I don't have any prior experience with this problem but I know your pain. My coop and run are on the north side of my woodshed and our property is FLAT so rainwater just stands until it dries up. I have been looking into the same type of setup and right now my plan is to cover the bottom of the run with hardware cloth, fill the bottom frame of the run, made of 1x6' pressure treated lumber, with a few inches of gravel, and add in enough topsoil to make a nice, semi-compacted "floor" before covering it with cardboard and then hay and attempting the deep litter method. I thought about putting down a weed barrier, but I believe it would be too easy to dislodge while raking out the composted hay and waste. Instead I'm going to use old cardboard because I've had good results using it in my garden. Last fall I covered my raised beds in cardboard, laid a light layer of mulch over them now the cardboard has completely disintegrated, and my beds are largely weed-free. So, I'm going to use them as a base for the deep litter, and by the time I need to deep-clean the coop they will most likely have degraded and been incorporated into the compost. Basically, I'm hoping to get the raised-floor effect while still being able to do the deep litter. I want BOTH the things! Hopefully there is someone here who has done this successfully!