The dynamics of droplet spreading on two-dimensional wicking surfaces were studied using square arrays of Si nanopillars. It was observed that the wicking film always precedes the droplet edge during the spreading process causing the droplet to effectively spread on a Cassie-Baxter surface composed of solid and liquid phases. Unlike the continual spreading of the wicking film, however, the droplet will eventually reach a shape where further spreading becomes energetically unfavorable. In addition, we found that the displacement-time relationship for droplet spreading follows a power law that is different from that of the wicking film. A quantitative model was put forth to derive this displacement-time relationship and predict the contact angle at which the droplet will stop spreading. The predictions of our model were validated with experimental data and results published in the literature.