# Determination of Unsaturated Hydraulic Properties of Seepage Flow Process in Municipal Solid Waste

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## Abstract

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## 1. Introduction

_{θ}of MSW are limited [2,9,18,19,20]. Moreover, using a single domain model to describe the hydraulic characteristics wastes for predicting the water migration inside landfills often has a specific error in engineering practice because of the broad pore characteristics of wastes.

## 2. Mathematical Model

#### 2.1. Single Permeability Model—Van Genuchten Model

^{3}L

^{−3}]; K(h) is the unsaturated hydraulic conductivity [LT

^{−1}]; h is the water head [L]; and S is the source-sink term [-].

_{e}is the effective saturation [-]; α is the inverse of the air-entry pressure [L

^{−1}]; n is the pore size distribution index, which determines the slope of the retention curve [-]; K

_{s}is the saturated hydraulic conductivity [LT

^{−1}]; and l is the tortuosity parameter [-].

#### 2.2. Dual Permeability Model

_{f,m}is the water volume of the fracture domain (matrix domain) divided by the total volume (L

^{3}L

^{−3}); K

_{f,m}is the hydraulic conductivity of the fracture domain (matrix domain) (LT

^{−1}); h

_{f,m}is the head of the fracture domain (matrix domain) (L); S

_{f,m}is the source and sink term (-); and w

_{f}is the volume of the fracture domain divided by the total flow domain volume (-). Mass exchange term Γ

_{w}is defined as follows:

_{w}is an empirical scaling factor, and K

_{a}is the effective hydraulic conductivity at the interface between the two domains (LT

^{−1}).

_{f}and w

_{m}, and variables θ, h, and K at any time and space, and are the weighted averages of the two local values:

## 3. Materials and Methods

#### 3.1. Experimental Setup

#### 3.2. Test Materials and Design

#### 3.3. Determination of Saturated Water Content and Saturated Hydraulic Conductivity

_{w}) into the reactor is recorded. Given that the pores are filled with water when saturated, the saturated water content is equal to the porosity:

_{1}, h

_{2}, and the distance l between the two sensors, the hydraulic gradient is i = (h

_{1}− h

_{2})/l, and at six hydraulic gradients (1.25, 1.5, 1.75, 2, 2.5, 3), the upper spout is connected to the electronic balance to measure the outflow. After the sensor reading stabilizes, the outflow volume (V) within Δt is recorded. According to Darcy’s law, the saturated hydraulic conductivity is

#### 3.4. Multi-Step Drainage Experiment

_{f}values of the four groups of refuse samples are 0.418, 0.263, 0.319, and 0.37, respectively.

## 4. Results and Discussion

#### 4.1. Saturated Water Content and Saturated Hydraulic Conductivity

^{−4}$\mathsf{\rho}$

_{d}+ 1.0399.

^{−2}and 3.2 × 10

^{−3}cm s

^{−1}, respectively. The permeability of waste decreases with the increase in density. When the density is high, the water migration path is small and the resistance is large, and vice versa. The fitting degree of the saturated hydraulic conductivity values of the four groups of waste samples measured in this experiment with the data of Chen et al. [31] and Beaven et al. [32] is 0.97 on the logarithmic coordinate, for which the functional relation is k = 5.1912 × 10

^{−0.01027}

^{$\mathsf{\rho}$}

^{d}. Group C’s saturated hydraulic conductivity with the same density and twice the particle size of the group B was 1.44 × 10

^{−2}cm s

^{−1}, which is 4.5 times that of the samples in group B. The increase in particle size causes the refuse skeleton and pore distribution to become more uneven, and the large pore channels guide the preferential flow to increase the permeability. Compared with that of group B, group D’s permeability with only one year of degradation has a very small increase, indicating that a single factor (i.e., degradation age) has little effect on the permeability of waste.

#### 4.2. Hydrus-1D Model Construction

_{s}) and saturated hydraulic conductivity (K

_{s}) are directly used. The tortuosity parameter (l) is assumed to be 0.5, like in the studies of most scholars [2,25,33,34]. For VGM model parameters α and n, residual water content θ

_{r}is inversed. The DPeM model has 17 parameters, which are nearly three times the number of parameters of the VGM model. Some reasonable assumptions about the model parameters are also made. That is, given the large size of the fracture domain, no other small particles exist, and the water is easily outflowed in the moderate capillary. The fracture domain’s residual and saturated water contents are 0 and 1, respectively [25,26,35]. Most of the water in the drainage test is directly discharged from the fracture domain under the action of the drainage experiment, and if fracture domain parameters α

_{f}and n

_{f}are equal to VGM models α and n [35]. Audebert et al. pointed out that the saturated hydraulic conductivity of the fracture domain is 10~103 times that of the single domain K

_{s}. Thus, this study assumes that K

_{sf}= 100 K

_{s}; double domain tortuosity parameters l

_{f}and l

_{m}are both 0.5; the saturated water content of the matrix domain can be calculated by the following formula:

#### 4.3. Fitting of Cumulative Outflow in Multi-Step Drainage Experiment

^{2}) reaches 0.99 because, in the early stage of the drainage test, the large pores’ water flows out preferentially. The fracture domains with large permeability and flow velocity in the DPeM model can just “capture” this dynamic superior flow characteristic. The fit of the VGM and DPeM models can improve as the pore water content of the waste decreases in the later stage of the drainage as well as the permeability. In theory, the DPeM model is more consistent with the pore characteristics of waste, and the errors of the DPeM model are smaller than that of VGM model. The result of numerical inversion is exactly the same. Although more parameters complicate the process of moisture migration, they also improve the accuracy of characterizing the spatial and temporal distribution of leachate.

_{m}= 0.63) will be higher. In group A (w

_{m}= 0.592), for the same average volumetric water content, the volumetric water content of the matrix domains of group D will be significantly less than that of the group A matrix domains.

#### 4.4. Analysis of Influencing Factors of Hydraulic Characteristic Parameters

#### 4.4.1. VGM Model Parameter Analysis

_{r}∈ [0.15,0.34]). The reason for the small value is that this test uses stale garbage samples with a degradation age of up to 13 years. Microbial degradation significantly reduces the organic component and content, reducing the residual moisture.

^{−1}smaller, the residual water content is 0.036 larger, and pore size distribution index n is basically the same. The higher the degradation age is, the higher the air entry pressure and the lower the residual moisture content are, indicating that the microbial degradation reaction consumes the rich organic components of the waste, increasing the number of fine particles and reducing the moisture retention.

^{−3}), which are within the confidence range. The values of α tend to decrease, corresponding to an increase in the air entry suction of MSW, as the dry unit weight of MSW increased in the laboratory specimens. This phenomenon suggests that increases in dry unit weight results in increases in air entry suction, which may occur because of the compression of the largest pores in MSW that governs the air entry suction.

#### 4.4.2. DPeM Model Parameter Analysis

_{m}and n

_{m}) of the four groups of refuse samples vary from 0.067 to 0.102 and 2.03 to 2.15, respectively. Residual moisture content θ

_{r}varies from 0.102 to 0.171, and the air entry pressure and residual moisture content of the B waste matrix domain are the largest. The comparison of the parameters of the DPeM and VGM models under the assumption that fracture domains α

_{f}, n

_{f}, and k

_{sf}are respectively equal to VGM model α, n, and 100 k, the residual water content of matrix domain θ

_{rm}is on average 13.7% larger than that of VGM model θ

_{r}, because the matrix domain mainly has small pores; the refuse skeleton and pores are evenly distributed; the leachate stored is difficult to release; and the moisture retention is excellent. The value of the residual water content of the matrix domain is similar to that in the study of Audebert et al. [26] by 10.2%. Matrix domains α

_{m}and n

_{m}are approximately 0.46 and 1.36 times VGM models α and n, indicating that the matrix domain has a uniform distribution of small pores and a sizeable air entry pressure, which is much smaller than that proposed by Han et al. and Audebert et al. (i.e., α

_{m}≈ 0.01 α). The particle size of stale waste is speculated to be smaller, and the fitting parameters are not unique. Further experimental research and demonstration are needed. The pore size distribution index of matrix domain n

_{m}is 1.29 times that of VGM model n, which is similar to that in this study.

_{m}

_{,f}, n

_{m}, and

_{f}are negatively correlated with dry density and degradation age and positively correlated with particle size. Residual moisture content θ

_{rm}is positively correlated with dry density and particle size and negatively correlated with degradation age.

## 5. Conclusions

^{−3}are as follows: When using the VGM model for characterization, the α and n change ranges are 0.181–0.21 cm

^{−1}and 1.45–1.618, respectively, and the residual moisture content θ

_{r}change range is 0.092–0.147. When using the DPeM model for characterization, the parameters of matrix domains θ

_{rm}and n

_{m}can be approximated to be 1.16 and 1.36 times those of α and n in the VGM model, respectively. The reciprocal of air intake value α

_{m}and K

_{sm}must be studied and verified further.

_{r}will be. The influence of particle size on the hydraulic characteristics is opposite to that of dry density. The larger the degradation age is, the smaller are the reciprocal of air entry pressure α, pore size distribution index n, and residual moisture content θ

_{r}.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Diagram of combined testing system: 1—base, 2—moisture content sensor, 3—porous permeable board, 4—screw, 5—tensiometer, 6—scale bucket, 7—pulley, 8—pump, 9—data acquisition instrument, 10—electronic balance.

**Figure 4.**Cumulative outflow data for groups (

**A**–

**D**) during drainage experiments with inverse modeling fits using best-fit parameters.

Component | % of Dry Mass |
---|---|

Plastic | 23.42 |

Textiles | 0.95 |

Wood | 1.80 |

Rubber | 0.96 |

Humus | 64.66 |

Miscellaneous | 8.21 |

Waste Group | Dry Density (kg m^{−3}) | Maximum Particle Size (cm) | Degradation Age (Years) |
---|---|---|---|

A | 205 | 2.575 | 13 |

B | 312.5 | 2.575 | 13 |

C | 312.5 | 5.15 | 13 |

D | 205 | 2.575 | 1 |

**Table 3.**Hydraulic parameters of waste VGM model with different properties (±values are 95% confidence interval).

Group | θ_{r} * | θ_{s} | α * (cm ^{−1}) | n * | k_{s} (cm min^{−1}) | l | R^{2} |
---|---|---|---|---|---|---|---|

A | 0.092 ± 0.0012 | 0.877 | 0.210 ± 0.034 | 1.610 ± 0.029 | 2.442 | 0.5 | 0.9852 |

B | 0.147 ± 0.052 | 0.790 | 0.173 ± 0.059 | 1.45 ± 0.061 | 0.192 | 0.5 | 0.9759 |

C | 0.139 ± 0.046 | 0.793 | 0.189 ± 0.037 | 1.50 ± 0.013 | 0.864 | 0.5 | 0.9824 |

D | 0.128 ± 0.091 | 0.880 | 0.181 ± 0.015 | 1.618 ± 0.081 | 2.55 | 0.5 | 0.9871 |

**Table 4.**Hydraulic parameters of waste DPeM model with different properties (±values are 95% confidence interval).

Matrix | θ_{rm} * | θ_{sm} | α_{m} * (cm^{−1}) | n_{m} * | k_{sm} * (cm min^{−1}) | l_{m} |

A | 0.102 ± 0.064 | 0.778 | 0.098 ± 0.0025 | 2.15 ± 0.371 | 0.198 ± 0.073 | 0.5 |

B | 0.171 ± 0.048 | 0.715 | 0.067 ± 0.0097 | 2.03 ± 0.539 | 0.072 ± 0.013 | 0.5 |

C | 0.152 ± 0.087 | 0.686 | 0.084 ± 0.0049 | 2.12 ± 0.754 | 0.133 ± 0.022 | 0.5 |

D | 0.165 ± 0.055 | 0.81 | 0.102 ± 0.0035 | 2.10 ± 0.215 | 0.203 ± 0.081 | 0.5 |

[25,26,35] | 0.15~0.22 | 0.65~0.83 | 0.0015~0.25 | 1.5~2.5 | 0.003~0.6 | 0.5 |

Fracture | θ_{rf} | θ_{sf} | α_{f} (cm^{−1}) | n_{f} | k_{sf} (cm min^{−1}) | l_{f} |

A | 0 | 1 | 0.21 | 1.62 | 244.2 | 0.5 |

B | 0 | 1 | 0.173 | 1.45 | 19.2 | 0.5 |

C | 0 | 1 | 0.189 | 1.50 | 86.4 | 0.5 |

D | 0 | 1 | 0.181 | 1.618 | 255 | 0.5 |

[25,26,35] | 0 | 1 | 0.08~0.72 | 1.5~2.0 | 6~178.55 | 0.5 |

Transfer Term | w_{f} | β | γ_{w} | a (cm) | k_{a} (cm min^{−1}) | R^{2} |

A | 0.418 | 8 | 0.4 | 10 | 10^{−6} | 0.9931 |

B | 0.263 | 8 | 0.4 | 10 | 10^{−6} | 0.9963 |

C | 0.319 | 8 | 0.4 | 10 | 10^{−6} | 0.9984 |

D | 0.37 | 8 | 0.4 | 10 | 10^{−6} | 0.9976 |

[25,26,35] | 0.1~0.657 | 3~15 | 0.4 | 2.5~10 | 10^{−6}~10^{−1} |

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**MDPI and ACS Style**

Zhang, C.; Liang, B.; Liu, L.; Wan, Y.; Zhu, Q.
Determination of Unsaturated Hydraulic Properties of Seepage Flow Process in Municipal Solid Waste. *Water* **2021**, *13*, 1059.
https://doi.org/10.3390/w13081059

**AMA Style**

Zhang C, Liang B, Liu L, Wan Y, Zhu Q.
Determination of Unsaturated Hydraulic Properties of Seepage Flow Process in Municipal Solid Waste. *Water*. 2021; 13(8):1059.
https://doi.org/10.3390/w13081059

**Chicago/Turabian Style**

Zhang, Chai, Bing Liang, Lei Liu, Yong Wan, and Qichen Zhu.
2021. "Determination of Unsaturated Hydraulic Properties of Seepage Flow Process in Municipal Solid Waste" *Water* 13, no. 8: 1059.
https://doi.org/10.3390/w13081059