numerical characterization of advective gas flow through gm/gcl composite liners having a circular...

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Numerical Characterization of Advective Gas Flowthrough GM/GCL Composite Liners Having a Circular Defect

in the GeomembraneHossam M. Abuel-Naga1 and Abdelmalek Bouazza2

Abstract: Numerical experiments were conducted to understand the effect of geometric and transport characteristics of a geomembrane-geosynthetic clay liner �GM/GCL� composite liner on gas leakage rate through a circular defect in the geomembrane �GM�. Theoriginality of the approach proposed in this paper rests on the use of a new conceptual two-layered system for modeling of GM/GCLcomposite liners where the interface zone between the GM and geosynthetic clay liner �GCL� has been merged with the GCL covergeotextile and handled as one layer; the GCL bentonite layer was considered the second layer. The role of the carrier geotextile layer wasignored since it can be considered as a no pressure loss layer. Analysis of numerical simulation results shows the existence of aconstitutive leakage flow surface which enables evaluation of the leakage flow state for different geometric and transport properties ofGM/GCL composite liners. Furthermore, the determined surface was also exploited to evaluate gas leakage rates under the framework ofthe Forchheimer’s analytical solution. The gas leakage rate predictions were found to be in good agreement with experimental resultsobtained at different GCL moisture content.

DOI: 10.1061/�ASCE�GT.1943-5606.0000116

CE Database subject headings: Gas flow; Leakage; Geomembranes; Defects; Landfills; Clay liners.

Introduction

Geosynthetic clay liners �GCLs� are most typically comprised ofa thin layer of bentonite contained between two layers of geotex-tile with the components being held together by needle punchingor stitch bonding. They are widely used in landfill applicationsand have been subject to considerable recent research �Barroso etal. 2006; Dickinson and Brachman 2006; Hurst and Rowe 2006;Touze-Foltz et al. 2006; Bouazza and Vangpaisal 2006; Bouazzaet al. 2006,2007; Bouazza and Rahman 2007; Nye and Fox 2007;Southen and Rowe 2007; Benson et al. 2007; Meer and Benson2007�. In this respect, there is a wide body of work available ontheir hydraulic performance and the measurements of their per-meability to fluids are well documented in literature. However,experimental measurements of their permeability to gases are lesswidely available and only recently has information on their gasadvective flow performance became available in the context oflandfill capping �Didier et al. 2000; Bouazza and Vangpaisal2003,2004,2007a; Vangpaisal and Bouazza 2004; Bouazza et al.2006�. The published data, mostly experimental, has shown thatthe gas permeability or advective gas flux of GCLs is dependant

1Senior Lecturer, Dept. of Civil and Environmental Engineering,Univ. of Auckland, Private Bag 92019, Auckland Mail Centre, Auckland1142, New Zealand �corresponding author�. E-mail: [email protected]

2Associate Professor, Dept. of Civil Engineering, Building 60,Monash Univ., Melbourne, Victoria 3800, Australia. E-mail: [email protected]

Note. This manuscript was submitted on August 10, 2007; approvedon March 25, 2009; published online on March 27, 2009. Discussionperiod open until April 1, 2010; separate discussions must be submittedfor individual papers. This paper is part of the Journal of Geotechnicaland Geoenvironmental Engineering, Vol. 135, No. 11, November 1,
2009. ©ASCE, ISSN 1090-0241/2009/11-1661–1671/$25.00.
JOURNAL OF GEOTECHNICAL AND GEOE

J. Geotech. Geoenviron. Eng.

on moisture content �gravimetric and volumetric�, the manufac-turing process of the GCL and operational conditions present.

A measure frequently used to minimize gas migration is toinstall a geomembrane �GM� on top of the GCL, since it is es-sentially impervious to gas flow when devoid of holes or defects.However, possible defects in GMs need to be considered, since ithas been shown that defects in GMs can occur even with carefullycontrolled manufacture and damages can be found even in siteswhere strict construction quality control and construction qualityassurance programs have been put in place �Bouazza et al. 2002�.These defects will obviously form preferential gas flow pathsthrough the GM.

A comprehensive body of experimental and theoretical litera-ture on liquid leakage rates through composite liners, where de-fects in the GM occurred, is available �Rowe 1998; Rowe andBooker 2000; Foose et al. 2001; Touze-Foltz and Giroud2003,2005; Cartaud et al. 2005a,b; Giroud and Touze-Foltz 2005;Touze-Foltz and Barroso 2006; Barroso et al. 2006; Saidi et al.2006,2008�. On the other hand, very limited studies on the effectthat GM defects have on gas flow rate through composite barriersystems are available in literature. Recently, Bouazza and Vang-paisal �2006� reported the results of an experimental investigationon gas leakage rate through a geomembrane-geosynthetic clayliner �GM/GCL� composite liner, where the GCL was partiallyhydrated and the GM contained a circular defect. The test resultsshowed that gas leakage rates were affected by differential gaspressure across the composite liner, the moisture content of GCL,contact conditions, and the diameter of the defect.

The objective of this paper is to investigate numerically theeffect of geometric and transport characteristics of GM/GCL com-posite liners having circular defects in the GM on gas leakagerates. Furthermore, the numerical simulation results were ex-ploited to propose an approach for predicting gas leakage under
the framework of the Forchheimer’s analytical solution and ad-
NVIRONMENTAL ENGINEERING © ASCE / NOVEMBER 2009 / 1661

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dress the limitations reported in Bouazza et al. �2008� when usinganalytical solutions. The validity of the proposed equation wasverified by comparing the gas leakage rate predictions to the ex-perimental test results reported by Bouazza and Vangpaisal�2006�.

Problem Configuration

A schematic diagram of a GM/GCL composite cover containing acircular defect in the GM is shown in Fig. 1, it is assumed that thecomposite cover is both overlain and underlain by highly perme-able media. A GM containing a circular defect of radius r0 isunderlain by a partially saturated GCL. The GCL consists of abentonite layer contained between two geotextile layers. Thepresence of an interface layer between the GM and the GCL hasbeen highlighted in previous studies �Rowe 1998,2005� to explainthe underestimation of the leakage rate using equations for a per-fect contact condition when compared to measured laboratory orfield leakage rate where a perfect contact condition rarely can beheld true. From the practical view point, the physical existence ofa GM/GCL interface transmissive zone is due to undulations oc-curring during the installation of the GCL, wrinkles in the GMs,etc. For sake of simplicity, the numerical experiments reported inthis study were conducted assuming uniform thickness as well asuniform transport properties of the GM/GCL transmissive inter-face zone. Therefore, the thin composite lining system shown inFig. 1 can be identified as a five-layer system, comprising fromtop to bottom, an impermeable layer �GM�, an interface layer, acover geotextile layer, a bentonite layer, and a carrier geotextilelayer.

Transport Behavior of Geomembrane-GeosyntheticClay Liners Composite Liner

As the GM can be considered as an impervious material, theleakage rate of a GM/GCL composite liner having a circular de-fect in the GM can only be controlled by the transport propertiesof the GCL and interface layers of the composite. In previousstudies �i.e., Foose et al. 2001; Saidi et al. 2006�, the leakageproblem through GM/GCL composite liners was handled by sim-plifying the four porous layer system to one having only twoporous layers as shown in Fig. 2�a�. The first layer from the topunder the GM, represents the interface layer whereas the secondlayer consists of the GCL where its three permeable layers areconceptually merged into one layer having an equivalent transportproperty. The validity of this simplification approach for the GM/

Cover geotextileBentonite

Carrier geotextile

tGM

tbt

GM

Defect

ti

ro

Lgtt

Ugtt

re

tetGCL

Interface

P1

Patm

P1 and Patm are the gas pressures, P1 > Patm

tgti

Fig. 1. Schematic diagram of GM/GCLs cover composite liners

GCL leakage problem will be discussed in the following sections

1662 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGIN


in the light of the transport behavior of GCLs as well as theinterface layer.

Transport Properties of Geosynthetic Clay Liners

As GCLs are layered-composite materials, their experimentallymeasured transport properties depend both on �1� the transportproperties and the volumetric proportions of their material com-ponents �bentonite and geotextile layer� and �2� the flow directionrelative to the orientation of the GCLs layer system. Assumingthat the gas flow through the GCL is laminar, gas permeability ofthe GCL components follows isotropic behavior, and thickness ofthe cover and carrier geotextile layers are equal and their respec-tive gas permeabilities are identical, the GCL equivalent gas per-meability coefficient can be expressed as shown in Fig. 3 wherethe upper and lower limits of the equivalent gas permeability,ke

parallel and keseries, at different volumetric proportions of GCLs

components can be determined using the parallel and series flowtheory, respectively, as follows:

keparallel = � tgt

tGCL�kgt + �1 −

tgt

tGCL�kbt �1�

�keseries�−1 = � tgt

tGCL�kgt

−1 + �1 −tgt

tGCL�kbt

−1 �2�

tgt = tgtU + tgt

L �3�

where kbt and kgt=gas permeability of bentonite and geotextilelayer of GCL, respectively. The term tgt and tGCL are the thicknessof the geotextile layers and GCL, respectively, where tgt is thesum of the thickness of the upper and lower geotextiles tgt

U and tgtL ,

respectively.The possible equivalent gas permeability zone in the ke

− �tgt / tGCL� plane reflects the influence of flow direction on trans-port behavior. At any given tgt / tGCL ratio, the equivalent gas per-meability of the GCL can vary between the two extreme limits�Eqs. �1� and �2�� based on the flow direction relative to the ori-entation of the GCL layer system. Therefore, the conventional gaspermeability test for GCL �Vangpaisal and Bouazza 2004� that is

a) Conventional two layered system

b) Modified two layered system

DefectLayer 1: Interface

Layer 2: GCL

GM

DefectLayer 1: Interface +GCL cover geotextile

GM

Carrier geotextile

Layer 2: GCL bentonite

Fig. 2. Conceptual layering system of GM/GCLs composite linershaving circular defect for leakage rate modeling purposes

usually conducted under unidirectional flow condition, where the

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flow lines are perpendicular to the orientation of the GCLs layersystem as shown in Fig. 3, will only represent the series equiva-lent permeability coefficient for the GCL components.

However, in the case of gas leakage through a circular defectin the GM of a GM/GCL composite, the flow lines can havedifferent behavior than the simple unidirectional behavior. Inorder to illustrate this issue, assume for simplicity, the perfectcontact condition �ti=0.0�. In this case, the flow line configurationin the GM/GCL will be mainly controlled by the ratio betweenthe permeabilities of the GCL components that are directly underthe GM with a circular defect �cover geotextile and bentonitelayer� as shown in Fig. 4. The two cases where kbt≪kgt and kbt

�kgt can be considered as the extreme flow pattern conditions.When kbt≪kgt, the gas flow pattern through a GM/GCL compos-ite liner will consist of a unidirectional flow through the underly-ing bentonite and carrier geotextile layers, as well as radialhorizontal flow toward the circular defect in the GM through thecover geotextile layer as shown in Fig. 4�a�. However, a multidi-rectional flow pattern is expected when kbt�kgt as shown in Fig.4�b�. In this case, the two-layered system changes to a single layersystem. Thus, the flow pattern changes gradually from multidirec-tional to unidirectional flow as kgt /kbt increases and vice-versa.Therefore, it can be concluded that dealing with the GCL as oneunit in the modeling of the leakage problem is not a valid ap-proach as the used conventional equivalent gas permeability�Vangpaisal and Bouazza 2004� does not represent any of theexpected flow pattern through a GM/GCL composite liner havinga defect in the GM. To overcome this problem it is recommendedto deal with the GCL as a three-layered material �interface, geo-textile, bentonite� where each layer will be assigned its actualtransport properties.

Transport Properties of Interface Layer

In the conventional two-layered system �Fig. 2�a��, an indirectapproach is usually used to determine either the thickness or thetransmissivity of the interface layer �Rowe 1998; Touze-Foltz etal. 1999; Foose et al. 2001; Saidi et al. 2006�. It involves back-calculating the interface properties using analytical models thatare based on the assumption of unidirectional flow as shown inFig. 4�a�. The limitation of this approach for gas leakage problemis explained in the following paragraph.

Although in most cases the gas permeability of the GCL geo-textile layer is much higher than the permeability of the bentoniteGCL layer, the conventional two-layered system assigns a similar

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

0 0.2 0.4 0.6 0.8

tgt/tGCL

keParallel FlowSeries Flow

Lgt

Ugtgt ttt +=

(m/s)

Fig. 3. Possible domain of eq

permeability value to the GCL cover geotextile layer and GCL



bentonite layer. In fact such assumption leads to the suppressionof likely horizontal flow through the GCL cover geotextile layertoward the defect in the GM. Consequently, to simulate the ob-served fluid flow through the defect, the transport properties ofthe interface layer will need to be overestimated, by the back-calculation approach, to compensate for the misconceptualizedlayered system of GM/GCLs composite liner.

To address this problem in the conventional conceptual two-layered system, a modified two-layer system is proposed in thisstudy, as shown in Fig. 2�b�, where the interface and cover geo-textile GCL layer are merged together and handled as one layer,the GCL bentonite layer is considered as the second layer. Therole of the carrier geotextile component of the GCL can be ne-glected as zero pressure losses can be assumed for gas flowthrough this layer. The proposed modified conceptual layered sys-tem satisfies the requirement of describing the GCL transportproperties shown in Fig. 3. Moreover, it allows for the possibilityof radial flow through the GCL cover geotextile layer.

The permeability of the merged layer, kgti, to gas can be relatedto the gas permeability of the GCL cover geotextile layer usingthe following form:

a) kbt<<<kgt (unidirectional flow net)

Geotextile

Geotextile

Bentonite

Circular defect

GM

b) kbt ≈ kgt (multi-directional flow net)

Geotextile

Geotextile

Bentonite

GM

Flow line Equipotential line

Circular defect

Fig. 4. Effect of difference in the gas permeability of GCL compo-nents on flow lines configuration

Geotextile

GeotextileBentonite

Flow lines

Impermeable boundary condition

i) Parallel flow

Geotextile

GeotextileBentonite

ii) Series flow

LgttbttUgtt

CL

Flowlines

nt gas permeability in GCLs

1

Gt

uivale


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log� kgti

kgt� = Cn �4�

where Cn=contact condition parameter between GM and GCLcover geotextile layer. For the condition of perfect contact, Cn

will be equal to 0.0. However, as the quality of the contact con-dition decreases, Cn increases. The contact condition parametercan be a function of the applied surcharge load and GCL structure�GCL with bentonite encased in geotextile or GCL with bentoniteglued to GM�.

Determination of Transport Properties ofGeomembrane-Geosynthetic Clay Liners CompositeLiners Layers

In order to determine, individually, the gas permeability of theGCL components �i.e., geotextile, and bentonite layer� with Eq.�2�, the equivalent series gas permeability of GCL as a wholeunit, ke

series, and the gas permeability of the merged layer kgti, mustbe first determined. The proposed approach assumes that the gaspermeability of the GCL components follows isotropic behavior,the thickness of the cover and carrier geotextile layers are equaland their respective gas permeabilities are identical. The equiva-lent series gas permeability of GCL can be measured experimen-tally using test apparatus and procedures described in Bouazzaand Vangpaisal �2003� and Vangpaisal and Bouazza �2004�. Gaspermeability of the merged layer, kgti, can be obtained experimen-tally at different surcharge loads using equipment and test proce-dure proposed by Bouazza �2004� and Bouazza and Vangpaisal�2007b�. By considering the contact condition between GM andGCL, kgt can be determined using Eq. �4�.

Numerical Experimental Program

The conducted numerical experiment study was aimed at under-standing the effect of geometric and transport characteristics ofGM/GCL composite liner on gas leakage rate through a circulardefect in the GM. To address this aim, numerical simulationswere conducted for different geometric and transport conditions.Dimensionless variables are used in this study to describe a widevariety of flow situations and assist with the generalization of thetest results. The adopted dimensionless variables are tgti / te, ro / te,re /ro, and log�kgti /kbt�. The first three variables describe geomet-ric conditions, whereas the last variable expresses the transportproperties of the GM/GCL composite liner. The numerical simu-lation conducted in this study encompasses a range of geometricand transport conditions relevant to practical conditions. Based onthe conventional structure configuration of the GM/GCL compos-ite liner, the numerical study is limited to conditions wheretgti / te�0.5. Moreover, Foose et al. �2001� indicated that the ratioof the transport properties for two-layer composite systems thatdistinguishes between the flow modes is 4.0. Therefore, the nu-merical simulation was limited to conditions where log�kgti /kbt��6. The adopted numerical modeling assumption as well as theused governing equation and boundary conditions are presentedin the following sections.

Numerical Modeling Assumption

The GM/GCL composite cover is assumed to be both overlainand underlain by highly permeable media. Gas pressure under the
GM/GCL composite is assumed to be constant, uniform and per-


pendicular to the GCL plane as shown in Fig. 1. The results oflaboratory gas leakage test conducted by Bouazza and Vangpaisal�2006� showed that gas flow rate reached steady state within 5–20min from the commencement of the test. As the early transientresponse is too short, it will be ignored in this study and onlysteady state conditions will be considered for numerical modelingpurpose. Furthermore, the following assumptions were alsoadopted:• There is only one circular defect in the GM of the GM/GCL

composite cover under consideration and there is no wrinkle inthe GM.

• No pressure is lost when gas flows through the GM defect.This implies that the thickness of the GM can be neglected inthe numerical modeling process.

• Gas flow is driven by pressure gradient and only gas advectiveflow mechanism is considered.

• Gas flow through the GCL is laminar and consequently, Dar-cy’s equation is applicable.

• Gas flow is in a constant temperature environment; therefore,gas density and gas viscosity are constant.

• The bulk movement of liquid phase under gas flow is negli-gible.

• The transport properties of the considered porous medium areisotropic and homogenous.

Governing Equation and Boundary Conditions

The finite-element air flow model SVAIRFLOW was used to con-duct the numerical gas leakage experimental tests. SVAIRFLOWsolves the following governing equation for the axisymmetric gasflow in the unsaturated zone which considers the validity of Dar-cy’s law, the applicability of ideal gas law.

I

r

�

�r�kgr

�P

�r� +

�

�z�kg

�P

�z� = 0 �5�

Eq. �5� ignores the Klinkenberg effect which accounts for thenonzero or slip flow velocity at the pore walls in low permeableporous media. Vangpaisal and Bouazza �2004� indicated that theKlinkenberg effect is not observed when the pressure difference islower than 20 kPa �similar to the range of differential pressuresencountered in landfills� and can, on this basis, be excluded fromthe modeling process from gas advective transport conditions.

The boundary conditions of the numerical experiments wereestablished considering the existence of multiple holes of thesame dimensions in the GM. Consequently, no-flow boundaryconditions will apply at locations halfway between the holes.Therefore, simulating the flow through a single hole with an ap-propriate placement of the no-flow boundary can be considered.Although, for closely packed holes in the GM, the no-flow bound-aries form a honeycomb shape, an approximated circular shape ofno flow boundary was adopted in this study to facilitate the solu-tion using Eq. �5�. Therefore, the boundary conditions for theconducted numerical simulation can be described in axisymmetriccylindrical form as follows:• Atmospheric pressure, Patm, at the circular defect �P= Patm at

0�r�ro and Z= te�.• No flow boundary is located at the GM surface ��P /�Z=0� at

ro�r�re and Z= te as well as at r=re and 0.0�Z� te.• Elevated pressure, P1, at the bottom surface of the bentonite

GCL layer �P= P1 at 0�r�re and Z=0.0�.Fig. 5 shows a typical configuration of the numerical mesh,

built by an automatic adaptive mesh approach, and the boundary
conditions used in the present study.

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Discussion of the Numerical Experiment Results

The numerical experiments results, over a range of different trans-port properties �kgti /kbt� and geometric configuration �tgti / te, ro / te,and re /ro�, for GM/GCL composite liners are presented in a di-mensionless form using the nondimensional flow factor F. Theflow factor F has been introduced by Foose et al. �2001� in orderto extend the applicability of Forchheimer’s analytical model topredict leakage rate through thin composite liners having perfectcontact condition. The equation from Foose et al. �2001� is asfollows:

Q = Fkeserieshro �6�

where Q and h=leakage flow rate and the total head drop acrossthe composite liner, respectively.

Fig. 6 shows typical numerical results plotted on the F−log�kgti /kbt� plane for different geometric configurations �tgti / te,ro / te, and re /ro�. The results indicate that unique leakage charac-teristic curves can be obtained on the F−log�kgti /kbt� plane fordifferent composite liner geometric configurations. Such curvescan be used to describe the effect of transport properties on theleakage rate. In this respect, these curves will be referred to,herein, as transport leakage characteristic curves �TLCCs�. Fig. 6shows that F increases as log�kgti /kbt� increases. Moreover, it alsoshows that the F domain is largely a function of geometric prop-erties �tgti / te, re /ro, ro / te�.

To more thoroughly explore the effect of geometric propertieson gas leakage behavior, the results were plotted on the F− �tgti / te� plane for different geometric properties �ro / te, re /ro�, butfor constant transport properties conditions, log�kgti /kbt�, asshown in Fig. 7. The curves obtained in Fig. 7 will be referred to,herein, as geometric leakage characteristic curves �GLCCs�. Fig.7 indicates that F increases as tgti / te increases. Moreover, at con-stant transport properties, log�kgti /kbt�, the F domain is largelycontrolled by changes in re /ro and ro / te.

Modeling of Leakage Flow Curves

The observed TLCCs on the F−log�kgti /kbt� planes, and GLCCson the F− �tgti / te� planes, have some common key features. Ingeneral, both types of curves can be idealized as depicted in Fig.8 where two characteristics flow limits can be identified, namely,the threshold limit and the residual limit. These two limits breakthe leakage curve into three parts. The first and third parts arewhere insignificant change in F factor can be observed as eitherlog�kgti /kbt� or �tgti / te� increases. The second part contains thesensitive part of the leakage flow curve where a significant

Z

r

rore

Patm

P1

No flow boundary condition (GM)

No flowboundarycondition

No flowboundarycondition

Merged interface and cover geotextile layer

Bentonite GCL layer

te

tgti

Fig. 5. Typical mesh configuration and boundary conditions �ro / te

=0.9, re /ro=5.55, kgti /kbt=1.0,0.25�

change in F factor can be induced as either log�kgti /kbt� or �tgti / te�



increases. As the leakage flow curve has the same key features asthe soil water characteristics curve, the normalized approach usedby Fredlund et al. �1995� can also be used to normalize bothTLCCs and GLCCs shown in Figs. 6 and 7 using the followingrelation:

FN =F − Fr

Fo − Fr�7�

where FN=normalized nondimensional flow parameter; Fo

=nondimensional flow parameter at zero value of log�kgti /kbt� or�tgti / te�; and Fr=nondimensional flow parameter at residual flowlimit as shown in Fig. 8. When Fr is taken as being zero, FN

=F /Fo. Figs. 9 and 10 show the adequacy of the used normaliza-tion approach for proposed leakage flow curves. Therefore, the

Fig. 6. Typical TLCCs obtained from numerical results

Fredlund and Xing �1994� equation can also be used to describe


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the normalized leakage flow curves. For TLCC, FN can be ex-pressed as follows:

FN = �1 −

ln�1 +log�kgti/kbt�log�kgti/kbt�r

�ln�1 +

c

log�kgti/kbt�r�� 1

ln�e + � log�kgti/kbt�a

�n�m

�8�

whereas for GLCC, FN can be expressed as follows:

FN = �1 −

ln�1 +�tgti/te��tgti/te�r

�ln�1 +

c

�tgti/te��

r

� 1

ln�e + � �tgti/te�a

�n�m

�9�

where log�kgti /kbt�r and �tgti / te�r=residual values in TLCCs andGLCCs, respectively and a, c, n, and m=equation constants. Thevalues of the equation constants that fit well with the normalizednumerical results for both TLCCs and GLCCs are listed in Table1. Figs. 9 and 10 show reasonable agreement between FN pre-dicted using Eqs. �8� and �9� and the numerically determinedvalues. The normalized leakage curves show that, at threshold andresidual flow limits, the value of log�kgti /kbt�TL and log�kgti /kbt�r

for TLCCs are 1.2 and 3.0, respectively, whereas the value of�tgti / te�TL and �tgti / te�r for GLCCs are 0.08 and 0.20, respectively.

Constitutive Leakage Flow Surface

The identified TLCCs and GLCCs provide evidence for the exis-tence of a constitutive leakage flow surface �CLFS� in F

Fig. 10. Normalization of GLCCs

Table 1. Values of Equation Constants

Parameter TLCC �Eq. �8�� GLCC �Eq. �9��

a 2.35 0.13

c 100 100

n 3.45 4.1

m 3.7 2.2

log�kgti /kbt�r 3.0 —

log�kgti /kbt�TL 1.2 —

�tgti / te�r — 0.2

�tgti / te�TL — 0.08

Fig. 8. Idealization of GLCC and TLCC

Fig. 9. Normalization of TLCCs

Fig. 7. Typical GLCCs obtained from numerical results


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−log�kgti /kbt�− �tgti / te� space, at constant ro / te and re /ro, as illus-trated in Fig. 11. Such a surface represents the locus of the pos-sible leakage flow state. The four sides of CLFS express theleakage flow for special cases. The sides AB and AC represent thecase of leakage in one-layer system having perfect contact withthe GM. Based on the numerical study conducted by Foose et al.�2001�, the nondimensional flow parameter F for this conditioncan be determined using the following equation where the effectof re /ro was not considered for sake of simplicity

F = 4.0 + 3.35�ro/te� �10�

The above equation indicates that F for the above special cases isindependent of log�kgti /kbt� and �tgti / te�. Consequently, the valuesof F at both sides, AB and AC, are constant as depicted in Fig. 11.The sides CD and BD represent the residual leakage flow curvesTLCCr and GLCCr, which characterize the leakage flow condi-tion at �tgti / te�� tgti / te�r and log�kgti /kbt�� log�kgti /kbt�r, respec-tively.

As the proposed CLFS was presented in F−log�kgti /kbt�− �tgti / te� dimensions at constant ro / te and re /ro ratio, it can bededuced that the extent of the CLFS in the direction of F iscontrolled by the nondimensional flow parameter at residual state,Frr, where log�kgti /kbt�� log�kgti /kbt�r and �tgti / te�� tgti / te�r.Therefore, several leakage simulations were conducted at differ-ent ro / te and re /ro ratio within the residual state in order to in-

Fig. 12. Effect of ro / te and re /ro on Frr

Fig. 11. CLFS at constant ro / te and re /ro �tgti�0.5te�



vestigate the effect of these ratios on Frr. The results given in Fig.12 indicate that, at constant re /ro ratio, Frr increases linearly withro / te and can be described in the following form:

Frr = ��ro/te� �11�

where �=slope index that depends on re /ro. The numerical re-sults reveal that the relation between � and ro / te can be expressedusing the following equation �Fig. 13�:

� = 3.4181�re/ro�1.6603 �12�

Therefore, an infinitive number of CLFS can be established inF−log�kgti /kbt�− �tgti / te� space for different ro / te and re /ro ratiosas shown in Fig. 14.

Generally, based on the variation of F in the F−log�kgti /kbt�− �tgti / te� space, CLFS can be divided into three different zones, asshown in Fig. 11, having the following boundary conditions:• Zone 1: Threshold zone, where 0.0� �tgti / te�� tgti / te�TL and

0.0� �kgti /kbt�� kgti /kbt�TL �F /��kgti /kbt��0.0 and�F /��tgti / te��0.0.

• Zone 2: Transitional zone, where �tgti / te�TL� �tgti / te�� tgti / te�r and �kgti /kbt�TL� �kgti /kbt�� kgti /kbt�r �F /��kgti /kbt��0.0 and �F /��tgti / te��0.0.

• Zone 3: Residual zone, where �tgti / te�� tgti / te�r and �kgti /kbt�� kgti /kbt�r �F /��kgti /kbt��0.0 and �F /��tgti / te��0.0.As the geometric characteristics of the flow net through com-

posite liner can be controlled by log�kgti /kbt� and �tgti / te�, it is

Fig. 13. Evolution of � with re /ro

Fig. 14. Evolution of CLFS at different ro / te and re /ro ratios


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believed that the above CLFS denotes zones that can representdifferent leakage flow modes �unidirectional and multidirectional�shown in Fig. 4.

In order to challenge this hypothesis, the development of theflow net geometrical characteristics for different two paths, runthrough log�kgti /kbt�-�tgti / te� plane, as shown in Fig. 15, were ex-plored numerically. The selected paths were chosen to cover thedifferent proposed zones in log�kgti /kbt�-�tgti / te� plane. Path 1crosses over the three zones, whereas Path 2 only passes throughthe first and the second zone. Fig. 16 illustrates the evolution ofthe equipotential lines following Path 1 as the log�kgti /kbt� ratioincreases. The results show that the prevailing flow mode can bedepicted as multidirectional where log�kgti /kbt�� log�kgti /kbt�TL,unidirectional where log�kgti /kbt�� log�kgti /kbt�r and transitionalflow mode between the threshold and the residual limits. How-ever, the development of the equipotential lines following Path 2shows only two flow modes, as illustrated in Fig. 17: a multidi-rectional flow mode where �tgti / te�� tgti / te�TL and a transitionalflow mode where �tgti / te�� tgti / te�TL. The unidirectional flowmode was not detected under Path 2. Therefore, it can be con-cluded that the flow mode in the threshold, transitional, and re-sidual zones is multidirectional, transitional, and unidirectional,respectively.

Predicting Gas Leakage Flow Rate

In the previous sections, the analysis of results of simulationsshowed that the leakage flow behavior of composite GM/GCLsliner having a circular defect can be described using the proposedCLFS. Moreover, governing equations were proposed to expressthe CLFS for different conditions in F−log�kgti /kbt� and F− �tgti / te� space using Eqs. �7�–�12�. To the authors’ knowledge,there is no simple analytical solution that can be used to verify thevalidity of the proposed CLFS. In fact, even for Forchheimer’sanalytical model, it can be only used for calculating leakagethrough a small circular defect in an infinitely thick liner havingperfect contact with the GM which is not our case. However, theproposed CLFS, under certain conditions shown in Fig. 11 �lines

Fig. 15. Numerical experiment paths

AB and AC�, recognizes the numerical model that has been de-



veloped by Foose et al. �2001� to predict leakage rate through thinliner having perfect contact with the GM using Eq. �10�.

The validity of the CLFS concept for predicting gas leakagerate through a GM/GCL composite liner containing a circulardefect in the GM was verified using the laboratory test resultsreported by Bouazza and Vangpaisal �2006�. In their study, gasleakage rate through a GM/GCL composite liner containing acircular defect in the GM with a radius, ro, of 2.5 mm was mea-sured, under a 20 kPa surcharge, at different hydration levels ofGCLs. Based on the size of the used test apparatus, the radius re

was equal to the radius of the gas permeability cell �50 mm�. Theproperties of the GCL used in their study are listed in Table 2. Thedetails of the test setup and procedures were explained thoroughlyin Bouazza and Vangpaisal �2006�.

The conventional �keseries� of the used GCL as well as GCL

thickness �tGCL� at different moisture contents �wc� was obtainedfrom Vangpaisal and Bouazza �2004� as shown in Fig. 18. As thegas permeability results show scatter with moisture content, upperand lower bound relationships were therefore used for modelingof gas leakage. The gas permeability of the merged interface andGCL cover geotextile layer, kgti, was obtained from the experi-mental results reported by Bouazza and Vangpaisal �2007b�. Asthe test results shown in Fig. 19 indicate, the effect of moisturecontent on kgti is not significant. Thus, an average kgti value of2.0�10−4 m /s was used in the predictive model where tgti was

Fig. 16. Path 1: Evolution of geometric characteristics of flow net�equipotentials line� as log�kgti /kbt� increases �ro / te=0.9, re /ro=5.55,tgti / te=0.25�

assumed as independent of moisture content and, based on several


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direct measurements, was given a constant value of 1.5 mm. ThenEq. �2� and the test results shown in Fig. 18�b� was used to de-termine the gas permeability of bentonite GCL layer, kbt, at dif-ferent moisture contents. The bentonite GCL layer thickness, tbt,and the effective GCL thickness, te, were calculated at differentmoisture contents using the results shown in Fig. 18�a� as follows:

tbt = tGCL − 2tgti �13�

Fig. 17. Path 2: Evolution of geometric characteristics of flow net�equipotentials line� as tgti / te=0.25 increases �ro / te=0.4, re /ro=12.5,log�kgti /kbt�=2�

Table 2. Properties of Tested GCL

Type of bentonite Sodium/powder

Type of bonding Needle punched

Upper geotextile Nonwoven

Lower geotextile Nonwoven+Silt film woven

Mass per unit area of GCL �Kg /m2� 3.8–4.5

Mass per unit area of bentonite �Kg /m2� 3.1–3.8

Thickness of GCL at dry state �mm� 7.8–8.7

As received moisture content �%� 9–14



te = tbt + tgti �14�

Based on Fig. 18 it can be inferred that te increases and kbt de-creases as moisture content increases and considering that tgti andkgti are independent of moisture content, this leads to a decrease intgti / te and increase in log�kgti /kbt� as moisture content increases.

Fig. 18. Gas permeability and thickness of partially hydrated GCLunder a 20 kPa surcharge �modified Vangpaisal and Bouazza �2004��

Fig. 19. Gas permeability of the merged interface and GCL covergeotextile layer versus moisture content �modified from Bouazza andVangpaisal �2007b��


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Moreover, increasing moisture content at constant re /ro also de-creases Fr as ro / te decreases �Fig. 12�. Consequently, the geom-etry of the CLFS surface can be considered as being dependent onmoisture content. Therefore, the path of gas leakage tests con-ducted by Bouazza and Vangpaisal �2006� at different moisturecontents can be then presented in F−log�kgti /kbt� and F− �tgti / te�space by the intersection of the tilted vertical gray plane with theCLFS �Fig. 20�, which represent given moisture contents at thecorresponding values of log�kgti /kbt� and �tgti / te�. The F values ofthis path can be used to determine the leakage flow rate using Eq.�6�. A comparison of predicted gas leakage rates using the pro-posed technique and measured gas leakage rate as reported inBouazza and Vangpaisal �2006� is shown in Fig. 21. The upperand lower limit prediction in Fig. 21 are related to upper andlower gas permeability of GCL as shown in Fig. 18�b�. Excellentagreement is observed between the prediction by the proposednumerical model and the experimentally measured gas leakagevalues.

Conclusions

The salient conclusions that can be drawn from this study of theproblem of gas leakage through a defective GM component of acomposite GM/GCL liner are as follows:• It is recommended to adopt the GCL as a three-layered system.

Therefore, the transport characteristics of the GCL compo-nents �i.e., geotextile, bentonite� should be determined sepa-

Fig. 20. Evolution of CLFS at different ro / te and re /ro ratios

Fig. 21. Comparison between the predicted gas leakage rates and theexperimental results reported by Bouazza and Vangpaisal �2006�



rately instead of using an equivalent series transportcharacteristic for the whole GCL as it is usually done.

• A new conceptual two-layered system was introduced wherethe possible interface zone between GM and GCL was mergedwith GCL cover geotextile layer and handled as one layer. TheGCL bentonite layer is considered as the second and thirdlayer. The proposed system considers the possibility of hori-zontal radial flow through the cover geotextile layer which wasneglected in the conventional two-layered system.

• The results of the numerical simulation point to the existenceof a CLFS in F−log�kgti /kbt� and F− �tgti / te� nondimensionalspaces. Mathematical forms that describe and govern theCLFS for different conditions were provided. The proposedCLFS can be used to characterize and evaluate the gas leakageflow rate under the framework of the Forchheimer’s analyticalsolution. The adequacy of the CLFS hypothesis was verifiedby comparing the gas leakage flow rates measured experimen-tally with the predicted results using the CLFS concept, a goodagreement was found between the two approaches.

Acknowledgments

The present study was supported by a Discovery grant from theAustralian Research Council. Our sincere appreciation is ex-tended to the Council.

Notation

The following symbols are used in this paper:a, c, n, m � equation constants;

Cn � contact condition parameter;F � flow factor;

FN � normalized nondimensional flow parameter;Fo � nondimensional flow parameter at zero value

of log�kgti /kbt� or �tgti / te�;Fr � nondimensional flow parameter at residual

limit of TLCCs or GLCCs;Frr � nondimensional flow parameter at residual

state of CLFS;h � total head drop across the composite liner;

kbt � gas permeability of bentonite layer;ke � equivalent gas permeability;

keparallel � equivalent parallel gas permeability;ke

series � equivalent series gas permeability;kg � gas permeability;kgt � gas permeability of geotextile layer;kgti � gas permeability of the merged upper

geotextile and interface layer;P � gas pressure;

Patm � atmospheric gas pressure;Q � leakage flow rate;re � radius of no flow boundary condition;ro � radius of defect;

tGCL � thickness of GCL;tGM � thickness of GM layer;

tbt � thickness of bentonite layer;te � tgti+ tbt;tgt � tgt

U + tgtL ;

t U
gti � ti+ tgt;

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tgtU, tgt

L � thickness of the upper and lower geotextilelayer;

ti � thickness of interface layer; and� � slope index.

References

Barroso, M., Touze-Foltz, N., von Maubeuge, K., and Pierson, P. �2006�.“Laboratory investigation of flow rate through composite liners con-sisting of a geomembrane, a GCL and soil liner.” Geotext.Geomembr., 24�3�, 139–155.

Benson, C. G., Thorstad, P. A., Jo, Y. Y., and Rock, S. A. �2007�. “Hy-draulic performance of geosynthetic clay liners in a landfill finalcover.” J. Geotech. Geoenviron. Eng., 133�7�, 814–827.

Bouazza, A. �2004�. “Effect of wetting on gas transmissivity of a nonwoven geotextile.” Geotext. Geomembr., 22�6�, 531–541.

Bouazza, A., Jefferis, S., and Vangpaisal, T. �2007�. “Analysis of degreeof ion exchange on atterberg limits and swelling of geosynthetic clayliners.” Geotext. Geomembr., 25�3�, 170–185.

Bouazza, A., and Rahman, F. �2007�. “Oxygen diffusion through partiallyhydrated geosynthetic clay liners.” Geotechnique, 57�9�, 767–772.

Bouazza, A., and Vangpaisal, T. �2003�. “An apparatus to measure gaspermeability of geosynthetic clay liners.” Geotext. Geomembr., 21�2�,85–101.

Bouazza, A., and Vangpaisal, T. �2004�. “Effect of straining on gas ad-vective flow of a needle punched GCL.” Geosynthet. Int., 11�4�, 287–295.

Bouazza, A., and Vangpaisal, T. �2006�. “Laboratory investigation of gasleakage rate through a GM/GCL composite liner due to a circulardefect in the geomembrane.” Geotext. Geomembr., 24�2�, 110–115.

Bouazza, A., and Vangpaisal, T. �2007a�. “Gas permeability of GCLs:Effect of poor distribution of needle punched fibres.” Geosynthet. Int.,14�4�, 248–252.

Bouazza, A., and Vangpaisal, T. �2007b�. “Gas transmissivity at the in-terface of a geomembrane and the geotextile cover of a partially hy-drated geosynthetic clay liner.” Geosynthet. Int., 14�5�, 316–319.

Bouazza, A., Vangpaisal, T., Abuel-Naga, H. M., and Kodikara, J. �2008�.“Analytical modelling of gas leakage rate through a geosynthetic clayliner-geomembrane composite liner due to a circular defect in thegeomembrane.” Geotext. Geomembr, 26�2�, 122–129.

Bouazza, A., Vangpaisal, T., and Jefferis, S. �2006�. “Effect of wet drycycles and cation exchange on gas permeability of geosynthetic clayliners.” J. Geotech. Geoenviron. Eng., 132�8�, 1011–1018.

Bouazza, A., Zornberg, J., and Adam, D. �2002�. “Geosynthetics in wastecontainments: Recent advances.” Proc., 7th Int. Conf. on Geosynthet-ics, Vol. 2, P. Delmas and J. P. Gourc, eds. A. A. Balkema, 445–507.

Cartaud, F., Goblet, P., and Touze-Foltz, N. �2005b�. “Numerical simula-tion of the flow in the interface of a composite bottom barrier.” Geo-text. Geomembr., 23�6�, 513–533.

Cartaud, F., Touze-Foltz, N., and Duval, Y. �2005a�. “Experimental inves-tigation of the influence of a geotextile beneath the geomembrane in acomposite liner on leakage through a hole in the geomembrane.” Geo-text. Geomembr., 23�2�, 117–143.

Dickinson, S., and Brachman, R. W. I. �2006�. “Deformations of a geo-synthetic clay liner beneath a geomembrane wrinkle and coarsegravel.” Geotext. Geomembr., 24�5�, 285–298.

Didier, G., Bouazza, A., and Cazaux, D. �2000�. “Gas permeability ofgeosynthetic clay liners.” Geotext. Geomembr., 18�2�, 235–250.

Foose, G. J., Benson, C. H., and Edil, T. B. �2001�. “Predicting leakage



through composite landfill liners.” J. Geotech. Geoenviron. Eng.,127�6�, 510–520.

Fredlund, D. G., and Xing, A. �1994�. “Predicting the permeability func-tion for unsaturated soils using the soil water characteristic curve.”Can. Geotech. J., 31�3�, 533–546.

Fredlund, D. G., Xing, A., Fredlund, M. D., and Barbour, S. L. �1995�.“The relationship of the unsaturated soil shear strength to the soil-water characteristic curve.” Can. Geotech. J., 32, 440–448.

Giroud, J. P., and Touze-Foltz, N. �2005�. “Equations for calculating therate of liquid flow through geomembrane defects of uniform widthand finite or infinite length.” Geosynthet. Int., 12�4�, 186–199.

Hurst, P., and Rowe, R. K. �2006�. “Average bonding peel strength ofgeosynthetic clay liners exposed to jet fuel A-1.” Geotext. Geomembr.,24�1�, 58–63.

Meer, S. R., and Benson, C. G. �2007�. “Hydraulic conductivity of geo-synthetic clay liners exhumed from landfill final covers.” J. Geotech.Geoenviron. Eng., 133�5�, 550–563.

Nye, C. J., and Fox, P. J. �2007�. “Dynamic shear behaviour of a needlepunched geosynthetic clay liner.” J. Geotech. Geoenviron. Eng.,133�8�, 973–983.

Rowe, R. K. �1998�. “Geosynthetics and the minimization of contaminantmigration through barrier systems beneath solid waste.” Proc., 6th Int.Conf. on Geosynthetics, Atlanta, Industrial Fabrics Association Inter-national, St. Paul, MN, 27–102.

Rowe, R. K. �2005�. “Long term performance of contaminant barriersystems.” Geotechnique, 55�9�, 631–678.

Rowe, R. K., and Booker, J. R. �2000�. “Theoretical solutions for calcu-lating leakage through composite liner systems.” Proc., Developmentsin Theoretical Geomechanics, Balkema, Rotterdam, 580–602.

Saidi, F., Touze-Foltz, N., and Goblet, G. �2006�. “2D and 3D numericalmodelling through composite liners involving partially saturatedGCLs.” Geosynthet. Int., 13�6�, 265–276.

Saidi, F., Touze-Foltz, N., and Goblet, G. �2008�. “Numerical modellingof advective flow through composite liners in case of two interactingadjacent square defects in the geomembrane.” Geotext. Geomembr.,26, 196–204.

Southen, J. M., and Rowe, R. K. �2007�. “Evaluation of the water reten-tion curve for geosynthetic clay liners.” Geotext. Geomembr., 25�1�,2–9.

Touze-Foltz, N., and Barroso, M. �2006�. “Empirical equations for calcu-lating the rate of liquid flow through GCL-geomembrane compositeliners.” Geosynthet. Int., 13�2�, 73–82.

Touze-Foltz, N., Duquennoi, C., and Gaget, E. �2006�. “Hydraulic andmechanical behaviour of GCLs in contact with leachate as part of acomposite liner.” Geotext. Geomembr., 24�3�, 188–197.

Touze-Foltz, N., and Giroud, J. P. �2003�. “Empirical equations for cal-culating the rate of liquid flow through composite liners due geomem-brane defects.” Geosynthet. Int., 10�6�, 215–233.

Touze-Foltz, N., and Giroud, J. P. �2005�. “Empirical equations for cal-culating the rate of liquid flow through composite liners due to largecircular defects in the geomembrane.” Geosynthet. Int., 12�1�, 205–207.

Touze-Foltz, N., Rowe, R. K., and Duquennoi, C. �1999�. “Liquid flowthrough composite liners due to geomembrane defects: analytical so-lutions for axi-symmetric and two-dimensional problems.” Geosyn-thet. Int., 6�6�, 455–479.

Vangpaisal, T., and Bouazza, A. �2004�. “Gas permeability of partiallyhydrated geosynthetic clay liners.” J. Geotech. Geoenviron. Eng.,130�1�, 93–102.


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