Even though the well water doesn't have chlorine in it, it may have a different ph than what he was in before. You might want to do some searching on the net and find out what ph bettas prefer. Then you can get a test kit at the
pet store and find out what ph your water has. I've never kept my bettas in a bowl before so I can't give you much advice on that. Here's a really good site to check out:
http://www.bettatalk.com/
Bettas like warm water too, so if there was a temp change, that could be affecting him. Good luck and I hope he's ok!