# 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

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**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 |

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**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