Environ Sci Pollut Res DOI 10.1007/s11356-015-4200-9

RESEARCH ARTICLE

Steady-state analytical models for performance assessment of landfill composite liners Haijian Xie & Yuansheng Jiang & Chunhua Zhang & Shijin Feng & Zhanhong Qiu

Received: 13 October 2014 / Accepted: 2 February 2015 # Springer-Verlag Berlin Heidelberg 2015

Abstract One-dimensional mathematical models were developed for organic contaminant transport through landfill composite liners consisting of a geomembrane (GM) and a geosynthetic clay liner (GCL) or a GM and a compacted clay liner (CCL). The combined effect of leakage through GM defects, diffusion in GM and the underlying soil liners, and degradation in soil liners were considered. Steady state analytical solutions were provided for the proposed mathematical models, which consider the different combinations of advection, diffusion, and degradation. The analytical solutions of the time lag for contaminant transport in the composite liners were also derived. The performance of GM/GCL and GM/ CCL was analyzed. For GM/GCL, the bottom flux can be reduced by a factor of 4 when the leachate head decreases from 10 to 0.3 m. The influence of degradation can be ignored for GM/GCL. For GM/CCL, when the leachate head decreases from 10 to 0.3 m, the bottom flux decreases by a factor of 2–4. Leachate head has greater influence on bottom flux in case of larger degradation rate (e.g., half-life=1 year) compared to the case with lower degradation rate (e.g., half-life= 10 years). As contaminant half-life in soil liner decreases from 10 to 1 year, bottom flux decreases by approximately 2.7

Responsible editor: Michael Matthies H. Xie (*) : Y. Jiang : C. Zhang College of Architecture and Civil Engineering, Zhejiang University, Hangzhou 310058, China e-mail: [email protected] S. Feng Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Z. Qiu College of Architecture and Civil Engineering, Taizhou University, Taizhou 317000, China

magnitudes of orders. It is indicated that degradation may have greater influence on time lag of composite liner than leachate head. As leachate head increases from zero to 10 m, time lag for GM/CCL can be reduced by 5–6 years. Time lag for the same composite liner can be reduced by 10–11 years as contaminant half-life decreases from 10 to 1 year. Reducing leachate head acting on composite liners and increasing the degradation capacity of the soil liner would be the effective methods to improve the performance of the composite liners. The proposed analytical solutions are relatively simple and can be used for preliminary design and performance assessment of composite liners. Keywords Landfill . Composite liner . Contaminant transport . Groundwater pollution . Analytical model . Time lag

Introduction Approximately 90 % of China’s shallow groundwater is polluted, according to the Ministry of Land and Resources, and an alarming 37 % of the shallow groundwater is heavily contaminated and cannot be treated for use as drinking water (Qiu 2011). Municipal solid waste landfills and hazardous waste landfills are important sources of groundwater pollution. The composite liners consisting of geomembrane (GM) and compacted clay layer (CCL) or geomembrane (GM) and geosynthetic clay liner (GCL) have been widely used as bottom barrier systems in the landfills to minimize groundwater contamination caused by landfill leachate (Varank et al. 2011; Park et al. 2012; Hoor and Rowe 2013). Four types of liner systems are recommended in the China landfill standard and three of them require the use of composite liners (MCC 2007). GCL has been widely used in landfill composite liner to be an alternative material of CCL (Bouazza and Bowders 2010).

Environ Sci Pollut Res

Geosynthetic clay liners have significant advantages for landfill barrier applications, e.g., easily control of construction quality, more available and cost effective compared to CCLs (Rowe et al. 2004; Koerner 2005; Guyonnet et al. 2009). GCLs also have some disadvantages. One of the most important disadvantages is that hydraulic conductivity of the GCL will be increased due to the increased porosity of bentonite and decreased swelling index when the GCLs are permeated by acidic solutions (Liu et al. 2013). Most regulations allow alternative liner designs provided one can demonstrate equivalence among the liner systems. Of practical interest in this article is the question as to whether a GM/GCL composite liner is equivalent to the standard GM/CCL composite liner under the same environmental conditions. However, assessment of equivalence may depend on what is being compared and how it is being compared (Rowe and Brachman 2004). The leakage rate of composite liners, bottom solute concentration, and flux of the composite liners are always chosen to be the assessment parameters. Foose et al. (2002) concluded that the equivalence of alternative liner designs should be based on contaminant transport rather than leakage rate, which is consistent with the recommendation of Rowe (1998). Thus, mathematical models regarding contaminant transport in the composite liners should be developed to assess the equivalence of the different liner systems. Numerical models are commonly used to analyze contaminant transport in the composite liners. Different boundary conditions and variability of parameters regarding contaminant transport in the composite liner can be considered in the numerical models. Rowe et al. (2004) developed the onedimensional semi-analytical model for contaminant transport in the composite liners. Finite element models were developed by El-Zein (2008), El-Zein and Rowe (2008), and El-Zein et al. (2012) to assess the performance of composite liners with intact and leaking geomembranes. One of the limitations of these numerical methods is that they are not tools readily available to engineering practitioners addressing issues of landfill liner designs. Recognizing this difficulty, analytical models for one-dimensional organic contaminant diffusion in the composite liners were presented for semi-infinite boundary conditions (Foose 2002; Xie et al. 2013b) and finite domains (Chen et al. 2009; Cleall and Li 2011). Xie et al. (2013a) presented a onedimensional analytical solution for organic pollutants diffusion in the composite liner considering degradation. Xie et al. (2010, 2014) presented one-dimensional analytical models for contaminant transport in the composite liners with geomembrane defects. Kandris and Pantazidou (2012) compared the analytical solutions used to analyze the performance of GM/CCL and GM/GCL. To the authors’ knowledge, the analytical solutions for contaminant transport in the composite liners considering the combined

effect of leakage, diffusion, degradation, and adsorption are not available in the literature. The leachate stabilization time of a typical landfill may be 15–30 years in China, and the operation period of the municipal solid waste landfills is typically 10 to 15 years (Zhao et al. 2000; Liu et al. 2004). The design of the landfill liner system should meet the requirement of groundwater protection during the landfill operation time plus the landfill leachate stabilization time. Therefore, it is of great importance to investigate the time required to reach steady state for contaminant transport in the composite liners. This time can be evaluated from the time lag for the contaminant transport in the composite liners (Müller 2009). The objective of this article is to present the analytical solutions for assessment of the performance of landfill composite liners. Four different cases were considered, i.e., (1) pure diffusion; (2) diffusion-adsorptiondegradation; (3) advection-diffusion; and (4) advectiondiffusion-degradation. The time lag for contaminant transport in the composite liners for the different cases will also be provided. The proposed solution would be very useful for the assessment of the performance of landfill composite liners and can be used as a preliminary design tool for the composite liners.

Mathematical model One-dimensional model for contaminant transport in the composite liner is developed (see Fig. 1). The origin of z-axis coordinate is on the top surface of GM. The thickness of GM and soil liner (GCL or CCL) is h1 and h2, respectively. The models are developed on the basis of the following assumptions: (1) Contaminant transport in the composite liner is onedimensional; (2) GM, GCL, and CCL are all homogeneous; (3) GCL and CCL are assumed to be saturated. The concentration of pollutants in the leachate was assumed to be a constant, C0. The bottom of the liner system is assumed to be a zero concentration boundary. This is the case when a composite liner overlies a leakage detection layer or layer conducting flow that instantaneously removes all of the solute from the base of the composite liner. This bottom boundary condition results in the highest possible long-term chemical gradient across the composite liner, which will result in the greatest long-term mass flux through the barrier. Therefore, the results obtained by using this bottom boundary condition are conservative.

Environ Sci Pollut Res Fig. 1 Schematic diagram of landfill leachate contaminant transport in composite liner

Leachate, C0 h1

Geomembrane



Diffusion, D

h2

Soil liner

Leakage, Q

Degradation, λ Groundwater, Cb

]

Governing equations and boundary conditions The analytical solutions for the models are derived under the four different cases: (1) pure diffusion; (2) diffusionadsorption-degradation; (3) advection-diffusion; and (4) advection-diffusion-degradation. The governing equations and boundary conditions for the different cases are discussed in the following sections. The case of pure diffusion Governing equation of organic pollutants diffusion in GM is as follows: ∂C g ðz; t Þ ∂2 C g ðz; t Þ ¼ Dg ∂t ∂z2

C g ð0; t Þ ¼ C 0 S 0;GM

ð5Þ

ð1Þ

ð2Þ

where H is the thickness of the composite liner. The interface continuous conditions between GM and the underlying soil liner should be obeyed (Rowe et al. 2004; Xie et al. 2013a, b):

C g ð h1 ; t Þ ¼ C r ð h1 ; t Þ S ML;GM

where Cr (z, t) is the concentration of the contaminant in the soil. D* is the effective diffusion coefficient of the contaminant through the soil liner, and Rd is retardation factor of the contaminant in the media: ρK d Rd ¼ 1 þ n

ð4Þ

where S0,GM is the partition coefficient between the geomembrane and leachate contaminants. The bottom boundary is assumed to be a constant concentration boundary: C r ðH; t Þ ¼ 0

where Dg is diffusion coefficient of pollutants transport in GM, and Cg (z, t) is the concentration of pollutants in the GM. Governing equation of organic pollutants diffusion in the underlying soil liner is as follows: ∂C r ðz; t Þ D* ∂2 C r ðz; t Þ ¼ ∂t ∂z2 Rd

The surface boundary condition for the mathematical model in this case is (Rowe et al. 2004; Xie et al. 2013a, b):

ð3Þ

where ρ is the dry density of the medium, and Kd is the distribution coefficient of medium assuming linear adsorption of the solute onto the soils.

Dg

∂C g ðh1 ; t Þ ∂C r ðh1 ; t Þ ¼ nD* ∂z ∂z

ð6Þ

ð7Þ

where SML,GM is the partition coefficient of organic contaminant between the geomembrane and the underlying soil liner.

Environ Sci Pollut Res

The case of diffusion-adsorption-degradation

Darcy velocity of the composite liner va can then be obtained by (Rowe and Brachman, 2004; Xie et al. 2010):

In this case, the governing equation of geomembrane is the same as (1). The governing equation for contaminant transport in the underlying soil liner is ∂C r ðz; t Þ D* ∂2 C r ðz; t Þ ¼ − λC r ðz; t Þ ∂t ∂z2 Rd

ð8Þ

where λ is a first-order degradation constant of the organic contaminants and can be determined from λ ¼ ln2=t 1=2

ð9Þ

where t1/2 is the degradation half-life of the contaminants. Boundary conditions and interface continuous conditions of this case are the same as Eqs. (4)–(7). The case of advection-diffusion In this case, the governing equation of organic contaminant transport in GM is as follows (Xie et al. 2010, 2014): ∂C g ðz; t Þ ∂C g 2 ðz; t Þ ∂C g ðz; t Þ ¼ Dg − va ∂t ∂z2 ∂z

ð10Þ

where va is Darcy velocity through the composite liner. It can be determined by the leakage rate through the defects of the geomembranes. The leakage rate of water flow through a single hole of the composite liner can be obtained from (Giroud et al. 1997) h i Q ¼ 0:976C q0 1 þ 0:1ðhw =h2 Þ0:95 d 0:2 hw 0:9 k 0:74 s

va ¼ mQ=A

where m is the number of the defects in GM per unit area, and A is the cross-section area of the flow. Foose et al. (2001) has compared the mass flow rate predicted by the equivalent one-dimensional equations (e.g., Eqs. 11 and 14) and that predicted by the threedimensional model. The results show that the mass flow rate predicted using the one-dimensional approximate system is 7 % greater than that predicted using the threedimensional model for GCL composite liner. For the CCL composite liner, the mass flow rate predicted using the one-dimensional model is substantially greater than that predicted by the three-dimensional model. However, the steady-state mass flow rates obtained by the two methods are almost the same. It is a limitation of the proposed model to use the equivalent one-dimensional model. But over-predicting the mass flow rate is conservative for evaluating the effectiveness of the composite liners. The governing equation for organic pollutant advectiondiffusion in the underlying soil liner is as follows (Rowe et al. 2004; Guan et al. 2014): ∂C r ðz; t Þ D* ∂2 C r ðz; t Þ vs ∂C r ðz; t Þ ¼ − ∂t ∂z2 Rd Rd ∂z

ð11Þ

where Q is the leakage rate of the composite liners, hw is the leachate head, d is the diameter of the circular defect, ks is the hydraulic conductivity of CCL or GCL, and Cq0 is the contact quality factor (dimensionless) regarding the contact conditions between the GM and GCL or CCL. For good contact conditions (Giroud et al. 1997), ð12Þ

and for poor contact conditions (Giroud et al. 1997), C q0 ¼ 1:16

ð15Þ

where vs is the seepage velocity of the soil liner and can be determined by vs ¼ va =n

C q0 ¼ 0:21

ð14Þ

ð13Þ

ð16Þ

where n is the porosity of the soil liner. The case of advection-diffusion-degradation In this case, the governing equation for geomembrane is the same as Eq. (10). The governing equation for contaminant transport in the underlying soil liner is as follows (Guan et al. 2014): ∂C r ðz; t Þ D* ∂2 C r ðz; t Þ vs ∂ C r ðz; t Þ ¼ − λC r ðz; t Þ − ∂t ∂z2 ∂z Rd Rd

ð17Þ

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Analytical solutions of the mathematical model

B21 ¼ −

nD* α fexp½ðh1 − 2H Þα þ expð− αh1 ÞgB22 Dg

ð27Þ

Steady-state bottom flux of the liner system The case of pure diffusion

B22 ¼

In the steady state, the general solutions to the governing Eqs. (1) and (2) are as follows: 

C g ðzÞ ¼ A11 þ B11 z C r ðzÞ ¼ A12 þ B12 z

ð18Þ

C 0 S 0;GM 1 h1 S ML;GM nD* α K 11 þ K 12 Dg h1

ð28Þ

where rffiffiffiffiffiffiffiffiffi λRd α¼ D*

ð29Þ

where A11 ¼ C 0 S 0;GM

ð19Þ

K 11 ¼ exp ð− αh1 Þ−exp½ðh1 − 2H Þα

ð30Þ

A12 ¼ −B12 H

ð20Þ

K 12 ¼ exp ð− αh1 Þ þ exp½ðh1 − 2H Þα

ð31Þ

ð21Þ

On the basis of the solute concentration solutions, the solute flux J at position z and any time t can be obtained by

B11 ¼

B12

nD* B12 Dg

C 0 S 0;GM ¼− nD* h1 h2 S ML;GM þ Dg

ð22Þ

∂C r ðH Þ ¼ nD* B12 ∂z

∂C r ðH Þ ∂z

¼ − nD* α½A22 expðαH Þ−B22 expð−αH Þ

On the basis of the solute concentration solutions, the solute flux J at position z and any time t can be obtained by J jz¼H ¼ − nD*

J jz¼H ¼ − nD*

ð23Þ

ð32Þ

The case of advection-diffusion The general steady-state solutions of governing Eqs. (10) and (15) are as follows: 

C g ðzÞ ¼ A31 þ B31 expðβ 1 zÞ C r ðzÞ ¼ A32 þ B32 expðβ2 zÞ

ð33Þ

The case of diffusion-adsorption-degradation In the steady state, the general solutions of governing Eqs. (1) and (8) are as follows:

where



β1 ¼

va Dg

ð34Þ

β2 ¼

vs D*

ð35Þ

C g ðzÞ ¼ A21 þ B21 z C r ðzÞ ¼ A22 expðαzÞ þ B22 expð−αzÞ

ð24Þ

where A21 ¼ A11

ð25Þ

A22 ¼ −B22 expð− 2αH Þ

ð26Þ

A31 ¼ C 0 S 0;GM −B31

ð36Þ

A32 ¼ ‐B32 exp ðβ2 H Þ

ð37Þ

Environ Sci Pollut Res

B31 ¼ −

C 0 D* β 22 S 0;GM 2 * D ½−1 þ expðβ 1 h1 Þβ 2 þ Dg expðβ1 h1 Þ½−1 þ expðβ2 h2 Þβ1 S ML;GM

ð38Þ

B32 ¼ −

C 0 Dg β 1 S 0;GM exp½ðβ1 −β2 Þh1  D* ½−1 þ expðβ 1 h1 Þβ22 þ Dg expðβ1 h1 Þ½−1 þ expðβ 2 h2 Þβ1 S ML;GM

vs − η2 ¼ 2D*

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2s λRd þ * *2 4D D

ð47Þ

K 21 ¼ η1 expðη1 h2 Þ−η2 expðη2 h2 Þ

ð48Þ

K 22 ¼ exp ðη1 h2 Þ−exp ðη2 h2 Þ

ð49Þ

ð39Þ

On the basis of the solute concentration solutions, the solute flux J at position z and any time t can be obtained as follows: J jz¼H ¼−nD*

∂C r ðH Þ ¼ −nD* β2 B32 expðβ2 H Þ ∂z

ð40Þ

The case of advection-diffusion-degradation

C g ðzÞ ¼ A41 þ B41 expðβ1 zÞ C r ðzÞ ¼ A42 expðη1 zÞ þ B42 expðη2 zÞ

J jz¼H ¼−nD* ½η1 A42 expðη1 H Þ þ η2 B42 expðη2 H Þ

ð41Þ

The case of pure diffusion The time-dependent governing Eqs. (1) and (2) can be linearized with respect to time by introducing a Laplace transformation: Z ∞ C ðz; pÞ ¼ e−pt C ðz; t Þdt ð51Þ 0

where A41 ¼ C 0 S 0;GM −B41

ð42Þ

B42 ¼ ‐A42 exp ½ðh1 þ h2 Þ  ðη1 −η2 Þ

ð43Þ

B41 ¼

ð50Þ

Time lag of contaminant transport in composite liner

The general steady-state solutions of governing Eqs. (10) and (15) are as follows: 

On the basis of the solute concentration solutions, the solute flux J at position z and any time t can be obtained by:

C 0 D* nK 21 S 0;GM * nD ½−1 þ expðβ 1 h1 ÞK 21 −Dg K 22 β 1 S ML;GM exp ðβ 1 h1 Þ

where C ðz; pÞ is the Laplace transform of C(z,t), and p is the Laplace parameter. The Laplace transformed forms of Eqs. (1) and (2) can thus be obtained as follows: pC g ðz; pÞ ¼ Dg

∂2 C g ðz; pÞ ∂z2

ð52Þ

ð44Þ pC r ðz; pÞ ¼ A42 ¼

C 0 Dg β 1 S 0;GM expðh1 β 1 −h1 η1 þ h2 η2 Þ nD ½1−expðβ1 h1 ÞK 21 −Dg K 22 β1 S ML;GM expðβ1 h1 Þ

D* ∂2 C r ðz; pÞ ∂z2 Rd

ð53Þ

*

ð45Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vs v2s λRd η1 ¼ þ þ * * 2D 4D*2 D

ð46Þ

The boundary conditions (4) and (5) as well as the continuous conditions (6) and (7) can be transformed into C g ð0; pÞ ¼ C 0 S 0;GM =p

ð54Þ

C r ðH; pÞ ¼ 0

ð55Þ

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C g ðh1 ; pÞ=S ML;GM ¼ C r ðh1 ; pÞ

Dg

∂C g ðh1 ; pÞ ∂C r ðh1 ; pÞ ¼ nD* ∂z ∂z

ð56Þ

approximately one third of the time required for steady state (Crank 1975).

ð57Þ

The case of diffusion-adsorption-degradation

The time lag can be obtained as follows (see Appendix A1 for the detailed derivation process): θ¼

  2 D* h1 3 n þ Dg 2 h2 3 Rd S ML;GM þ 3Dg D* h1 h2 h2 nRd þ h1 S ML;GM   6Dg D* D* h1 n þ Dg h2 S ML;GM

Using the method of Laplace transformation, Eq. (8) can be transformed into: pC r ðz; pÞ ¼

D* ∂2 C r ðz; pÞ −λC r ðz; pÞ ∂z2 Rd

ð59Þ

ð58Þ The time lag has a close relationship with the required time to reach steady state and the time lag is

The time lag in this case was obtained as follows (see Appendix A2 for the detailed derivation process):

 rffiffiffiffiffiffiffiffiffi   rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi λRd λRd λRd u1 cosh h2 þ u2 sinh2 exp h2 * * D D D* θ¼  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  rffiffiffiffiffiffiffiffiffi   rffiffiffiffiffiffiffiffiffi

λRd λRd * 3Dg λ h1 n D λRd 1 þ exp 2h2 þ Dg S ML;GM −1 þ exp 2h2 * D D*

where

where

rffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λRd 3 *2 * 3 u1 ¼ h1 n D λ Rd þ 3D h2 S ML;GM D*

ð61Þ

   u2 ¼ 3Dg −Dg S ML;GM þ h1 λ h2 nRd þ h1 S ML;GM

ð62Þ

       h2 v s h1 va  u1 ¼ Rd v2a −1 þ exp þ exp v −nv nv 2S s ML;GM a s Dg D*

ð66Þ 

   h1 va h2 vs  h2 Rd v2a þ −Dg þ h1 va vs u2 ¼ S ML;GM va exp − D g D*

ð67Þ

The case of advection-diffusion The governing Eqs. (10) and (15) in the Laplace domain are as follows: ∂2 C g ðz; pÞ ∂C g ðz; pÞ −va pC g ðz; pÞ ¼ Dg 2 ∂z ∂z

ð63Þ

D* ∂2 C r ðz; pÞ vs ∂C r ðz; pÞ − ∂z2 ∂z Rd Rd

ð64Þ

pC r ðz; pÞ ¼

ð60Þ

The time lag in this case can be obtained as follows (see Appendix A3 for the detailed derivation process):      h2 vs h1 va S −D* u1 þ vs Dg exp v v þ u þ u þ u exp ML;GM a s 2 3 4 Dg D*     

θ¼ h v h v 1 a 2 s −1 þ exp S v þ nv va 2 vs 2 −nvs þ exp ML;GM a s Dg D*

ð65Þ

   u3 ¼ vs Dg −S ML;GM va þ 2nvs þ nva ðh2 Rd va þ h1 Rd vs Þ ð68Þ   u4 ¼ h2 Rd va 2 S ML;GM va −nvs   þ vs Dg S ML;GM va −h1 S ML;GM va 2 −2Dg nvs þ h1 nva vs ð69Þ

The case of advection-diffusion-degradation The governing Eq. (17) in the Laplace transformation domain takes the form as pC r ðz; pÞ ¼

D* ∂2 C r ðz; pÞ vs ∂C r ðz; pÞ −λC r ðz; pÞ ð70Þ − ∂z2 ∂z Rd Rd

Environ Sci Pollut Res

The time lag in this case can be obtained as follows (see Appendix A4 for the detailed derivation process):     va h1 va h1 4u1 va w1 cosh þ w2 sinh 2Dg 2Dg    

θ¼  *  32 va h1 u6 þ −1 þ nðu7 −u8 Þexp va 2 4D λRd þ vs 2 Dg

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !#  " va h1 h2 4D* λRd þ vs 2 u6 ¼ −2exp S ML;GM va −1 þ exp Dg D*

ð79Þ



ð71Þ

" u7 ¼ −1 þ exp

h2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# 4D* λRd þ vs 2 vs D*

ð80Þ

where " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # h2 4D* λRd þ vs 2 w1 ¼ u2 cosh 2D* " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 4D* λRd þ vs 2 4D* λRd þ vs 2 ð72Þ þ u3 sinh 2D* " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # h2 4D* λRd þ vs 2 w2 ¼ u4 cosh 2D* " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 4D* λRd þ vs 2 4D* λRd þ vs 2 −u5 sinh 2D*

v a h1 h2 u1 ¼ exp þ 2Dg

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 4D* λRd þ vs 2 2D*

u8 ¼ 1 þ exp

h2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !#qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4D* λRd þ vs 2 4D* λRd þ vs 2 D* ð81Þ

Comparisons with numerical method ð73Þ

ð74Þ

   u2 ¼ − 4D* λRd þ vs 2 4D* λRd nh1 þ h2 Rd S ML;GM va 2 þ h1 nvs 2

ð75Þ   u3 ¼ h1 vs 2 −S ML;GM va þ nvs  þ 2D* Rd S ML;GM va ð−2h1 λ þ va Þ þ 2h1 λnvs

"

ð76Þ

  u4 ¼ 4D* λRd þ vs 2      h2 Rd va 2 −S ML;GM va þ nvs þ 2Dg n 4D* λRd þ vs 2 ð77Þ

h     u5 ¼ −2Dg S ML;GM va −nvs 4D* λRd þ vs 2 þ va 2 h2 nRd þ h1 S ML;GM vs 2 i   þ2D* Rd 2h2 λnRd þ 2h1 λS ML;GM −S ML;GM va þ nvs

ð78Þ

The results obtained by the proposed analytical solutions were compared with those obtained by using the numerical software Pollute V7.0 (Rowe and Booker 2005). Dichloromethane (DCM) was selected as the representative organic pollutants in the leachate. DCM concentration was assumed to be C0 =1.0 mg/L. A composite liner consisting of a GM and a 75cm CCL was considered. The initial concentration of the liner system was assumed to be zero. The frequency of holes present in the GM was assumed to be 20/ha, and the holes are circular with the area of 1 cm2. The transport parameters of DCM in GM, GCL, and CCL were obtained from Rowe et al. (2004) (see Table 1). The leachate head was assumed to be 2 m for the cases considering advective transport. The degradation half-life was assumed to be 10 years for the case considering degradation. The pore water concentration profiles in the CCL for the four cases, i.e., pure diffusion, advectiondiffusion, diffusion-degradation, and advection-diffusiondegradation were predicted by the proposed analytical methods and Pollute V7.0, respectively (see Fig. 2). It is shown that the concentration profiles obtained by the proposed analytical solutions have a good agreement with those obtained by the numerical software.

Analysis of the performance of the composite liners The performance of the composite liners GM/GCL and GM/ CCL was analyzed on the basis of the proposed analytical solutions. The physical and transport properties of the GM, GCL, and CCL were the same as Table 1. The frequency of holes present in the GM was also assumed to be 20/ha, and the holes are circular with the area of 1 cm2. The composite liner consisting of a GM and a 75-cm CCL is the standard liner

Environ Sci Pollut Res

Parametric analysis of concentration and the bottom flux in composite liner Analytical solutions obtained in the BMathematical model^ section were used to analyze the performance of the composite liners for the different cases. Figure 3 shows the effect of degradation on contaminant concentration profiles of GCL and CCL for the composite liners GM/GCL and GM/CCL, respectively. The concentration profiles of the organic pollutants in the GCL with 0.3 m leachate head are shown in Fig. 3a. It is shown that the influence of biodegradation on concentration profiles is negligible due to the fact that the GCL is very thin (e.g., about 10 cm). Degradation has greater influence on the concentration profiles of CCL than that on the GCL (see Fig. 3b). The concentrations of chemical in CCL for the case with t1/2 =10 years (case 2) and 1 year (case 3) are much lower than the corresponding cases (i.e., cases 1 and 4) which do not consider degradation. Meanwhile, the concentrations in the Table 1

Material and chemical characteristics of the liner media

Thickness (m) Porosity Dry density (g/cm3) Permeability (m/s) Diffusion coefficient (m2/s) Partition coefficient Kd (mL/g)

GM

GCL

CCL

0.0015 – – – 5.0×10−13 5.0 –

0.0138 0.86 0.79 5.0×10−11 3.3×10−10 – 1.5

0.75 0.32 1.79 1.0×10−9 8.0×10−10 – 1

Pore water contaminant concentration, C (mg/L) 0.0 0.0

0.2

0.4

0.6

0.8

1.0

GM+75 cm CCL t1/2=10 years

Distance to GM (m)

system recommended by the China specification (MCC 2007). In China and the other developing countries, the leachate head of landfill is often very high and can reach 10–20 m due to the clogging of drainage system and poor water management of landfill water in the waste bodies (Xie et al. 2009; Zhang and Qiu 2010; Guan et al. 2014; Zhan et al. 2014). In this case, the leachate head was assumed to be 10 and 0.3 m, respectively, to investigate its effect on the time lag and concentration profiles. Two values of half-lives of the pollutant in the CCL, i.e., 10 and 1 year are chosen to investigate the effect of degradation. Seven different cases are selected to investigate the relative effects of the leachate head and the half-life of the contaminant on the performance of the composite liner (see Table 2). The pure diffusion case is chosen to be the reference case. The half-life of pollutants is infinite and the leachate head is zero for the reference case. The leachate head of cases 1–3 is 0.3 m and the half-lives of the pollutant for these cases are infinite, 10 years, and 1 year, respectively. The leachate head of cases 4–6 is 10 m, and the half-lives of the pollutant for the three cases are infinite, 10 years, and 1 year, respectively. The other conditions are the same for all the seven cases.

0.2

hw =2 m

0.4

Pollute v7.0-Pure diffusion Pollute v7.0-Diffusion-adsorption-degradation Pollute v7.0-Advection-diffusion Pollute v7.0-Advection-diffusion-degradation Analytical solution-Pure diffusion Analytical solution-Diffusion-adsorption-degradation Analytical solution-Advection-diffusion Analytical solution-Advection-diffusion-degradation

0.6

0.8

Fig. 2 Comparisons of the results of proposed analytical solution and those of Pollute v7.0 in terms of pore water concentration profiles in CCL

CCL with t1/2 =1 year (case 3) are much lower than the case with t1/2 =10 years (case 2). For example, the 0.2-m-depth steady-state concentration of case 3 is five times greater than that of the case 2. This result indicates that the concentrations in the compacted clay liner can be reduced greatly when the degradation capacity of the CCL increases. Figure 4 shows the effect of degradation on DCM concentration distribution in the GCL and CCL in cases with hw = 10 m. The concentration values in the GCL or CCL for the case 6 with t1/2 =1 year are lower than those for the cases 4 and 5. For example, the 0.4-m-depth contaminant concentration in the CCL for case 4 (t1/2 =∞) is four times greater than that for the case 6 (t1/2 =1 year) and two times greater than that for the case 5 (t1/2 =10 year). It is indicated that degradation can have great influence on the concentration profiles in both the CCL and much thinner layer GCL in the cases with hw =10 m. When compared to Fig. 3, it is shown that under the same degradation half-life, the higher the leachate head is, the greater the concentration values in the liners are. This indicates that effect of degradation on the performance of the composite liner would be more important in cases of higher leachate head (e.g., cases 4–6). For example, under the same half-life, e.g., 10 years, the 0.2-m-depth contaminant concentration in the CCL for the case with hw =10 m (i.e., case 5) is 1.7 times greater than that of the case with hw =0.3 m (i.e., case 2). When the contaminant half-life in the CCL was reduced to 10 years (case 5) and 1 year (case 6), the contaminant concentrations in

Table 2

Leachate head and contaminant half-life for different cases Reference Case Case Case Case Case Case case 1 2 3 4 5 6

Half-life t1/2 (a) Leachate head hw (m)

∞ 0

∞ 0.3

10 0.3

1 0.3

∞ 10

10 10

1 10

Environ Sci Pollut Res

0.00 0.000

Pore water contaminant concentration, C (mg/L) 0.02 0.04 0.06 0.08

0.10

Distance from GM (m)

GM+13.8 mm GCL

0.004

Pore water contaminant concentration, C (mg/L)

Distance from GM (m)

0.0 0.000

0.4

0.6

0.8

1.0

0.004

0.008 GM+13.8 mm GCL 0.012

(a) Pore water contaminant concentration, C (mg/L) 0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4 GM+75 cm CCL

0.6

0.8

0.008

0.2

0.016

Distance from GM (m)

the CCL are lower than that for the pure diffusion case. This result indicates that the degradation of the contaminant in the CCLs can be more important than the effect of the leachate head. The steady-state bottom fluxes of the composite liners for the six cases were also calculated and presented in Table 3. The steady-state bottom flux of GM/GCL for case 3 is 23.7 times greater than that for case 4, while the steady-state bottom flux of GM/GCL for the case 3 can be 7,604 times greater than that for the case 4. This indicates that degradation has a significant effect on the performance of the composite liners. Increasing biodegradation capacity of the liner materials for the organic contaminants can be an effective method to improve the performance of the composite liners. The concept of biobarrier systems have been proposed to enhance the effectiveness of the remediation method for polluted groundwater by organic contaminant (Kao et al. 2001; Kao et al. 2004; Liang et al. 2013). For example, microorganism was injected to the aquifer to increase the degradation capacity and decrease the hydraulic conductivity of the soils

(b)

Fig. 4 Effect of degradation on solute profiles in liner: with hw =10 m: a GM/GCL; b GM/CCL 0.012

(a) Pore water contaminant concentration, C (mg/L)

Distance from GM (m)

0.0 0.0

0.2

0.4

0.6

0.8

1.0

0.2

GM+75 cm CCL 0.4

(Kim et al. 2006b). Kim et al. (2006a) found that attenuation of benzene in the aquifer increased in the presence of bacteria due to biodegradation. Varank et al. (2011) also found that phenol concentration in the groundwater can be reduced greatly due to the degradation of phenol by methanogenic bacteria in the consortium. Therefore, the landfill liner can be possibly treated by injection of effective microorganism into the soil particles to increase the capacity of degradation of the organic contaminants in the leachate. However, more research works should be carried out to investigate the performance of the treated liner materials, such as the change of hydraulic conductivity, before they can be used in landfill practice.

0.6

Time lag analysis of contaminant transport in the composite liners 0.8

(b)

Fig. 3 Effect of degradation on solute profiles in the liner with hw = 0.3 m: a GM/GCL; b GM/CCL

The time lags for contaminant transport in GM/GCL and GM/ CCL for the six different cases are presented in Table 4. The time lags of GM/GCL are very short (about 14 days), which

Environ Sci Pollut Res Table 3 year)

Base flux of the composite liner for different cases (mg/ha/

Cases

Reference case Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Base flux of composite liners (mg/ha/year) GM/GCL

GM/CCL

4.86×105

8.93×104

5

9.16×104 1.76×104 30.25 2.29×105 5.33×104 117.9

4.89×10 4.88×105 4.82×105 2.03×106 2.03×106 2.01×106

may be due to the fact that GCL layer is very thin. Instead, the time lags of the thicker GM/CCL range from 6 to 33 years for the different cases, which is much longer than those for GM/GCL. Leachate head and biodegradation have little effect on the time lags of GCL due to the thin thickness of GCL. The time lags of GM/ GCL for the different cases were almost the same. Leachate head and biodegradation have greater effects on the time lags of GM/CCL. The time lag is 32.9 years for the pure diffusion case. When the leachate head hw is 0.3 m and contaminant half-life is 1 year (case 3), the time lag is evaluated to be 7.08 years. When the leachate head hw increases to 10 m, the time lag is reduced to 6.93 years with t1/2 =1 year (see case 6 in Table 4). It is shown that with the other conditions remain unchanged, the shorter the degradation half-life is or the higher the leachate head is, the shorter the time lag is. It is also shown that degradation has greater effect on the time lag than the leachate head does. For example, the lag time decreases by a factor of 1.21 times when the leachate head increase from 0.3 m (case 1) to 10 m (case 4). The time lag decreases by a factor of 4.63 when the contaminant half-life decreases from 10 years (case 1) to 1 year (case 3).

Table 4 Time lags of contaminant transport in composite liners for different cases

Cases

Reference case Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Time lag (year) GM/GCL

GM/CCL

0.048 0.048 0.048 0.048 0.037 0.037 0.037

32.9 32.79 18.71 7.08 27.17 17.20 6.93

Conclusions One-dimensional transport models of organic pollutants transport through the composite liner were developed on the basis of the assumption that contaminant transport through the composite liner system is a steady-state process. Four different cases were considered including the following: (1) pure diffusion case; (2) diffusion, adsorption, and degradation case; (3) advection and diffusion case; and (4) advection, diffusion, and degradation case. The analytical solutions for contaminant transport in the composite liners were derived for the different cases, and the time lag for contaminant transport in the composite liners was also presented for those cases. The results obtained by the proposed analytical solutions are in good agreement with those obtained by the numerical model. The influences of contaminant half-life and the leachate head on organic pollutants transport through the composite liner system were investigated on the basis of the proposed solutions. The main conclusions are as follows: (1) For GM/GCL, the bottom flux can be approximately reduced by a factor of 4 when the leachate head decreases from 10 to 0.3 m. The influence of degradation can be ignored for GM/GCL. For GM/ CCL, when the leachate head decrease from 10 to 0.3 m, the bottom flux decreases by a factor of 2–4 and the decreases greatly for the case with lower half-lives. When the contaminant half-life decreases from 10 to 1 year, the bottom flux decreases approximately 2.7 magnitudes of orders. (2) The time lag for contaminant transport in GM/GCL is very short, and both degradation and the leachate head have negligible effect on it. For GM/CCL, the effect of degradation has greater effect on the performance of the composite liner than the leachate head. When the leachate head increases from 0.3 to 10 m and the other conditions remain unchanged, the time lag can be reduced by 0.1–6 years. When the degradation half-life decreases from 10 to 1 year and the other conditions remain unchanged, the time lag can be reduced by 10–11 years. (3) Decreasing the leachate head and improving soil liner degradation capacity of organic pollutants are both effective control methods to improve the performance of the composite liners. (4) The proposed analytical solution is relatively simple and can be easily applied to engineering practice in landfill liner system design. It can also be used to evaluate the effectiveness and validity of complex numerical models.

Environ Sci Pollut Res Acknowledgments The authors would like to express the sincere gratitude to National Natural Science Foundation of China (Grant Nos. 51478427, 51278452, 51108293, and 51008274), the National Basic Research Program of China (973 program) (Grant No. 2012CB719806), the Fundamental Research Funds for the Central Universities (Grant No. 2014QNA4019), and Zhejiang Provincial Natural Science Foundation (Grant No. LY13D060003) for the financial support to this study.

face continuous conditions, i.e., Eqs. (54)–(57), are as follows: C g ðz; pÞ ¼ A1 expðm1 zÞ þ B1 expðm2 zÞ

ðA1Þ

C r ðz; pÞ ¼ A2 expðn1 zÞ þ B2 expðn2 zÞ

ðA2Þ

Appendix The appendix gives the derivation of the analytical solutions for the time lag for organic contaminants transport in the composite liners for the four cases.

where rffiffiffiffiffiffi p m1 ¼ −m2 ¼ Dg

ðA3Þ

rffiffiffiffiffiffiffiffi pRd n1 ¼ −n2 ¼ D*

ðA4Þ

A1 The case of pure diffusion In the case of pure diffusion, the general solutions of C g ðz; pÞ and C r ðz; pÞ subject to the boundary and inter-

    C 0 S 0;GM expðh1 m2 Þ D* nn2 −Dg m2 S ML;GM expðh2 n1 Þ þ Dg m2 S ML;GM −D* nn1 expðh2 n2 Þ A1 ¼ pnD* G1 −pS ML;GM Dg G2

B1 ¼ C 0 S 0;GM =p−A1

ðA6Þ

ðA5Þ

G2 ¼ ½m1 expðh1 m1 Þ−m2 expðh1 m2 Þ½expðh2 n1 Þ−expðh2 n2 Þ ðA10Þ

2C 0 S 0;GM Dg m1 expðHn2 Þ A2 ¼ pnD* G1 −pS ML;GM Dg G2

ðA7Þ

B2 ¼ −A2 expð2Hn1 Þ

ðA8Þ

The accumulated flux at the bottom of the composite liner can be obtained as follows:

Z Qt ¼ −

where

t

nD* 0

G1 ¼ ½n2 expðh2 n1 Þ−n1 expðh2 n2 Þ½expðh1 m1 Þ−expðh1 m2 Þ ðA9Þ

∂C r ðh1 þ h2 ; t Þ dt ∂z

The above equation can be transformed into the following equation in the Laplace domain:

 1 ∂C r ðh1 þ h2 ; pÞ C 0 S 0;GM nRd QP ¼ −  nD* ¼ p ∂z npm1 Rd coshðh2 n1 Þsinhðh1 n1 Þ þ pn1 S ML;GM coshðh1 m1 Þsinhðh2 n1 Þ

The limit of QP as p approaching zero can be determined as follows

lim QP ¼

p→0

  C 0 Dg D* nS 0;GM 1 θ − D* h1 n þ Dg h2 S ML;GM p2 p

ðA13Þ

ðA11Þ

ðA12Þ

The above equation can be transformed into the following expression using the inverse Laplace transformation:

lim Qt ¼

t→∞

C 0 Dg D* nS 0;GM ðt−θÞ D h1 n þ Dg h2 S ML;GM *

ðA14Þ

Environ Sci Pollut Res

where θ is the time lag. Equation (A14) demonstrates that the curve regarding variation of base accumulated flux of composite liner with time is a straight line when the contaminant transport in the system reaches steady state and the intercept of the straight line in the time coordinate is the time lag for the system (Carslaw and Jaeger 1959). The expression of time lag (i.e., Eq. 58) can then be obtained by the method described in the Carslaw and Jaeger (1959) and Barrie et al. (1963).

A2 The case of diffusion-adsorption-degradation In this case, the expressions of the general solutions of C g ðz; pÞ and C r ðz; pÞ have the same form as Eqs. (A1) and (A2). The parameters m1 and m2 in this case are the same as those in the case of pure diffusion, i.e., Eq. (A3). The other parameters to be determined are as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λRd pRd n1 ¼ n2 ¼ − þ * ðA15Þ D* D

    C 0 S 0;GM expðh1 m2 Þ D* nn2 −Dg m2 S ML;GM expðh2 n1 Þ þ expðh2 n2 Þ −D* nn1 þ Dg m2 S ML;GM A1 ¼ −D* npK 1 þ Dg pS 0;GM K 2

B1 ¼ C 0 S 0;GM =p−A1

ðA17Þ

ðA16Þ

where q1 ¼ ½expð2h1 m1 Þ−1½expð2h1 n1 þ 2h2 n1 Þ þ expð2h1 n1 Þ ðA24Þ

A2 ¼

C 0 S 0;GM Dg exp½ð−h1 n1 þ h2 n2 Þðm1 −m2 Þ D* npK 1 −Dg pS 0;GM K 2

B2 ¼ −exp½2ðh1 þ h2 Þn1 A2

ðA18Þ

q2 ¼ ½1 þ expð2h1 m1 Þ½expð2h1 n1 þ 2h2 n1 Þ−expð2h1 n1 Þ ðA25Þ

ðA19Þ

where

The limit of QP as p approaching zero is rffiffiffiffiffiffiffiffiffi  rffiffiffiffiffiffiffiffiffi λRd λRd   exp ð2h1 þ h2 Þ 2C 0 Dg D nS 0;GM 1 θ D* D* rffiffiffiffiffiffiffiffiffi lim QP ¼ − p→0 p2 p λRd þ Dg L2 S ML;GM D* h1 nL1 * D *

K 1 ¼ ½expðh2 n1 Þn2 −expðh2 n2 Þn1 ½expðh1 m1 Þ−expðh1 m2 Þ ðA20Þ

ðA26Þ K 2 ¼ ½expðh1 m1 Þm1 −expðh1 m2 Þm2 ½expðh2 n1 Þ−expðh2 n2 Þ ðA21Þ The accumulated flux at the bottom of the composite liner in this case is as follows: Z Qt ¼ −

t

* ∂C r ðh1

nD 0

þ h2 ; t Þ dt ∂z

ðA22Þ

The above equation in the Laplace domain is " # * ∂C r ðh1 þ h2 ; pÞ QP ¼ −1=p  nD ∂z ¼

C 0 Dg D* S 0;GM nm1 n1 expð2h1 n1 þ h2 n1 þ h1 m1 Þ   p2 D* nn1 q1 þ Dg m1 S ML;GM q2 ðA23Þ

where "

" rffiffiffiffiffiffiffiffiffi# rffiffiffiffiffiffiffiffiffi# λRd λRd L1 ¼ exp 2ðh1 þ h2 Þ þ exp 2h1 D* D* " rffiffiffiffiffiffiffiffiffi# rffiffiffiffiffiffiffiffiffi# λRd λRd L2 ¼ exp 2ðh1 þ h2 Þ −exp 2h1 * D D*

ðA27Þ

"

ðA28Þ

Using the inverse Laplace transformation, the Eq. (26) in the time domain can be obtained as follows: rffiffiffiffiffiffiffiffiffi  rffiffiffiffiffiffiffiffiffi λRd λRd 2C 0 Dg D* nS 0;GM ð þ h Þ exp 2h 1 2 D* D* rffiffiffiffiffiffiffiffiffi lim Qt ¼ ðt−θÞ t→∞ λRd D* h1 nL1 þ D L S g 2 ML;GM D* ðA29Þ

Environ Sci Pollut Res

The time lag in this case (i.e., Eq. 60) can also be obtained by the method described in A1.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4p va 2 þ 2 Dg Dg

1 m2 ¼ 2

va − Dg

1 n1 ¼ 2

vs þ D*

ðA31Þ

A3 The case of advection-diffusion The forms of the general solutions of C g ðz; pÞ and C r ðz; pÞ for the Eqs. (63) and (64) are also the same as Eqs. (A1) and (A2). The parameters are obtained as follows:

m1 ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! va 4p va 2 þ þ 2 Dg Dg Dg

1 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4pRd vs 2 þ 2 * D D*

0

ðA30Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 vs 4pRd v2s A n2 ¼ @ * − þ *2 * 2 D D D

     C 0 expðh1 m2 ÞS 0;GM expðh2 n1 Þ D* nn2 −Dg m2 S ML;GM þ −D* nn1 þ Dg m2 S ML;GM expðh2 n2 Þ A1 ¼ −D* npK 1 þ Dg pS ML;GM K 2

B1 ¼ C 0 S 0;GM =p−A1

ðA32Þ

ðA35Þ

ðA33Þ

ðA34Þ

where K 1 ¼ ½expðh1 m1 Þ−expðh1 m2 Þ½−n1 expðh2 n2 Þ þ n2 expðh2 n1 Þ

C 0 Dg exp½h1 ðm1 þ m2 −n1 Þ þ h2 n2 ðm1 −m2 ÞS 0;GM A2 ¼ D* npK 1 −Dg pS ML;GM K 2

B2 ¼ −exp½ðh1 þ h2 Þðn1 −n2 ÞA2

ðA38Þ K 2 ¼ ½expðh2 n1 Þ−expðh2 n2 Þ½m1 expðh1 m1 Þ−m2 expðh1 m2 Þ

ðA36Þ

ðA39Þ

ðA37Þ

The accumulated flux at the bottom of the composite liner in the Laplace domain in this case is as follows:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Dg p þ va 2 4D* pRd þ vs 2 i D2g D*2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#) Q P ¼ − (" " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# 4Dg p þ v2a 4D* pRd þ v2s q þ nq −1 þ exp h p2 1−exp h2 S ML;GM 2 1 3 *2 Dg 2 D 2C 0 Dg D* q1 nS 0;GM

ðA40Þ

where 2 * * 6 D h1 va þ Dg D h1 6 q1 ¼ exp6 6 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4Dg p þ va 2 4D* pRd þ vs 2 * h v þ D D h þ D g 2 s g 2 7 2 Dg D*2 7 7 7 * 2Dg D 5

(A41)

Environ Sci Pollut Res

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! # 4Dg p þ va 2 q2 ¼ exp h1 −1 va D2g sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!#sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " 4Dg p þ va 2 4Dg p þ va 2 þ Dg 1 þ exp h1 2 Dg D2g "

ðA42Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #   1 vs λRd pRd vs 2 − 4 þ * þ 2 n2 ¼ 2 D* D* D D*

A1 ¼

C 0 S 0;GM K 1 expðh1 m2 Þ −D* np½expðh1 m1 Þ−expðh1 m2 ÞK 2 þ Dg K 3 pS 0;GM ðA48Þ

2

0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 4D* pRd þ v2s A5 vs q3 ¼ 4−1 þ exp@h2 D*2 2 *4

−D

ðA47Þ

ðA43Þ

0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4D* pRd þ v2s A5 4D* pRd þ v2s 1 þ exp@h2 D*2 D*2

B1 ¼ C 0 S 0;GM =p−A1

ðA49Þ

The limit of QP as p approaching zero is lim QP ¼

p→0





h1 va h2 vs   C 0 nS 0;GM va vs exp þ * Dg 1 θ D     2− h1 va h 2 vs p p −nvs þ exp −S ML;GM va þ S ML;GM va exp þ nvs Dg D*

A2 ¼

C 0 Dg exp½h1 ðm1 þ m2 Þ þ Hn2 ðm1 −m2 ÞS 0;GM −D* np½expðh1 m1 Þ−expðh1 m2 ÞK 2 þ Dg K 3 pS 0;GM ðA50Þ

ðA44Þ Using the inverse Laplace transformation, the following equation can be obtained: lim Qt ¼

t→∞

 h1 va h2 vs C 0 nS 0;GM va vs exp þ * Dg D     ðt−θÞ h 1 va h2 vs −nvs þ exp −S ML;GM va þ S ML;GM va exp þ nv s Dg D*

ðA45Þ

B2 ¼ −exp½ðh1 þ h2 Þðn1 −n2 ÞA2

ðA51Þ

where   K 1 ¼ expðh1 n1 þ Hn2 Þ D* nn1 −Dg m2 S ML;GM   þ expðHn1 þ h1 n2 Þ −D* nn2 þ Dg m2 S ML;GM ðA52Þ

The time lag in this case (i.e., Eq. 65) can also be obtained by the method described in A1. K 2 ¼ n1 expðh1 n1 þ Hn2 Þ−n2 expðh1 n2 þ Hn1 Þ

A4 The case of advection-diffusion-degradation The forms of the general solutions of C g ðz; pÞ and C r ðz; pÞ in this case are also the same as Eqs. (A1) and (A2). The parameters m1 and m2 are the same as given in Eqs. (A30) and (A31). The other parameters are obtained as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #   1 vs λRd pRd vs 2 n1 ¼ þ 4 þ * þ 2 2 D* D* D D*

ðA46Þ

ðA53Þ

K 3 ¼ ½expðh1 n1 þ Hn2 Þ−expðh1 n2 þ Hn1 Þ½m1 expðh1 m1 Þ−m2 expðh1 m2 Þ

ðA54Þ

The accumulated flux at the bottom of the composite liner in the Laplace domain in this case is as follows:

"

# qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dg h2 vs þ D* q1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi * 2 4D 2nS 0;GM C 0 exp p þ v g a 4D ðλ þ pÞRd þ vs 2Dg D* " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#! " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# ) Qp ¼ − ( * 2 4Dg p þ v2a 4D ð λ þ p ÞR þ v d s 1−exp h2 p2 S ML;GM q2 þ −1 þ exp h1 nq3 *2 D2g D

ðA55Þ

Environ Sci Pollut Res

where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q1 ¼ h1 va þ 4Dg p þ va 2 þ Dg h2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4D* ðλ þ pÞRd þ v2s D*2

ðA56Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!# " 4Dg p þ v2a 4Dg p þ v2a 4Dg p þ v2a q2 ¼ −1 þ va exp h1 exp h1 þ Dg 1 þ 2 2 Dg Dg D2g 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * 2 4D ðλ þ pÞRd þ vs A 4D* ðλ þ pÞRd þ vs 2 A5 − 4D* ðλ þ pÞRd þ vs 2 41 þ exp@h2 q3 ¼ −vs þ vs exp@h2 *2 D D*2

ðA57Þ

ðA58Þ

The limit of Qp as p approaching zero is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2C 0 nS 0;GM va L1 4D* λRd þ vs 2 1 θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# lim QP ¼  "    p2 − p p→0 h1 v a h2 4D* λRd þ vs 2 h1 v a 2va S 0;GM exp −1 þ exp −nL3 −1 þ exp Dg Dg D*

ðA59Þ

where

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 h v þ 4D* λRd þ vs 2 2 s h v 1 a 4 5 L1 ¼ exp þ Dg 2D*

"

2

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4D* λRd þ vs 2 h2 5 L2 ¼ −1 þ exp4 D*

ðA60Þ

ðA61Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# 4D* λRd þ vs 2 vs L3 ¼ −1 þ exp D* " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# ðA62Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4D* λRd þ vs 2 h 2 − 1 þ 4D* λRd þ vs 2 exp D* h2

Using the inverse Laplace transformation, the following equation can be obtained:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C 0 nS 0;GM va L1 4D* λRd þ vs 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# lim Qt ¼  "    ðt−θÞ t→∞ h1 v a h2 4D* λRd þ vs 2 h1 v a 2va S 0;GM exp −nL3 −1 þ exp −1 þ exp Dg Dg D*

ðA63Þ

The time lag in this case (i.e., Eq. 71) can also be obtained by the method described in A1.

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