A phenomenological model of grain-boundary diffusion and electromigration in thin-film diffusion barriers for microelectronic device metallizations is developed by using the transmission line matrix method. Both diffusion and drift effects are included in the numerical analysis with a multidimensional variable mesh structure. Special attention is paid to the "vertical" electromigration effect and a comparison of the two-dimensional (2D) and three-dimensional (3D) simulation results. Two fundamental driving forces for impurity grain-boundary diffusion, the concentration gradient across a diffusion barrier, and the current flow perpendicular to the deposited metal films can reinforce each other, leading to an enhanced impurity diffusion and a reduced barrier efficiency. Such material transport through the diffusion barrier is quantified with a 2D model of parallel planar grain boundaries and a 3D model of rectangular columnar microstructures. Concentration profiles along all directions are visualized in graphs for the cases with and without the elecromigration driving force. Average atom penetration as a function of time and the underestimation by the 2D model are also illustrated.