In addition to an enhancement in mechanical properties, dispersing clay platelets within a polymer matrix also significantly improves the gas barrier properties. Reducing the gas permeability of polymer in this way has applications particularly in the food and pharmaceutical industries. We present an automated Finite Element modelling scheme to predict the gas transfer properties of polymer reinforced with thin sheet-like clay nanoparticles. The method is based on a Representative Volume Element approach, where particles are randomly placed into a periodic cube of model polymer material according with a set of desired statistical distributions (particle position, size, orientation etc). The statistical distributions are based on measurements of the real material, or on idealized forms. The effective permeability of the heterogeneous system is determined via a mass diffusion problem in the commercial Finite Element package Abaqus, where it is assumed that the diffusivity of the matrix is the same as that of the pure polymer. For each set of particle statistics, the effective permeability is averaged over a large ensemble of randomly generated realisations. We investigate the effect of particle filling fraction, orientation (with respect to the diffusion direction), aspect ratio and particle agglomeration on the overall gas barrier properties. The Finite Element predictions are compared with those of simple analytical models, and with experimental observations.