Drainage issues!

Hello, I need some advice regarding bogged up grassy areas. The lawn in my back garden has become waterlogged in parts and I put that down to the lack of drainage due to the concrete path that surrounds the grass. There is a run off drain in the garden which is 8 inches below the grass level but as can be seen in the pictures, the water cannot run to it. Would a French drain using pvc pipe and gravel be a good solution to this issue? I would dig a 8" trench through the concrete and have the pipe run straight to the drain and put a network of pipes under the grass in that area. Please find attached some images for clarification! Thanks

 

There's a few pics of a French Drain under construction in my gallery here

 

Would a French drain be an ideal solution for this situation? I don't mind the work involved as long as it fixes the issue. This is my other halfs parents house and they've been having this issue for years but they recently got a dog which because of the mud has been trailing paw prints throughout the house and causing enough frustration now to warrant trying to sort the problem. I've looked into them and they seem like the right path to with, however I wanted some clarification befor hand. Thanks!

 

As long as there's a fall into the drain, then yes. At the moment the water is backing up against the path with no where to go to.

 

The level that gully's set at looks odd (too low), but it may of had a plinth/raising piece around it which no longer exists and the existing paving has been dished around it. But it's enough to make me wonder whether ground/paving levels have been raised around the house (and whether that's contributing to existing problems or was a response to pre-existing problems).

Do you know where the dpc level is in relation to the top of that external insitu concrete paving?