No chicken run has grass in it for very long. In any confined space, chickens will eat grass down to the roots (and below) in no time. It is important to make sure the run is large enough for the number of chickens in it (not less than 10 square feet per bird, and more is much better), and that you provide "enrichment" activities. For chickens, I think this primarily means providing them opportunities to scratch and look for food. I bag and save leaves in the fall and have found that a bag of dry leaves dumped in the run is excellent chicken entertainment. I also use a day tractor so our flock does get out on grass most days. And for the days they don't get out, I clip up leaves of romaine or swiss chard in the run for them to peck at. Since we are now getting way more eggs than we can eat, I also feed back hardboiled egg to the flock in their treat pan and toss down a handful of scratch in the run for them to hunt for.