Considering both fracture toughness and rock strength criteria, a numerical model of an elliptical-circular composite wellbore was developed. The displacement discontinuity method (DDM) was applied to investigate the fracture initiation pressure and initiation location under various in-situ stress regimes. Simulation results reveal that minor variations in wellbore geometry significantly influence fracture initiation behavior. Under near-isotropic stress conditions (deviatoric stress to mean stress ratio S/P<0.1), initiation characteristics are predominantly governed by wellbore geometry. In high deviatoric stress regimes (S/P>0.5), the stress state becomes the dominant controlling factor for initiation locations. In intermediate stress conditions (0.10≤S/P≤0.50), both wellbore geometry and stress state jointly influence initiation positions, with X-shaped multiple fracture branches being more prone to initiate from non-circular wellbores under such stress configurations. Furthermore, X-shaped fractures preferentially initiate from enlarged borehole sections. It`s provides a novel methodology for characterizing fracture initiation behavior in non-circular wellbores under complex stress fields, offering significant implications for optimizing fracturing designs and enhancing reservoir stimulation efficiency.