You put a large protective sheet on the drive.
Popular way of cleaning rugs is full submersion, and you can do this on a driveway if you build the sides of your protective sheet - form a paddling pool.
That way you can flood it, scrub it, extract it, but then you have the problem of drying it.
The drying process needs to be reasonably quick, which can be a problem, but if you got a large thick pile rug it will weigh a lot even after extraction.
Unless there is at least 2 of you and you have a frame of some sort to hang it then think carefully before you do this.
It is very effective process though and an eye opener when you see it done for the first time.