Depends. If it drifted long enough all the tannins may have leached out already; usually wood in this state is gray and bone-like in appearance. If your water is very hard this may also prevent the water from changing color, by causing the tannins to precipitate out.
I think it also matters what type of wood it is . . . IME, some woods leach more than others . . . generally speaking, I have found those that are self-sinking (malaysian in particular) leach more than those that need to be weighted down