I think the mulch might be a good idea if you want to dig it out every now and again to compost it. With the river sand, the rain comes and the poop goes.
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So, is mulch safe then? I've been spending the last couple weeks shoveling out mulch from where the run will be in our new coop area thinking that they wouldn't like all those big wood chips. If it's ok then I'll just leave it. I'm not sure what type it is, but I know it's been down for over a year now.
I highlyrecommend sand. It's what we use in both of our runs & we have the same set-up you're describing with the plastic sheeting in the winter. It does freeze if it has gotten wet but each spring we simply till it and add a fresh layer of new sand. I have read horror stories about gravel, particularly gravel under sand as it becomes a mixture very quickly, with the gravel working its way up through the sand. Gravel is also tougher on the chickens' feet. Plus, chickens love to dustbathe in the sand (most months of the year when it is not frozen).
You can see our coop set-up and the slope of our property in my coop tour here:
http://eggcartonlabels.blogspot.com/2011/05/virtual-tour-de-coop.html
Best wishes with your project!