AL composite liners

Geotextiles and Geomembranes xxx (xxxx) xxx–xxx

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Geotextiles and Geomembranes journal homepage: www.elsevier.com/locate/geotexmem

Fully transient analytical solution for degradable organic contaminant transport through GMB/GCL/AL composite liners Shi-Jin Feng∗, Ming-Qing Peng, Hong-Xin Chen, Zhang-Long Chen Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China

ARTICLE INFO

ABSTRACT

Keywords: Geosynthetics Biodegradation Advection Diffusion Composite liner Analytical solution

In this study, analytical solution for degradable organic contaminant transport through a composite liner consisting of a geomembrane (GMB) layer, a geosynthetic clay liner (GCL) and an attenuation layer (AL) is derived by the separation of variables method. The transient contaminant transport in the whole composite liner can be well described avoiding some weird phenomena in existing analytical solutions. The results of parametric study show that GCL has significant effect on improving the barrier efficiency especially for scenarios with high leachate head. The biodegradation and adsorption in GCL have significant influence on the contaminant transport through the composite liner when the half-life of contaminant in GCL is less than 5 years. Otherwise, the effect can be neglected.

1. Introduction Liner system is an important part of modern Municipal Solid Waste (MSW) landfill to protect surrounding environment and groundwater from landfill leachate contaminants (Rowe and Brachman, 2004; Rowe et al., 2004). Composite liner, which generally consists of a geomembrane (GMB) layer, a compacted clay liner (CCL) or a geosynthetic clay liner (GCL) and an attenuation layer (AL), has been widely used in practices (Rowe and Brachman, 2004; Barroso et al., 2006; Bouazza and Bowders, 2010; El-Zein et al., 2012; Park et al., 2012; Xie et al., 2015a, b; Wu and Shi, 2017). Since organic compounds are among the most hazardous constituents in landfill leachate (Edil, 2003; Islam and Rowe, 2008), it is substantially essential to study the transport of organic contaminants in composite liners. Defects may occur in GMB for lining landfills, including holes, patches, and cracks due to inadequate seaming, punctures and tears (Touze-Foltz et al., 1999; Rowe et al., 2003; Giroud, 2005). Rowe and Abdelatty (2012, 2013) reported when modeling the transport of organic contaminants through composite liners, both leakage through GMB defects and diffusion through intact GMB should be considered, especially in cases with high leachate head and numerous GMB defects (Xie et al., 2010, 2015a; Chen et al., 2015; Feng et al., 2019). Analytical methods can offer fundamental insight into the physical mechanisms of contaminant transport and can be used as a tool for preliminary

assessment. Some researches so far have focused on analytical solutions considering diffusion and advection for predicting contaminant migration through composite liners (Xie et al., 2010, 2011; 2015a; Chen et al., 2015; Feng et al., 2019). Apart from diffusion and advection, biodegradation has proved to be one of the governing processes for deciding the mobility and persistence of organic contaminants in soils (Davis et al., 2009; Xu et al., 2009; Xie et al., 2013; Wu and Shi, 2017). Plenty of organic contaminants can be significantly degraded by microbes in soil liner, such as CCL and AL (Rowe et al., 1997; Bright et al., 2000; Kim et al., 2001; Head et al., 2003; Mitchell and Santamarina, 2005; Singh et al., 2008; García-Delgado et al., 2015; Palsikowski et al., 2018). Furthermore, recent studies indicate that many organic contaminants significantly degrade not only in AL, but also in GCL (Xie et al., 2013; Guan et al., 2014; Wu et al., 2016). Two analytical solutions have been developed to predict the performance of composite liners considering the effect of biodegradation (Xie et al., 2013; Guan et al., 2014). However, these solutions assume that the transport of organic contaminants in GMB or GCL is a steady-state process. More importantly, these solutions are only applicable for double-layer liners and cannot be directly used in liners with multiple layers. In fact, composite liner with multiple layers (e.g., GMB/GCL/AL) has been widely used in landfill (Rowe et al., 2004; Barroso et al., 2006; El-Zein et al., 2012; Xie et al., 2015a, b; Wu et al., 2016; Wu and Shi,

Corresponding author. E-mail addresses: [email protected] (S.-J. Feng), [email protected] (M.-Q. Peng), [email protected] (H.-X. Chen), [email protected] (Z.-L. Chen). ∗

https://doi.org/10.1016/j.geotexmem.2019.01.017 Received 9 August 2018; Received in revised form 24 January 2019; Accepted 28 January 2019 0266-1144/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Shi-Jin Feng, et al., Geotextiles and Geomembranes, https://doi.org/10.1016/j.geotexmem.2019.01.017

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2017). However, scanty transient analytical solution for composite liner with multiple layers considering degradation is available at present. Wu et al. (2016) presented analytical solution for steady-state diffusion of degradable organic contaminant through a triple-layer composite liner. However, the solution cannot consider the leakage through GMB defects (i.e., the effect of advection). Fortunately, Xie et al. (2018) have developed an analytical model that considers the coupling effect of advection, diffusion and degradation for triple-layer composite liner, which is the state-of-the-art of this issue. However, the transport in GMB and GCL is still assumed to be a steady-state process. In fact, the transport of organic contaminant is transient in the whole composite liner (Foose, 2002). In addition, the steady-state assumption also causes that the adsorption effect in GCL cannot be considered. To the best of our knowledge, no fully transient analytical solution for degradable contaminant transport through a triple-layer liner system synthetically considering the effects of diffusion, advection and adsorption is available, where the “fully transient” denotes that the contaminant concentration changes with time in the whole composite liner. This paper tries to propose a new analytical solution to address the above problem. The proposed solution is first verified against an existing analytical solution and a numerical model. Comparison with the state-of-the-art of this issue is then conducted to show the significance of the present solution. Finally, since the influence of biodegradation and adsorption in GCL cannot be considered in the existing analytical models for triple-layer liners, particular attention is paid to this issue.

Cgmb (z , t ) t

= Dgmb

2C gmb (z , z2

hole Cgmb (z , t )

t) a

(1)

z

where Cgmb (z, t) is the contaminant concentration in GMB at any pohole sition z and any time t; Dgmb is the diffusion coefficient of GMB; Cgmb (z, t) is the contaminant concentration in the flow through the holes in GMB, which is assumed to be 1/Kg times Cgmb (z, t) according to Henry's law; Kg is the partition coefficient between GMB and the leachate, which is the ratio of the contaminant concentration at equilibrium in GMB to that in the leachate; va is Darcy velocity of the leakage flow through the composite liner. Rowe (1998) developed a simple equation to predict leakage rate through a hole in a GMB coincident with a wrinkle, which well agrees with the results of numerical analysis (Foose et al., 2001):

Q=

2h w L w (kb + l

kl )

(2)

where Q is the leakage rate through a hole in a GMB coincident with a wrinkle; Lw is the length of wrinkle; 2b is the width of wrinkle; l is the total thickness of GCL and AL; θ is the transmissivity of the interface between GMB and GCL; k is the harmonic mean hydraulic conductivity of GCL and AL. The Darcy velocity in the composite liner can then be obtained by (Rowe and Brachman, 2004; Xie et al., 2010, 2015a, 2018) a

(3)

= mh Q / A

where mh is the number of holes in GMB of the area under consideration; A is the area of the liner under consideration. Here k is expressed as (Rowe, 2012; Xie et al., 2015a)

2. Mathematical model

k=

The composite liner consists of three individual layers, namely a GMB layer, a GCL layer and an AL (see Fig. 1). Advection and diffusion occur in all the three layers, and biodegradation and adsorption occur in GCL and AL. The transport of organic contaminant through the whole composite liner is transient. z is the spatial dimension in the direction of contaminant transport, and the origin is on the top surface of the GMB. Lgmb, Lgcl, Lal and Lcl are the thicknesses of GMB, GCL, AL and the whole composite liner, respectively. hw is the hydraulic leachate head mounding on GMB surface. Cb represents the contaminant concentration at the bottom of AL layer. In order to facilitate developing the mathematical model, the following assumptions are adopted: (1) GCL and AL are both saturated and homogeneous (Benson et al., 1999; Benson, 2015; McWatters et al., 2016); (2) the concentration of contaminant in leachate is assumed to be constant at C0 (Foose, 2002; Xie et al., 2013), which would result in conservative predictions of contaminant transport (Shackelford, 1990; Foose, 2002); (3) the adsorption of contaminant in GCL and AL is a linear and equilibrium process (Edil, 2003); (4) the first-order kinetic biodegradation model is used (Williams and Tomasko, 2008; Guan et al., 2014). The governing equation for transient transport of organic contaminant through GMB is (Feng et al., 2019)

Lgcl + Lal (4)

Lgcl / kgcl + Lal / kal

where kgcl, kal are the hydraulic conductivities of GCL and AL, respectively. The governing equation for transient transport of biodegradable organic contaminant through GCL is (Guan et al., 2014; Wu et al., 2016; Feng et al., 2019)

Cgcl (z, t ) t

=

Dgcl R d,gcl

2 C (z , gcl z2

t)

gcl

Cgcl (z, t )

Rd,gcl

z

gcl Cgcl (z ,

t)

(5)

where Cgcl (z, t) is the concentration of contaminant in GCL at any position z and any time t; Dgcl is the dispersion coefficient of GCL; Rd,gcl is the retardation factor of GCL, which can be determined by Rd,gcl = 1 + ρd,gclKd,gcl/ngcl, where ρd,gcl is the dry density of GCL, Kd,gcl is the distribution coefficient of GCL, reflecting the adsorption capacity of the GCL, and ngcl is the porosity of GCL; νgcl is the seepage velocity in GCL; λgcl is the first-order degradation constant in GCL, which can be determined by λgcl = ln2/t1/2,gcl, where t1/2,gcl is the half-life of contaminant in GCL. Similarly, the governing equation for the transient transport of biodegradable organic contaminant through AL is (Xie et al., 2018; Feng et al., 2019)

Cal (z, t ) D = al t R d,al

2 C (z , al z2

t)

al

Rd,al

Cal (z, t ) z

al Cal (z ,

t)

(6)

The parameter meanings are similar to those for GCL. The relationship among the Darcy velocity and the seepage velocities in GCL and AL is νa = ngclνgcl = nalνal. Initially, there is no contaminant in the composite liner. Then the initial condition is

Cgmb (z , 0) Cgcl (z, 0) Cal (z, 0)

=0 (7)

The top boundary condition at the interface between GMB and leachate is (Sangam and Rowe, 2005; Xie et al., 2015a):

Fig. 1. Mathematical model for contaminant transport through composite liner. 2

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S.-J. Feng, et al.

(8)

Cgmb (0, t ) = C0 K g

The continuity conditions of concentration and mass flux at the interface between GMB and GCL, and that between GCL and AL are

Cgmb (z , t ) Kg

va

= Cgcl (Lgmb , t )

z = Lgmb

z = Lgmb

Cgmb (z , t )

Dgmb

Kg

z

z = Lgmb

= ngcl vgcl Cgcl (z, t )

Cgcl (z , t )

ngcl Dgcl

z = Lgmb + Lgcl

= Cal (z, t )

Cgcl (z , t ) z

z

z = Lgmb

= nal Dal

z = Lgmb + Lgcl

z = L cl

Cal (z, t ) z

= 0 (for the Dirichlet boundary)

w 2 (z , t )

w3 (z , t )

i

Rd, i

Ci (z, t ) z

i Ci (z ,

ni Di

= Ci + 1 (z , t )

Ci (z , t ) z

z = Li ,

= ni + 1 Di + 1 z = Li

Ci + 1 (z, t ) z

i = 1, 2

z = Li

i = 1, 2, 3

(23)

z = L1 / Kg + L2

= n3 D3 z = L1 / K g + L2

w 3 (z , t ) z

z = L1 / K g + L2

(24) (25)

= 0 (for the Neumann boundary)

(26)

z=H

(27) (28)

i = 1, 2, 3

Cm gm, i (z ) e ai z

mt,

i = 1, 2, 3

(29)

m=1

Cm, ai are coefficients; βm are eigenvalues of the composite liner. When βm − i2 /(4DiRd,i) − λi ≥ 0,

gm, i (z ) = Am, i sin µi When βm −

m, i

z + Bm, i cos µi H

2 i /(4DiRd,i) − λi

gm, i (z ) = Am, i sinh µi

m, i

m, i

z , i = 1, 2, 3 H

(30a)

< 0,

z + Bm, i cosh µi H

m, i

z , i = 1, 2, 3 H (30b)

Substituting Eqs. (29) and (30) into Eq. (19), the coefficients ai and λm,i can be described as

ai =

m, i

vi , 2Di =

i = 1, 2, 3

R d,1 D1

m

vi2 4Di Rd, i

(31) i

,

i = 1,2,3

(32)

and the coefficients μi are defined as

µi = H (17)

Rd, i /Di , Rd,1/ D1

i = 1, 2, 3

(33)

Substituting Eqs. (29) and (30) into the continuity conditions (Eqs. (23) and (24)), the coefficients Am,i and Bm,i are determined by the following recursive equation:

The problem can be transformed into two sub-problems by substituting Eq. (18) into Eq. (15):

Ci (z, t ) = wi (z, t ) + ui (z ) C0,

= w 3 (z , t )

= 0 (for the Dirichlet boundary)

wi (z , t ) =

(16)

,

(22)

z = L1 / Kg

The solution to the sub-problem 1 satisfying all the relevant conditions can be expressed as

t ), i = 1, 2, 3

i = 1, 2

w 2 (z , t ) z

The initial condition is

where i = 1, 2, 3 represents the parameters for GMB, GCL and AL, respectively. The normalized general continuity conditions between GMB, GCL, and AL are z = Li

z=H

wi (z , 0) = 0,

(15)

Ci (z, t )

= n2 D2 z = L1 / Kg

H = L1 / K g + L 2 + L3

The form of the governing equation for GMB is different from that for GCL and AL, resulting in much difficulty in solving the problem. Thus, a normalized general governing equation that is applicable for all the three layers is acquired to solve the problem, which is enclosed in the Appendix. The normalized general governing equation is (the normalization mark, asterisk, is omitted in the following derivation for simplicity)

t)

(21)

z = L1 / K g

where

3. Derivation of the analytical solutions

2C (z , i z2

= w 2 (z , t )

z = L1/ K g + L2

w 3 (z , t ) z

The above two equations represent the two most widely adopted ideal boundary conditions. In fact, the real boundary condition may be between them. So Eqs. (13) and (14) bracket the range of possible boundary conditions that may exist beneath a composite liner. The Dirichlet boundary can well describe the scenario of an aquifer or a leachate detection layer bellow the bottom of the composite liner where the contaminant can instantaneously be removed. The Neumann boundary can well describe the scenario of an aquitard bellow the bottom of the composite liner.

Ci (z, t ) D = i t Rd, i

z = L1/ K g

The potential bottom boundary conditions are

(14)

z = Lcl

(20)

=0

w 2 (z , t ) z

n2 D2

(12)

(13)

= 0 (for the Neumann boundary)

t ), i = 1, 2, 3

The continuity conditions at the interface between GCL and AL are

where K g is the partition coefficient between GMB and GCL, which is assumed to be equal to Kg (Chen et al., 2009). The boundary conditions are

Cal (z , t )

z=0

w1 (z, t ) z

D1

(10) (11)

Cal (z , t ) z

i wi (z ,

The continuity conditions at the interface between GMB and GCL

w1 (z, t )

z = Lgmb

z = Lgmb + Lgcl

z = Lgmb + Lgcl

w1 (z, t )

Cgcl (z, t )

ngcl Dgcl

vi wi (z, t ) R d, i z

(19)

are z = Lgmb

t)

The top boundary condition is

(9)

Cgmb (z , t )

2w (z , i z2

wi (z , t ) D = i t Rd, i

(18)

Am, i = Ti Bm, i

The governing equation for the sub-problem 1 is 3

1

Am, i Bm, i

1 1

,

i = 2, 3

(34)

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S.-J. Feng, et al.

where

Ti =

(4Di+1Rd,i+1) − λi+1 < 0,

B¯i G¯ i + C¯i F¯i D¯ i G¯ i C¯i H¯ i , i = 1, 2 B¯i E¯i A¯ i F¯i A¯ i H¯ i + D¯ i E¯i

He (ai ai +1 ) zi ni + 1 Di + 1 µi + 1

m, i + 1

when βm − i2 /(4DiRd,i) − λi ≥ 0 (4Di+1Rd,i+1) − λi+1 ≥ 0,

( B¯ = sin (µ C¯ = cos (µ D¯ = cos (µ A¯ i = sin

z µi + 1 m , i + 1 H

)

i

z i m, i H

i

z i + 1 m, i + 1 H

i

z i m, i H

)

(

z m, i + 1 H

z m, i + 1 H

)

z m, i H

)

i

z i m, i H

z i m, i H

i i

i

i i

when βm − (4Di+1Rd,i+1) − λi+1 < 0, z m, i + 1 H

( ¯ D = cos (µ

z m, i + 1 H

C¯i = cosh µi + 1 i

E¯i =

z i m, i H

) (

z m, i + 1 H

cosh µi + 1

(

z µi + 1 m , i + 1 H

z m, i H

i

z i m, i H

i

)

i

z i m, i H

)

+

βm −

z i m, i H

i

i i

z i m, i H

)

(37)

in which, for the Dirichlet bottom boundary

[sin(µ3

m,3)

[sinh(µ3

µ3

, i = 1, 2 z i m, i H

T3 =

)

m,3)

cos(µ3

m,3)],

cosh(µ3

when

m,3)],

m

when

vi2 4Di R d, i

m

vi2 4Di R d, i

i

0

i

<0

i i

m,3

µ3 µ3 µ3

cos (µ3

m,3 sin (µ3

cosh (µ3 m,3 sinh (µ3

m,3

m,3)

+ Ha3 sin (µ3

(

z m, i H

sin

(

)

z m, i + 1 H

z m, i + 1 H

m,3 )

(

z i m, i H

+ Ha3 sinh (µ3 m,3 ) + Ha3 cosh (µ 3

z µi m , i H

)

2 i /(4DiRd,i) − λi

βm

–

)

2 i

+

vi2 4Di Rd, i

m

T

m,3 )

, when

m

vi2 4Di Rd, i

0

i

i

<0

(40)

The coefficient Cm can be determined by the orthogonality condition (Guerrero et al., 2013; Feng et al., 2019):

1/

z1

0

(

)

z m, i H

)

z i m, i H

and

βm

a1 zdz

z2 u2 (z ) n2 R d,2 S1 gm,2 (z ) e a2 zdz z1 z3 u (z ) n3 Rd,3 S2 gm,3 (z ) e a3 zdz z2 3 z z + z 2 n2 R d,2 S1 gm2 ,2 (z ) dz + z 3 n3 Rd,3 S2 gm2 ,3 (z ) dz 1 2

in which

S1 = e 2(a2 S2 = e

i i

n1 Rd,1 gm2 ,1 (z ) dz

u1 (z ) n1 Rd,1 gm,1 (z ) e

(41)

, i = 1, 2

)

<0

, when

T0 = [Am,1 Bm,1]T = [1 0]T

(36b)

z1

) + n D a cosh (µ i

m,3)

T

Cm

ni Di ai sinh µi

z m, i + 1 H

m,3 )

T0 can then be obtained by substituting Eqs. (29) and (30) into the top boundary condition (Eq. (20)):

and

)

z µi + 1 m , i + 1 H

m,3)

m,3 ) + Ha3 cos (µ3

(39)

a1) z1

(42)

2z1 a1+ 2(z1 z2) a2 + 2z2 a3

In this way, all the coefficients in the solution to the sub-problem 1 are determined. The governing equation for the sub-problem 2 is

)

Di d 2ui (z ) Rd, i dz 2

(36c) when

z i + 1 m, i + 1 H

i+1

+

+ ni + 1 Di + 1 ai + 1 cos µi + 1

(

)

z i m, i H

i i

z i + 1 m, i + 1 H

+

(

µ m, i + 1 ni + 1 Di + 1 i +1 H

, i = 1, 2

)

T3 T2 T1 T0 = 0

1/

)

(

i

µ m, i + 1 ni + 1 Di + 1 i +1 H

=

ni + 1 Di + 1 ai + 1 sin µi + 1

sinh

–

)

)

i i

i

µ m, i + 1 E¯i = ni + 1 Di + 1 i +1 H cos µi + 1

H¯ i =

βm

)

µ F¯i = ni Di i Hm, i cosh µi

)

) n D a sinh (µ sinh (µ ) a cosh (µ ) sinh (µ ) + n D a cosh (µ z m, i H

0

z i + 1 m, i + 1 H

µ ni Di i Hm,i

(36a) 2 i

z i + 1 m, i + 1 H

i

G¯ i =

)

z i + 1 m, i + 1 H

i+1

z m, i + 1 H

z m, i + 1 H

(38)

when βm − i2 /(4DiRd,i) − λi < 0 (4Di+1Rd,i+1) − λi+1 ≥ 0,

( B¯ = sinh (µ C¯ = cos (µ D¯ = cosh (µ

(

z m, i + 1 H

and for the Neumann bottom boundary

z i m, i H

A¯ i = sin µi + 1

(

cosh µi + 1

Substituting Eqs. (29) and (30) into the bottom boundary conditions (Eqs. (25) and (26)), the eigenvalues βm are determined by the characteristic equation:

)

µ m, i + 1 G¯ i = ni + 1 Di + 1 i +1 H

µ H¯ i = ni Di i Hm,i

z i m, i H

and

) n D a sin (µ sinh (µ ) a cosh (µ ) sin (µ ) + n D a cos (µ

+ ni + 1 Di + 1

)

(36d)

T3 =

ni + 1 Di + 1 ai + 1 sinh µ F¯i = ni Di i Hm, i cos µi

)

µ m, i + 1 ni + 1 Di + 1 i +1 H

µ H¯ i = ni Di i Hm,i

)

)

(

µ m, i + 1 ni + 1 Di + 1 i +1 H

i

z i m, i H

+ ni + 1 Di + 1

z i + 1 m, i + 1 H

)

z i + 1 m, i + 1 H

G¯ i =

z i + 1 m, i + 1 H

i+1

z m, i H

i

, i = 1, 2

2 i /(4DiRd,i) − λi ≥ 0

(

)

i

(

µ m, i + 1 ni + 1 Di + 1 i +1 H

B¯i = sin µi

1/

µ F¯i = ni Di i Hm, i cosh µi

) n D a sin (µ sin (µ ) a cos (µ ) sin (µ ) + n D a cos (µ

(

+

)

ni + 1 Di + 1 ai + 1 sinh µi + 1

(

(

A¯ i = sinh µi + 1

2 i

)

µ F¯i = ni Di i Hm, i cos µi

µ H¯ i = ni Di i Hm,i

–

z m, i + 1 H

z i m, i H

E¯i =

ni + 1 Di + 1 ai + 1 sin µi + 1

+ ni + 1 Di + 1

βm

)

µ m, i + 1 cos µi + 1 E¯i = ni + 1 Di + 1 i +1 H

G¯ i =

and

( B¯ = sinh (µ C¯ = cosh (µ D¯ = cosh (µ

A¯ i = sinh µi + 1

(35)

–

2 i

+

1/ 4

vi dui (z ) R d, i dz

i u i (z )

= 0, i = 1, 2, 3

(43)

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S.-J. Feng, et al.

The top boundary condition is

u1 (z ) are

(44)

= C0

The continuity conditions at the interface between GMB and GCL

u1 (z )

D1

z=0

z = L1/ K g

u1 (z ) z

= u2 (z )

(45)

z = L1 / K g

= n2 D2 z = L1 / Kg

u2 (z ) z

(46)

z = L1/ K g

The continuity conditions at the interface between GCL and AL are

u2 (z )

z = L1/ K g + L2

u2 (z ) z

n2 D2

= u3 (z )

(47)

z = L1 / Kg + L2

= n3 D3 z = L1 / Kg + L2

u2 (z ) z

(48)

z = L1 / K g + L2

The potential bottom boundary conditions are

u3 (z )

z=H

u3 (z ) z

(49)

= 0 (for the Dirichlet boundary)

= 0 (for the Neumann boundary)

Fig. 2. Comparison between this paper and Wu et al. (2016) in terms of the concentration profiles of DCM in GMB/GCL/AL composite liner at t = 20 years (considering the effect of diffusion and degradation).

(50)

z=H

Obviously, the solution to the sub-problem 2 satisfying all the relevant conditions is

ui (z ) = ki,1 eri,1 z + ki,2 e ri,2 z ,

Dichloromethane (DCM) through a GMB/GCL/AL composite liner reported by Wu et al. (2016) is adopted. The effect of biodegradation occurring in GCL and AL are considered. Notably, diffusion is assumed to be steady-state in GMB layer in the solution proposed by Wu et al. (2016). C0 = 100 μg/L and the contaminant concentration gradient at the bottom boundary is fixed at zero. The concentration profiles at t = 20 years calculated by the present solution and that proposed by Wu et al. (2016) are compared in Fig. 2. Even though the results almost coincide, some slight differences can still be observed with naked eye, especially for the scenario of t1/ 2,gcl = 10 years and t1/2,al = 20 years. The reason is that Wu et al. (2016) assumed that diffusion is steady-state in the GMB layer, leading to that diffusion reaches steady state slightly earlier in the composite liner. Since the maximum allowable concentration in drinking water for many organic contaminants is very low (e.g., 0.005 mg/L for DCM in USEPA, 2001 and 0.002 mg/L for phenol in MCPRC, 2007), the difference should not be ignored despite the magnitude of difference is not large.

(51)

i = 1, 2, 3

where ki,1, ki,2, ri,1 and ri,2 can be obtained (from boundary and continuity conditions Eqs. (44)–(50)) following the similar procedures as those for sub-problem 1. Finally, substituting the solution to the two sub-problems into Eq. (18), the solution to the original problem is obtained. 4. Model verification Because there exists no fully transient analytical solution for diffusion and advection of degradable contaminant through triple-layer composite liner, the present solution is compared with an existing analytical solution and a numerical model to verify its accuracy and validity from different respects. If not specified, the required parameters are given in Table 1. Case 1. quasi-steady-state diffusion through a GMB/GCL/AL composite liner

Case 2. transient transport through a GMB/GCL/AL composite liner

In order to verify the performance of the present solution in describing degradable organic contaminant in triple-layer composite liner, an analytical solution to quasi-steady-state diffusion of

To verify the performance of the present solution more comprehensively, the concentration breakthrough curves and concentration profiles of benzene for a GMB/GCL/AL composite liner obtained by the present solution are compared with those obtained by COMSOL Multiphysics 5.4 (a commercial finite element software). Different values of the first-order degradation half-lives of GCL and AL are considered. The adopted model is shown in Fig. 1 and the underlying layer is aquitard. C0 = 1 μg/L and the calculated Darcy velocity is 5.62 × 10−10 m/s. The comparison of concentration breakthrough curves and concentration profiles are shown in Fig. 3. It is shown that the results obtained by the present fully transient analytical solution and the numerical method are in good agreement.

Table 1 Material parameters and transport properties of the composite liners. Properties Thickness, L (m) Porosity, n Dry density, ρd (g/cm3) Hydraulic conductivity, k ( × 10−10 m/s) Effective diffusion coefficient, D ( × 10−10 m2/s) Distribution coefficient, Kd (mL/g) Partition coefficient or Henry's coefficient, Kg

GMB

GCL a

DCM Benzene DCM Benzene DCM Benzene

0.0015 – – – 0.005a,d 0.002c,d – – 5a 19e

AL a

0.0138 0.86a 0.79a 0.5b 3.30a 3.30a 0.3d 0.3d – –

2a 0.4a 1.62a 1000b 8.90a 8.90a 0.28a 0.28a – –

5. Comparison with the state-of-the-art of this issue Benzene is selected as the representative degradable organic contaminant in leachate in this study. C0 = 1 μg/L, the length of the wrinkle Lw, the width of wrinkle 2b, the interface transmissivity θ, the hydraulic conductivity kgcl and kal are assumed to be 1000 m, 0.2 m, 5 × 10−11 m2/s, 5 × 10−11 m/s, 1 × 10−7 m/s, respectively (Xie et al., 2018). The suggested range of the first-order kinetics anaerobic half-life

Note. a Xie et al. (2009). b Xie et al. (2018). c Sangam and Rowe (2005). d Xie et al. (2013). e Nefso and Burns (2007). 5

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Fig. 4. Calculated concentration profiles of benzene without considering the effect of degradation (note different horizontal and vertical scales).

Fig. 3. Comparison of present analytical solution with COMSOL Multiphysics 5.4 in terms of: (a) concentration of benzene at the bottom of GMB/GCL/AL composite liner; (b) relative concentration profiles of benzene.

derived from the tests with different types of soil or sediment is 0.6 year to infinity (Hrapovic, 2001). Without special mentioning, the leachate head is 1.5 m, and the required parameters for GMB, GCL and AL are given in Table 1. 5.1. Concentration profiles of degradable organic contaminant The concentration profiles calculated by the present method are given in Figs. 4 and 5 for scenarios of t1/2,al = +∞ and 30 years, respectively, ignoring the biodegradation in GCL here. Zero concentration gradient at the bottom of composite liner is adopted. As shown in Fig. 4, the contaminant transport in GMB or GCL or AL has not reached a steady state even until 100 years for scenario of t1/2,al = +∞. If considering the effect of biodegradation in AL (t1/2,al = 30 years, see Fig. 5), the time to reach steady state for GMB and GCL is nearly 50 years, and that for AL is nearly 100 years. Therefore, contaminant transport in the whole composite liner is a transient process within a considerable elapsed time. Xie et al. (2018) developed an analytical model for degradable organic contaminant transport through the GMB/GCL/AL system, which is the-state-of-art of this issue. As aforementioned, the model assumes that the transport in GMB and GCL is steady-state. These assumptions may cause some problems, which are discussed in the rest part of this section.

Fig. 5. Calculated concentration profiles of benzene with t1/2,al = 30 years (note different horizontal and vertical scales). 6

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Fig. 6. Comparison between this paper and Xie et al. (2018) in terms of: (a) the relative concentration of benzene at the base of GMB for scenario of zero concentration gradient bottom boundary condition; (b) the mass flux at the base of GMB for scenario of zero concentration bottom boundary condition.

Fig. 7. Comparison between this paper and Xie et al. (2018) in terms of: (a) the relative concentration of benzene at the base of GCL for scenario of zero concentration gradient bottom boundary condition; (b) the mass flux at the base of GCL for scenario of zero concentration bottom boundary condition.

5.2. Comparison of this paper and the-state-of-the-art in terms of breakthrough curves

relatively high value at the initial elapsed time (e.g., the relative concentration is 7.1 at 0.0001 year for Lgmb = 1 mm). (3) The initial mass flux of this paper is zero, whereas that of Xie et al. (2018) is a relatively large value. The mass flux of Xie et al. (2018) gradually decreases from a high value to a stable value with time. The variation tendency is governed by the method itself, which will be explained as follows.

The results calculated by this paper and Xie et al. (2018) are compared at the bases of GMB (see Fig. 6), GCL (see Fig. 7) and AL (see Fig. 8) in terms of concentration for scenario of zero concentration gradient condition at the bottom boundary, and mass flux for scenario of zero concentration condition at the bottom boundary. Since the solution of Xie et al. (2018) is quasi-steady, the adsorption in GCL cannot be considered. Hence the distribution coefficient of benzene in GCL is adjusted to 0 in this section. The half-life of contaminant in the composite liner is taken as infinity. Three thicknesses of GMB (1, 1.5, 3 mm) are chosen. As shown in Fig. 6, the results of Xie et al. (2018) are insensitive to the thickness of GMB, whereas those of this paper indicate that the thickness of GMB has significant effect on the variation of relative concentration and mass flux. Same conclusion can be obtained from the results for GCL (see Fig. 7) and AL (see Fig. 8). Apart from the above difference, there are three important points to which special attention should be paid. Take Fig. 6 as an example.

The mass flux at the bottom of GMB is given by

Jgmb(Lgmb,t )= va

Cgmb (Lgmb,t ) Kg

Dgmb

Cgmb (Lgmb,t ) z

(52)

The concentration in GMB is (Xie et al., 2018)

Cgmb (z ) = A1 + A2 exp

va z Dgmb

(53)

where

A2 =

(1) Apart from the results calculated by this paper and Xie et al. (2018), the results calculated by the present method with the assumption of hole (z, t) = Cgmb (z, t), namely a modified model, are also given in Cgmb hole (z, t) = Cgmb (z, t) is the assumption of Xie et al. (2018) Fig. 6. Cgmb hole whereas the assumption of Cgmb (z, t) = Cgmb (z, t)/Kg is adopted in this paper. (2) For the results of this paper, the relative concentration at the bottom of GMB gradually increases from zero to a stable value, which is clearly reasonable. However, the results of Xie et al. (2018) in terms of concentration at bottom of GMB strangely start from a

K g Cal (Lgmb + Lgcl , t ) 1 + exp (Lgmb va / Dgmb )[1

C0 K g

K g + K g exp (Lgcl va /Dgcl / ngcl )]

(54)

Substituting Eq. (53) into Eq. (52), combined with the inlet boundary condition, Cgmb (0) = C0Kg, it can be deduced that

Jgmb(Lgmb,t )

A2

(55)

It can be concluded from Eq. (54) that

A2

Cal (Lgmb + Lgcl , t )

(56)

Because Cal(Lgmb + Lgcl, t) obviously is a monotonous non-reducing function with time, it is not strange that Jgmb (Lgmb,t) is a monotonous non-increasing function with time. Therefore, the mass flux of Xie et al. 7

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Fig. 9. Influence of the existence of GCL on the breakthrough curve.

in organic contaminant transport through composite liner will be deeply investigated in this part. Zero concentration gradient is adopted at the base of the composite liner in the following study. 6.1. Influence of the existence of GCL In order to investigate the influence of the existence of GCL, GMB/ GCL/AL and GMB/AL composite liners are selected. Two levels of leachate head (0 and 1.5 m), representing the scenarios of no advection and strong advection, respectively, are chosen in this part. Herein, t1/ 2,gcl = +∞ and t1/2,al = +∞. The comparison of GMB/AL and GMB/GCL/AL composite liners in terms of breakthrough curve is shown in Fig. 9. The breakthrough curves of the two composite liners almost overlap for the scenario of zero leachate head. Whereas for scenario of 1.5 m leachate head, the time for the bottom concentration to reach 0.5C0 are 80 and 11 years for the GMB/GCL/AL and the GMB/AL composite liners, respectively. Thus, for the scenario of no or weak advection, GCL's role in improving the performance of the composite liner is negligible. Whereas, for the scenario of strong advection, the GCL can significantly improve the performance of composite liner.

Fig. 8. Comparison between this paper and Xie et al. (2018) in terms of: (a) the relative concentration of benzene at the base of AL for scenario of zero concentration gradient bottom boundary condition; (b) the mass flux at the base of AL for scenario of zero concentration bottom boundary condition.

(2018) keeps at a stable value at the beginning, then gradually decreases to a new stable value. Similar phenomenon can be observed in GCL (see Fig. 7), but Xie et al. (2018) overestimate the results of both relative concentration and mass flux all the time. Similarly, it is not strange that the mass flux of GCL at the base, Jgcl (Lgmb + Lgcl, t), is a monotonous non-increasing function with time due to

Jgcl(Lgmb + Lgcl,t )

Cal (Lgmb + Lgcl , t )

(57)

For breakthrough curves of AL (see Fig. 8), the results of Xie et al. (2018) in terms of concentration and mass flux are smaller before reaching the steady state, even though the above weird phenomena disappear. Moreover, the mass flux of Xie et al. (2018) is abnormally bellow zero at the beginning and this part is not shown here (see Fig. 8b). The results obtained using the existing analytical solutions are reasonable when the transport processes reach steady-state, of which the elapsed time is around 200 years in the case (see Figs. 6–8). Even so, the above comparison reveals that it is essential to propose the present solution to overcome the drawbacks of the existing methods.

6.2. Influence of biodegradation in GCL Four different t1/2,gcl are selected to analyze the effect of biodegradation in GCL on the effectiveness of the triple-layer composite liner. t1/2,al is assumed to be infinity or 30 years and the breakthrough curves are shown in Figs. 10 and 11, respectively. Given the same t1/2,gcl, with the leachate head increasing from 0 to 1.5 m, the stable relative concentration increases a little for cases with t1/2,al = +∞ (see Fig. 10) whereas significantly increasing for cases with t1/2,al = 30 years (see Fig. 11). For example, for the scenario with t1/2,gcl = 1 year, t1/ 2,al = +∞, the stable relative concentration slightly increases from 0.88 (see Fig. 10a) to 0.89 (see Fig. 10b). However, it significantly increases from 0.10 (see Fig. 11a) to 0.23 (see Fig. 11b) for the scenario with t1/2,gcl = 1 year, t1/2,al = 30 years. It indicates that the biodegradation in AL has great influence on the contaminant transport through the composite liner. In Figs. 10 and 11, if t1/2,gcl is less than 5 years, the stable relative concentration significantly increases with increasing t1/2,gcl. With t1/2,gcl increasing from 5 years to infinity, the increment is quite small. In conclusion, if t1/2,gcl is less than 5 years, the degradation in GCL has a significant effect on the contaminant transport in the composite liner.

6. Results and discussion The biodegradation of organic contaminants is usually only considered in soil liner for the sake of conservative estimation. However, Guan et al. (2014) pointed out that biodegradation of organic contaminants (e.g., benzene) in GCL has great influence on its migration if t1/2,gcl is less than 1 year. To the best of our knowledge, there exists no analytical solution for organic contaminant transport through a triplelayer composite liner considering the biodegradation in GCL. In addition, the steady-state assumption also causes that the adsorption effect in GCL cannot be considered in the existing works. Thus, the role of GCL 8

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Fig. 11. Breakthrough curves with t1/2,al = 30 years: (a) hw = 0 m; (b) hw = 1.5 m.

Fig. 10. Breakthrough curves with t1/2,al = +∞: (a) hw = 0 m; (b) hw = 1.5 m.

Otherwise, this effect is negligible.

2, 4 and 8, respectively. The difference becomes quite small. For t1/ 2,gcl = +∞, the difference is negligible. In summary, if t1/2,gcl is less than 5 years, the adsorption in GCL has a significant impact on the contaminant transport. Otherwise, this effect is negligible.

6.3. Influence of adsorption in GCL In fact, experimental test for adsorption of benzene in GCL is limited. Fortunately, Rowe et al. (2005) experimentally investigated the distribution coefficient of GCL for benzene. The results show that the value of distribution coefficient is affected by temperature and approaches 4.4 mL/g at 22 °C. Thus the retardation factor (Rd,gcl = 1 + ρd,gclKd,gcl/ngcl) equals to 5.04 at 22 °C, where ρd,gcl = 0.79 g/cm3 and ngcl = 0.86 (see Table 1). Considering the influence of temperature and GCL properties on Rd,gcl, four values of Rd,gcl are considered here: 1, 2, 4, and 8. The breakthrough curves with different Rd,gcl are shown in Fig. 12. To consider the coupling effect of biodegradation and adsorption of GCL, four values of t1/2,gcl are considered. For the case of t1/2,gcl = 0.5 year (see Fig. 12a), the time required for the bottom concentration to reach 0.2C0 are 41, 46, 64 and 1280 years for Rd,gcl = 1, 2, 4 and 8, respectively. Meanwhile, the stable relative concentration decreases from 0.84 to 0.38 as Rd,gcl increases from 1 to 8. It indicates that the effect of adsorption in GCL is significant on the contaminant transport. However, as t1/2,gcl gradually increases, the influence of Rd,gcl weakens. For t1/2,gcl = 5 years (see Fig. 12c), the time required for the bottom concentration to reach 0.2C0 are 37, 38, 39 and 41 years for Rd,gcl = 1,

7. Summary and conclusions An analytical method for fully transient transport of degradable organic contaminants in GMB/GCL/AL composite liner is developed, which is applicable for typical bottom boundary conditions. Normalized governing equation for the whole composite liner is established to accommodate the difference between GMB and GCL and AL, which is then solved by the separation of variables method. The solution is effectively verified against an existing analytical solution and a numerical model. Particular attention is paid to the role of GCL in the composite liner. Some major conclusions can be drawn from the analysis results. (1) The transport of degradable organic contaminant through the whole composite liner is a transient process within a considerable elapsed time. (2) The present solution is more reasonable and avoids some weird phenomena in existing analytical solutions, such as quite large initial concentration and mass flux and no-increasing trend of the 9

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Fig. 12. Breakthrough curves with different Rd,gcl for hw = 1.5 m and t1/2,al = +∞: (a) t1/2,gcl = 0.5 year; (b) t1/2,gcl = 1 year; (c) t1/2,gcl = 5 years; (d) t1/2,gcl = +∞.

mass flux at the GMB/GCL and GCL/AL interfaces. (3) GCL has significant effect on improving the barrier efficiency especially for scenarios with high leachate head. The biodegradation and adsorption in GCL have significant effect on the contaminant transport through the composite liner when t1/2,gcl is less than 5 years. Otherwise, the effect is negligible. (4) The present solution is a practical tool for assessing the transient breakthrough processes of organic contaminants in the whole GMB/ GCL/AL composite liner system. In fact, the method can easily be applied to other typical multilayered composite liner if some minor modifications are made.

Acknowledgments Much of the work described in this paper was supported by the National Natural Science Foundation of China under Grant Nos. 41725012 and 41572265, the Shanghai Shuguang Program under Grant No. 16SG19, the Fundamental Research Funds for the Central Universities, and the Newton Advanced Fellowship of the Royal Society under Grant No. NA150466. The writers would like to greatly acknowledge all these financial supports and express their most sincere gratitude.

Appendix. Normalized governing equation for the whole composite liner Based on Eq. (1), the governing equation for contaminant transport through GMB is

Cgmb (z , t ) t

= Dgmb

2C gmb (z , z2

t)

a

Cgmb (z , t )

Kg

z

gmb Cgmb (z ,

t)

(A1)

where λgmb = 0. The contaminant concentration in GMB can be normalized by the partition coefficient as * Cgmb (z , t ) =

Cgmb (z, t ) (A2)

Kg

By substituting Eq. (A2) into Eq. (8), the top boundary condition in the form of Cgmb (z , t ) can be described as * Cgmb (z

(A3)

= 0, t ) = C0

By substituting Eq. (A2) into Eq. (9), the concentration continuity condition between GMB and GCL in the form of Cgmb (z , t ) can be described as * Cgmb (z

(A4)

= Lgmb , t ) = Cgcl (z = Lgmb , t )

Eq. (A1) can also be rewritten in the form of Cgmb (z , t ). Notably, the contaminant mass flux through GMB computed using the normalized concentration must be equal to that calculated using the actual concentration. Consequently, the spatial dimension of GMB, z (0 ≤ z ≤ Lgmb), must also be normalized by the partition coefficient:

10

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z =

z , Kg

0

z

Lgmb

(A5)

Moreover, the calculated contaminant mass flux through GMB by the actual parameters and that by the normalized parameters should be identical:

va

Cgmb (z, t )

Dgmb

Kg

Cgmb (z, t ) z

* = va Cgmb (z *, t )

Dgmb

* Cgmb (z * , t )

(A6)

z*

Finally, Eq. (A1) can be rewritten in the form of the normalized concentration and normalized dimension as

Cgmb (z , t ) t

=

Dgmb K g2

2C gmb (z z 2

, t)

a

Cgmb (z , t )

K g2

z

gmb Cgmb (z

, t)

(A7)

The top boundary condition and the continuity conditions between GMB and GCL in the form of Cgmb (z , t ) can also be obtained by substituting Eq. (A2) and (A5) into Eqs. (8)–(10). In this way, the expressions of the contaminant transport through GMB have the same form as those through GCL and AL, including governing equations, boundary and continuity conditions by replacing ngmb, Rd,gmb, z (0 ≤ z ≤ Lgmb) and Cgmb (z, t) with 1, K g2 , z∗ (0 ≤ z∗ ≤ Lgmb/Kg) and Cgmb (z , t ) , respectively. Thus, the normalized general governing equation for the whole composite liner can be expressed as 2C i

Ci (z , t ) D = i t R d, i

(z , t ) z 2

i

Rd, i

Ci (z , t ) z

i Ci

(z , t ), i = 1, 2, 3

(A8)

where i = 1, 2, 3 represents the parameters for GMB, GCL and AL, respectively, and Ci (z, t ) , Kg

Ci* (z , t ) =

z =

z Kg

z+

0

Ci (z, t ),

z

Lgmb

z > Lgmb

0

(

1 Kg

)

1 Lgmb

z

(A9)

Lgmb

z > Lgmb

(A10)

List of notations z t Cgmb(z, t) Dgmb Kg va Q hw Lw 2b l θ k mh A kgcl kal Cgcl(z, t) Dgcl Rd,gcl ρd,gcl Kd,gcl ngcl νgcl λgcl t1/2,gcl Cal(z, t) Dal Rd,al ρd,al Kd,al nal

distance from the contaminant origin along the transport orientation (L) elapsed time for contaminant transport (T) contaminant concentration in GMB at position z and time t (M L−3) diffusion coefficient of GMB (L2 T−1) partition coefficient between GMB and leachate (dimensionless) Darcy velocity of the leakage flow through the composite liner (L T−1) leakage rate through a hole in a GMB coincident with a wrinkle (L3 T−1) leachate hydraulic head (L) length of wrinkle (L) width of wrinkle (L) total thickness of GCL and AL (L) transmissivity of the interface between GMB and GCL (L2 T−1) harmonic mean hydraulic conductivity (permeability) of GCL and AL (L T−1) number of holes in GMB of the area under consideration (dimensionless) area of the liner under consideration (L2) hydraulic conductivity (permeability) of GCL (L T−1) hydraulic conductivity (permeability) of AL (L T−1) contaminant concentration in GCL at position z and time t (M L−3) dispersion coefficient of GCL (L2 T−1) retardation factor of GCL (dimensionless) dry density of GCL (M L−3) distribution coefficient of GCL (L3 M−1) porosity of GCL (dimensionless) seepage velocity in GCL (L T−1) first-order degradation constant in GCL (T−1) half-life of contaminant in GCL (T) contaminant concentration in AL at position z and time t (M L−3) dispersion coefficient of AL (L2 T−1) retardation factor of AL (dimensionless) dry density of AL (M L−3) distribution coefficient of AL (L3 M−1) porosity of AL (dimensionless) 11

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seepage velocity in AL (L T−1) first-order degradation constant in AL (T−1) half-life of contaminant in AL (T) partition coefficient between GMB and GCL (dimensionless) constant contaminant concentration in leachate (M L−3) contaminant concentration at the bottom of composite liner (M L−3) thickness of GMB (L) thickness of GCL (L) thickness of AL (L) thickness of composite liner (L) normalized thickness of composite liner, given in Eq. (27) (L) contaminant concentration in the i-th layer at position z and time t (M L−3) dispersion coefficient of the i-th layer (L2 T−1) retardation factor of the i-th layer (dimensionless) dry density of the i-th layer (M L−3) distribution coefficient of the i-th layer (L3 M−1) porosity of the i-th layer (dimensionless) seepage velocity in the i-th layer (L T−1) first-order degradation constant in the i-th layer (T−1) half-life of contaminant in the i-th layer (T) factor of concentration for sub-problem 1 (dimensionless) factor of concentration for sub-problem 2 (dimensionless) coefficients given in Eq. (41) (dimensionless) function defined by Eq. (30) (dimensionless) coefficient given in Eq. (42) (dimensionless) coefficient given in Eq. (42) (dimensionless) coefficient given in Eq. (31) (L−1) eigenvalues of the composite liner (T−1) coefficient given in Eq. (32) (L−1) coefficient given in Eq. (33) (L) coefficient in Eq. (30) (dimensionless) coefficient in Eq. (30) (dimensionless) matrix given in Eq. (35) (dimensionless) matrix given in Eq. (40) (dimensionless) coefficient given in Eq. (36) (dimensionless) coefficient given in Eq. (36) (dimensionless) coefficient given in Eq. (36) (dimensionless) coefficient given in Eq. (36) (dimensionless) coefficient given in Eq. (36) (L T−1) coefficient given in Eq. (36) (L T−1) coefficient given in Eq. (36) (L T−1) coefficient given in Eq. (36) (L T−1) ki,1 coefficient in Eq. (51) (dimensionless) ki,2 coefficient in Eq. (51) (dimensionless) ri,1 coefficient in Eq. (51) (L−1) ri,2 coefficient in Eq. (51) (L−1) Jgmb(z, t) contaminant mass flux in GMB at position z and time t (M L−2 T−1) A1 coefficient in Eq. (53) (M L−3) A2 coefficient given in Eq. (54) (M L−3) Jgcl(z, t) contaminant mass flux in GCL at position z and time t (M L−2 T−1) Cgmb (z , t ) normalized contaminant concentration, given by Eq. (A9) (M L−3) z* normalized z by partition coefficient, given by Eq. (A10) (L)

νal λal t1/2,al Kg C0 Cb Lgmb Lgcl Lal Lcl H Ci(z, t) Di Rd,i ρd,i Kd,i ni νi λi t1/2,i wi(z, t) ui(z) Cm gm,i(z) S1 S2 ai βm λm,i μi Am,i Bm,i Ti T0 A¯ i B¯i C¯i D¯ i E¯i F¯i G¯ i H¯ i

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.geotexmem.2019.01.017.

Phenomena in Environmental Geotechnics. CRC Press/Balkema, Taylor and Francis Group, Leiden, Netherlands. Bouazza, A., Bowders, J.J., 2010. Geosynthetic Clay Liners for Waste Containment Facilities. CRC Press, Taylor and Francis Group, London, UK. Bright, M.I., Thornton, S.F., Lerner, D.N., Tellam, J.H., 2000. Attenuation of landfill leachate by clay liner materials in laboratory columns, 1. Experimental procedures and behaviour of organic contaminants. Waste Manag. Res. 18 (3), 198–214. Chen, Y., Xie, H., Ke, H., Chen, R., 2009. An analytical solution for one-dimensional contaminant diffusion through multi-layered system and its applications. Environ. Geol. 58 (5), 1083–1094.

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