Scattering Hay or woodchips on the really muddy areas can help absorb the moisture and avoid this in the future. Stones or gravel might also be good.Sketch of the run
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Below is the West side and where I started the trench. Partial there because I wanted to save my energy to do the North side. Very hard work, cutting into shale, but that it was raining helped - it makes the stones easier to dig into, and then I could see the muddy water move and clear water entering the trench. I had an adze, a shovel and a big long metal pointy thing to pound and break up big stones. I couldn't find our pickax.
Looking North-East, the driveway on the left, opening up to the turn-around /parking lot sort of straight ahead
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We just had more shale put on the driveway but not all the flat area beyond, you can see that darker spot is water pooled because it was just raining. The flat area was carved out of the hillside, and a French drain put in where the area meets the base of the hillside.
Continuing around to the North side. I tried to make a little trench out from the run because right in front of the dustbath pool was pretty wet. Water just filled in as I dug, so I stopped for now.
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The North-East really wet corner, and the trench meeting the French drain stones. That black pipe is the gutter take-away, just lying there right now. I found water pooling where it met the French drain stones so I thought it could be clogged there and dug into them a good six inches down and a foot in, and then water began draining away.
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Closeup on how wet it is there. This spot had the chicken run on it two years ago. I cleared away most of the litter when the next Spring came and I moved the run to the woodsy area. The run now has been there since last October. So the bottom of the new litter has composted and now it's all muddy muck?
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As you go around, the East side gets drier further South. The gutter take-away pipe has let out there and that water just goes down into the stones. So thinking the clogged French drain corner was a big part of the problem. I worry or wonder if the trench needs to be bigger and deeper maybe, in case water is flowing through the ground under the shallow trench and could seep up into the run corner? That corner under the predator skirt is mucky, muddy soil and decomposed litter.
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Here's the long view looking South more or less (the right side of the run toward the light is West). Water closer to the left side drains south and east, toward the French drain which the dismantled gutter is pointing down at. But where the water in the foreground is, covered by the new shale, I couldn't detect any water motion, but it does seep away eventually. Around where I was standing the water flows more West.
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Continue digging the trench, making it deeper, and then fill with stones / gravel? Clear out the muck under the predator skirt, replace with small gravel? What about inside the run, at least in that corner - gravel topped with bricks or pavers, and then litter on top? I'm thinking something easily removed in case that doesn't work well and I need to do something else?
To add: Cute Buff Orpingtons!
. Maybe if possible, build your structure higher up the hill, or build some sort of wall around your complex, channeling the water out and away from the run. I realize your substrate shale is hard to break up and dig into. Try to find that pick axe 
?
