# Two-Dimensional Model for Consolidation-Induced Solute Transport in an Unsaturated Porous Medium

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

#### 2.1. Two-Dimensional Consolidation Model

#### 2.2. Two-Dimensional Solute Transport Model

#### 2.3. Special Case 1: Two-Dimensional Saturated Porous Model

#### 2.4. Special Case 2: One-Dimensional Unsaturated Porous Model

#### 2.5. Special Case 3: One-Dimensional Saturated Model

## 3. Model Validation

## 4. Results and Discussions

#### 4.1. Homogeneous Soil and Uniform Contaminant Source (2D)

#### 4.2. Point-Source Pollution Study (2D)

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**(

**a**) Excesspore pressure (${p}^{e}$ (kPa)) and (

**b**) vertical displacement (w (m)) at the top point ($x={L}_{l}/2$, $z={H}_{l}$) vs. t and (

**c**) solute concentration (${c}_{f}$ (kg/m${}^{3}$)) at the bottom point ($x={L}_{l}/2$, $z=0$) from the 2D model with various soil domain lengths and the 1D model [9].

**Figure 4.**The comparison of solute concentration distribution in the vertical direction for the present model (solid lines) and that of Zhang and Fang [15] (triangles and squares) with a constant loading. Note: results are present for simulation times of 5 years (red) and 10 years (black).

**Figure 7.**Excess pore pressure (${p}^{e}$ (kPa)) distribution along (

**a**) center vertical cut line where $x=$ 30 m; (

**b**) top horizontal cut line where $z=$ 3 m; and (

**c**) ${p}^{e}$ (kPa) vs. t (years) at top cut point $x=$ 30 m, $z=$ 3 m, middle cut point $x=$ 30 m, $z=$ 1.5 m and bottom cut point $x=$ 30 m, $z=$ 0 m.

**Figure 10.**(

**a**) Vertical displacement (w (m)) distribution along center vertical cut line ($x=$ 30 m); (

**b**) w (m) vs. t (years) at top cut point $x=$ 30 m, $z=$ 3 m, middle cut point $x=$ 30 m, $z=$ 1.5 m and bottom cut point $x=$ 30 m, $z=$ 0 m; (

**c**) horizontal displacement (u (m)) distribution along middle horizontal cut line at $z=$ 1.5 m.

**Figure 11.**Distribution of solute concentration (${c}_{f}$ (kg/m${}^{3}$)) from 1 year to 50 years; note that all sub-figures share the same colour legend.

**Figure 12.**Solute concentration (${c}_{f}$ (kg/m${}^{3}$)) distribution along (

**a**) center vertical cut line where $x=$ 30 m; (

**b**) horizontal cut line where $z=$ 15 m; (

**c**) top horizontal cut line where $z=$ 3 m; (

**d**) middle horizontal cut line where $z=$ 1.5 m; and (

**e**) ${c}_{f}$ (kg/m${}^{3}$) vs. t (years) at top cut point $x=$ 30 m, $z=$ 3 m, middle cut point $x=$ 30 m, $z=$ 1.5 m and bottom cut point $x=$ 30 m, $z=$ 0 m.

**Figure 14.**Contour diagram of solute concentration (${c}_{f}$ (kg/m${}^{3}$)) at Locations A, B and C after (

**a**) 5 years, (

**b**) 20 years and (

**c**) 50 years.

**Figure 15.**Solute concentration (${c}_{f}$ (kg/m${}^{3}$)) distribution along horizontal cut lines (

**a**) $z=$3 m; (

**b**) $z=1.5$ m and (

**c**) $z=0$ m from 1 year to 50 years.

Boundary Index | BC of ${\mathit{p}}^{\mathit{e}}$ | BC of $\mathit{u},\mathit{w}$ | BC of ${\mathit{c}}_{\mathit{f}}$ |
---|---|---|---|

1 | $\frac{\partial {p}^{e}}{\partial z}=0$ | $\left[\begin{array}{c}0\\ {\sigma}_{z}\end{array}\right]=\left[\begin{array}{c}0\\ Q\left(t\right)+{p}^{e}\end{array}\right]$ | $\frac{\partial {c}_{f}}{\partial z}=\frac{{D}_{G}}{{n}^{0}h{D}_{m}}({c}_{f}-{c}_{0})$ |

2 | ${p}^{e}=0$ | $\left[\begin{array}{c}u\\ w\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$ | $\frac{\partial {c}_{f}}{\partial z}=0$ |

3 & 4 | $\frac{\partial {p}^{e}}{\partial x}=0$ | $u=0$ | $\frac{\partial {c}_{f}}{\partial x}=0$ |

IC: ${p}^{e}=0$, $\left[\begin{array}{c}u\\ w\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$ and ${c}_{f}=0$ |

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

$Q\left(t\right)$ | Referring to Figure 2 | Waste loading |

h | 0.0015 m | Thickness of geomembrane |

${}^{*}{H}_{l}$ | 3 m | Depth of soil domain |

${L}_{l}$ | 1 m, 3 m, 5 m, 8 m, 10 m and 20 m | Width of soil domain |

${S}_{r}^{0}$ | 0.88 | Degree of saturation |

${n}^{0}$ | 0.33 | Initial porosity |

G | $2.75\times {10}^{6}$ Pa | Shear modulus |

$\mu $ | 0.33 | Poisson’s ratio |

${}^{*}{K}_{x}$ | $1\times {10}^{-10}$ m/s | Hydraulic conductivity in the x-direction |

${K}_{z}$ | $1\times {10}^{-10}$ m/s | Hydraulic conductivity in the z-direction |

${\rho}_{w}$ | $1\times {10}^{3}$ kg/m${}^{3}$ | Density of the pore fluid, |

varied due to fluid compressibility | ||

${\rho}_{s}$ | $2.6\times {10}^{3}$ kg/m${}^{3}$ | Density of the solid phase |

${K}_{d}$ | 0 | Partitioning coefficient |

${r}_{h}$ | 0.02 m | Volumetric fraction of dissolved air |

within pore water | ||

${D}_{G}$ | $1.5\times {10}^{-4}$ m${}^{2}$/s | Mass transfer coefficient of geomembrane |

${D}_{m}$ | $5\times {10}^{-9}$ m${}^{2}$/s | Molecular diffusion coefficient in the clay |

${\alpha}_{L}$ | 0.1 m | Longitudinal dispersion coefficient |

${}^{*}{\alpha}_{T}$ | 0.1 m | Transverse dispersion coefficient |

${c}_{0}$ | 0.1 kg/m${}^{3}$ | Reference solute concentration |

**Note 1:**The symbol * indicates that the parameter is only applied in the 2D model, while the rest of the parameters remain same in both models.

**Note 2:**Total of six 2D models were computed with various ${H}_{l}$ as listed.

**Table 3.**Parameters for the validation with Zhang and Fang [15].

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

$Q\left(t\right)$ | 100 kPa | Constant waste loading |

${S}_{r}^{0}$ | 1.0 | Initial degree of saturation |

${n}^{0}$ | 0.44 | Initial porosity |

G | $2.6\times {10}^{6}$ Pa | Shear modulus |

$\mu $ | 0.3 | Poisson’s ratio |

${K}_{x}$ | $1.7\times {10}^{-9}$ m/s | Hydraulic conductivity in x-direction |

${K}_{z}$ | $1.7\times {10}^{-9}$ m/s | Hydraulic conductivity in z-direction |

${\rho}_{s}$ | $2.6\times {10}^{3}$ kg/m${}^{3}$ | Density of the solid phase |

${K}_{d}$ | $8.142\times {10}^{-4}$ kg/m | Partitioning coefficient |

${D}_{m}$ | $6.76\times {10}^{-9}$ m${}^{2}$/s | Molecular diffusion coefficient in the clay |

${\alpha}_{L}$ | 0.5 m | Longitudinal dispersion coefficient |

${\alpha}_{T}$ | 0.05 m | Transverse dispersion coefficient |

${c}_{0}$ | 0.5 kg/m${}^{3}$ | Reference solute concentration |

Location | Pollution Region | Description |
---|---|---|

A | $x=$ 130 m to $x=$ 132 m | single pollution for 2 m |

B | $x=$ 149 m to $x=$ 151 m | two 2 m pollution points |

and $x=$ 152 m to $x=$ 154 m | that are close to each other | |

C | $x=$ 170 m to $x=$ 174 m | single pollution for 4 m |

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**MDPI and ACS Style**

Wu, S.; Jeng, D.-S.
Two-Dimensional Model for Consolidation-Induced Solute Transport in an Unsaturated Porous Medium. *Sci* **2023**, *5*, 16.
https://doi.org/10.3390/sci5020016

**AMA Style**

Wu S, Jeng D-S.
Two-Dimensional Model for Consolidation-Induced Solute Transport in an Unsaturated Porous Medium. *Sci*. 2023; 5(2):16.
https://doi.org/10.3390/sci5020016

**Chicago/Turabian Style**

Wu, Sheng, and Dong-Sheng Jeng.
2023. "Two-Dimensional Model for Consolidation-Induced Solute Transport in an Unsaturated Porous Medium" *Sci* 5, no. 2: 16.
https://doi.org/10.3390/sci5020016