## 1 Introduction  
Currently, energy demand is constantly increasing due to the increase in population. To meet the growing energy demand, the extraction of coalbed methane has become necessary. Globally, the total amount of coalbed methane in coal (1730 T tons) is approximately 30,000 TCF [1]. As is well known, coalbed methane is present in the adsorbed state within the pores of a coal matrix and in a free state in coal pores and fractures, along with a small amount in a dissolved state in coal seam water [1; 2; 3]. The accumulation of coalbed methane is a prerequisite and foundation for its exploration and development [4]. From the 1980s to the early 21 st century, the exploration of coalbed methane with depths ranging from 1500 to 3000 m was conducted in North America [5; 6; 7]. Scientists estimated the resources of coalbed methane in the major basins of North America and pointed out that the development of deep coalbed methane was an important direction for developing unconventional natural gas in the future [8]. The test project consisting of 65 coalbed methane wells (with coal seams buried at depths of 1635-2591 m) achieved success in the Picence Basin [7; 9], indicating the prospects for the exploration and development of deep coalbed methane. In China, a large-scale deep coalbed methane block was built on the southern margin of Yanchuan [10]. Since 2021, several coabled gas wells with a depth of more than 2000 m have produced tens of thousands of cubic meters of gas per day in the Junggar Basin and the eastern margin of the Ordos Basin [11; 12; 13]. Therefore, the study of methane adsorption characteristics in deep coal reservoirs is a hot topic, which can help predict the total exploitable methane content in deep coal reservoirs [14; 15]. Experimental testing and numerical simulation are the main approaches used to study methane adsorption on coal. Previous studies have shown that the methane adsorption capacity on coal is controlled by internal factors such as coal properties, macroals, rank, deformation, and moisture [16; 17; 18; 19; 20; 21] and various other external factors such as temperature and pressure [22; 23; 24]. [PERSON] et al. [16] found that the adsorption capacity of dry coal for methane varied during various stages of the rank of coal. For example, in the low-rank stage of coal, the methane adsorption capacity on coal increased with the increase in the rank of coal. In the middle- to high-rank stage of coal, the methane adsorption capacity on coal showed a U-shaped variation trend with the increase in the coal rank. There is a minimum adsorption capacity near the highly volatile bituminous coal. Vitrinite has the strongest methane adsorption capacity, while exinite has the weakest methane adsorption capacity among the three petrography compositions [17; 18; 19]. The micropores (<2 nm) often result in the largest proportion of the total specific surface area and are the major contributors to the adsorption of methane [17; 20]. Coal with a high content of aromatic carbon has a strong adsorption capacity, whereas coal with aliphatic structures and oxygen-containing functional groups exhibits a weaker capability for methane adsorption [25; 26]. Pressure has a positive effect, whereas temperature has a negative effect on the adsorption of methane [13]. There are various models used to describe the adsorption of methane on coal, including the Langmuir model, BET model, D-A model, and D-R model [1; 3; 4; 27; 28]. The Langmuir model has been widely applied due to its simplicity and the practical value of its parameters. With the continuous deepening of coalbed methane exploration and development, the adsorption characteristics and reservoir formation mechanism of deep coalbed methane have become increasingly valued. However, changes in pressure and temperature with the increase in burial depths affect the adsorption characteristics of coal and the occurrence state of coalbed methane.