<p>As the drilling depth increases in nearly horizontal drilling, the drill-string system experiences great external forces, leading to heightened drill-string vibrations that occur in lateral, axial and torsional dimensions simultaneously. A comprehensive understanding of the multi-dimensional coupled effects within these vibrations is essential to disclose the drill-string dynamic for effective control. This paper establishes a multi-degrees-of-freedom drill-string vibration model through Lagrange analytical mechanics, explicitly characterizing the multi-dimensional coupled effects in nearly horizontal drilling. The model incorporates nonlinear coupled mechanisms between axial cutting and torsional friction during rock breaking process, while introducing an lateral contact model with a judgment function to quantify the intermittent contact behavior. Numerical simulations with actual drilling data verify that our drill-string vibration model outperforms the existing torsional, lateral-torsional coupled and axial-torsional coupled models. It is found that lateral borehole wall contact destroys axial-torsional periodicity through multidimensional coupled mechanisms, while increasing depth induces simultaneous vibration intensification across all dimensions.</p>