## 1 Introduction  
Coalbed methane (CBM) in coal seams is usually stored as free gas in cracks and pores and adsorbed gas on organic surfaces [1]. CBM production not only eliminates the threat of hazardous coal mine production and prevents gas over-limits, but also offers economic benefits [2, 3]. CO2 is a major greenhouse gas and the biggest contributor to climate change [4, 5, 6, 7, 8], because its adsorption capacity is higher than that of CH4. CO2 displacement can be adopted for CBM recovery [9]. CO2-enhanced coalbed methane (CO2-ECBM) technology involves the injection of CO2 into a coal seam rich in CBM, to sequester CH4, promotes clean green energy, and is widely used in the production of deep ultra-low permeability coal seams [10, 11, 12, 13, 14]. [PERSON] reported that CH4 could be effectively extracted from crushed coal by injecting flowing CO2 at ambient temperature [15]. [PERSON] injected CO2 waste from CBM power plants into a coal seam and produced more CH4, obtaining a competitive adsorption CH4:CO2 ratio of approximately 2:1 [16]. [PERSON] and [PERSON] found that H2S, CO2, CH4, and N2 expand to different degrees in coal seams, and the strain caused by CO2 adsorption was the largest [17]. [PERSON] conducted CH4 and CO2 adsorption experiments on coal at different temperatures and found that the equilibrium time of coal adsorption of CH4 and CO2 is negatively correlated with pressure and temperature and that matrix expansion leads to a decrease of pore width within coal seam fractures [18]. This leads to a significant decrease in permeability. The degree of coal expansion caused by CO2 is greater than that caused by CH4[19]. [PERSON] established a multi-component gas diffusion kinetic model based on a Bidisperse diffusion mechanism and Maxwell-Stefan (MS) diffusion theory to verify model effectiveness under pure gas diffusion conditions [20]. [PERSON] studied the feasibility of storing CO\({}_{2}\) underground while extracting CH\({}_{4}\) from coal seams and showed that CO\({}_{2}\) injection can increase gas production [21]. The results of Luo showed that vertical permeability heterogeneity can improve the transport capacity of CO\({}_{2}\) to a production well [22]. [PERSON] used numerical simulations to study the production effect of coal with different sorption times under CO\({}_{2}\) action. Their results indicate that the CO\({}_{2}\) injection capacity of coal with high sorption time is higher than that of coal with low sorption time [23]. [PERSON] established a fully coupled gas flow model based on double permeability diffusion, adsorption strain, and geomechanics and showed that the double-pore diffusion model better describes the diffusion process than the single-pore diffusion model [24]. [PERSON] established the hydraulic-mechanical-thermal coupling model of CO\({}_{2}\)-ECBM, considering the diffusion of CO\({}_{2}\) and CH\({}_{4}\) gas and non-isothermal initial temperature absorption of coal, and simulated the effect of injection pressure and initial reservoir temperature on CO\({}_{2}\)-ECBM [25]. [PERSON] established a fully coupled equation of gas diffusion, adsorption, seepage, and heat transfer and simulated the displacement process and effective influence radius of injected CO\({}_{2}\) under different pressure and temperature conditions [26]. The abovementioned studies on CO\({}_{2}\)-ECBM established the developmental foundation CBM displacement by CO\({}_{2}\). However, most scholars have assumed a single temperature or pressure parameter in the study of numerical simulation of CO\({}_{2}\)-ECBM, and most of them set the coal seam as a homogeneous body. In this paper, we established the THM coupling equation of CO\({}_{2}\)-ECBM for coal seams with non-uniform porosity and that are non-isothermal, and non-isobaric adsorption of binary gas is considered while combining percolation and diffusion.