Carbon Dioxide-Enhanced Coalbed Methane (CO2-ECBM), a progressive technique for extracting coalbed methane, substantially boosts gas recovery and simultaneously reduces greenhouse gas emissions. In this process, the dynamics of coalbed fractures, crucial for CO2 and methane migration, significantly affect carbon storage and methane retrieval. However, the extent to which fracture roughness, under the coupled thermal-hydro-mechanic effects, impacts engineering efficiency remains ambiguous. Addressing this, our study introduces a pioneering, cross-disciplinary mathematical model. This model innovatively quantifies fracture roughness, incorporating it with gas flow dynamics under multifaceted field conditions in coalbeds. This comprehensive approach examines the synergistic impact of CO2 and methane adsorption/desorption, their pressure changes, adsorption-induced coalbed stress, ambient stress, temperature variations, deformation, and fracture roughness. Finite element analysis of the model demonstrates its alignment with real-world data, precisely depicting fracture roughness in coalbed networks. The application of finite element analysis to the proposed mathematical model reveals that (1) fracture roughness ξ markedly influences residual coalbed methane and injected CO2 pressures; (2) coalbed permeability and porosity are inversely proportional to ξ; and (3) adsorption/desorption reactions are highly sensitive to ξ. This research offers novel insights into fracture behavior quantification in coalbed methane extraction engineering.