This section firstly introduces the bio-hydro coupled model for waste that was reported and validated by Lu et al. (2020). On this basis, the stability analysis method for landfill slope and the numerical implementation of the entire process are presented in this study to investigate the effects of leachate recirculation and aeration. The following assumptions were adopted in the numerical modelling: (1) landfill gas and leachate were immiscible, and their migrations can be described by Darcy’s law; (2) leachate was incompressible, and landfill gas was assumed as ideal gas; (3) the intermediate products of waste biodegradation processes were neglected; (4) and the small-strain assumption was applied to calculate waste deformation.

### Bio-hydro coupled model

#### Fluid flow equations

Gas and leachate flow in landfills can be described by the classic two-phase flow model for unsaturated porous materials as follows:

$$\frac{{\partial (n\rho_{\alpha } S_{\alpha } )}}{\partial t} + \nabla \cdot (\rho_{\alpha } {\mathbf{v}}_{\alpha } ) = \begin{array}{*{20}c} {Q_{\alpha } } & {\left( {\alpha = {\text{g}},{\text{l}}} \right)} \\ \end{array}$$

(1)

where *n* is the porosity of waste; *S*_{α}, *ρ*_{α} (kg/m^{3}) and **v**_{α} (m/s) are the saturation degree, density, and Darcy velocity of phase *α* (g for gas and l for leachate), respectively, and *S*_{g} + *S*_{l} = 1; and *Q*_{α} (kg/m^{3}/s) is the source term of phase *α* due to biodegradation. The Darcy velocities of gas and leachate can be calculated according to Darcy’s law as:

$${\mathbf{v}}_{\alpha } = - \frac{{{\mathbf{K}}_{\alpha } }}{{\mu_{\alpha } }}\left( {\nabla p_{\alpha } - \rho_{\alpha } {\mathbf{g}}} \right)$$

(2)

where the tensor field **K**_{α} (m^{2}) is the permeability of phase *α* in waste; *μ*_{α} (kg/m/s) and *p*_{α} (Pa) are the dynamic viscosity and pore pressures of phase *α*, respectively; and **g** (m/s^{2}) is the gravitational acceleration.

The leachate and gas permeability of landfilled waste can be expressed as:

$${\mathbf{K}}_{\alpha } = \left( {\begin{array}{*{20}c} {AK_{{\text{v}}} } & {} & {} \\ {} & {AK_{{\text{v}}} } & {} \\ {} & {} & {K_{{\text{v}}} } \\ \end{array} } \right)k_{{{\text{r}}\alpha }}$$

(3)

where *A* is the anisotropic coefficient of waste; *K*_{v} (m^{2}) is the vertical intrinsic permeability of waste; and *k*_{rα} is the relative permeability of phase *α*, which can be estimated by van Genuchten–Mualem model as follows (Mualem 1976; van Genuchten 1980):

$$\left\{ {\begin{array}{*{20}l} {k_{rg} = (1 - S_{le} )^{1/2} \left( {1 - S_{le}^{1/m} } \right)^{2m} } \hfill \\ {k_{rl} = S_{le}^{1/2} \left[ {1 - \left( {1 - S_{le}^{1/m} } \right)^{m} } \right]^{2} } \hfill \\ {S_{le} = \left( {S_{l} - S_{r} } \right)/(S_{m} - S_{r} )} \hfill \\ \end{array} } \right.$$

(4)

where *S*_{le}, *S*_{m}, and *S*_{r} are the effective, maximum and residual leachate saturations, respectively; and *m* is a dimensionless constant for the model. The relationship between *S*_{le} and pore pressures (*p*_{l} and *p*_{g}) can be expressed by van Genuchten model as (Lu et al. 2020; van Genuchten 1980):

$$p_{{\text{c}}} = \left\{ {\begin{array}{*{20}l} {p_{{\text{g}}} - p_{{\text{l}}} } \hfill & {(p_{{\text{g}}} > p_{l} )} \hfill \\ 0 \hfill & {(p_{{\text{g}}} \le p_{{\text{l}}} )} \hfill \\ \end{array} } \right\} = p_{{{\text{c0}}}} \left( {S_{{{\text{le}}}}^{ - 1/m} - 1} \right)^{1 - m}$$

(5)

where *p*_{c} (Pa) and *p*_{c0} (Pa) are the capillary pressure and the entry capillary pressure of gas, respectively.

The density of leachate was assumed as constant, while the density of gas mainly depends on the gas pressure, and its time derivative of density can be written as:

$$\frac{1}{{\rho_{{\text{g}}} }}\frac{{\partial \rho_{{\text{g}}} }}{\partial t} = \frac{1}{{p_{{\text{g}}} }}\frac{{\partial p_{{\text{g}}} }}{\partial t} + \frac{1}{{M_{{\text{g}}} }}\frac{{\partial M_{{\text{g}}} }}{\partial t} - \frac{1}{T}\frac{\partial T}{{\partial t}}$$

(6)

where *M*_{g} (g/mol) is the average molecular weight of gas mixture; and *T* (K) is the temperature.Substituting Eqs. (2) and (6) into Eq. (1) yields:

$$\frac{1}{{\rho_{{\text{l}}} }}\nabla \cdot \left[ {\rho_{{\text{l}}} \frac{{{\mathbf{K}}_{{\text{l}}} }}{{\mu_{{\text{l}}} }}\left( {\rho_{{\text{l}}} {\mathbf{g}} - \nabla p_{{\text{l}}} } \right)} \right] + \frac{{\partial \left( {nS_{{\text{l}}} } \right)}}{\partial t} = \frac{{Q_{{\text{l}}} }}{{\rho_{{\text{l}}} }}$$

(7)

$$\frac{{nS_{{\text{g}}} }}{{p_{{\text{g}}} }}\frac{{\partial p_{{\text{g}}} }}{\partial t}{ + }\frac{1}{{\rho_{{\text{g}}} }}\nabla \cdot \left[ {\rho_{{\text{g}}} \frac{{{\mathbf{K}}_{{\text{g}}} }}{{\mu_{{\text{g}}} }}\left( {\rho_{{\text{g}}} {\mathbf{g}} - \nabla p_{{\text{g}}} } \right)} \right] - \frac{{\partial \left( {nS_{{\text{l}}} } \right)}}{\partial t}{ + }\frac{{nS_{{\text{g}}} }}{{M_{{\text{g}}} }}\frac{{\partial M_{{\text{g}}} }}{\partial t} - \frac{{nS_{{\text{g}}} }}{T}\frac{\partial T}{{\partial t}} = \frac{{Q_{{\text{g}}} }}{{\rho_{{\text{g}}} }}.$$

(8)

#### Oxygen transport equations

In aeration scenarios, the oxygen concentration can significantly affect the reaction modes and the degradation rate of waste (Kim et al. 2007; Omar and Rohani 2017), thereby affecting the distribution of pore pressures. To evaluate the oxygen distribution in landfills, its mass conservation in gas phase can be written as:

$$\frac{{\partial \left( {nS_{{\text{g}}} C_{{\text{O}}} } \right)}}{\partial t} + \nabla \cdot \left( {C_{{\text{O}}} {\mathbf{v}}_{{\text{g}}} } \right) - \nabla \cdot \left( {nS_{{\text{g}}} {\mathbf{J}}_{{\text{O}}} } \right) = Q_{{\text{O}}}$$

(9)

where *C*_{O} (kg/m^{3}), **J**_{O} (kg/m^{2}/s), and *Q*_{O} (kg/m^{3}/s) are the concentration, diffusive flux and source term of oxygen in gas phase, respectively. **J**_{O} can be calculated using Fick’s law:

$${\mathbf{J}}_{{\text{O}}} = - \tau D_{{\text{O}}} \nabla C_{{\text{O}}}$$

(10)

where *D*_{O} (m^{2}/s) is the diffusion coefficient of oxygen in gas; and *τ* is the tortuosity factor for gas diffusion considering the effect of porosity and saturation, and is defined as (Millington and Quirk 1961):

$$\tau = S_{{\text{l}}}^{7/3} n^{1/3}$$

(11)

The time derivative of *C*_{O} can be written as:

$$\frac{1}{{C_{{\text{O}}} }}\frac{{\partial C_{{\text{O}}} }}{\partial t} = \frac{1}{{\rho_{{\text{g}}} Y_{{\text{O}}} }}\frac{{\partial \left( {\rho_{{\text{g}}} Y_{{\text{O}}} } \right)}}{\partial t} = \frac{1}{{Y_{{\text{O}}} }}\frac{{\partial Y_{{\text{O}}} }}{\partial t} + \frac{1}{{p_{{\text{g}}} }}\frac{{\partial p_{{\text{g}}} }}{\partial t} + \frac{1}{{M_{{\text{g}}} }}\frac{{\partial M_{{\text{g}}} }}{\partial t} - \frac{1}{T}\frac{\partial T}{{\partial t}}$$

(12)

where *Y*_{O} is the mass fraction of oxygen (*C*_{O} = *ρ*_{g}*Y*_{O}). Incorporating Eq. (12) into Eq. (9) yields:

$$\begin{aligned} & nS_{{\text{g}}} \rho_{{\text{g}}} \frac{{\partial Y_{{\text{O}}} }}{\partial t} + \nabla \cdot \left( {\rho_{{\text{g}}} Y_{{\text{O}}} {\mathbf{v}}_{{\text{g}}} } \right) - \nabla \cdot \left( {nS_{{\text{g}}} \rho_{{\text{g}}} \tau D_{{\text{O}}} \nabla Y_{{\text{O}}} } \right) \\ & \quad { + }\rho_{{\text{g}}} \left( {\frac{{nS_{{\text{g}}} }}{{p_{{\text{g}}} }}\frac{{\partial p_{{\text{g}}} }}{\partial t} + \frac{{nS_{{\text{g}}} }}{{M_{{\text{g}}} }}\frac{{\partial M_{{\text{g}}} }}{\partial t} - \frac{{nS_{{\text{g}}} }}{T}\frac{\partial T}{{\partial t}} - \frac{{\partial nS_{{\text{l}}} }}{\partial t}} \right)Y_{{\text{O}}} = Q_{{\text{O}}} \\ \end{aligned}$$

(13)

#### Biodegradation equations

The reaction mode of waste in landfills depends on the local concentration of oxygen. Kim et al. (2007) suggested a threshold pressure of oxygen required for aerobic biodegradation:

$${\text{Reaction mode}} = \left\{ {\begin{array}{*{20}l} {\text{Aerobic degradation}} \hfill & {{\text{If }}p_{{{\text{Og}}}} \ge {\text{100 Pa}} \Rightarrow RM = 1} \hfill \\ {\text{Anaerobic degradation}} \hfill & {{\text{If }}p_{{{\text{Og}}}} {\text{ < 100 Pa}} \Rightarrow RM = 0} \hfill \\ \end{array} } \right.$$

(14)

where *p*_{Og} (Pa) is the oxygen partial pressure and *RM* is the parameter used to represent different reaction modes. To calculate the source term due to biodegradation mentioned in Eqs. (7), (8) and (13), a Monod-type biodegradation sub-model considering the effect of leachate saturation was adopted in this study (El-Fadel et al. 1996; Fytanidis and Voudrias 2014; Lu et al. 2020; Omar and Rohani 2017), which can be expressed as:

$$R_{{\text{A}}} = \frac{{\partial X_{{\text{A}}} }}{\partial t} = k_{{{\text{A}},\max }} f_{{\text{s}}} \frac{S}{{S_{{\text{A}}} + S}}\frac{{p_{{{\text{Og}}}} /p_{{\text{g}}} }}{{k_{{\text{O}}} + p_{{{\text{Og}}}} /p_{{\text{g}}} }}X_{{\text{A}}} - R_{A,D}$$

(15)

$$R_{{\text{N}}} = \frac{{\partial X_{{\text{N}}} }}{\partial t} = k_{{{\text{N}},\max }} f_{{\text{s}}} \frac{S}{{S_{{\text{N}}} + S}}X_{{\text{N}}} - R_{N,D}$$

(16)

where *R*_{A} (kg/m^{3}/day) and *R*_{N} (kg/m^{3}/day) are the growth rates of aerobic and anaerobic species, respectively; *X*_{A} (kg/m^{3}) and *X*_{N} (kg/m^{3}) are the concentration of aerobic and anaerobic species, respectively; *k*_{A,max} (day^{−1}) and *k*_{N,max} (day^{−1}) are the maximum biodegradation rates under aerobic and anaerobic conditions, respectively; *f*_{s} is the leachate saturation correction factor; *S* (kg/m^{3}) is the biodegradable substrate concentration; *S*_{A} (kg/m^{3}) and *S*_{N} (kg/m^{3}) are the half-saturation constants of the substrates for aerobic and anaerobic species, respectively; *k*_{O} (kg/m^{3}) is the oxygen half-saturation constant; and *R*_{A,D} (kg/m^{3}/day) and *R*_{N,D} (kg/m^{3}/day) are the decay rates of aerobic and anaerobic species, respectively. *R*_{A,D} and *R*_{N,D} can be expressed as:

$$R_{{\text{A,D}}} = 0.05k_{{{\text{A}},\max }} (X_{{\text{A}}} - X_{{{\text{A}},{0}}} )$$

(17)

$$R_{{\text{N,D}}} = 0.05k_{{{\text{N}},\max }} (X_{{\text{N}}} - X_{{\text{N,0}}} )$$

(18)

where *X*_{N,0} (kg/m^{3}) and *X*_{A,0} (kg/m^{3}) are the initial concentrations of anaerobic and aerobic species, respectively. The leachate saturation correction factor *f*_{s} can be expressed as (Meima et al. 2008):

$$f_{{\text{s}}} = \left\{ {\begin{array}{*{20}l} 0 \hfill & {\left( {S_{{\text{l}}} < 0.2} \right)} \hfill \\ {\frac{{S_{{\text{l}}} - 0.2}}{0.3}} \hfill & {\left( {0.2 \le S_{{\text{l}}} \le 0.5} \right)} \hfill \\ 1 \hfill & {\left( {0.5 < S_{{\text{l}}} } \right)} \hfill \\ \end{array} } \right.$$

(19)

The process of waste biodegradation can be described using the following chemical equations (Tchobanoglous et al. 1993):

$$\left\{ {\begin{array}{*{20}l} {\text{Aerobic:}} \hfill & {{\text{C}}_{a} {\text{H}}_{b} {\text{O}}_{c} + \left( {\frac{4a + b - 2c}{4}} \right){\text{O}}_{2} \mathop{\longrightarrow}\limits^{{{\text{Biomass}}}}a{\text{CO}}_{2} + \frac{b}{2}{\text{H}}_{{2}} {\text{O}}} \hfill \\ {\text{Anaerobic:}} \hfill & {{\text{C}}_{a} {\text{H}}_{b} {\text{O}}_{c} + \left( {\frac{4a - b - 2c}{4}} \right){\text{H}}_{2} {\text{O}}\mathop{\longrightarrow}\limits^{{{\text{Biomass}}}}\left( {\frac{4a - b + 2c}{8}} \right){\text{CO}}_{2} + \left( {\frac{4a + b - 2c}{8}} \right){\text{CH}}_{4} } \hfill \\ \end{array} } \right.$$

(20)

where C_{a}H_{b}O_{c} is the molecular formula of waste depending on its chemical composition; and *a*, *b* and *c* are the constants representing the contents of carbon, hydrogen and oxygen, respectively (Feng et al. 2021b). According to the law of mass conservation and Eq. (20), the production rates of methane *R*_{M} (kg/m^{3}/day) and carbon dioxide *R*_{C} (kg/m^{3}/day), and the consumption rates of oxygen *R*_{O} (kg/m^{3}/day) and water *R*_{H} (kg/m^{3}/day) can be calculated as follows:

$$R_{{\text{M}}} { = }\left[ {\left( {\frac{4a + b - 2c}{8}} \right)\frac{{R_{{\text{N}}} }}{{Y_{{\text{N}}} }} \times \left( {1 - RM} \right)} \right]\frac{{M_{{\text{M}}} }}{{M_{{{\text{MSW}}}} }}$$

(21)

$$R_{{\text{C}}} { = }\left[ {\left( {\frac{4a - b + 2c}{8}} \right)\frac{{R_{{\text{N}}} }}{{Y_{{\text{N}}} }} \times \left( {1 - RM} \right) + a\frac{{R_{{\text{A}}} }}{{Y_{A} }} \times RM} \right]\frac{{M_{{\text{C}}} }}{{M_{{{\text{MSW}}}} }}$$

(22)

$$R_{{\text{O}}} = \left[ { - \left( {\frac{4a + b - 2c}{8}} \right)\frac{{R_{{\text{A}}} }}{{Y_{A} }} \times RM} \right]\frac{{M_{{\text{O}}} }}{{M_{{{\text{MSW}}}} }}$$

(23)

$$R_{{\text{H}}} { = }\left[ { - \left( {\frac{4a - b - 2c}{4}} \right)\frac{{R_{{\text{N}}} }}{{Y_{{\text{N}}} }} \times \left( {1 - RM} \right) + \frac{b}{2}\frac{{R_{{\text{A}}} }}{{Y_{A} }} \times RM} \right]\frac{{M_{{\text{H}}} }}{{M_{{{\text{MSW}}}} }}$$

(24)

where *Y*_{N} and *Y*_{A} are the biomass/substrate yield coefficients of anaerobic and aerobic conditions respectively. Therefore, the aforementioned source terms *Q*_{l}, *Q*_{g}, and *Q*_{O} can be written as:

$$\left\{ {\begin{array}{*{20}l} {Q_{{\text{l}}} = R_{{\text{H}}} } \hfill \\ {Q_{{\text{g}}} = R_{{\text{C}}} + R_{{\text{M}}} + R_{{\text{O}}} } \hfill \\ {Q_{{\text{O}}} = R_{{\text{O}}} } \hfill \\ \end{array} } \right.$$

(25)

### Slope stability analysis

The finite volume method (FVM) is usually used to deal with computational fluid dynamics problems. The introduced bio-hydro coupled model for waste has been programmed into the OpenFOAM platform based on FVM by Lu et al. (2020). This section is to add a sub-program for slope stability analysis based on the strength reduction method, which can expand the application of FVM in the deformation analysis of solid phase.

The pore pressures calculated by the bio-hydro coupled model will be directly imported into the slope stability analysis. The effective stress can be calculated based on the effective stress principle of unsaturated soil (Khoei and Mohammadnejad 2011):

$${\text{d}}{{\varvec{\upsigma}}}^{\prime} = {\text{d}}{{\varvec{\upsigma}}} + b{\mathbf{I}}{\text{d}}p$$

(26)

where **σ**′ (Pa) and **σ** (Pa) denote the effective stress and total stress, respectively; *b* is the Biot coefficient; **I** is the identity tensor; and *p* (Pa) is the average pore pressure, which can be expressed as:

$$p = p_{{\text{l}}} S_{{\text{l}}} + p_{{\text{g}}} S_{{\text{g}}}$$

(27)

This study adopts the Mohr–Coulomb model to calculate the mechanical plastic strain of waste, as has been applied in several studies (Feng et al. 2020; Lu et al. 2019, 2020):

$$\left\{ {\begin{array}{*{20}l} {f = \left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right) + \left( {\sigma^{\prime}_{1} + \sigma^{\prime}_{3} } \right)\sin \varphi - 2c\cos \varphi } \hfill \\ {g = \left( {\sigma^{\prime}_{1} - \sigma^{\prime}_{3} } \right) + \left( {\sigma^{\prime}_{1} + \sigma^{\prime}_{3} } \right)\sin \psi } \hfill \\ \end{array} } \right.$$

(28)

where *f* and *g* are the yield function and plastic potential function, respectively; *σ*_{1}′ (Pa) and *σ*_{3}′ (Pa) are the maximum and minimum principal stresses, respectively; and *c* (Pa), *φ* (°), and *ψ* (°) are the cohesion, friction angle and dilation angle of waste, respectively. The plastic strain increment can be expressed as:

$$\begin{array}{*{20}c} {{\text{d}}{{\varvec{\upvarepsilon}}}^{{\text{p}}} = \Lambda \frac{\partial g}{{\partial {{\varvec{\upsigma}}}{^{\prime}}}},} & {\Lambda = \left\{ {\begin{array}{*{20}l} {0,} \hfill & {f < 0} \hfill \\ {\Lambda ,} \hfill & {f = 0} \hfill \\ \end{array} } \right.} \\ \end{array}$$

(29)

where Λ is the plastic factor that can be calculated using the local return mapping method proposed by Clausen et al. (2007).

Based on the small-strain theory, Lu et al. (2020) derived the relationship between the plastic strain increment and deformation increment of waste as:

$$\begin{aligned} & \nabla \cdot \left\{ {\mu \nabla \left( {{\text{d}}{\mathbf{u}}} \right) + \mu \nabla \left( {{\text{d}}{\mathbf{u}}} \right)^{{\text{T}}} + \lambda {\mathbf{\rm I}}tr\left[ {\nabla \left( {{\text{d}}{\mathbf{u}}} \right)} \right]} \right\} - \nabla \cdot \left[ {2\mu {\text{d}}{{\varvec{\upvarepsilon}}}^{{\text{p}}} + \lambda {\mathbf{\rm I}}tr\left( {{\text{d}}{{\varvec{\upvarepsilon}}}^{{\text{p}}} } \right)} \right] \\ & \quad = \nabla \cdot \left( {b{\mathbf{I}}{\text{d}}p} \right) - {\text{d}}\left( {\rho {\mathbf{g}}} \right) \\ \end{aligned}$$

(30)

where *μ* and* λ* are Lame constants which can be derived from Young’s modulus *E*_{s} and Poisson’s ratio *ν*; and **u** (m) is the displacement vector.

This study adopts the strength reduction method (SRM) to analyze the stability of a landfill slope during recirculation and aeration. The SRM is suitable for linear Mohr–Coulomb failure criterion and has been widely used in the stability analysis of rock slopes (Yuan et al. 2020), soil slopes (Dawson et al. 1999; Griffiths and Lane 1999), and landfill slopes (Feng et al. 2018, 2020). The factor of safety is defined as the number by which the original shear strength parameters are divided to bring the slope to the point of failure, which is the same as that used in the traditional limit equilibrium method (Griffiths and Lane 1999). Therefore, the response of waste deformation can be examined by reducing the shear strength parameters in Eq. (28) following:

$$\left\{ {\begin{array}{*{20}l} {c_{{\text{r}}} = \frac{c}{FS}} \hfill \\ {\varphi_{{\text{r}}} = \arctan \left( {\frac{\tan \varphi }{{FS}}} \right)} \hfill \\ \end{array} } \right.$$

(31)

where *FS* is the factor of safety. By gradually increasing the value of *FS*, the slope will eventually reach an unstable state at which the displacement calculated by Eqs. (26)–(30) shows a dramatic increase. The *FS* at this moment reaches the real *FS*.

### Numerical model implementation

Equations (7), (8), (13), and (30) are the main governing equations for the landfill coupled model established in this study. There are four independent unknown variables: leachate pressure *p*_{l}, gas pressure *p*_{g}, mass fraction of oxygen in gas phase *Y*_{O}, and displacement increment d**u**. Other variables can be derived based on the above basic variables.

Based on FVM, the governing equations of the coupled model are discretized with the Gauss divergence theorem which converts the volume integrals of the governing equations over a specific volume into surface integrals. The detailed information on the discretization process can be found in Lu et al. (2020) and Weller and Tabor (1998). Finally, the implicit term is converted into a matrix of unknown variables; and the explicit terms such as source terms can be calculated using the results obtained in the previous iteration or time step.

After discretizing, the governing equations in the centroid *C* of each control volume can be transformed and rearranged into linear algebraic equations as follows:

$$\left\{ {\begin{array}{*{20}l} {\beta_{C}^{{p_{\alpha } }} \left( {p_{\alpha } } \right)_{C} + \sum\limits_{N} {\beta_{N}^{{p_{\alpha } }} \left( {p_{\alpha } } \right)_{N} } = R_{C}^{{p_{\alpha } }} } \hfill \\ {\beta_{C}^{{Y_{{\text{O}}} }} \left( {Y_{{\text{O}}} } \right)_{C} + \sum\limits_{N} {\beta_{N}^{{Y_{{\text{O}}} }} \left( {Y_{{\text{O}}} } \right)_{N} } = R_{C}^{{Y_{{\text{O}}} }} } \hfill \\ {\beta_{C}^{{{\text{d}}{\mathbf{u}}}} \left( {{\text{d}}{\mathbf{u}}} \right)_{C} + \sum\limits_{N} {\beta_{N}^{{{\text{d}}{\mathbf{u}}}} \left( {{\text{d}}{\mathbf{u}}} \right)_{N} } = R_{C}^{{{\text{d}}{\mathbf{u}}}} } \hfill \\ \end{array} } \right.$$

(32)

where *β*_{C}, *β*_{N}, and *R*_{C} are the diagonal coefficients, neighbor coefficients, and source terms, respectively. Assembling Eq. (32) for all control volumes as a whole, a system of linear equations can be obtained as:

$${\mathbf{B}}{\varvec{F}} = {\varvec{b}}$$

(33)

where **B** is a symmetric matrix; *F* is the unknown vector for leachate pressure, gas pressure, mass fraction of oxygen, and displacement increment; and *b* is the right-hand side vector.

Figure 1 shows the overall procedures of the solver based on the sequential iteration method. At each iteration of time step *t*, the fluid flow equations, the oxygen transport equation, and the displacement equation are solved in sequence, with the related parameters and source terms being updated. The convergence criteria of the calculation are defined as:

$$\left\{ \begin{gathered} \begin{array}{*{20}l} {\max \left| {F^{t + 1,n + 1} - F^{t + 1,n} } \right| \le \varepsilon_{F} } \hfill \\ {\max \left| {\frac{{F^{t + 1,n + 1} - F^{t + 1,n} }}{{F^{t + 1,n + 1} }}} \right| \le \zeta_{F} } \hfill \\ \end{array} \hfill \\ \hfill \\ \end{gathered} \right.$$

(34)

where *ε*_{F} and *ζ*_{F} are the absolute and relative convergence tolerances for variable *F*, respectively.The local return mapping method is adopted to obtain the modified plastic strain increment using the pore pressure calculated in the previous section. The detailed information about the algorithm can be found in Clausen et al. (2007) and Tang et al. (2015). The strength parameters *c* and *φ* are gradually reduced to examine the response of deformation until a dramatic increase in the displacement. Based on FVM, the code for the waste bio-hydro coupled model and landfill slope stability analysis are implemented in the C++ based open-source platform OpenFoam. The code is flexible in dimensionality and can solve problems in one dimension, two dimensions, and three dimensions (Lu et al. 2019, 2020).