Thanks for the reply and suggestions.

I really wish I could go to the street, but there is no way. It would be way too costly, too much cement and walls between my back yard and front. Also, our housing track is depressed below the street that runs behind my back yard.

I like the idea of a dry well, but it has to be located in the back yard. The only spot for that would be to the right of the new area. How deep would the dry well need to be and would it be filled with crushed stone?

I also thought, why not form a cement trough, but bottomless and under that have about a foot of crushed stone and top it with a grate. This would travel the entire length of the new area rather than just the one spot of the dry well. The water would end up being more dispersed. Would this work?