Can you cut a swale leading the water to drain somewhere else?

Where's a picture that shows the overall problem and surroundings?

Do you know why it's collecting there in the first place?

Do you know what the development/county/city plan is for the surface drainage of your property and the surrounding property? If so, you should drain to there.

As for calculating gravel needs for a french drain, it's pretty much length * depth * width, minus the volume of the drainage pipe. Your edges aren't going to be perfect, so add in some extra for that.

However, a french drain may not be the right solution, and even if it is, you'll still need a way to keep silt and debris from building up inside it and/or a way to clean it out.

Drainage is complicated, and I would recommend either a ton of reading or consulting a professional if you want it done right.

This is one of the best comprehensive writeups that covers drainage. Don't be put off by the fact that it says it's written for golf courses. Drainage Systems for Golf Courses Dr. Ed McCoy Ohio State University