Understanding the leakage current caused by charge transport and local accumulation in dielectric oxides is critical for predicting and extending the lifetime of dielectric-based electronic devices. The internal interfaces such as grain boundaries (GBs) inside a dielectric induce local strain and charge segregation and thus further influence the charge transport behavior. In this work, we employ computational modeling based on the Schottky barrier model and nonlinear Nernst-Planck transport equationis used to study the oxygen vacancy transport and leakage current evolution in a SrTiO3 thin film under a DC bias with planar electrodes. It is found that in polycrystalline SrTiO3, the GB-bounded donors create an electric potential barrier and a local depletion region near the GBs, impeding the oxygen vacancy transport and suppressing the leakage current increase compared to a single crystal SrTiO3 thin film. The effects of temperature, the magnitude of an applied field, the number density of GBs, the GB-bounded donor concentration, and the depletion layer width on the leakage current evolution are systematically investigated. The simulation results are compared with the analytical solutions, as well as with existing theoretical and experimental reports. This work thus helps shed light to the grain-structure dependent electrostatic behaviors in dielectric thin films under different intrinsic and extrinsic conditions.