## 1 Introduction  
The urbanization and increasing growth of the planet's population accumulate significant volumes of disposed waste and increase risks on human health and environmental safety [1, 2]. Municipal solid waste (MSW) landfills are a major source of methane (CH\({}_{4}\)) emissions, which significantly contribute to global warming [3, 4, 5]. CH\({}_{4}\), a potent greenhouse gas, has a global warming potential approximately 28-34 times greater than that of carbon dioxide (CO\({}_{2}\)) over a 100-year period [6]. The fugitive emissions of CH\({}_{4}\) from landfills pose serious climate threats and increase the risk of fires and explosions due to their high flammability. To mitigate CH\({}_{4}\) emissions and utilize them as a resource, landfill gas (LFG) collection systems are typically installed. However, these systems may exhibit low collection efficiency, and installing them in older landfills is frequently not economically viable [7, 8]. Consequently, an alternative approach involves utilizing microbial aerobic CH\({}_{4}\) oxidation (MAMO) in soil covers to degrade CH\({}_{4}\)[9]. This biological CH\({}_{4}\) oxidation process consumes CH\({}_{4}\) and oxygen (O\({}_{2}\)), yielding water, heat, CO\({}_{2}\), and biomass [10]. Optimizing soil properties, such as porosity, moisture content, and nutrient levels, can enhance the activity and efficiency of microbial communities responsible for CH\({}_{4}\) oxidation,thereby improving CH\({}_{4}\) mitigation in landfill cover systems [11; 12]. Previous studies have demonstrated that MAMO occurring within the top 1 m of landfill cover soil can reduce CH\({}_{4}\) emissions by 10% to 70% [6; 10]. Investigating LFG transport in landfill covers requires a dual consideration of both MAMO and soil properties. The MAMO in landfill covers significantly alters the gas composition, primarily reducing CH\({}_{4}\) concentrations while increasing CO\({}_{2}\) and water vapor levels. The efficiency of MAMO is influenced by various factors, with soil temperature and soil moisture content being the most critical [13; 14]. Soil temperature typically ranges between 25 and 35 \({}^{\circ}\)C, which is optimal for the metabolic activity of methanotrophic bacteria [10]. This temperature range ensures that microbial enzymes function effectively, facilitating the oxidation process. Similarly, soil moisture content must be maintained within an optimal range to support microbial activity. Adequate moisture ensures that bacteria have the necessary water for metabolic processes without creating anaerobic conditions that inhibit aerobic oxidation. Excessive moisture can lead to waterlogging, reducing O\({}_{2}\) availability, while insufficient moisture can limit microbial activity due to desiccation [9]. The permeability characteristics of cover soil directly influence the transport of water and gases, which in turn indirectly affect the supply of moisture and O\({}_{2}\) necessary for CH\({}_{4}\) oxidation activities [15; 16; 17]. The gas permeability coefficient of soil is regulated by its moisture content. Research conducted in laboratory and field settings indicates that when soil saturation exceeds 85%, the gas permeability coefficient significantly decreases [15; 18]. This reduction impedes gas convection and diffusion within the cover, thus impacting the efficiency of gas transport and the CH\({}_{4}\) oxidation process [19; 20; 21]. Landfill covers encounter complex environmental conditions. The upper section is directly exposed to atmospheric influences, while the lower section faces erosion from the high temperatures and elevated humidity produced by the degradation of waste [22; 23; 24; 25]. A theoretical model provides a feasible approach for studying the intricate interactions of water, gas, and heat transport, as well as the associated mechanisms of MAMO [26]. For now, a variety of theoretical models have been developed to analyze LFG transport in landfill covers. [PERSON] et al.