We study the patterns observed in the vicinity of a Faraday instability, in the limit of a very thin layer of viscous fluid. We numerically solve our previous model [N.O. Rojas et al., Phys. Rev. Lett. 104, 187801 (2010)] and compare our predictions to experiments. Our model captures quantitatively the threshold of instability. The direct simulation of our system permits us to predict the patterns observed in experiments.