Who owns the Bottom of the River?

That makes no sense. The kidney doesn’t filter contaminants out of your pee, it filters contaminants out of your blood and INTO your pee. Are they implying that the river water consists largely of human blood?

Similarly, the Ohio river is not in Ohio (or Indiana or Illinois) because the Northwest Ordinance defined its namesake territory as north of the river.

Maybe you shouldn’t take statements about what happens on a flat circular planet supported by four (large) elephants, which in turn ride on the back of a giant turtle too seriously.

One shouldn’t expect Terry Pratchet’s writings to make sense. And I’m not sure that the inhabitants of the Discworld should be expected to know much about the function of the kidneys, either.