2.3.1 Assumptions
In order to better analyze the insitu pumping experiments, the air pressure model based on the dualporosity medium seepage theory was developed (see Fig. 2b). The main assumptions of the model are as follows:
(1) The flow direction of the gas is parallel to the horizontal axis (r axis) and the vertical axis (y axis);
(2) Gas entering the gas well are from the matrix pores and the fractures;
(3) The flow of gas obeys Darcy's law;
(4) The gas is an ideal gas, and it satisfies the ideal gas equation of state.
2.3.2 Governing equations
Considering the nonhomogeneity of the MSWs, pores in MSWs are divided into fractures and matrix. The gas flow in the fractures and matrix pores are both considered to be anisotropic. Firstly, the pressure in the pumping well would decrease. Then, the gas in the fractures and matrix pores would flow to the well under the pressure gradient. Gas mass exchange between fractures and matrix pores would occur during this process.
Considering the nonhomogeneity of waste in the vertical direction, the waste was assumed to be composed of n layers. The gas production rate was assumed to be a constant. Combining with Liu and Zheng’s model (Liu et al., 2016; Zheng et al., 2019), a dualporosity model for gas transport in the waste and the pumping well was developed:
$${K_{fri}}\left( {\frac{{{\partial ^2}{p_{fi}}^{2}}}{{\partial {r_i}^{2}}}+\frac{1}{{{r_i}}}\frac{{\partial {p_{fi}}^{2}}}{{\partial {r_i}}}} \right)+{K_{fzi}}\frac{{{\partial ^2}{p_{fi}}^{2}}}{{\partial {z_i}^{2}}}+{\beta _i}{K_{mi}}\left( {{p_{mi}}^{2}  {p_{fi}}^{2}} \right)  2\mu \frac{{\partial {n_{gi}}{p_{fi}}}}{{\partial t}}+\frac{{2RT\mu }}{\omega }{a_i}=0$$
1
$${K_{mri}}\left( {\frac{{{\partial ^2}{p_{mi}}^{2}}}{{\partial {r_i}^{2}}}+\frac{1}{{{r_i}}}\frac{{\partial {p_{mi}}^{2}}}{{\partial {r_i}}}} \right)+{K_{mzi}}\frac{{{\partial ^2}{p_{mi}}^{2}}}{{\partial {z_i}^{2}}}  {\beta _i}{K_{mi}}\left( {{p_{mi}}^{2}  {p_{fi}}^{2}} \right)  2\mu \frac{{\partial {n_{gi}}{p_{mi}}}}{{\partial t}}+\frac{{2RT\mu }}{\omega }{a_i}=0$$
2
pi = wfpfi + (1wf)pmi (3)
$${K_{ri}}={w_f}{K_{fri}}+\left( {1  {w_f}} \right){K_{mri}}$$
4
$${K_{zi}}={w_f}{K_{fzi}}+\left( {1  {w_f}} \right){K_{mzi}}$$
5
where βi is the mass exchange coefficient of the ith layer; Kmi is the air permeability of the fracture and matrix exchange of the ith layer; Kfri is the horizontal air permeability of the fractures flow of the ith layer; Kfzi is the vertical air permeability of the fractures flow of the ith layer; Kmri is the horizontal air permeability of the matrix pores flow of the ith layer; Kmzi is the vertical air permeability of the matrix pores flow of the ith layer; T is Temperature; R is the gas constant; µ is the viscosity coefficient; ω is the molar mass of the gas; ai is the gas production rate of the ith layer; ngi is the void gas content of the ith layer of landfill; pfi is the absolute pressure of the fractures flow in the ith layer; pmi is the absolute pressure of the matrix pores flow in layer i; pi is the absolute pressure of the ith layer; wf is the volume of fractures divided by the total pores (0༜wf༜1); Kfri is the horizontal air permeability of the MSW of the ith layer; and Kfzi is the vertical air permeability of the MSW of the ith layer.
The landfill gas production rate can be calculated by (Mccarty, 1975; Thompson et al., 2009):
$${a_i}=\frac{{\text{1}}}{{{\text{3}}{\text{.15}} \times {\text{1}}{{\text{0}}^{{\text{10}}}}}}\frac{{{Q_{ti}}}}{{{M_{\text{1}}}}} \times {\rho _l} \times {\rho _g}$$
6
$${Q_t}{\text{=}}{M_{\text{1}}} \cdot {L_0} \cdot k \cdot {e^{  kt}}$$
7
$${L_0}=1867 \times \sum\limits_{{i=1}}^{4} {[{W_i} \times (1  {d_i}) \times DO{C_i}]}$$
8
where ρl is the density of the landfill; ρg is the density of landfill gas; Qt is gas production rate of waste in the ith year; M1 is the mass of the landfilled waste; L0 is the potential LFG generation capacity; and k is the average gas production rate constant of the landfill waste; DOCi is the content of degradable organic carbon of the i component; Wi is the wet weight content of the i component; and di is the moisture content of the i component.
According to the degradable components in the fresh solid waste of the landfill from 2009 to 2011, as well as the wet weight content and moisture content of each component, L0 can be calculated through the Christensen biodegradable model (Christensen and McCarty, 1975).
2.3.3 Initial and boundary conditions
The upper boundary of the proposed model is the cover layer. The radial gas flow at this boundary can be ignored. The gas flow is mainly vertical flow (Yu et al., 2009). The upper boundary conditions can then be
$$\frac{{\partial {p_f}}}{{\partial z}}=Lc\frac{{{p_f}{\text{}}{p_0}}}{H}\left( {0 \leqslant r \leqslant {R_b},z=H} \right)$$
9
$$\frac{{\partial {p_m}}}{{\partial z}}=Lc\frac{{{p_m}{\text{}}{p_0}}}{H}\left( {0 \leqslant r \leqslant {R_b},z=H} \right)$$
10
where Lc is the cover layer coefficient. Eqs. (9) and (10) belong to the second type of boundary conditions. Yu et al. (2009) defined the cover layer coefficient as
$$Lc=\frac{{{k_l}H}}{{{d_l}{K_{fz1}}}}$$
11
where, kl is the vertical air permeability of the cover layer; dl is the thickness of the cover layer; and H is the thickness of the landfill. The cover layer coefficient Lc ranges from 0.001 to 10 (Yu et al., 2009).
When the cover layer is composed of materials with relatively high air permeability, such as loess, the landfill gas can exchange with the atmosphere freely. In this case, the landfill gas pressure and atmospheric pressure tend to be the same (Liu et al., 2016; Townsend et al., 2005; Chen et al., 2003). In this case, the upper boundary conditions are
$${p_f}={p_0}\left( {{r_w} \leqslant r \leqslant {R_b},z=H} \right)$$
12
$${p_m}={p_0}\left( {{r_w} \leqslant r \leqslant {R_b},z=H} \right)$$
13
where p0 is the atmospheric pressure. The Eqs. (12) and (13) belong to the first type of boundary conditions.
The boundary at r = Rb is (Liu et al., 2016; Vigneault et al., 2004; Chen et al., 2003; Yu et al., 2009):
$$\frac{{\partial {p_f}}}{{\partial r}}=0\left( {r={R_b},\;0 \leqslant z \leqslant H} \right)$$
14
$$\frac{{\partial {p_m}}}{{\partial r}}=0\left( {r={R_b},\;0 \leqslant z \leqslant H} \right)$$
15
The lower boundary condition is also assumed to be zero flux (Liu et al., 2016; Vigneault et al., 2004; Chen et al., 2003; Yu et al., 2009; Young et al., 1989):
$$\frac{{\partial {p_f}}}{{\partial z}}=0\left( {{r_w} \leqslant r \leqslant {R_b},z=0} \right)$$
16
$$\frac{{\partial {p_m}}}{{\partial z}}=0\left( {{r_w} \leqslant r \leqslant {R_b},z=0} \right)$$
17
When the pumping pressure is constant, the pressure at the well wall is the pumping pressure (Liu et al., 2016; Yu et al., 2009). The boundary condition near the pumping well is
$${p_f}={p_0}+{p_1}\left( {r={r_w},0 \leqslant z \leqslant H} \right)$$
18
$${p_m}={p_0}+{p_1}\left( {r={r_w},0 \leqslant z \leqslant H} \right)$$
19
where p1(kPa) is the pumping pressure of the extraction well.
At the interfaces between the ith layer and the i + 1th layer, the pressure and the gas flow rate are assumed to be continuous:
$$\frac{{\partial {p_{fi}}}}{{\partial z}}=\frac{{\partial {p_{f,i+1}}}}{{\partial z}}\left( {{r_w} \leqslant r \leqslant {R_b},z={z_i}} \right)$$
20
\(\frac{{\partial {p_{mi}}}}{{\partial z}}=\frac{{\partial {p_{m,i+1}}}}{{\partial z}}\left( {{r_w} \leqslant r \leqslant {R_b},z={z_i}} \right)\) (21)
\({p_{fi}}={p_{f,i+1}}\left( {{r_w} \leqslant r \leqslant {R_b},z={z_i}} \right)\) (22)
$${p_{mi}}={p_{m,i+1}}\left( {{r_w} \leqslant r \leqslant {R_b},z={z_i}} \right)$$
23
The initial conditions of this model are assumed as follows:
$${p_f}={p_0}\left( {t=0} \right)$$
24
$${p_m}={p_0}\left( {t=0} \right)$$
25
2.3.4 Parameterization
According to the radius of influence of the insitu pumping well, the radius of the landfill is set to be 25 m. The diameter of the pumping well is 800 mm. The values of temperature, ideal gas constant, viscosity, and the molar mass of landfill gas are set as 303 K, 8.31 kg·m2·s− 2·mole− 1·K− 1, 1.76⋅10− 5 kg·m− 1·s− 1, and 0.03 kg/mole, respectively (Zhan et al.,2015). The constant air permeability in the middle of the landfill layer was used.
The specific wet weight content, moisture content and the content of degradable organic carbon of each component of the MSW are shown in the Table 1. The value of L0 of food waste is 5.8 times larger than that of wood. According to equations 6–8, L0 and k were determined to be 163.2 m3/t, and 0.347 a− 1, respectively. The ages of the three layers of waste were 2.5 months, 1 year and 2 years, respectively. The corresponding gas production rates are 1.75×10− 5, 1.33×10− 5, and 9.43×10− 6 kg/(m3∙s), respectively. The value of L0 is 1.3 times larger than that of the Tianziling landfill, which is mainly because the food waste content of the two is similar (Feng et al., 2017). The gas production rates of these highkitchen food content wastes were quite high just after several months after the wastes were landfilled (Chen et al., 2010; Shen et al., 2018). The value of L0 is 1.633 times larger than that of a landfill in the European countries such as Greece and Denmark. This is mainly due to the fact that because the proportion of kitchen waste at this landfill is much greater than those at the landfills in the Europe (Chalvatzaki et al., 2010; Cho et al., 2012).
Table 1
Components and properties of the MSWs
Properties

Food waste

Paper

Textile

Wood

The wet weight content Wi (%)

55.3

12.3

2.7

1.5

The moisture content di, (%)

85.0

20.0

15.0

25.0

The content of degradable organic carbon DOCi, (%)

38.0

44.0

30.0

50.0

The potential LFG generation capacity L0, (m3/t)

58.8

81.1

13.1

10.2

The values of all parameters are summarized in Table 2. The partial differential Eqs. (1) and (2) and the above parameters are used to establish a gas migration analysis model. The model was solved by COMSOL Multiphysics 5.5 (COMSOL, 2014). The complete mesh contains 548 domain elements and 81 boundary elements, and each cell ranges from 0.007 m to 1.3 m.
Table 2
Parameter values used in the comparison of numerical model with field tests
Parameters

Value

rw (mm)

400

rb (m)

25

R(kg·m2·s− 2·mole− 1·K− 1)

8.31

T(K)

303

µ(10− 5kg·m− 1·s− 1)

1.76

ω(kg/mole)

0.03

β(m− 2)

0.01

p1(kPa)

101.33

Km(10− 13m2)

4.5
