Fill the ruts with salvaged pieces of drywall from a building site (typical house builders discard over a ton of drywall pieces), and then cover everything with wood chips (a tree trimming company will be happy to have a place to dump their loads). If you work it right, this solution won't cost you a cent, and depending on tipping fees at the landfill, you might even make a buck.
Over time, as the drywall decomposes, the calcium in it will help to loosen up all that clay.
Here is a promo piece by people selling gypsum, but since building sites are throwing it away (as drywall), why pay for it?