Monte Carlo calculations of diffusion across membranes containing impermeable flakes show effects of tortuous paths around the flakes, of diffusion through slits between flakes, and of constricted transport from entering these slits. The calculations show that a simple analytical equation developed by Aris reliably predicts these three effects. Both the calculations and this equation show the separate conditions when each of these effects is most important. The results have value in barrier packaging, in clothing useful in chemical warfare, and in transport across human skin.