How do you dispose of nuclear waste?

You don't, once it is created it exists forever (or close to forever depending on the amount of radioactive material involved). There is a process called the "half life', some isotopes degrade rapidly, others last for thousand of years. All you can do is isolate it, entomb it, and hope that it remains undistrubed for the next zillion years, just like the pharohs thought when they were entombed. The nuclear wastes cannot enter any underground river or water supply, since it would carry it elsewhere.