A two-dimensional analytical solution for organic contaminant diffusion through a composite geomembrane cut-off wall and an aquifer

Computers and Geotechnics 119 (2020) 103361

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Research Paper

A two-dimensional analytical solution for organic contaminant diffusion through a composite geomembrane cut-off wall and an aquifer

T

Chun-Hui Peng (PHD candidate)a,b, Shi-Jin Feng (PHD)a, , Qi-Teng Zheng (PHD)a, Xiang-Hong Ding (PHD candidate)a, Zhang-Long Chen (PHD candidate)a, Hong-Xin Chen (PHD)a ⁎

a

Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China b School of Architecture and Civil Engineering, Jinggangshan University, Ji’an, Jiangxi 343009, China

ARTICLE INFO

ABSTRACT

Keywords: Composite geomembrane cut-off wall Analytical model Two-dimensional diffusion Breakthrough time Organic contaminant

A composite geomembrane cut-off wall (CGCW) is one of the most efficient technologies for impeding the horizontal migration of contaminants. A two-dimensional analytical model is developed for describing the transient diffusion of an organic contaminant in a CGCW and a contiguous aquifer with adsorption and degradation. This model considers a non-uniformly distributed contaminant source with respect to depth, which is common in practice. The diffusion behavior of an organic contaminant in a CGCW-aquifer system and the service performance of a CGCW are comprehensively investigated. For a normally distributed TOL source with a distribution range σ of 0.2 m, the breakthrough time of a CGCW is 2.95 times longer than that for σ = +∞ (uniform distribution). In addition, the performance of a CGCW can be improved by embedding a thicker geomembrane (GMB) with a lower diffusion coefficient into an optimal location in a bentonite cut-off wall (BCW). Finally, based on a 50-year designed service life, the equivalency between two types of CGCWs with HDPE GMB and EVOH GMB is assessed using the proposed solution. The solution proves to be a suitable tool for the preliminary design of a CGCW.

1. Introduction Soil and groundwater pollution that is caused by leaking solid waste landfills, chemical storage tank accidents and mining activities has become a global issue [1,2]. To prevent contaminant migration in groundwater and to protect the surrounding environment, effective measures are required, among which vertical cut-off walls are extensively used in situ [3]. Vertical cut-off walls are filled with soilbentonite (SB), cement bentonite (CB) or soil-bentonite-cement (SCB), as each has low permeability, high adsorption and self-healing features [4–8]. However, when subjected to extreme or complex conditions such as cyclic wetting-drying, cyclic freezing-thawing and severe chemical pollution, these filled materials might experience a prominent increase in hydraulic conductivity, thereby leading to a failure to impede contaminant migration [9–11]. To solve this problem, geomembranes (GMBs) with extremely low permeability and high chemical resistance are currently embedded into bentonite cut-off walls (BCWs) in practice, thereby forming composite geomembrane cut-off walls (CGCWs, Fig. 1) [12–15]. The new CGCW is a three-layered system that involves an

upstream BCW, a GMB and a downstream BCW. If a CGCW with an intact GMB is keyed into an aquifuge, advection is typically negligible; however, organics can still migrate through a CGCW via molecular diffusion [12,15]. Thus, it is essential to accurately predict the diffusion of organic contaminants into aquifers, which is also affected by degradation and adsorption, and can be modeled using numerical software and via analytical methods. Although numerical methods can model a more realistic scenario, analytical solutions are of substantial value in revealing the mechanism and studying the sensitivity and uncertainty of parameters. In the past decades, numerous analytical solutions have been proposed for contaminant migration in a single-layered system of porous media [16–21]; however, a multilayered system is more reasonable since media differ in terms of their transport properties and geometrical characteristics. Analytical contaminant migration solutions in a multilayered ground [22], a threelayered composite liner [23–26], a CGCW without an aquifer [12], and a BCW with an infinite aquifer [27] have been developed; however, they are all one-dimensional (1D) and cannot describe a non-uniformly distributed contaminant source with respect to depth, as measured by

Corresponding author. E-mail addresses: [email protected] (C.-H. Peng), [email protected] (S.-J. Feng), [email protected] (Q.-T. Zheng), [email protected] (X.-H. Ding), [email protected] (Z.-L. Chen), [email protected] (H.-X. Chen). ⁎

https://doi.org/10.1016/j.compgeo.2019.103361 Received 25 May 2019; Received in revised form 25 October 2019; Accepted 20 November 2019 0266-352X/ © 2019 Published by Elsevier Ltd.

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[L3 M−1] K' g partition coefficient between GMB and downstream BCW [L3 M−1] Cin (z, t) influent source concentration at position (0, z) at time t [M L−3] P(z) arbitrary depth-dependent function [M L−3] Q(t) continuous time-dependent function X, Z, ζ, ξi, ψi, φi coefficient given in Eq. (14) Ci (X, Z, t) transient concentration in i-th domain at position (X,Z) [M L−3] Si,i+1 coefficient given in Eq. (19) s Laplace transform parameter in respective to the time t Ci (X ,¯ Z , s ) Laplace transform of Ci(X, Z, t) [M L−3] Ai, Bi, αi coefficient given in Eq. (24) Cin,max maximum concentration at the influent boundary [M L−3] Cw,max maximum contaminant concentration in aquifer [M L−3] ΔC value of Cw,max/Cin,max μ, σ location of Cin,max and distribution range of influent contaminant [L] tb breakthrough time of CGCW [T] Tw,s minimum thickness of BCW [L] Hi function given in Eq. (A.1) matrix given in Eq. (A.9) Ri bi, ci, fi, gi, coefficient given in Eq. (A.10)

Notation x horizontal distance from the influent face [L] z vertical distance from the top face [L] t elapsed time for contaminant transport [T] Tg, Tw, Ta thicknesses of GMB, BCW and aquifer [L] Lg distance from the influent face to GMB [L] h height of CGCW-aquifer system [L] i code of each domain Ci (x, z, t) transient concentration in the i-th domain at position (x,z) [M L−3] effective diffusion coefficient in the i-th domain [L2 T−1] Di D2 diffusion coefficient in GMB [L2 T−1] molecular diffusion coefficient in water [L2 T−1] D0 ni porosity of the i-th domain Rd,i sorption retardation factor of the i-th domain λi first-order degradation rate constant in the i-th domain [T−1] Gs,i specific gravity of the i-th domain density of water [M L−3] ρw Kd,i distribution coefficient of the i-th domain [L3 M−1] t1/2,i half-life of contaminant in the i-th domain [T] θi parameter set of parameters of the i-th domain partition coefficient between upstream BCW and GMB Kg

Fig. 1. Conceptual model of contaminant transport in a CGCW-aquifer system.

Parker et al. [28]. Thus, it is necessary to develop a two-dimensional (2D) multilayered solution for contaminant migration through vertical barriers. Recently, Chen et al. [29] presented a remarkable attempt at solving this 2D problem; however, their model is limited to a dual-domain soil system and cannot resolve the discontinuity in concentration between media, such as BCW and GMB in a CGCW. The current paper extends this work and develops a 2D analytical solution for the diffusion, adsorption and degradation of an organic contaminant in a multidomain system that involves various media, such as a CGCW-aquifer system.

Then, the established solution is evaluated using a laboratory experiment of a double-layer composite liner, which was conducted by Park et al. [30], and a numerical model of a CGCW-aquifer system that is based on COMSOL. The migration behavior of an organic contaminant in a 2D CGCW-aquifer system is characterized, together with the influence of a non-uniform contaminant source with depth. Several guidelines will be developed via a throughout investigation of the parameter sensitivity, which can be used for designing and constructing a CGCW.

2

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2. Mathematical model of a CGCW-aquifer system

and it is known as the effective diffusion coefficient [31]; and Rd,i (dimensionless) and λi (T−1) are the sorption retardation factor and the first-order degradation rate constant, respectively, of an organic contaminant in the i-th domain. The effective diffusion coefficients (Di) of the saturated BCW and the aquifer can be conservatively estimated via the following equations [32,33]:

A CGCW-aquifer system consists of four domains in the horizontal direction with interfaces that are parallel to the influent face and perpendicular to the ground: an upstream BCW, a GMB, a downstream BCW and an aquifer (Fig. 2). The origin is located at the top-left corner of the upstream BCW with the x-axis rightwards and the z-axis downwards. The thicknesses of the GMB, BCW and aquifer are denoted as Tg, Tw, and Ta, respectively, and the distance from the influent face to GMB is denoted as Lg. Several assumptions were imposed to facilitate the development of a mathematical model: (1) the BCW and aquifer were both saturated, homogeneous and isotropous, and an intact GMB without holes was embedded vertically into the BCW; (2) a non-uniform distribution of the contaminant concentration with depth was assumed at the influent face (x = 0, z) and the possible change horizontally along the BCW was neglected, which simplified the problem to 2D; (3) the contaminant migration below the CGCW-aquifer system can be neglected because a CGCW is typically keyed into an aquifuge with extreme low permeability and diffusion coefficient [12]; (4) a sufficiently thick aquifer was assumed to neglect the effect of the downstream boundary (x = Tw + Ta, z) on contaminant transport; and (5) the adsorption equilibrium of contaminants in the BCW and aquifer was assumed as linear, instantaneous and noncompetitive sorption [12,27].

Di = D0 × ni4/3

where D0 is the molecular diffusion coefficient of an organic contaminant in water at ambient temperature and ni is the porosity of the ith domain. In addition, the sorption retardation factor of a saturated porous medium (Rd,i) can be described as [31]:

R d,i = 1 + (1

i

=

t)

Ci (x , z , t ) t

+

2C (x , z , i z2

(i = 1, 3, 4)

t)

w

× Gs,i K d,i / ni

(i = 1, 3, 4)

(3)

= ln(2)/ t1/2,i

(4)

(i = 1, 3, 4)

where t1/2,i is the half-life of an organic contaminant in the i-th domain. The upstream BCW and downstream BCW share the same properties: 3

Embedding a GMB into the BCW would substantially reduce the groundwater flow due to its extremely low permeability; hence, in a CGCW-aquifer system, advection and mechanical dispersion that is proportional to water flow can be neglected and only isotropic molecular diffusion is considered. Based on Fick’s second law, the governing equation of organic contaminant diffusion in a soil medium can be expressed as: 2C (x , z , i x2

ni ) ×

where Kd,i and Gs,i are the distribution coefficient and specific gravity, respectively, of the i-th domain and ρw is the density of water. The first-order degradation rate constant in the i-th domain (λi) can be estimated as [34]:

2.1. Mathematical model development

Di R d,i

(2)

(i = 1, 3, 4)

=

(5)

1

where θi represents a parameter set of Di, Rd,i, λi, ni, Kd,i, Gs,I and t1/2,i. The transient diffusion of an organic contaminant through the GMB can be described as:

D2

2C ( x , z , 2 x2

t)

+

2C (x , z , 2 z2

t)

=

−3

C2 (x , z , t ) t 2

(6)

−1

where C2 (x, z, t) (M L ) and D2 (L T ) are the concentration and diffusion coefficient, respectively, of an organic contaminant in the GMB. Assuming that there is no contaminant in the CGCW-aquifer system initially (t = 0) yields:

i Ci (x , z , t )

(1)

Ci (x , z, t )|t = 0 = 0

where i = 1, 3, 4 refer to the upstream BCW, the downstream BCW and the aquifer, respectively; Ci (x, z, t) (M L−3) is the contaminant concentration in the i-th domain at time t (T); Di (L2 T−1) is the diffusion coefficient for an organic contaminant that is diffusing in the i-th domain, which is a porous medium, namely, soil in this studied problem,

(i = 1, 2, 3, 4)

(7)

The influent boundary condition was assumed as:

C1 (x , z, t )|x = 0 = Cin (z , t )

(8)

where Cin (z, t) is the influent source concentration at the position (0, z)

Fig. 2. Mathematical model of contaminant transport in a CGCW-aquifer system. 3

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at time t, and it can be expressed as the product of an arbitrary depthdependent function P(z) and a continuous time-dependent function Q (t), namely, Cin (z, t) = P(z) × Q(t). The continuity conditions of the concentration and the mass flux at the interfaces between the upstream BCW and the GMB (x = Lg), between the GMB and the downstream BCW (x = Lg + Tg), and between the downstream BCW and the aquifer (x = Tw) are expressed as:

K g C1 (x , z, t )|x = Lg = C2 (x , z, t )|x = Lg

n1 D1

C1 (x , z, t ) x

C2 (x , z , t ) x

= D2 x = Lg

x = Lg

C2 (x , z, t )|x = Lg + Tg = K g C3 (x , z , t )|x = Lg + Tg

D2

C2 (x , z , t ) x

= n3 D3 x = Lg + Tg

C3 (x , z, t ) x

= n4 D4 x = Tw

i

z=0

Ci (x , z, t ) z

z=h

= 0 (i = 1, 2, 3, 4)

(9b)

(19b)

C4 (X , Z , t ) |X = 1 = 0 Z

(19c)

where S1,2 = Kg, S2,3 = 1/ Kg, and S3,4 = 1. Subsequently, a procedure that combines the Laplace transform, the Fourier cosine transform and the numerical inverse Laplace transformation is used to solve the general governing equation of Eq. (15) [29,35,36]. Applying Laplace transform to Eq. (15) yields the following governing equations: 2C (X ,¯ Z , i X2

C4 (x , z, t ) z

2

2C (X ,¯ Z , i Z2

s)

(

i

+ i s ) Ci (X ,¯ Z , s ) (20)

C1 (X ,¯ Z , s )|X = 0 = P (Z ) q (s )

(21)

Si, i + 1 Ci (X ,¯ Z , s )|X = Xi = Ci + 1 (X¯, Z , s )|X = Xi

(12c)

x = Tw + Ta

+

where s is the Laplace transform parameter with respect to time t and Ci (X ,¯ Z , s ) is the Laplace transform of Ci(X, Z, t). The Laplace transform procedure eliminates the time-dependency of the original partial differential equation. Applying the Laplace transform to the normalized initial condition, boundary conditions and continuity conditions of Eqs. (17)–(19) yields:

(12b)

=0

s)

= 0 (i = 1, 2, 3, 4)

(12a)

= 0 (i = 1, 2, 3, 4)

i

Ci (X ,¯ Z , s ) |X = Xi = X

(i = 1, 2, 3) (22a)

Ci + 1 (X =¯ Xi , Z , s ) |X = Xi X

i+1

(i = 1, 2, 3) (22b)

2.2. Derivation of the analytical solution The governing equation for the GMB (Eq. (6)) can be re-expressed in a form that is similar to that for a soil medium:

D2 Rd,2

2C (x , z , 2 x2

t)

2C (x , z , 2 z2

+

t)

2 C2 (x ,

z, t ) =

C2 (x , z , t ) t

(13)

=

x , Tw + Ta

z

Z = h,

(Tw + Ta )2Rd, i , Di

2C ( X , i

Z, t)

+

2

X2 C (X , Z , t ) = i i t

= i

=

Tw + Ta , h i i

i

=

ni Di , Tw + Ta

(i = 1, 2, 3, 4)

Ci (X ,¯ Z , s ) |Z = 1 = 0 Z

(i = 1, 2, 3, 4)

(23b)

+ 2

k= k=1

+ B i (k , s ) e

(23c)

i (k = 0, s ) X

[(Ai (k, s ) e

+ Bi (k = 0, s ) e

i (k = 0, s ) X

i (k , s ) X

i (k, s ) X ) cos (k

Z )]

(i = 1, 2, 3, 4)

(24)

where Ai, Bi and αi are coefficients that can be obtained by applying the Fourier cosine transform to Eqs. (20)–(23); the detailed derivation is presented in Appendix A. Since the coefficients Ai, Bi and αi in the general solution of Eq. (24) are functions of s, it is virtually impossible to invert this solution from the Laplace domain to the real time domain via an analytical method [35]. Therefore, a numerical Laplace inversion, namely, Talbot’s algorithm [37], which performed well in the diffusion problems that were studied by Wang and Zhan [38], is adopted in this paper.

i Ci (X , Z , t )

(15)

The normalized initial condition, boundary conditions and continuity conditions (Eqs. (7)–(12)) can be re-expressed as:

Ci (X , Z , t = 0) = 0 (i = 1, 2, 3, 4)

(23a)

Ci (X ,¯ Z , s ) = Ai (k = 0, s ) e

i

(i = 1, 2, 3, 4)

t)

(i = 1, 2, 3, 4)

The general solution to Eq. (21) in the Laplace domain can be expressed as:

(14) 2C (X , Z , i Z2

Ci (X ,¯ Z , s ) |Z = 0 = 0 Z

C4 (X ,¯ Z , s ) |X = 1 = 0 X

where n2 = 1, Rd,2 = 1, and λ2 = 0. To facilitate the solution process, several normalized coefficients, which are listed in Eq. (14), are introduced to enable the development of a general governing equation for all domains, which is presented as Eq. (15):

X=

(18b)

Ci (X , Z , t ) |Z = 1 = 0 (i = 1, 2, 3, 4) Z

where Kg is the partition coefficient between the upstream BCW and the GMB, and K' g is the partition coefficient between the GMB and the downstream BCW, which is equal to Kg [30]. Following the previous assumptions, the boundary conditions at the top (z = 0), at the bottom (z = h) and at the downstream face (x = Tw + Ta) were assumed to be no-flux conditions. In the absence of advection, these conditions can be reduced to a zero gradient of concentration:

Ci (x , z, t ) z

(i = 1, 2, 3)

(19a)

(11b)

x = Tw

Ci + 1 (X , Z , t ) |X = Xi X

i+1

(18a)

Ci (X , Z , t ) |Z = 0 = 0 (i = 1, 2, 3, 4) Z

(11a)

C4 (x , z, t ) x

Ci (X , Z , t ) |X = Xi = X

(i = 1, 2, 3)

(9a)

(10b)

C3 (x , z, t )|x = Tw = C4 (x , z , t )|x = Tw n3 D3

Si, i + 1 Ci (X , Z , t )|X = Xi = Ci + 1 (X , Z , t )|X = Xi

(10a)

C3 (x , z, t ) x x = Lg + Tg

(17)

C1 (X , Z , t )|X = 0 = P (Z ) Q (t )

(16) 4

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3. Verification

(termed as Case 1) and 1D horizontal diffusion (termed as Case 2) in a CGCW-aquifer system, and 2D diffusion (termed as Case 3) and 1D horizontal advection-2D diffusion (termed as Case 4) in a BCW-aquifer system without the GMB in Figs. 5 and 6. MTBE and TOL are chosen as target contaminants to represent polar (hydrophilic) and nonpolar (hydrophobic) organics, respectively. Relevant parameters are listed in Table 1 with a contaminant source that is described by Eq. (25), and the degradation of an organic contaminant in both the BCW and the aquifer is not considered in this or in the following sections, except in Section 5.2. In a BCW-aquifer system without the GMB (Case 4), the Darcy velocity of the groundwater was assumed as 5 × 10−10 m/s along the positive direction of the x-axis [27]. Case 4 is simulated using COMSOL, and the other three cases are solved using the proposed analytical solution. As expected, compared to Case 2 (1D diffusion in a CGCW-aquifer system), Case 1 (2D diffusion in a CGCW-aquifer system) for MTBE has lower relative concentrations maximally by 0.16 in the vicinity of the plume centerline (z = 5 m) but has higher relative concentrations maximally by 0.07 in the area away from the plume centerline (Figs. 5a and 6a). For TOL, a similar phenomenon is observed (Figs. 5b and 6b). This is caused by contaminant dilution, which is due to the diffusion along the depth direction. The 1D diffusion assumption might underestimate the service life of a CGCW, which depends on the influent contaminant source and breakthrough concentration, which will be discussed in Section 5.1. When an HDPE GMB is embedded in the BCW (Case 1), there is a substantial concentration mutation between the sides of GMB for MTBE, and compared to Case 3 (2D diffusion in a BCW-aquifer system without GMB), the MTBE concentration in Case 1 is higher at x < 0.5 m (upstream BCW) but lower at x > 0.502 m (downstream BCW and aquifer) (Figs. 5a and 6a). For another target contaminant of TOL, its concentrations on both sides of the HDPE GMB in Case 1 are equivalent but slightly lower than those in Case 3 without the GMB; hence, the GMB has little effect on the TOL diffusion through a CGCW (Figs. 5b and 6b). The difference in migration behavior between the MTBE and TOL is mainly due to the substantial difference in the values of the partition coefficient Kg between the GMB and the BCW (Eqs. (9) and (10)), namely, 0.57 for MTBE and 86.7 for TOL. Thus, the HDPE GMB tends to outperform the TOL in impeding the migration of MTBE due to lower surface sorption and desorption of MTBE on the GMB, even though its diffusion coefficient in the GMB (7.74 × 10−13 m2/s) is slightly higher than that for the TOL (3.77 × 10−13 m2/s). For Case 4 of a BCW-aquifer system, advection is considered due to the lack of

3.1. Comparison with a 1D experiment Park et al. [30] conducted an experiment that involved the migration of five organic compounds through a composite liner of a 1.5-mmthick high-density polyethylene (HDPE) GMB and a 120-mm-thick compacted clay layer (CCL, porosity of 0.38 and density of 1700 kg/ m3). The measured concentrations of methyl tertiary butyl ether (MTBE) at 60-mm-deep and 90-mm-deep sampling ports with respect to time are considered in this section for comparison. The proposed 2D analytical solution can be degenerated to a 1D solution by assuming a uniform concentration in the influent boundary, namely, 100 mg/L in this case. The following parameters, which were obtained via a backfitting analysis that was conducted by the source paper of the experiments [30], are used for calculation: a partition coefficient of 0.57 for MTBE in HDPE GMB, effective diffusion coefficients of 7.74 × 10−13 m2/s in HDPE GMB and 1.18 × 10−10 m2/s in CCL for MTBE, and a distribution coefficient of 0.053 mL/g for MTBE in CCL. The calculated breakthrough curves of the MTBE concentrations at depths of 60 and 90 mm accord with the experimental data (Fig. 3), thereby demonstrating the correctness of the proposed analytical solution. 3.2. Comparison with a 2D numerical model In this section, the proposed solution is compared with a 2D CGCWaquifer model that was established via the finite-element-method-based COMSOL Multiphysics 5.3a [39]. This CGCW-aquifer system contains a 1-m-thick BCW, a 2-mm-thick HDPE GMB that was embedded in the middle of the BCW, and a 20-m-thick aquifer, which is sufficiently thick for ignoring the effect of the finite boundary based on a sensitivity study. MTBE is selected as the source contaminant and its molecular diffusion coefficient (D0) is 7.9 × 10−10 m2/s in 25 ℃ water [40]. The effective diffusion coefficients for MTBE in the BCW (D1) and the aquifer (D4) can be estimated via Eq. (2), with the other geometric parameters and migration parameters are summarized in Table 1. The half-life of MTBE in both the BCW (t1/2,1) and the aquifer (t1/2,4) was assumed to be 50 years. A time-stabilized but depth-dependent concentration source of

Cin = Cin,max × exp

µ) 2

(z 2

2

(25)

is specified at the influent face, where Cin,max is the maximum concentration at the influent boundary; μ is the mathematical expectation, which corresponds to the location of Cin,max; and σ is the standard deviation, which corresponds to the distribution range of the influent contaminant. In this section, σ = 0.5 m and μ = 5 m are used. The concentration distributions along the center line of the influent pollution plume (z = 5 m) at 10, 20, and 50 years that were obtained from the proposed analytical solution and COMSOL accord with each other (Fig. 4), thereby supporting the accuracy of the proposed solution. Then, the analytical solution is applied in the following for analysis.

Leachate (MTBE) 100 mg/L

10

Geomembrane Point A Compacted clay liner

1.5 mm

120 mm

As discussed previously, 1D horizontal diffusion through a vertical barrier was assumed in the previous work without consideration of contaminant diffusion in the depth direction. However, the influent contaminant is typically non-uniformly distributed with respect to the depth in practice; hence, the above assumption would inevitably cause substantial errors in predicting the migration behavior of an organic contaminant through a vertical barrier. Together with studying the influence of the embedded GMB, this section compares 2D diffusion

Point A Point B

90 mm

4. Diffusion behavior of organic contaminants in a CGCW-aquifer system

This study Park et al. (2011)

60 mm

Concentration, C (mg/L)

15

Point B

5

0

0

100

200

300

400

Time, t (days) Fig. 3. Comparison between the proposed analytical model and a 1D experiment that was conducted by Park et al. (2011) in terms of the time-dependent MTBE concentration in a GMB-CCL composite liner. 5

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which can be expressed as a relative concentration of ΔC = Cw,max/ Cin,max = 0.1 in this study. In the following, only the variables of interest are changed, and the other variables remaining unchanged unless otherwise specified

Table 1 Material properties of a CGCW-aquifer system. Geometrical parameters

BCW

GMB

aquifer

Thickness, T (m) Height, h (m) Specific gravity, Gs Porosity, n

1 10 2.68 0.5b

0.002 10 – –

20 10 2.7 0.52

Migration parameters

BCW

GMB

aquifer

0.57c 86.7c

– –

– –

7.92 × 10−10 9.68 × 10−10

a

5.1. Influence of the influent concentration distribution

Partition coefficient, Kg

MTBE TOL

– –

Molecular diffusion coefficient, D0 (m2/s)

MTBE TOL

7.92 × 10−10 9.68 × 10−10

Effective diffusion coefficient, D (m2/s)

MTBE TOL

3.48 × 10−10 3.84 × 10−10

7.74 × 10−13c 3.77 × 10−13c

3.31 × 10−10 4.05 × 10−10

Distribution coefficient, Kd (mL/g)

MTBE TOL

0.2 0.54f

– –

0 0

Retardation factor, Rd

MTBE TOL

1.54 2.45

– –

1 1

d e

Note: a Hong et al. [7]; b Zhan et al. [12]; c Park et al. [30]; Yaws [45]; f Park et al. [46]; g Xie et al. [27]. 1.0

C / Cin,max

0.8

d

As previously discussed, the contaminant concentration at the upstream face of a CGCW (Cin) may be non-uniformly distributed with respect to the depth, and the proposed analytical solution can be applied to any arbitrary distribution of an influent contaminant source with the depth. To provide insight into its influence, a representative influent source of Eq. (25) is adopted in this section for values of the three parameters that were considered previously, namely, Cin,max, μ and σ. For various values of mathematical expectation μ, the breakthrough curves overlap with each other if the distribution range of the influent contaminant remains unchanged (identical σ) (Fig. 7); hence, the breakthrough time is independent of the location of maximum influent concentration Cin,max, namely, μ. As the distribution range of σ increases from 0.2 to +∞, the relative concentration ΔC shows a substantial decrease, together with a shorter breakthrough time, at ΔC = 0.1 (115 years for σ = 0.2 m, 50 years for σ = 0.5 m, 41 years for σ = 1 m, and 39 years for σ = +∞). A scenario with σ = +∞ corresponds to a uniformly distributed influent concentration with depth, which is equivalent to a 1D diffusion scenario in which contaminant dilution in the depth direction is ignored. The assumption of 1D diffusion would underestimate the breakthrough time and the service life of a CGCW, and it is only reasonable when the influent contaminant is widely distributed along the depth, such as if σ ≥ 1 m. However, for a much more narrowly distributed influent concentration, such as σ = 0.2 m, the design of a CGCW that is based on a 1D diffusion assumption would be too conservative. In the following studies, a time-stabilized TOL source (Eq. (25)) with σ = 0.5 m and μ = 5 m is used.

d e

g g

Xu et al. [40];

e

10 years 20 years 50 years This study COMSOL 5.3a

0.6

5.2. Influence of contaminant degradation

0.4

Various values of the TOL half-life in the BCW (t1/2,1) and the aquifer (t1/2,4) are used in this section to characterize the levels of nutrients, organic matter and electron acceptors in various soils. The relative concentration ΔC increases over time and asymptotically approaches a steady-state value with consideration of t1/2,1 and t1/2,4 (Fig. 8). A smaller half-life in the BCW or aquifer results in a shorter time for reaching a smaller stable value of ΔC. For example, for t1/ 2,4 = 20 years, the time required for reaching a steady state decreases by 46% as the TOL half-life in the BCW, namely, t1/2,1, decreases from 100 to 20 years, and the relevant stable value of ΔC decreases by 62% from 0.138 to 0.052. In addition, the degradation of TOL in BCW (t1/2,1) has a stronger impact on the breakthrough time of a CGCW than the TOL degradation in an aquifer (t1/2,4). For t1/2,1 = 100 years, the breakthrough time of a CGCW increases from 60 to 68 years as t1/2,4 decreases from 100 to 20 years, while for t1/2,4 = 100 years, it increases from 60 to an infinite number of years as t1/2,1 decreases from 100 to 20 years. In this case, the CGCW will satisfy the requirement of a resistance to TOL all the time when t1/2,1 = 20 years and t1/ 2,4 ≤ 100 years. Therefore, shortening the half-life of an organic contaminant in the BCW can substantially enhance the service performance of a CGCW.

0.2

0.0 0.0

0.5

x (m)

1.0

1.5

Fig. 4. Comparison of the concentration profiles along the plume centerline (z = 5 m) at t = 10, 20, and 50 years that were obtained from the proposed analytical solution and numerical software COMSOL 5.3a.

extremely low permeability GMB, and a much wider contaminant plume is observed compared to Case 1. Hence, a CGCW has a higher resistance to an organic contaminant than a traditional BCW. 5. Evaluation of the service performance of a CGCW Although the leakage of contaminants into an aquifer can be substantially delayed by a CGCW, it would eventually occur and endanger the surrounding environment and residents as time goes on. Hence, it is important to evaluate the service performance of a CGCW in terms of the maximum contaminant concentration in the aquifer (denoted as Cw,max) at the interface between the CGCW and the aquifer. In the following study, TOL is the targeted contaminant, with a typical value of 10 mg/L for its maximum influent concentration (Cin,max) in landfill leachate [41]. The breakthrough time (denoted as tb) is the time that is required for Cw,max to reach its allowable value, namely, 1 mg/L for TOL, which is based on United States drinking water regulations [42],

5.3. Study on the design parameters of a CGCW 5.3.1. Influence of the GMB thickness and location An HDPE GMB has poor resistance to the migration of hydrophobic organics. To address this weakness, new types of GMB have been developed, such as a coextruded GMB with an inner core of ethylene-vinyl alcohol (denoted as EVOH GMB) [43,44]. An EVOH GMB has a 6

Computers and Geotechnics 119 (2020) 103361

C.-H. Peng, et al.

2D diffusion

0

GMB

C/Cin,max

Exit face of CGCW

1D diffusion

0

1.000

GMB

0.500

2

C/Cin,max

1.000

Exit face of CGCW

0.500

2

0.200

0.200 0.100

0.100 0.050 0.020 0.010

6

4

z (m)

z (m)

4

0.050 0.020

6

0.010

0.005

10

0.005

0.002

8

1

2

0.002

Case 2

0.001

Case 1 0

8

0.000

10

3

0.000

0

1

2

3

x (m)

x (m)

2D diffusion

0

0.001

C/Cin,max

Exit face of BCW

1D advection-2D diffusion

0

1.000

1.000

Exit face of BCW

0.500

2

C/Cin,max

0.500

2

0.200

0.200

0.100

0.100

0.050 0.020

6

4

z (m)

z (m)

4

0.010

0.050 0.020

6

0.010

0.005

8

Case 3 10

0.005

8

0.002

0.000

0

1

2

1

2

1D diffusion

0

1.000

GMB

0.500

2

3

x (m)

C/Cin,max

Exit face of CGCW

0.001 0.000

0

(a)

2D diffusion GMB

10

3

x (m)

0

0.002

Case 4

0.001

C/Cin,max

1.000

Exit face of CGCW

0.500

2

0.200

0.200

0.100

0.100

0.050 0.020

6

4

z (m)

z (m)

4

0.010

0.050 0.020

6

0.010

0.005

8 10

0.005

8

0.002

Case 1

Case 2

0.000

0

1

2

0.002

0.001

10

3

0.000

0

1

2

x (m)

3

x (m)

2D diffusion

0

0.001

C/Cin,max

1D advection-2D diffusion

0

1.000

Exit face of BCW

0.500

2

C/Cin,max

1.000

Exit face of BCW

0.500

2

0.200

0.200

4

0.100

0.050

z (m)

z (m)

0.100

0.020

6

0.010

4

0.050 0.020

6

0.010

0.005

8 10

0.005

8

0.002

Case 3

Case 4

0.000

0

1

2

x (m)

0.002

0.001

10

3

(b)

0.001 0.000

0

1

2

3

x (m)

Fig. 5. Comparison of Cases 1-4 in terms of (a) MTBE and (b) TOL concentration contours at t = 50 years (In a CGCW-aquifer system: Case 1 of 2D diffusion and Case 2 of 1D horizontal diffusion; In a BCW-aquifer system: Case 3 of 2D diffusion and Case 4 of 1D horizontal advection-2D diffusion).

7

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C.-H. Peng, et al.

1.0

0.8

0.6

GMB

0.4

C

0.6

0.2

0.4

0.500

0.2

0.0

0.501

0.502

Case 1 Case 2 Case 3 Case 4 0.0

0.5

1.0

1.5

Concentration, C (mg/L)

0.5 0.4

μ+σ

0.3 0.2

Cin

0.1 0.0

Breakthrough standard 0

50

100

0.8

10

0.6

GMB

1

C

0.1

0.500

0.501

0.502

Case 1 Case 2 Case 3 Case 4 1.0

t1/2,1 (years) t1/2,4 (years)

0.30

Relative concentration,

C/Cin,max

Enlargement

0.5

200

Fig. 7. Effect of the influent concentration distribution of TOL (various values of μ and σ) on the breakthrough time of a CGCW (Tw = 1 m, Kd,1 = 0.54 mL/g, n1 = 0.5, Tg = 2 mm, Lg = Tw/2, D2 = 3.77 × 10−13 m2/s, Kg = 86.7, t1/2,1 = +∞, and t1/2,4 = +∞).

100

0.0

150

Time, t (years)

1.0

0.0

σ=+

μ

(a)

0.2

σ=1m

Cin,max

2.0

x (m)

0.4

σ = 0.5 m

μ-σ

z (m)

0.6

Enlargement

Relative concentration,

C/Cin,max

0.8

σ = 0.2 m

μ=5m μ = 2.5 m μ=0m

1.5

2.0

x (m)

+ +

100 100

100 20

20 100

20 20

0.25 0.20 0.15 Breakthrough standard 0.10 0.05

(b)

0.00

Fig. 6. Concentration profiles of (a) MTBE and (b) TOL along the plume centerline (z = 5 m) at t = 50 years in Cases 1, 2, 3 and 4.

1

10

100

Time, t (years)

1000

10000

Fig. 8. Breakthrough curves for various value pairs of the TOL degradation rate in a BCW and an aquifer (Tw = 1 m, Kd,1 = 0.54 mL/g, n1 = 0.5, Tg = 2 mm, Lg = Tw/2, D2 = 3.77 × 10−13 m2/s, and Kg = 86.7).

partition coefficient Kg for TOL that is close to that of an HDPE GMB (97.4 for an EVOH GMB and 86.7 for an HDPE GMB); however, its diffusion coefficient is much smaller, namely, 0.037 times that of an HDPE GMB (1.4 × 10−14 m2/s for an EVOH GMB and 3.77 × 10−13 m2/s for an HDPE GMB) [30,44]. Therefore, a CGCW with an EVOH GMB (denoted as EVOH CGCW) can outperform a CGCW with an HDPE GMB (denoted as HDPE CGCW) in resisting TOL, as shown in Fig. 9, where t1/2,1 = 0 years and t1/2,4 = 0 years are conservatively adopted. The breakthrough time tb of an EVOH CGCW or an HDPE CGCW that is subjected to TOL linearly increases with the GMB thickness Tg (Fig. 9). As the GMB thickness increases from 1 to 4 mm, the breakthrough time of an HDPE CGCW increases by approximately 20% from 46 to 56 years, whereas for an EVOH CGCW it increases by approximately 82% from 54 to 99 years. Hence, increasing the thickness of an EVOH GMB is much more effective in improving the resistance to TOL than increasing the thickness of an HDPE GMB. The location of a GMB in the BCW has a weaker effect on the breakthrough time than the GMB thickness (Fig. 10). As the distance between the influent boundary and the GMB, which is denoted as Lg, increases from 0.1 to 0.9 m, the breakthrough time of an HDPE CGCW increases from 47 to 50 years, whereas the breakthrough time of an EVOH CGCW drops from 72 to 67 years. This phenomenon suggests that the lifetime of a CGCW can be

improved by selecting a better embedding location for the GMB in the barrier, and this location mainly depends on the partition coefficient and the diffusion coefficient of an organic in the GMB. 5.3.2. Influence of the thickness, distribution coefficient and porosity of the BCW The breakthrough time tb of a CGCW can be increased by increasing the BCW thickness Tw due to an increase in the migration path of a contaminant, and Fig. 10 shows an exponential increase in tb with Tw. Another effective strategy is to increase the distribution coefficient Kd,1 of BCW (Fig. 10a). For example, when n1 = 0.5 and Tw = 1 m, the value of tb for TOL increases substantially from 50 years to 94 and 222 years as the distribution coefficient of BCW Kd,1 increases from 0.54 mL/g to 1.62 and 4.86 mL/g, respectively. Hence, adding a highly effective adsorption amendment for a specified contaminant into a BCW could improve the performance of a CGCW, as has been suggested by Malusis et al. [5]. Reducing the porosity of BCW n1 would decrease its effective 8

Computers and Geotechnics 119 (2020) 103361

C.-H. Peng, et al.

The thickness of GMB, Tg (mm) 0

1

HDPE CGCW EVOH CGCW

90

2

Tg

3

4

The minimum thickness of BCW (Tw,s)

5

Lg

The breakthrough time, tb (years)

The breakthrough time, tb (years)

100

300

80 70

Design Life: 50 years 60

Tg=2 mm

Lg=0.5 m

250

Kd,1=0.54 mL/g Kd,1=1.62 mL/g Kd,1=4.86 mL/g

200

n1=0.5

150

Design Life: 50 years 100

50

50

0 0.4 40 0.0

0.2

0.4

0.6

0.8

0.6

0.8

1.0

1.2

The thickness of BCW, Tw (m)

1.0

(a)

The distance between influent boundary and GMB, Lg (m) Fig. 9. Effects of the thickness and location of an HDPE GMB and an EVOH GMB on the breakthrough time of a CGCW that is subjected to TOL (Tw = 1 m, Kd,1 = 0.54 mL/g, and n1 = 0.5).

The breakthrough time, tb (years)

300

diffusion coefficient D1 and increase the sorption retardation factor Rd,1, thereby increasing the breakthrough time tb of a CGCW (Fig. 10b and Cases 1, 5 and 6 in Table. 2). When n1 decreases from 0.5 in Case 1 to 0.4 in Case 5 (or 0.3 in Case 6) by 20% (or 40%) for Kd,1 = 0.54 mL/ g and Tw = 1 m, tb increases by approximately 96% (420%), which is mainly attributed to a 26% (or 49%) decrease in D1 and a 30% (or 79%) increase in Rd,1. These values of D1 and Rd,1 were calculated via Eqs. (2) and (3) and to further study their effects on the breakthrough time, two additional cases with fixed values of D1 and Rd,1 (Cases 7 and 8) were modeled. Compared to Case 1, a 26% decrease in D1 with unchanged Rd,1 in Case 7 yields an increased value of tb by 42%, while a 30% increase in Rd,1 with unchanged D1 in Case 8 increases tb by 22% (Table 2). Thus, the barrier performance of a CGCW can be effectively improved by compacting the BCW as densely as possible, and the decreased effective diffusion coefficient D1 that is induced by the reduced porosity of the BCW plays a more important role than the incidental increase in the sorption retardation factor Rd,1. In the design of a vertical barrier, a key requirement is that the breakthrough time is not less than the service life, which is typically 50 years. Therefore, it is necessary to estimate the minimum thickness of the BCW (denoted as Tw,s) to satisfy the above requirement. For example, when n1 = 0.5 and Kd,1 = 0.54 mL/g, the value of Tw,s is 1 m (see the dashed line in Fig. 10a) and if Tw,s < 1 m, the barrier would fail prior to 50 years. The minimum thickness of the BCW will be discussed in the next section for reference.

The minimum thickness of BCW (Tw,s)

Case 1 (n1=0.5) Case 5 (n1=0.4) Case 6 (n1=0.3)

250

200

Kd,1=0.54 mL/g 150

Design Life: 50 years 100

50

0 0.4

0.6

0.8

1.0

1.2

The thickness of BCW, Tw (m)

(b) Fig. 10. Effect of BCW thickness on the breakthrough time of an HDPE CGCW that is subjected to TOL for various values of (a) the BCW distribution coefficient and (b) the BCW porosity (Tg = 2 mm, Lg = Tw/2, D2 = 3.77 × 10−13 m2/s, and Kg = 86.7).

required for an EVOH CGCW to reach tb = 50 years, compared to 0.87 for an HDPE CGCW. When Tw,s = 0.6 m and n1 = 0.5, the distribution coefficient Kd,1 of an HDPE CGCW (or an EVOH CGCW) should be larger than 3.1 mg/L (or 1.9 mg/L). Replacing an HDPE GMB with an EVOH GMB can reduce the requirements on the thickness and the distribution coefficient of a BCW. Fig. 11 presents an equivalence between an HDPE CGCW and an EVOH CGCW, which can help engineers decide which CGCW to use.

5.3.3. Equivalence between an HDPE CGCW and an EVOH CGCW During the period of service, the performance of a CGCW in resisting contaminant migration depends on both GMB and BCW, and it can be significantly improved via measures such as using an EVOH GMB, adding adsorptive materials into the BCW and increasing the thickness of the BCW. An equivalence between an HDPE CGCW and an EVOH CGCW, which have a same breakthrough time of 50 years, can help optimize the design of a vertical barrier. Many calculations were conducted to investigate the relationship between the minimum thickness Tw,s, the porosity n1 and the distribution coefficient Kd,1 of a BCW, followed by an equivalent design between an HDPE CGCW and an EVOH CGCW. The minimum thickness Tw,s of an EVOH CGCW is substantially lower than that of an HDPE CGCW (Fig. 11). For example, when Kd,1 = 1 mL/g and n1 = 0.5, a minimum thickness Tw,s of 0.74 is

Table 2 Values of breakthrough time tb for different combinations of n1, D1, and Rd,1. D1 (m2/s)

n1 Case Case Case Case Case

Note:

9

1 (Ref.) 0.5 5 0.4 (20% ↓) 6 0.3 (40% ↓) 7 0.5 8 0.5 a

Eq. (2);

b

Eq. (3).

Rd,1 −10 a

3.84 × 10 2.85 × 10−10 a (26% ↓) 1.94 × 10−10 a (49% ↓) 2.85 × 10−10 (26% ↓) 3.84 × 10−10

tb (years) b

2.45 3.17b (30% ↑) 4.38b (79% ↑) 2.45 3.17 (30% ↑)

50 98 (96% ↑) 260 (420% ↑) 71 (42% ↑) 61 (22% ↑)

Computers and Geotechnics 119 (2020) 103361

The distribution coefficient of BCW, Kd,1, mL/g

C.-H. Peng, et al.

5

n1 HDPE CGCW EVOH CGCW

0.3

0.4

0.5

0.6

6. Conclusions

0.7

A 2D analytical model was developed for describing the transient diffusion of an organic contaminant in a multidomain system that involves various media, which considers adsorption and degradation. It was applied to study the performance of a composite geomembrane cutoff wall, which consists of four domains: an upstream bentonite cut-off wall, a GMB, a downstream bentonite cut-off wall and an aquifer. The proposed solution was evaluated via a 1D laboratory experiment that was conducted by Park et al. [30] and a 2D numerical solution that was based on COMSOL 5.3a. The following conclusions are drawn:

4

3

2

(1) The assumption of 1D contaminant diffusion in the horizontal direction neglects the contaminant dilution along the depth, thereby yielding an overestimated concentration in the vicinity of the plume centerline and an underestimated concentration in the area away from that. (2) For a non-uniformly distributed contaminant source with depth, the breakthrough time of a CGCW is independent of the location of its maximum concentration. For a normally distributed TOL source with σ = 0.2 m, the breakthrough time of a CGCW is 2.95 times longer than that for the case with σ = +∞. Thus, it would be too conservative to use a 1D horizontal diffusion model to design a CGCW against a much more narrowly distributed contaminant source. (3) The degradation of an organic contaminant has substantial effects on the concentration distribution within the CGCW and on the breakthrough time. The degradation rate in the BCW has a stronger influence than that in the aquifer. If the half-life of an organic contaminant is less than a threshold value, the contaminant concentration in the aquifer will never exceed the threshold. (4) In the case of 4-mm-thick GMB, the breakthrough time of an HDPE CGCW (EVOH CGCW) can be 20% (82%) longer than that in the case of 1-mm-thick GMB for TOL. As Lg increases from 0.1 to 0.9 m, the breakthrough times of an HDPE CGCW and an EVOH CGCW undergo a 6.4% increase and a 7% decrease, respectively. Thus, the performance of a CGCW can be improved by embedding a thicker GMB with a lower diffusion coefficient into an optimal location in the BCW. Based on a designed service life of 50 years, the equivalence between an HDPE CGCW and an EVOH CGCW in terms of the thickness, porosity and distribution coefficient of the BCW was proposed as a reference for design.

1

0 0.4

0.6

0.8

1.0

1.2

The minimum thickness of BCW, Tw,s (m) Fig. 11. Equivalence assessment chart of an HDPE CGCW and an EVOH CGCW for a designed service life of 50 years (Tg = 2 mm and Lg = Tw/2).

5.4. Limitations The proposed two-dimensional analytical solution depends on the assumptions that are discussed in the section on model development with several limitations. One limitation is that the proposed analytical solution cannot correctly describe the migration behavior of a contaminant in three dimensions; however, it can provide a conservative design scheme for CGCW and facilitate the development of analytical solutions for contaminant transport through a multilayered barrier. Another limitation is the simplification of the boundary conditions. A zero-flux boundary condition was imposed at the top and bottom surfaces based on the assumptions that the liquid-phase organic contaminants cannot penetrate the ground plane and cannot bypass the vertical barrier through an underlying stratum. A zero-flux downstream boundary condition was imposed in this study instead of a more accurate boundary condition due to a severe lack of field data. When the downstream surface is far away from the CGCW, the effect of the downstream boundary condition on the concentration distribution can be ignored within the service life of the CGCW. An additional limitation is that the proposed solution is restricted to the evaluation of the performance of a vertical barrier against contaminant migration in saturated and homogeneous soils, and it cannot consider the layered characteristics of natural soils that are caused by sedimentation or the variational properties of the BCW in space and time that are due to consolidation. Moreover, the approach that is presented in this paper is limited to the analysis of one contaminant at a time and neglects the competitive sorption of various compounds. The contaminant sources, such as the leachate from landfills, contain a variety of chemicals and processes, such as preferential sorption and chemical reactions, may effect solute transport. Furthermore, the sorption process in the soil is assumed to be a linear and equilibrium process. However, this assumption is suitable only when the concentration of a contaminant source is relatively low. Thus, further experimental investigations are required for moderating the recommendations that are made in this study; however, the proposed analytical solution remains valuable for investigating the migration mechanism of an organic contaminant in a CGCW-aquifer system and for providing guidelines for the design of a CGCW.

Declaration of Interest Statement We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgments Much of the work described in this paper was supported by the National Natural Science Foundation of China under Grant Nos. 41572265, 41931289 and 41725012, the Shanghai Shuguang Scheme under Grant No. 16SG19, the Fundamental Research Funds for the Central Universities under Grant No. 0200219152, Shanghai Science and Technology Innovation Action Plan under Grant No. 18DZ1204402, the Department of Education Science and Technology Research Project of Jiangxi Province in China under Grant No. GJJ180586, and the Science and Technology Research Project of Jinggangshan University under Grant No. JZ1905. The writers would like to greatly acknowledge all these financial supports and express their most sincere gratitude.

10

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Appendix A According to Fourier cosine inverse transform, the following solution is proposed for C¯i :

Ci (X ,¯ Z , s ) = Hi (X , k = 0, s ) + 2

k=

[Hi (X , k, s ) cos(k Z )]

(i = 1, 2, 3, 4)

k=1

(A.1)

where Hi is the function related to the x coordinate that needs to be determined. Similarly, the depth-dependent function P(z) can be transformed into the following form: k=

P (Z ) = p (k = 0) + 2

[p (k )cos(k Z )]

(A.2)

k=1

where p(k) is the undetermined coefficient. Substituting Eq. (A.1) and Eq. (A.2) into Eqs. (20)–(23) results in the following equation:

d 2Hi dX 2

( 2k 2

2

+

is

+

i ) Hi

=0

(i = 1, 2, 3, 4)

(A.3) (A.4)

H1 (X = 0, k, s ) = p (k ) q (s )

Si, i + 1 Hi (X , k, s )|X = Xi = Hi+ 1 (X , k, s )|X = Xi i

dHi (X , k, s ) |X = Xi = dX

i+1

(A.5a)

(i = 1, 2, 3)

dHi + 1 (X , k, s ) |X = Xi dX

(i = 1, 2, 3)

(A.5b)

dH4 (X , k, s ) |X = 1 = 0 dX

(A.6)

Obviously, the solution to Eq. (A.3) satisfying all the relevant boundary conditions is:

Hi (X , k, s ) = Ai (k, s )e

i (k , s ) X

+ Bi (k, s )e

i (k , s ) X

(A.7)

(i = 1, 2, 3, 4)

The coefficients αi can be obtained: i (k ,

2k 2 2

s) =

+

is

+

i

(A.8)

(i = 1, 2, 3, 4)

Substituting Eq. (A.7) into the continuity conditions (Eq. (A.5)) and introducing transfer matrix Ri, the coefficients Ai and Bi can be determined by the following recursive equation:

Ai = Ri Bi

1

Ai Bi

1

(i = 2, 3, 4)

1

(A.9)

where Ri can be determined by

1

Ri =

2

i+1

Si, i + 1

i

Si, i + 1

bi

ci i + 1 gi i

fi

fi Si, i + 1 +

i+1

+

bi Si, i + 1

i

gi i + 1 ci i

(i = 1, 2, 3) (A.10)

i+1

in which

bi = ci = fi =

( i + 1 i) Xi i+1 e ( i + 1 i ) Xi ie ( i + 1+ i ) Xi i+1 e

gi =

ie

(i = 1, 2, 3)

( i + 1+ i) Xi

(A.11)

A4 and B4 can be obtained by substituting Eq. (A.7) into the influent boundary condition (Eq. (A.4)) and the downstream boundary condition (Eq. (A.6)) as:

A4 = B4 =

p (k ) q (s ) e G (1) + G (2)

2 4

(A.12)

p ( k ) q (s ) G (1) + G (2)

(A.13)

in which

G= e

2 4

1

R3 1 R2 1 R1

1

(A.14)

Substituting Eqs. (A.12)–(A.13) into the recursive equation (Eq. (A.9)), A1, A2, A3 and B1, B2, B3 can be obtained.

11

Computers and Geotechnics 119 (2020) 103361

C.-H. Peng, et al.

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