If you were using the bark mulch and it got thin, then just add some more. If it is deep enough, then your drainage will be under control. If you use wood shavings, they may float away if you get a puddle in your run. The wood shavings are good inside your coop where they stay dry. The bark mulch is slightly heavier and I think will not float away, but rather absorb the water and stay put. DON'T BE WORRIED ABOUT THE LEAVES FALLING INTO YOUR RUN.. It is all part of nature and the leaves will break up in time.. You will not be able to know they were ever there. I know you are concerned about mold. Remember that mold is present and occurs in nature on a continuous basis. NOT ALL MOLD IS BAD. We try to avoid it when possible, as a precaution. That is all good. As to your chickens,.... just let them live a natural life. Don't try to raise them as if they were (the boy in the bubble).. Chickens are more resistant than many people believe them to be.
