You don't say where you are at, but around where I live, Tree removal services chip all the branches from their jobs. The chips are available FREE OF CHARGE. You can create what is known as DEEP LITTER in your run. The top will be dry because the water will drain down. Also handles chicken droppings naturally. For more Ideas read up on deep litter in some of these threads. I have every so often a load of these wood chips dropped off at my yard,, in the low area where grass does not want to grow well, I put a layer of chips about 6 inches or so. It keeps weeds from growing, Looks clean, dogs and cats can perform business there, chickens free ranging find all sort of activity scratching, looking for worms. In time all changes and turns back to earth. Consider something like this for your run. I think it may be just the solution.
