I would strongly urge you to ask your questions of the authorities in charge of that land. They will gladly tell you what you can and cannot do. It is a lead-pipe cinch that if you do something that is illegal, they will be all over you when they find out, as they are likely to do if they are actually doing their job, and when they do, they will have no patience for your ignorance. Nor will the judge if/when your case comes up in court.
There is a widely held, but demonstrably false, belief that you can get away with things if "they" don't have any inkling you are doing them. This occasionally works, but boy, if they catch you anyway, they will throw the book at you.
The simple truth is that the legality of dumping gray water on the ground varies from state to state and location to location within the state, as well as with the amount of gray water we are talking about. 500 gallons a week is a LOT, and will be noticeable in most locations. Far, far better for you to find out that you can't do that and make alternate arrangements than to get caught doing it.
