Don't know if it's an option for you, but my hen's coop and run are sited under a big old fir tree at the bottom of the garden. This is is very effective at keeping the run mostly dry in all but the most consistent of wet periods.
The ground had grass (all gone now
) but nothing else really grew there because it was so dry - anyway now it's just soil which I periodically rake and from which I regularly remove droppings. So far it seems to be ok!
