## 1 Introduction
Among them, the CSEMP (CO\({}_{2}\) Storage and Enhanced Methane Production) project on which this study is based may be a relatively rare shallow low-rank coal CO\({}_{2}\) injection test project. From the Langmuir adsorption curve, we have known that the CO\({}_{2}\) adsorption capacity of low-rank coal is very small when the pressure is low, and when the pressure increases, the adsorption capacity increases slowly, but the adsorption capacity of high-rank coal is very large when the pressure is low, and the adsorption capacity increases greatly with the increase of pressure. However, it is worth mentioning that the CO\({}_{2}\)/CH\({}_{4}\) adsorption ratio of low-rank coal is much larger than that of high-rank coal, which highlights the superiority of CO\({}_{2}\) replacement of CH\({}_{4}\) in low-rank coal. The gas migration in the CO\({}_{2}\)-ECBM process is a very complex process, including the competitive adsorption, diffusion, seepage, coal deformation and temperature exchange of coalbed methane and injected CO\({}_{2}\) gas. Based on the laboratory scale, previous scholars have carried out many studies on the adsorption characteristics of CO\({}_{2}\) and CH\({}_{4}\), the mechanism of gas flow in coal as well as the replacement efficiency and utilization rate of CO\({}_{2}\) by coal. [PERSON] [8] carried out an experimental research into the sorption properties of coal briquettes under the various states of stress, proved that the existence of a relation between the stress state, sorption capacity and volumetric deformation of coal. [PERSON] et al. [9] carried out an experimental study on the displacement of methane by CO\({}_{2}\) using eight bituminous and anthracite coals, to evaluate the displacement efficiency and utilization ratio of CO\({}_{2}\). [PERSON] et al. [10] explored the characterizations of methane desorption and CO\({}_{2}\) displacement under different CO\({}_{2}\) injection pressures or different experimental temperatures using the self-designed NMR-based CO\({}_{2}\) displace methane equipment. [PERSON] et al. [11] proposed a new multi-scale dynamic diffusion-percolation model to compare the old and new model analysis, as well as carefully studying the mechanism of gas flow in coal. [PERSON] et al. [12] studied the the influence of coal water content and gas injection pressure on CH\({}_{4}\) replacement rate and CO\({}_{2}\) injection ratio based on the adsorption characteristics of coal seam gas injection. In addition to conducting physical experiments, many scholars have conducted in-depth research on CO\({}_{2}\)-ECBM using numerical simulation. [PERSON] et al. [25] carried out research work to understand the technical feasibility of CO\({}_{2}\) driven enhanced CBM recovery in Indian coals using a commercial reservoir simulator, COMET. [PERSON] et al. [26] performed a historical fit to the Pump Canyon ECBM field test in the United States using PSU-COALCOMP, where the anisotropy of the reservoir was considered in the model, and the study showed that a more accurate set of parameters could be determined using numerical simulations. The gas migration in the CO\({}_{2}\)-ECBM is a process of multi-phase fluid and multi-physical field coupling. In recent years, scholars have also done some work on CO\({}_{2}\)-ECBM multi-physics coupling. The effect of the numerical case is different based on the difference in coal reservoir structure and physical properties. [PERSON] et al. [27] established a gas-solid coupling model to study the migration of binary gases in coal reservoirs, and clarified the distribution of coabled methane concentration in CO\({}_{2}\)-ECBM, but did not consider gas diffusion and competitive adsorption of binary gases. [PERSON] et al. [28] carried out large-scale laboratory tests and deduced the seepage diffusion control equation, but did not consider the deformation of coal. [PERSON] et al. [29; 30] established the coupling equations of the diffusion-adsorption-seepage-heat transfer fields of gas.