This paper proposes a high accuracy numerical method to model oil spill trajectories using a particle-tracking algorithm. The Euler method, used to calculate oil trajectories, can give adequate solutions in most open ocean applications. However, this method may not predict accurate particle trajectories in certain highly non-uniform velocity fields near coastal zones or in river problems. Simple numerical experiments show that the Euler method may also introduce artificial numerical dispersion that could lead to overestimation of spill areas. This article proposes a fourth-order Runge-Kutta method with fourth-order velocity interpolation to calculate oil trajectories that minimize these problems. The algorithm is implemented in the OilTrack model to predict oil trajectories following the "Nissos Amorgos" oil spill accident that occurred in the Gulf of Venezuela in 1997. Despite lack of adequate field information, model results compare well with observations in the impacted area.