# Analytical Solution of the One-Dimensional Transport of Ionic Contaminants in Porous Media with Time-Varying Velocity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Assumptions and Calculation Model

#### 2.1. Basic Assumptions

#### 2.2. Governing Equations and Solution Conditions

#### 2.3. Solution of the Model

## 3. Verification

## 4. Parameter Impact Analysis

- (1)
- Influence coefficient $a$ varies, maximum velocity ${v}_{e}$ varies, and the time to maximum velocity ${t}_{e}$ is constant.
- (2)
- Influence coefficient $a$ varies, maximum velocity ${v}_{e}$ is constant, and the time to maximum velocity ${t}_{e}$ varies.
- (3)
- Influence coefficient $a$ is constant, maximum velocity ${v}_{e}$ varies, and the time to maximum velocity ${t}_{e}$ varies.

#### 4.1. Analysis of Parameter Influence in Group 1

#### 4.2. Analysis of Parameter Influence in Group 2

#### 4.3. Analysis of Parameter Influence in Group 3

#### 4.4. Analysis of Pollutant Concentration Differences

## 5. Example Analysis

#### 5.1. Analysis of a Soil–Attapulgite Barrier

^{2+}was studied. With the confining pressure of 100 kpa, the analysis was carried out in three cases according to different mixing proportions of attapulgite (A) and sand (S). At a temperature of 30–60 °C, the permeability coefficient ${k}_{s}$ of the wall was basically in the range of $0~4\times {10}^{-9}\mathrm{m}/\mathrm{s}$, the liquid limit of attapulgite was ${W}_{1}=11.41\%$, the plastic limit was ${W}_{p}=5.6\%$, the dry density was $\rho =808{\mathrm{kg}/\mathrm{m}}^{3}$, the porosity was $n=0.2~0.64$, the distribution coefficient was ${k}_{d}=43\mathrm{mL}/\mathrm{g}$, the barrier thickness was $L=1\mathrm{m}$, and $t=280\mathrm{d}$ for the temperature of the landfill to rise from 30 °C to 60 °C under the action of biodegradation, that is, it took 280 d to reach the maximum velocity. From $u(t)={u}_{0}V(at)$, we obtained $a$.

^{2+}for the barrier.

^{2+}ions in the lower layer gradually decreased with the increase in the doping amount. By comparing the variable flow rate with the constant flow rate, it can be seen that the increase in the permeability coefficient of the barrier caused by the increase in the landfill temperature weakened the barrier capacity.

^{2+}in the lower layer of the barrier became smaller; the concentration at the bottom in the 10% case was the highest, and was 0.27 different from that in the 60% case. This showed that with the increase in attapulgite content, the adsorption capacity of the lower layer of the partition barrier became larger, but the main factor affecting the decrease in its concentration was not the change in the retardation factor because the retardation factor R and permeability coefficient became smaller with the increase in attapulgite content; $R=81$ at 10%, $R=61$ at 30%, and $R=49$ at 60%. At the same time, the flow rate was also gradually reduced, so it can be seen that in this case of contaminant migration, the concentration change was dominated by the flow rate, so it was not necessary for the adsorption capacity to be strong when the retardation factor was large, and more attention should be paid to the permeability of the barrier. In engineering practice, measures such as increasing the content of attapulgite and controlling the temperatures of landfills can enhance barriers’ effects.

#### 5.2. Analysis of a Sand–Bentonite Mixture (SBM) Barrier

_{3})

_{2}-Zn(NO

_{3})

_{2}with initial metal concentrations ${C}_{0}$ of 50, 100, and 500 mmol/L, a retardation factor of $R=10$, and $z=1\mathrm{m}$ were selected, and the measured permeability coefficients were $4.1\times {10}^{-10}\mathrm{m}/\mathrm{s}$, $3.5\times {10}^{-9}\mathrm{m}/\mathrm{s}$, and $5\times {10}^{-9}\mathrm{m}/\mathrm{s}$, which increased to $3.8\times {10}^{-9}\mathrm{m}/\mathrm{s}$, $5.5\times {10}^{-9}\mathrm{m}/\mathrm{s}$, and $7.4\times {10}^{-9}\mathrm{m}/\mathrm{s}$ after one year, with influence coefficients of ${a}_{1}=2.67\times {10}^{-7}$, ${a}_{2}=1.81\times {10}^{-8}$, and ${a}_{3}=1.52\times {10}^{-8}$, respectively. The breakthrough criterion was a contaminant concentration at the lower boundary of 10% of that of the upper boundary.

## 6. Conclusions

^{2+}ions in a soil–attapulgite barrier was analyzed as the temperature of a landfill increased from 30 °C to 60 °C. With the increase in the temperature, the permeability coefficient of the barrier increased. The permeability coefficient and retardation factor decreased with the increase in the content of attapulgite, and the flow rate decreased, so the barrier’s effect on the pollutants improved. A breakthrough concentration of 10% was defined. In the case of increasing permeability coefficients with time, the breakthrough time was correspondingly advanced by 9.8% and 62.01%. In engineering practice, measures such as increasing the content of attapulgite and controlling the temperature of a landfill can enhance a barrier’s effect.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, Y.; Xiang, Y.; Chen, W. Heavy metal content in the bark of camphora tree in Xiangtan and its environmental significance. Appl. Ecol. Environ. Res.
**2019**, 17, 9827–9835. [Google Scholar] [CrossRef] - Cao, J.; Zhang, J.; Zhang, W.; Liu, M.; Shi, Z. Research progress of remediation technology for heavy metal chromium(VI) contamination in soils. Soil Bull.
**2022**, 53, 1220–1227. [Google Scholar] - Zeng, X.; Su, J.; Wang, H.; Gao, T. Centrifuge Modeling of Chloride Ions Completely Breakthrough Kaolin Clay Liner. Sustainability
**2022**, 14, 6976. [Google Scholar] [CrossRef] - Du, Y.-J.; Jin, F.; Liu, S.-Y.; Chen, L.; Zhang, F. Review of stabilization/solidification technique for remediation of heavy metals contaminated lands. Yantu Lixue = Rock Soil Mech.
**2011**, 32, 116–124. [Google Scholar] - Liu, S. Geotechnical investigation and remediation for industrial contaminated sites. Chin. J. Geotech. Eng.
**2018**, 40, 1–37. [Google Scholar] - Liu, Y.; Bouazza, A.; Gates, W.; Rowe, R. Hydraulic performance of geosynthetic clay liners to sulfuric acid solutions. Geotext. Geomembr.
**2015**, 43, 14–23. [Google Scholar] [CrossRef] - Zhang, Y.; Su, J.; Jiang, W.; Huang, Z.; Xiang, Y.; Zeng, F. Study on the Heavy Metal Pollution Evaluation and Countermeasures of Middle Size and Small Cities in Typical Drainage Area-Taking Xiangtan Reach of Xiangjiang River as an Example. Res. J. Chem. Environ.
**2012**, 16, 172–179. [Google Scholar] - Zhang, Y.; Xiang, Y.; Yu, G.; Yuan, K.; Wang, X.; Mo, H. Classification of environmental disaster in Hunan Province. Disaster. Adv.
**2012**, 5, 1756–1759. [Google Scholar] - Zhang, Y.; Huang, F. Indicative significance of the magnetic susceptibility of substrate sludge to heavy metal pollution of urban lakes. ScienceAsia
**2021**, 47, 374. [Google Scholar] [CrossRef] - Fetter, C.W.; Boving, T.; Kreamer, D. Contaminant Hydrogeology; Waveland Press: Long Grove, IL, USA, 2017. [Google Scholar]
- Zeng, X.; Li, Y.; Liu, X.; Yao, J.; Lin, Z. Relationship between the Shear Strength and the Depth of Cone Penetration in Fall Cone Tests. Adv. Civ. Eng.
**2020**, 2020, 8850430. [Google Scholar] [CrossRef] - Zeng, X.; Liu, X.; Li, Y.-H. The breakthrough time analyses of lead ions in CCL considering different adsorption isotherms. Adv. Civ. Eng.
**2020**, 2020, 8861866. [Google Scholar] [CrossRef] - Li, Y.; Zeng, X.; Lin, Z.; Su, J.; Gao, T.; Deng, R.; Liu, X. Experimental study on phosphate rock modified soil-bentonite as a cut-off wall material. Water Supply
**2022**, 22, 1676–1690. [Google Scholar] [CrossRef] - Gao, G.; Feng, S.; Ma, Y.; Zhan, H.; Huang, G. Kinetic model and semi-analytical solution for reactive solute transport considering dispersive scale effects and immobile water bodies. Hydrodyn. Res. Prog. A Ser.
**2010**, 25, 206–216. [Google Scholar] - Xie, H.; Tang, X.; Chen, y. One-dimensional model for contaminant diffusion through layered media. J.-Zhejiang Univ. Eng. Sci.
**2006**, 40, 2191. [Google Scholar] - Selim, H. Transport of Reactive Solutes during Transient, Unsaturated Water Flow in Multilayered SOILS1. Soil Sci.
**1978**, 126, 127–135. [Google Scholar] [CrossRef] - Foose, G.J. Transit-time design for diffusion through composite liners. J. Geotech. Geoenviron. Eng.
**2002**, 128, 590–601. [Google Scholar] [CrossRef] - Chen, Y.-M.; Xie, H.-J.; Ke, H.; Tang, X.-W. Analytical solution of one-dimensional diffusion of volatileorganic compounds (VOCs) through composite liners. Yantu Gongcheng Xuebao
**2006**, 28, 1076–1080. [Google Scholar] - Rowe, R.; Booker, J.R. The analysis of pollutant migration in a non-homogeneous soil. Geotechnique
**1984**, 34, 601–612. [Google Scholar] [CrossRef] - Yu, C.; Wang, H.; Fang, D.; Ma, J.; Cai, X.; Yu, X. Semi-analytical solution to one-dimensional advective-dispersive-reactive transport equation using homotopy analysis method. J. Hydrol.
**2018**, 565, 422–428. [Google Scholar] [CrossRef] - Zeng, X.; Wang, H.; Yao, J.; Li, Y. Analysis of Factors for Compacted Clay Liner Performance Considering Isothermal Adsorption. Appl. Sci.
**2021**, 11, 9735. [Google Scholar] [CrossRef] - Zhan, L.-T.; Zeng, X.; Li, Y.; Chen, Y. Analytical solution for one-dimensional diffusion of organic pollutants in a geomembrane–bentonite composite barrier and parametric analyses. J. Environ. Eng.
**2014**, 140, 57–68. [Google Scholar] [CrossRef] - Liu, X.; Chen, Y.; Zhang, W.; Chi, Y.; Guo, Z.; Xiao, J.; Hu, J. Effects of pH, ionic strength, time and temperature on the sorption of Cd (II) to illite. J. Nucl. Radiochem.
**2012**, 34, 358–363. [Google Scholar] - Zhang, J. Thermodynamic and Mechanistic Study on the Adsorption of Concave Barite Clay for the Treatment of Lead Containing Wastewater and Highly Fluorinated Water. Master’s Thesis, Peking University, Beijing, China, 2008. [Google Scholar]
- Rao, S.N.; Mathew, P.K. Effects of exchangeable cations on hydraulic conductivity of a marine clay. Clays Clay Miner.
**1995**, 43, 433–437. [Google Scholar] [CrossRef] - Zhu, W.; Xu, H.-Q.; Wang, S.-W.; Fan, X.-H. Influence of CaCl
_{2}solution on the permeability of different clay-based cutoff walls. Rock Soil Mech.**2016**, 37, 1224–1230. [Google Scholar] - Malusis, M.A.; McKeehan, M.D. Chemical compatibility of model soil-bentonite backfill containing multiswellable bentonite. J. Geotech. Geoenviron. Eng.
**2013**, 139, 189–198. [Google Scholar] [CrossRef] [Green Version] - Bohnhoff, G.L.; Shackelford, C.D. Hydraulic conductivity of polymerized bentonite-amended backfills. J. Geotech. Geoenviron. Eng.
**2014**, 140, 04013028. [Google Scholar] [CrossRef] - Xu, H.; Shu, S.; Wang, S.; Zhou, A.; Jiang, P.; Zhu, W.; Fan, X.; Chen, L. Studies on the chemical compatibility of soil-bentonite cut-off walls for landfills. J. Environ. Manag.
**2019**, 237, 155–162. [Google Scholar] [CrossRef] - Pickens, J.F.; Grisak, G.E. Modeling of scale-dependent dispersion in hydrogeologic systems. Water Resour. Res.
**1981**, 17, 1701–1711. [Google Scholar] [CrossRef] - Singh, M.K.; Mahato, N.K.; Kumar, N. Pollutant’s horizontal dispersion along and against sinusoidally varying velocity from a pulse type point source. Acta. Geophys.
**2015**, 63, 214–231. [Google Scholar] [CrossRef] [Green Version] - Barry, D.; Sposito, G. Analytical solution of a convection-dispersion model with time-dependent transport coefficients. Water Resour. Res.
**1989**, 25, 2407–2416. [Google Scholar] [CrossRef] - Zamani, K.; Bombardelli, F.A. Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers. Environ. Fluid Mech.
**2014**, 14, 711–742. [Google Scholar] [CrossRef] - Guerrero, J.P.; Pontedeiro, E.; van Genuchten, M.T.; Skaggs, T. Analytical solutions of the one-dimensional advection–dispersion solute transport equation subject to time-dependent boundary conditions. Chem. Eng. J.
**2013**, 221, 487–491. [Google Scholar] [CrossRef] - Basha, H.; El-Habel, F. Analytical solution of the one-dimensional time-dependent transport equation. Water Resour. Res.
**1993**, 29, 3209–3214. [Google Scholar] [CrossRef] - Singh, M.K.; Ahamad, S.; Singh, V.P. Analytical solution for one-dimensional solute dispersion with time-dependent source concentration along uniform groundwater flow in a homogeneous porous formation. J. Eng. Mech.
**2012**, 138, 1045–1056. [Google Scholar] [CrossRef] - Yates, S. An analytical solution for one-dimensional transport in heterogeneous porous media. Water Resour. Res.
**1990**, 26, 2331–2338. [Google Scholar] [CrossRef] - Kumar, A.; Jaiswal, D.K.; Kumar, N. Analytical solutions to one-dimensional advection–diffusion equation with variable coefficients in semi-infinite media. J. Hydrol.
**2010**, 380, 330–337. [Google Scholar] [CrossRef] - Crank, J. The Mathematics of Diffusion; Oxford University Press: Oxford, UK, 1979. [Google Scholar]
- Ogata, A.; Banks, R.B. A Solution of the Differential Equation of Longitudinal Dispersion in Porous Media; US Government Printing Office: Washington, DC, USA, 1961.
- Li, Y.C.; Cleall, P.J. Analytical solutions for advective–dispersive solute transport in double-layered finite porous media. Int. J. Numer. Anal. Methods Geomech.
**2011**, 35, 438–460. [Google Scholar] [CrossRef] - Chen, Z.; Luo, B.; Wang, Z.; Huang, Z. Effect of temperature on the impermeability of landfill impermeable layers. Environ. Eng.
**2015**, 33, 133–136. [Google Scholar] - Dong, J.; Wang, C.-L.; Yin, Y.; Lou, Q.-Z.; Wang, X.-S.; Yang, Z. Influences of the freezing and thawing action on the performance of landfill liners. J. Jilin Univ.
**2011**, 41, 541–544. [Google Scholar] - Chi, Y. Study on the Modification of Albite and Its Adsorption of Heavy Metal Ions. Master’s Thesis, Qinghai Normal University, Xining, China, 2013. [Google Scholar]
- Huang, R. Study on the Adsorption Characteristics and Soil Remediation Efficacy of Modified Bumpy Clay for Heavy Metals. Master’s Thesis, Guangdong University of Technology, Guangzhou, China, 2020. [Google Scholar]
- Lei, H.; Shi, J.Y.; Wu, X. Prediction of landfill temperature considering biodegradation. Henan Sci.
**2018**, 36, 584–592. [Google Scholar] - Guo, T. Experimental Study on the Influence of Temperature on the Permeability of Anti-Seepage Walls of Sand-Attapulgite Soil. Master’s Thesis, Yangzhou University, Yangzhou, China, 2022. [Google Scholar]
- Fan, R.; Du, Y.; Liu, S.; Yang, Y. Experimental study on the chemical compatibility of sand-bentonite vertical barrier materials under the action of inorganic salt solutions. Geotechnics
**2020**, 41, 736–746. [Google Scholar] [CrossRef]

**Figure 1.**Comparison of the outflow curves between the present solution and the analytical solution.

**Figure 14.**Peak concentration relationships. (

**a**) Relationship between peak concentration and influence coefficient. (

**b**) Relationship between peak concentration and maximum flow velocity.

Group | Case | $\mathit{a}$ $\left({\mathbf{s}}^{-1}\right)$ | ${\mathit{v}}_{\mathit{e}}$ $(\mathbf{m}/\mathbf{s})$ | ${\mathit{t}}_{\mathit{e}}$ $\left(\mathbf{d}\right)$ |
---|---|---|---|---|

1 | 1-1 | $1.46\times {10}^{-8}$ | $6.78\times {10}^{-8}$ | 1365 |

1-2 | $2.01\times {10}^{-8}$ | $1.06\times {10}^{-7}$ | 1365 | |

1-3 | $3.23\times {10}^{-8}$ | $1.44\times {10}^{-7}$ | 1365 | |

2 | 2-1 | $4.39\times {10}^{-8}$ | $1.44\times {10}^{-7}$ | 1365 |

2-2 | $2.19\times {10}^{-8}$ | $1.44\times {10}^{-7}$ | 2365 | |

2-3 | $1.46\times {10}^{-8}$ | $1.44\times {10}^{-7}$ | 3365 | |

3 | 3-1 | $1.46\times {10}^{-8}$ | $6.78\times {10}^{-8}$ | 1365 |

3-2 | $1.46\times {10}^{-8}$ | $1.06\times {10}^{-7}$ | 2365 | |

3-3 | $1.46\times {10}^{-8}$ | $1.44\times {10}^{-7}$ | 3365 |

Parameter Value | |
---|---|

$\mathrm{Partition}\mathrm{coefficient}{k}_{p}$ | 7.5 mL/g |

$\mathrm{Height}L$ | 1 m |

$\mathrm{Hydraulic}\mathrm{gradient}i$ | 5 |

$\mathrm{Dispersion}\alpha $ | 0.1 m |

$\mathrm{Porosity}n$ | 0.5 |

$\mathrm{Permeability}\mathrm{coefficient}{k}_{s}$ | $3\times {10}^{-9}\mathrm{m}/\mathrm{s}$ |

${t}_{0}$ | 365/d |

${u}_{0}$ | $3\times {10}^{-8}\mathrm{m}/\mathrm{s}$ |

${D}_{h}$ | $3\times {10}^{-9}{\mathrm{m}}^{2}/\mathrm{s}$ |

$R$ | 20 |

Case | Ratio % | Porosity | Permeability Coefficient m/s | $\mathit{a}$ | $\mathit{R}$ | |
---|---|---|---|---|---|---|

30 °C | 60 °C | |||||

1 | A10S90 | 0.286 | $1.66\times {10}^{-9}$ | $3.32\times {10}^{-9}$ | 0.0036 | 81 |

2 | A30S70 | 0.405 | $1.02\times {10}^{-9}$ | $2.64\times {10}^{-9}$ | 0.0056 | 61 |

3 | A60S40 | 0.5 | $2.22\times {10}^{-10}$ | $6.54\times {10}^{-10}$ | 0.0068 | 49 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zeng, X.; Gao, T.; Xie, L.; He, Z.
Analytical Solution of the One-Dimensional Transport of Ionic Contaminants in Porous Media with Time-Varying Velocity. *Water* **2023**, *15*, 1530.
https://doi.org/10.3390/w15081530

**AMA Style**

Zeng X, Gao T, Xie L, He Z.
Analytical Solution of the One-Dimensional Transport of Ionic Contaminants in Porous Media with Time-Varying Velocity. *Water*. 2023; 15(8):1530.
https://doi.org/10.3390/w15081530

**Chicago/Turabian Style**

Zeng, Xing, Tong Gao, Linhui Xie, and Zijian He.
2023. "Analytical Solution of the One-Dimensional Transport of Ionic Contaminants in Porous Media with Time-Varying Velocity" *Water* 15, no. 8: 1530.
https://doi.org/10.3390/w15081530