We had a similar problem with ground water (after heavy rain) leaching into the run. The best solution is to run some type of french drain around it to channel the water down and away from the run. Our soil here is 50% clay and 50% granite boulders (at least it seems like it :-) so digging is EXTREMELY difficult. I wound up putting down treated 6x6's around the base of the run and am in the process of raising the dirt/grass around the 6x6's to try and channel the water down and away. It's working, but if we have a week or so of constant rain, which happened a lot this year, I still get some seepage into the run (maybe 5 to 10% gets wet)
You have a great covered roof, so you may also want to look at gutters to get the water away from the edges of the run.
Our run is on about a 15 degree slope going towards the back of the property so I put down a base layer of pea gravel (two truck loads) to try and raise/level the area, then have about 2000 pounds of pdz on top of it in our 10x20 run. Excellent in theory, but what I'm finding (and should've realized this before) is that the pdz is slowly working it's way through the gravel, so the gravel is now sitting on top, defeating what I'm trying to do with the pdz. When I scoop/sift the poop in the run now I get a LOT of gravel on the sifter with poop in it...so it all goes in a bucket and out to the trash. I figure in about 15 years I will have scooped all the gravel out and all will be good :-)
In your case, there will be a mixing of pebbles and shavings at some point, especially when the girls start digging/scratching out a dust bath. Not sure if that will be good or bad in your situation but something to consider.
I got one of these http://www.amazon.com/gp/product/B001HL3ZAI?psc=1&redirect=true&ref_=oh_aui_detailpage_o01_s00 to quickly fill holes and level the gravel/pdz in the run...it's worked great so far. With pebbles you may need to get one as well.