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Article

Analytical Model for Contaminant Transport in the CGCW and Aquifer Dual-Domain System Considering GMB Holes

by
Long Ran
1,2,
Guijun Wan
3,
Hao Ding
3,* and
Haijian Xie
3
1
School of Civil Engineering and Architecture, Zhejiang University of Science and Technology, Hangzhou 310023, China
2
Zhejiang-Singapore Joint Laboratory for Urban Renewal and Future City, Hangzhou 310023, China
3
The Architectural Design and Research Institute of Zhejiang University Co., Ltd., 148 Tianmushan Rd., Hangzhou 310058, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(22), 10280; https://doi.org/10.3390/app142210280
Submission received: 24 September 2024 / Revised: 28 October 2024 / Accepted: 1 November 2024 / Published: 8 November 2024
(This article belongs to the Special Issue Advanced Technologies in Landfills)
Browse Figures
Figure 1
<p>Mathematical model of contaminant transport in the CGCW and aquifer system.</p> "> Figure 2
<p>Comparison with the results predicted by COMSOL.</p> "> Figure 3
<p>Comparison with the existing analytical model by Peng et al. (2021) [<a href="#B30-applsci-14-10280" class="html-bibr">30</a>].</p> "> Figure 4
<p>Leakages for different hydraulic conductivities and head losses when (<b>a</b>) <span class="html-italic">T<sub>G</sub></span> = 0.001 m and (<b>b</b>) <span class="html-italic">T<sub>G</sub></span> = 0.002 m.</p> "> Figure 5
<p>Leakages for different hydraulic conductivities and head losses with different GMB thicknesses.</p> "> Figure 6
<p>Dimensional drawing of different GMB hole shapes.</p> "> Figure 7
<p>Influence of the hole radius on the breakthrough curve.</p> "> Figure 8
<p>Influence of the hole shape on the breakthrough curve.</p> "> Figure 9
<p>Influence of the cut-off wall hydraulic conductivity on the breakthrough curve.</p> "> Figure 10
<p>Influence of the head loss on the breakthrough curve.</p> ">
Versions Notes

Abstract

:
Composite geomembrane cut-off walls (CGCW) have been widely used for the remediation of polluted sites, especially where the environmental conditions are complex. Accurate predictions of the GMB hole leakage and CGCW performance are essential for engineering design and cost control. This paper establishes empirical equations to predict the leakages through the CGCWs based on the numerical models. Additionally, an analytical solution for contaminant migration through the CGCW is proposed considering the effects of GMB holes. The accuracy of the established equations and analytical solution is verified by the numerical models. The key effects of the GMB thickness (TG), head loss (HG), cut-off wall hydraulic conductivity (kG), hole radius (rG) and shape on the leakage and CGCW performance are investigated. The results show that compared with other hole shapes, the leakage through the circular hole is lowest. This is mainly because the shape factor for the circular hole is 1.15–1.3 times lower than that for other shapes of holes with the same area. Additionally, the effects of the hole geometric properties and head loss on the CGCW performance can be more significant when the cut-off wall hydraulic coefficient is small. For example, the breakthrough time differences between the cases with rG = 0.005 m and 0.05 m are 0.8 and 5.0 years when kG = 10−10 and 10−9 m/s, respectively. This is because the impermeability of the CGCW is good when kG is small. This will weaken the impacts of the hole geometric properties on the leakage. The proposed empirical equations and analytical solution can provide effective suggestions for the design of the CGCW in different GMB hole cases.

1. Introduction

Groundwater pollution caused by industrial sewage, pesticides and waste has become a worldwide concern [1,2]. The cut-off wall made of soil-bentonite (SB), cemented bentonite (CB), or soil-bentonite-cement (SBC) is widely to prevent groundwater from being polluted by contaminants [3,4]. However, the performance of SB, CB, or SBC can be easily influenced by extreme or complex conditions such as wet-dry cycles and freeze-thaw cycles [5]. This will lead to an increase in the hydraulic conductivity of the cut-off wall [6,7]. It has been reported that the composite geomembrane cut-off walls (CGCWs) with low permeability and high chemical resistance geomembranes (GMBs) embedded can effectively solve this problem [8,9]. This type of barrier has been widely used for the remediation of polluted sites, especially where the environmental conditions are complex [10].
Accurate predictions of the CGCW performance are essential for engineering design and cost control [11]. Experiments, numerical simulations and analytical solutions are the primary tools to analyze the contaminant migration process and predict the service times of the CGCWs [12,13]. For example, Wang et al. [14] compared the transport behaviors of different types of contaminants in the CGCWs. Xu et al. [13] conducted the permeability test to evaluate the hydraulic performance of the CGCWs based on the flexible wall permeameter. However, though experiments can permit direct observation, incorrect operations and low-precision apparatuses will lead to errors in experimental results [15]. Additionally, the modeling and computational steps of numerical simulations are always time-costly and software-dependent [16,17]. Unlike the above methods, analytical solutions have been frequently used in recent years due to their ease of calculation [18,19]. For example, Peng et al. [20] considered the effects of diffusion, adsorption and degradation of organic contaminants on the performance of the CGCW and established a 2D analytical solution.
GMB defects are the main cause of GMB failure and will affect the performance of the CGCW. It is found that a GMB hole (even if it is smaller than 10 mm) will lead to a significant leakage [21]. The increase in the hole diameter can also cause a large leakage increase [22]. There are several empirical equations for the hole leakage calculation [23]. For example, Rowe et al. [24] established an empirical equation to predict leakages of different wrinkle heights and widths and tested it with landfills observed data in North America. Rowe and Fan [25] proposed new empirical equations to evaluate GMB hole leakage covered by saturated tailings based on experiments. However, all mentioned analytical studies on the performance of the CGCW didn’t take into consideration the impact of GMB holes on contaminant transport. This will overestimate the service time and hydraulic performance of the CGCW [26]. Additionally, the field investigations and experiments show that the shapes of GMB holes are not always circular and water flow rates vary with GMB hole shapes [27,28]. For example, Sharma and Fang [29] found that the leakage of a triangular hole is 5% higher than that of a circular hole with the same area. There is no simple method to calculate the leakage of different hole shapes in the CGCW and aquifer system, currently. Thus, it is necessary to propose a new simple method to evaluate the performance of the CGCW under different hole shapes and sizes.
This paper aims to obtain empirical equations to predict leakages through the CGCWs using different hole shapes based on the numerical model. Additionally, an analytical solution is developed to evaluate the transient convection and diffusion of contaminants in the CGCW and aquifer system with adsorption. The key effects of the hole radius, hole shape, cut-off wall hydraulic coefficient and head loss on the performance of the CGCW are illustrated.

2. Materials and Methods

2.1. Numerical Model

A 3D numerical model consisting of a CGCW and aquifer system is established to evaluate the effects of GMB holes based on COMSOL Multiphysics 6.0. A 0.0015 m HDPE GMB is embedded into the SB cut-off wall (SBCW). The upstream SBCW, downstream SBCW and the aquifer thicknesses are assumed to be equal to 0.6 m, 0.6 m and 10 m, respectively. The cross-section of the model is a 1 m2 square and the GMB hole is located in the center of the cross-section. The Transport of Diluted Species in Porous Media and Darcy’s Law Interfaces of COMSOL are used to simulate leachate migration in the CGCW and aquifer system. The influences of GMB thickness (TG), cut-off wall hydraulic conductivity (kG), hole radius (rG) and head loss (HG) on leakages through the hole are investigated.
To obtain a simple method for predicting leakages for different shapes of holes, the Parametric Sweep option in COMSOL is used. Additionally, the empirical equation proposed in the study of Fan and Rowe [22] is used as a basis equation. In other words, the empirical equations for different shapes of holes all follow the form:
Q = a k G H G T G b r G c ln ⁡ T + d
where, a, b, c and d are all empirical coefficients and Q is the leakage.

2.2. Mathematical Model

A CGCW consisting of an upstream SBCW, an HDPE GMB and a downstream SBCW is installed to prevent the aquifer from being polluted by contaminants, as shown in Figure 1. A one-dimensional model considering the effect of advection, hydrodynamic dispersion and adsorption is established to evaluate the performance of the CGCW. The thicknesses of the upstream SBCW, HDPE GMB, downstream SBCW and aquifer are denoted as L1, L2, L3 and L4, respectively. The following assumptions are made to develop the mathematical model:
(1)
The SBCWs, GMB and aquifer are saturated, homogeneous and isotropic.
(2)
The groundwater flows in a horizontal direction at a constant velocity.
(3)
The thickness of the aquifer is large enough to ignore the effect of contaminant outflow.
(4)
The adsorption of contaminants only occurs in SBCWs.
The governing equations for contaminant transport in the CGCW and aquifer system can be given as follows [30]:
R d i ∂ C i ∂ t = D i ∂ 2 C i ∂ x 2 − v i ∂ C i ∂ x ( i = 1 ,   2 ,   3 ,   4 )
where, i = 1, 2, 3, 4 refer to the upstream SBCW, HDPE GMB, downstream SBCW and aquifer; Ci is the contaminant concentration; x is the contaminant migration direction; Di is the diffusion coefficient; Rdi is the retardation factor; vi is the average linear velocity of contaminants.
It should be noted that Rd2 and Rd4 are equal to 1 [31]. The retardation factors, diffusion coefficients and average linear velocities in the upstream SBCW and downstream SBCW are equal.
It is assumed that there is no contaminant in the system initially [18]:
C i x , t = 0 = 0
The constant contaminant concentration inlet boundary is assumed at x = 0 [19]:
C 1 0 , t = C 0
The outflow boundary is a semi-infinite boundary and can be expressed as [31]:
∂ C 4 L , t ∂ x = 0
The concentration and mass flux continuity conditions at the upstream SBCW and the GMB interface, the GMB and the downstream SBCW interface, the downstream SBCW and the aquifer interface can be expressed as [12,13]:
C 1 x , t x = L 1 = C 2 x , t S g f x = L 1
D 1 ∂ C 1 x , t ∂ x x = L 1 = S g f D 2 ∂ C 2 x , t ∂ x x = L 1
C 2 x , t S g f x = L 1 + L 2 = C 3 x , t x = L 1 + L 2
S g f D 2 ∂ C 2 x , t ∂ z x = L 1 + L 2 = D 3 ∂ C 3 x , t ∂ z x = L 1 + L 2
C 3 x , t x = L 1 + L 2 + L 3 = C 4 x , t x = L 1 + L 2 + L 3
D 3 ∂ C 3 x , t ∂ x x = L 1 + L 2 + L 3 = D 4 ∂ C 4 x , t ∂ x x = L 1 + L 2 + L 3
where, Sgf is the partition coefficient between the SBCW and the GMB.
The following parameters are introduced to simplify equations:
X i = x i L T = D 1 t R 1 L 2 P e i = v i L D i ζ i = D 1 R i R 1 D i f i = D z i L
Equations (2)–(6) can be transformed into the following equations:
ζ i ∂ C i X , T ∂ T = ∂ 2 C i X , T ∂ X 2 − P e i ∂ C i X , T ∂ T
C X , T = 0 = 0
C 0 , T = C 0
∂ C X = 1 , T ∂ X = 0
C 1 X = L 1 L , T = C 2 X = L 1 L , T S g f
f 1 ∂ C 1 X = L 1 L , T ∂ X = S g f f 2 ∂ C 2 X , T ∂ X
C 2 X , T S g f = C 3 X , T
S g f f 2 ∂ C 2 X , T ∂ X = f 3 ∂ C 3 X , T ∂ X
C 3 X , T = C 4 X , T
f 3 ∂ C 3 X , T ∂ X = f 4 ∂ C 4 X , T ∂ X
where, Sgf is the partition coefficient between the SBCW and the GMB.
The numerical Laplace inversion proposed by Abate et al. [32] is used to obtain the analytical solution:
f t , M = 2 5 t ∑ k = 0 M − 1 Re ⁡ γ k f δ k t ¯
and
γ 0 = 1 2 e δ 0
γ k = 1 + i k π M 1 + cot 2 ⁡ k π M − i cot ⁡ k π M e δ k , 0 < k < M
δ 0 = 2 M 5
δ k = 2 k π 5 cot ⁡ k π M + i , 0 < k < M
It should be noted that i is the imaginary number and M is assumed to be 64.

2.3. Parameter Analysis

Toluene (TOL) is chosen as the typical contaminant to investigate the effects of the hole radius, hole shape, cut-off wall hydraulic coefficient and head loss on the performance of the CGCW. According to the study by Giao [33], the concentrations of TOL in the polluted sites can be 10 mg/L. Additionally, the maximum allowable concentration for TOL behind the CGCW is assumed to be 1 mg/L. Thus, the relative breakthrough concentration for TOL is regarded as 0.1. All other parameters used are shown in Table 1. It should be noted that the GMB is embedded in the center of the cut-off wall. The shape of the GMB hole is assumed to be circular in Section 4.2.1, Section 4.2.2 and Section 4.2.3.

3. Verification of the Analytical Solution

3.1. Compared with the Numerical Model

The 1D numerical model is established to verify the proposed analytical solution in this section. Darcy’s Law and Transport of Diluted Species in Porous Media modulus are used to simulate the contaminant transport through CGCW and aquifer system. The thicknesses of the SBCW, GMB, and aquifer are assumed to be 0.6 m, 0.0015 m and 10 m [12,20]. The extra fine physics-controlled meshing sequences are invoked and 166 computational units are generated. Methyl tertiary butyl ether (MTBE) is considered as the contaminant in the source [33]. The parameters used in the calculation are all listed in Table 1. It should be noted that the leakage through the GMB is calculated to be 2.7 × 10−10 m/s in this section. It can be concluded from Figure 2 that the results obtained from the analytical solution are in agreement with those from the numerical simulations. The differences between the results predicted by the proposed analytical solution and the numerical model for t = 25 and 50 years are only 0.1% and 0.12%, respectively. This indicates that the analytical solution can accurately predict contaminants transport through the CGCW and aquifer system. Additionally, the CGCW can effectively block the water flow and decrease the contaminant concentration in the groundwater. For example, the contaminant concentrations at x = 0.6 and 0.6015 are 0.73 and 0.31 when t = 25 years, respectively. The main reason for this phenomenon is that the diffusion coefficients (7.7 × 10−13 m2/s) and partition coefficients (0.6) in GMB for MTBE are small. GMB can effectively impede the transport of MTBE. This will lead to a huge contaminant concentration mutation between two sides of GMB [30].

3.2. Compared with the Existing Analytical Model

A pure diffusion analytical model proposed by Peng et al. [30] is selected to verify the proposed solution. The values of parameters used for comparison are the same as those in Section 3.1. The breakthrough curves of the CGCW under different leakages are shown in Figure 3. It can be concluded that the difference between the results predicted by the pure diffusion model and the proposed model is only 1.01% when there is no leakage (v2 = 0). This demonstrates the accuracy of the proposed solution. Additionally, it can be found that the leakage can pose significantly effects on the CGCW performance. For example, the contaminant concentrations for v2 = 10−9 m/s can be 7.43 and 1.79 times larger than that for v2 = 10−10 m/s and v2 = 5 × 10−10 m/s when t = 50 years, respectively. It indicates that the effects of leakage caused by the GMB holes should be taken into consideration. Additionally, the proposed solution can offer more accurate suggestions for the design of the CGCW.

4. Results and Discussion

4.1. Leakage Prediction

4.1.1. Influence of GMB Thickness, Head Loss, and Cut-Off Wall Hydraulic Conductivity on Leakage

Increasing the GMB thickness and decreasing the head loss and cut-off wall hydraulic conductivity can decrease the leakage (Figure 4). For example, the leakage for TG = 0.002 m is 1.52 × 10−10 m3/s while it for TG = 0.001 m is 1.85 × 10−10 m3/s when kG = 1 × 10−9 m/s, rG = 0.005 m, and HG = 10 m. The values of the leakage and head loss are linearly related. For example, increasing the HG from 1 m to 10 m leads to a 10-time increase in the leakage. Experiments conducted by Rowe and Fan [25] have also proved this conclusion. Additionally, it can be concluded that changing the TG has an insignificant influence on the leakage when the kG is small. For example, the leakage for TG = 0.002 m is 1.22 times smaller than that for TG = 0.001 m when kG = 1 × 10−9 m/s, rG = 0.005 m. However, the difference decreases to 1.02 times when kG = 1 × 10−11 m/s. This is mainly because the SBCWs are the dominant parts to prevent leakage, though increasing the GMB thickness can lead to a decrease in the hydraulic gradient [25].

4.1.2. Influence of the Hole Radius on Leakage

Increasing the hole radius leads to an increase in the leakage (Figure 5). Additionally, it can be concluded based on the results that there is no linear relationship between the leakage and GMB radius. For example, the leakage for rG = 0.05 m is 5.8 and 3.1 times larger than that for rG = 0.005 and 0.025 m, respectively. However, according to the study of Fan and Rowe [25], the increase in the leakage is slightly larger than linearly with the hole radius. The main reason for this difference is that Fan and Rowe [25] considered the influence of the GMB hole when covered by tailings rather than embedded in the cut-off wall. The cut-off wall has low permeability and can prevent leakage. Thus, the effects of the geometric properties of the hole on the leakage will be weakened. Moreover, an interesting conclusion is found that the change of the leakage caused by the change of the cut-off wall hydraulic coefficient is related to the hole radius. For example, the leakages for rG = 0.005, 0.025 and 0.05 m when kG = 1 × 10−11 m/s are 2.4, 12.3 and 21.1 times lower than those when kG = 1 × 10−9 m/s. The increases in leakages are approximately linearly with the hole radiuses.

4.1.3. Influence of the Hole Shape on Leakage

The effects of different hole shapes such as circular, triangular, rhombic, square, and rectangular on the leakage are investigated. The dimensional drawing of the designed GMB holes are shown in Figure 6. To avoid the influence of the different hole areas on the leakage, hole areas of all shapes are assumed to be 1960 mm2. The cut-off wall hydraulic coefficient, head loss and GMB thickness are assumed to be 1 × 10−9 m/s, 1 m and 0.0015 m, respectively [31]. It can be calculated that the leakage in the circular hole is 5.3 × 10−11 m3/s while leakages for other hole shapes are in the range of 1.28 × 10−10 m3/s–1.32 × 10−10 m3/s. This is mainly because the shape factor for the circular hole is 1.15–1.3 times lower than that for other shapes of holes with the same area [25]. It leads to the appearance of a lower leakage in the circular hole. Similar conclusions have also reported by Xia et al. [37]. They found that compared with other hole shapes, the stress distribution around the circular hole is more uniform. This can help minimize tearing or failure of the GMB caused by the stress concentration at any single point.

4.1.4. Empirical Equations for the Prediction of the Leakages

2500 conditions with rG in the range of 0.005–0.05 m, TG in the range of 0.001–0.0025 m, kG in the range of 1 × 10−9 m/s–1 × 10−11 m/s and HG in the range of 0.6 m–10 m are analyzed. Empirical equations to predict leakages for different shapes of holes are all shown in Table 2. It should be noted that in the empirical equations, the parameter rG represents the radius of circular holes with the same area for different hole shapes. It can be concluded that R2 for all equations are larger than 0.91. It indicates that the proposed empirical equations are suitable for evaluating the leakage in a CGCW and aquifer system.

4.2. CGCW Performance Analysis

4.2.1. Influence of the Hole Radius

The influence of the hole radius (rG) on the breakthrough curve of the CGCW is shown in Figure 7. The HG is assumed to be 3.0 m. It can be concluded that increasing rG can lead to a decrease in the breakthrough time. However, this effect can be insignificant when kG is small (e.g., <10−10 m/s). For example, the breakthrough times for rG = 0.005 m and 0.05 m are 47.3 and 48.1 years when kG = 10−10 m/s, respectively. Additionally, the breakthrough time for rG = 0.005 m is 5 years longer than that for rG = 0.05 m when kG = 10−9 m/s. The main reason is that the impermeability of the CGCW is good when kG is small (as mentioned in Section 3.2). This will weaken the impacts of the hole geometric properties on the leakage. Moreover, it can also be observed from the empirical equations in Table 2 that leakage is directly proportional to both kG and rG0.192, respectively. The influence of kG is more significant than that of rG. In other words, reducing kG can minimize the negative influence of GMB holes on the service time of the CGCWs. Therefore, it is essential to use the cut-off wall materials with lower hydraulic conductivities in the actual projects.

4.2.2. Influence of the Hole Shape

The influence of the hole shape on the breakthrough curve of the CGCW is shown in Figure 8. Two different hole shapes including circular and rectangular holes are analyzed. The HG, rG and TG are assumed to be 3 m, 0.05 m and 0.0015 m, respectively. Compared with other hole shapes, the performance of the CGCW is better when the GMB holes are circular. For example, the breakthrough time for a circular hole is 1.1 times larger than that for a rectangular hole when kG = 10−9 m/s. It is mainly because the leakage through a circular hole is lower than that through other shapes of holes as the shape factor for the circular hole is lowest. Additionally, it can be found that the influence of the hole radius and shape are similar. The same trend that the parameter impacts will be weakened when the impermeability of the CGCW is good. For example, the breakthrough time for the circular hole is only 0.9 years longer than that for the rectangular hole when kG = 10−10 m/s.

4.2.3. Influence of the Cut-Off Wall Hydraulic Coefficient

The influence of the cut-off wall hydraulic coefficient (kG) on the breakthrough curve of the CGCW is shown in Figure 9. The HG, rG and TG are assumed to be 3.0 m, 0.05 m and 0.0015 m, respectively. Increasing the kG can result in a worse performance of the CGCW. For example, the breakthrough time for kG = 10−10 m/s is 1.3 and 1.5 times larger than that for kG = 5 × 10−10 m/s and 1 × 10−9 m/s, respectively. This is mainly because decreasing kG can significantly decrease the leakage. This has also been mentioned in Section 4.1. Additionally, it can be concluded that kG has little effect on the CGCW performance when it is small (e.g., <10−10 m/s). The breakthrough curves of kG = 10−10 m/s and 10−11 m/s are almost coincident. The breakthrough time for kG = 10−11 m/s is only 1.06 times larger than that for kG = 10−10 m/s. The dispersion can be the dominant mechanism for contaminant transport as the Peclet number in the cut-off wall is lower than 1 (e.g., <0.26) when kG < 10−10 m/s. Thus, there is a threshold for kG in the CGCW (e.g., 10−10 m/s) that can not only satisfy economic and safety needs. In other words, decreasing kG cannot help much to the CGCW performance when it is lower than the threshold. It can be very important to find the threshold in the design of the CGCW.

4.2.4. Influence of the Head Loss

The influence of the head loss (HG) on the breakthrough curve of the CGCW is shown in Figure 10. The rG, TG and are assumed to be 0.05 m and 0.0015 m, respectively. Increasing HG can lead to a decrease in the breakthrough time. For example, the breakthrough time for HG = 0.3 m is 30 years longer than that for HG = 10 m when kG = 10−9 m/s. The main reason is that the contaminant transport in the CGCW can be accelerated with the increase of HG. More contaminants can break through the barrier and migrate to the groundwater. Thus, the methods that can control the head loss (e.g., a pumping well) should be taken to improve the performance of the CGCW. Additionally, it can be concluded that the effect of HG can be weakened when the impermeability of the barrier wall is relatively good (e.g., kG = 10−10 m/s). For example, although HG increases by 33-fold (from 0.3 m to 10 m), the breakthrough time of the CGCW is only 1.2 times larger than before when kG = 10−10 m/s. It can also be demonstrated from the empirical equation for circular holes in Table 2 that even if the head loss is large (e.g., 10 m), the leakage is lower than 1.2 × 10−10 m3/s when kG = 10−10 m/s.

5. Limitation

The proposed analytical model is not without limitation. One of the major limitations is that the heterogeneity of CGCW cannot be taken into consideration since the model is one-dimensional. It has been proved that the compaction and uneven density of the CGCW materials generated during the production and installation can significantly affect the porosity and hydraulic conductivity at different locations in the wall [38]. The actual contaminant concentrations behind the CGCW can be larger or lower than the results predicted by the model. Another limitation is that the parameters of the SBCW are assumed not to change with time. However, the materials of the cut-off wall can be easily affected by contaminants during the service life [39]. This can result in a gradual increase in the hydraulic conductivity. Further numerical simulations are recommended to investigate the effects of the heterogeneity and parameter time-variation on the CGCW performance.
The empirical equations proposed in this manuscript also have limitations. The parametric sweep feature is used to simulate the leakage of the GMB holes under different situations. However, the effects of the parameter uncertainty on the leakage and CGCW performance cannot be taken into consideration. It has been proved that the analysis of parameter uncertainty can offer significant helps in the engineering design since it can help in further identifying critical factors that influence the leakage behavior [31]. Uncertainty quantification feature of COMSOL is expected to be used in the future study to quantify the uncertainty in model outcomes caused by uncertainties in input parameters.

6. Conclusions

This paper proposes the empirical equations and analytical model to evaluate the performance of the CGCW while considering the effects of GMB holes. The accuracy of the established equations and analytical solution is confirmed by the numerical models. The key effects of the GMB thickness, head loss, cut-off wall hydraulic conductivity, hole radius and shape on the leakage and CGCW performance are investigated.
(1)
The proposed solution can offer more accurate suggestions for the design of the CGCW than the pure diffusion model, especially when the leakage is large (e.g., >10−10 m/s). For example, the contaminant concentrations for v2 = 10−9 m/s can be 7.43 and 1.79 times larger than that for v2 = 10−10 m/s and v2 = 5 × 10−10 m/s when t = 50 years, respectively.
(2)
The values of the leakage and head loss are linearly related. For example, increasing the head loss from 1 m to 10 m leads to a 10-time increase in the leakage. Additionally, changing the GMB thickness has an insignificant influence on the leakage when the kG is small. Thus, the pumping method is suggested to be used to control the head loss and improve the CGCW performance in practical engineering.
(3)
Various hole shapes can lead to different performances of the CGCW. Additionally, the performance of the CGCW is better when the GMB hole is circular. For example, the breakthrough time for the circular hole is 1.1 times larger than that for the rectangular hole when kG = 10−9 m/s. This is mainly because the leakage through the circular hole is low as the shape factor for the circular hole is 1.15–1.3 times lower than that for other shapes of holes.
(4)
The influences of the hole radius, shape and head loss on the breakthrough curve of the CGCW seem to be more significant when hydraulic conductivity is large. For example, although the head loss increases by 33-fold (from 0.3 m to 10 m), the breakthrough time of the CGCW is only 1.2 times larger than before when kG = 10−10 m/s. Thus, the hydraulic conductivity of the cut-off wall is suggested to be controlled lower than 10−9 m/s.

Author Contributions

L.R.: Conceptualization, Methodology, Validation, Visualization, Writing—original draft, Writing—review & editing. G.W.: Validation, Investigation, Writing—review & editing. H.D.: Conceptualization, Project administration, Resources, Supervision, Writing—original draft, Writing—review & editing. H.X.: Funding acquisition, Investigation, Writing—review & editing. All authors have read and agreed to the published version of the manuscript.

Funding

The financial supports from the National Key R&D research and development Program of China (Grant No. 2023YFC3709602), National Natural Science Foundation of China (Grant Nos. 52278375, 41931289, and 41977223), and Zhejiang Provincial Natural Science Foundation (Gran No. LR20E080002) are greatly acknowledged.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

Data are available from the authors upon reasonable request.

Conflicts of Interest

Authors Guijun Wan, Haijian Xie and Hao Ding were employed by the company The Architectural Design and Research Institute of Zhejiang University Co., Ltd.; the remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Figure 1. Mathematical model of contaminant transport in the CGCW and aquifer system.
Figure 1. Mathematical model of contaminant transport in the CGCW and aquifer system.
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Figure 2. Comparison with the results predicted by COMSOL.
Figure 2. Comparison with the results predicted by COMSOL.
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Figure 3. Comparison with the existing analytical model by Peng et al. (2021) [30].
Figure 3. Comparison with the existing analytical model by Peng et al. (2021) [30].
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Figure 4. Leakages for different hydraulic conductivities and head losses when (a) TG = 0.001 m and (b) TG = 0.002 m.
Figure 4. Leakages for different hydraulic conductivities and head losses when (a) TG = 0.001 m and (b) TG = 0.002 m.
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Figure 5. Leakages for different hydraulic conductivities and head losses with different GMB thicknesses.
Figure 5. Leakages for different hydraulic conductivities and head losses with different GMB thicknesses.
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Figure 6. Dimensional drawing of different GMB hole shapes.
Figure 6. Dimensional drawing of different GMB hole shapes.
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Figure 7. Influence of the hole radius on the breakthrough curve.
Figure 7. Influence of the hole radius on the breakthrough curve.
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Figure 8. Influence of the hole shape on the breakthrough curve.
Figure 8. Influence of the hole shape on the breakthrough curve.
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Figure 9. Influence of the cut-off wall hydraulic conductivity on the breakthrough curve.
Figure 9. Influence of the cut-off wall hydraulic conductivity on the breakthrough curve.
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Figure 10. Influence of the head loss on the breakthrough curve.
Figure 10. Influence of the head loss on the breakthrough curve.
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Table 1. Main parameters for the simulation.
Table 1. Main parameters for the simulation.
SBCWGMBAquifer
Porosity0.42 [31]-0.47 [31]
Hydraulic conductivity (m/s)1.0 × 10−9 [34]1.0 × 10−14 [34]1.0 × 10−5 [34]
Thickness (m)0.6 [12]0.0015 [20]10 [12]
Cross-sectional area (m2)111
Partition coefficientMTBE-0.6 [35]-
TOL-100 [35]-
Retardation factorMTBE1.5 [20]-1 [20]
TOL2.5 [20]-1 [20]
Diffusion coefficient (m2/s)MTBE3.5 × 10−10 [21]7.7 × 10−13 [21]3.3 × 10−10 [21]
TOL3.8 × 10−10 [36]3.8 × 10−13 [36]4.1 × 10−10 [36]
Table 2. Empirical equations to predict leakages for different shapes of holes.
Table 2. Empirical equations to predict leakages for different shapes of holes.
Hole ShapeEmpirical EquationR2
Circular Q = 0.2769 k G H G T G 0.0462 r G 0.0971 ln ⁡ T G + 0.8235 0.99
Triangular Q = 0.9572 k G H G T G 0.4854 r G 0.08 ln ⁡ T G + 0.1419 0.91
Rhombic Q = 0.1966 k G H G T G 0.2511 r G 0.616 ln ⁡ T G + 0.4733 0.93
Square Q = 0.3804 k G H G T G 0.5678 r G 0.0759 ln ⁡ T G + 0.054 0.91
Rectangular Q = 0.7094 k G H G T G 0.7547 r G 0.276 ln ⁡ T G + 0.6797 0.92
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Ran, L.; Wan, G.; Ding, H.; Xie, H. Analytical Model for Contaminant Transport in the CGCW and Aquifer Dual-Domain System Considering GMB Holes. Appl. Sci. 2024, 14, 10280. https://doi.org/10.3390/app142210280

AMA Style

Ran L, Wan G, Ding H, Xie H. Analytical Model for Contaminant Transport in the CGCW and Aquifer Dual-Domain System Considering GMB Holes. Applied Sciences. 2024; 14(22):10280. https://doi.org/10.3390/app142210280

Chicago/Turabian Style

Ran, Long, Guijun Wan, Hao Ding, and Haijian Xie. 2024. "Analytical Model for Contaminant Transport in the CGCW and Aquifer Dual-Domain System Considering GMB Holes" Applied Sciences 14, no. 22: 10280. https://doi.org/10.3390/app142210280

APA Style

Ran, L., Wan, G., Ding, H., & Xie, H. (2024). Analytical Model for Contaminant Transport in the CGCW and Aquifer Dual-Domain System Considering GMB Holes. Applied Sciences, 14(22), 10280. https://doi.org/10.3390/app142210280

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