## 2 Theoretical Model and Numerical Methods  
### Theoretical Model  
The theoretical model consists of the governing equations for the transport of moisture, heat, and LFG. Trace gases in LFG are disregarded, and primary consideration is given to the transport of CH\({}_{4}\), CO\({}_{2}\), O\({}_{2}\), and nitrogen (N\({}_{2}\)) within the cover. A detailed model development is given in [PERSON] et al. [26]. The theoretical model is capable of describing the transport of moisture, LFG, and heat within the landfill cover under non-isothermal conditions, driven by the effects of water vapor and MAMO. However, the model is constrained by the assumption of a rigid soil skeleton, which precludes the consideration of deformation within the cover.  
Moisture flow in soil consists of both liquid water flow and vapor flow. According to the moisture mass balance, the moisture flow could be depicted as follows [33; 35]:  
\[\frac{\partial(\rho_{l}\theta_{l}+\rho_{v}\theta_{v})}{\partial t}=-\
abla(\rho _{l}v_{l})-\
abla\big{(}\rho_{l}v_{v}+\rho_{v}v_{g}\big{)}+\rho_{d}M_{H_{2}O}R_ {w} \tag{1}\]  
where \(\rho_{l}\) is liquid water density; \(\theta_{l}\) is volumetric water content; \(\rho_{v}\) is vapor density; \(\theta_{v}\) is the volumetric vapor content (volume percentage of gas in the soil); \(t\) is time; \(v_{v}\) is the flow velocity of vapor; \(v_{l}\) is liquid water flow velocity; \(v_{g}\) is the flow velocity of the LFG mixture; \(\rho_{d}\) is dry soil density; \(M_{H_{2}O}\) is the molar mass of water; and \(R_{w}\) is the water generation rate by MAMO per unit mass of dry soil.  
The governing equation for each gas component (CH\({}_{4}\), CO\({}_{2}\), O\({}_{2}\), and N\({}_{2}\)) derived by [PERSON] et al. [33] is adopted as follows:  
\[\frac{\partial}{\partial t}\Big{[}(\phi-\theta_{l})c_{g}^{k}+\theta_{l}H_{l}^ {k}\Big{]}=-\
abla\Big{(}v_{g}c_{g}^{k}\Big{)}-\
abla\Big{(}v_{l}H_{l}^{k} \Big{)}-\
abla N_{g}^{k}\pm\rho_{d}r_{g}^{k} \tag{2}\]  
where \(\phi\) is soil porosity; \(c_{g}^{k}\) is the molar concentration of gas \(k\) in a gaseous phase (\(k=1\), 2, 3, and 4 represent CH\({}_{4}\), CO\({}_{2}\), O\({}_{2}\), and N\({}_{2}\), respectively); \(H_{l}^{k}\) is the molar concentration of gas \(k\) dissolved in water; \(N_{g}^{k}\) is the diffusive flow of gas \(k\) in a gaseous phase; and \(r_{g}^{k}\) is the amount of gas \(k\) involved in MAMO.  
According to the energy balance, heat transport is modeled as follows [33; 35]:  
\[\frac{\partial}{\partial t}[E(T-T_{r})+(\phi-\theta_{l})\rho_{v}L]=-\
abla(- \lambda_{T}\
abla T)-\
abla Q_{conv}-\
abla Q_{hv}+Q_{oxi} \tag{3}\]  
where \(E\) is the heat capacity of soil; \(T\) is soil temperature; \(T_{r}\) is the reference temperature; \(L\) is the latent heat of vaporization of water; \(\lambda_{T}\) is the thermal conductivity of soil; \(Q_{conv}\) is sensible heat convection; \(Q_{ln}\) is latent heat convection; and \(Q_{oxi}\) is heat generation by MAMO.  
According to [PERSON] et al. [33], the biochemical reaction involved in CH\({}_{4}\) oxidation can be described as follows:  
\[\text{CH}_{4}+1.5\text{O}_{2}\to 0.5\text{CO}_{2}+1.5\text{H}_{2}\text{O}+0.5 \text{CH}_{2}\text{O}+\text{Heat} \tag{4}\]  
The CH\({}_{4}\) oxidation rate is mainly affected by water content and temperature in soil. The formulation proposed by [PERSON] et al. [14] is adopted to depict the CH\({}_{4}\) oxidation rate:  
\[r_{g}^{CH_{4}}=-f_{V,T}f_{V,m}\frac{V_{\text{max}}y_{\text{CH}_{4}}}{K_{\text {m}}+y_{\text{CH}_{4}}}\times\frac{y_{\text{O}_{2}}}{K_{\text{O}}+y_{\text{O} _{2}}} \tag{5}\]  
where \(f_{V,T}\) and \(f_{V,m}\) determine the effects of temperature and water content on the CH\({}_{4}\) oxidation rate, respectively; \(V_{max}\) is the maximum CH\({}_{4}\) oxidation rate per unit mass of dry; \(y_{\text{O}_{2}}\) and \(y_{\text{CH}_{4}}\) are molar fractions of O\({}_{2}\) and CH\({}_{4}\), respectively; and \(K_{\text{m}}\) and \(K_{o}\) are half saturation constants for CH\({}_{4}\) and O\({}_{2}\), respectively.