I 

KJ 2 

(H-3) 

A relationship between the geonet relative open area, R GN , and the geonet porosity 

can be established as follows. The geonet porosity can be calculated using the model 

presented in Figure H-1b where the geonet strands are assumed to have a circular cross 

section. Each of the two layers of strand has the same porosity. Therefore, the porosity 

can be calculated for one layer only. From Figure H-1c, it appears that: 

hence: 

hence: 

1- n = 

GN 

dGN 

O + d 

GN 

2 

π dGN 

4 

O + d d 

b 

GN 

g 

GN GN GN 

b 

41- n 

= 

π 

GN 

g 

(H-4) 

(H-5) 


579


(a) 

O GN 

O GN 

d GN 

θ 

(b) 

(c) 

d GN 

O 

GN 

+ d 

GN 

Figure H-1. Geonet model: (a) view from top; (b) perspective; (c) cross section 

perpendicular to geonet strands. 

d O 

1- 

= = 1- 

41 - n 

GN 

GN 

O + d O + d 

π 

GN 

GN 

GN 

GN 

b 

GN 

g 

(H-6) 

Combining Equations H-3 and H-6 gives: 



R 

GN 

L 

N 

M 

b 

gO 

Q 

P 

- nGN 

= 1- 

41 2 

π 

(H-7) 

Therefore, the probability that a bentonite particle will meet a geonet strand is: 

1 1 1 41 2 

L - 

- = -M 

b n O 

- 

81 - 

P 

L 

= 1- 

21 

GNg b nGNg - n 

M 

b GNgO 

R 

P (H-8) 

GN 

π π π 

With n GN = 0.8, a typical geonet porosity: 

N 

Q 

1- R GN 

= 0. 

444 

In other words, for a bentonite particle migrating from the GCL, the probability that 

the particle will meet a geonet strand is 44.4%. It should be noted that only a fraction 

of the bentonite particles that meet a geonet adhere to the geonet; the bentonite particles 

that meet the geonet and continue to move downward will eventually accumulate on 

the secondary liner. 

Therefore, the probability that the bentonite particles will adhere on geonet strands 

is less than 44.4%; whereas, the probability that the bentonite particles will accumulate 

on the secondary liner is greater than 55.6%. 

N 

Q 


581

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