An earlier paper demonstrated a methodology for modeling the spreading process with a Gaussian random walk procedure, but was limited to the gravity-viscous spreading regime. Here we extend the methodology of re-presenting spread and transport of oil slicks on calm sea surface by updated Voronoi diagrams to account for the surface tension-viscous spreading regime as well. We have utilized the analogy between diffusion and spreading processes by defining a step length for the particle-based random walk scheme. In this study, calculation of the diffusive length is improved by including the surface tension term in the numerical solution method. The results from the numerical simulation of the spreading oil slick agree very well with the analytical solutions. The solution is robust in that good agreement is achieved for a large range of model and numerical solution parameters. This modeling procedure remains valid only for passive, quiescent spreading. The inclusion of spreading due to important horizontal and vertical turbulent shear processes in the Voronoi diagram paradigm remains a challenge for future work.