I am putting lime in the barrel, I made lime from marble so I don't even have limestone. (and it's not quicklime either)
I crushed the rocks to make lime and put the lime inside the barrel in the ratio I described earlier. At first I thought that maybe I need to wait for the water to mix and waited for two in game days just to be sure, but by doing a bit of research it seems that I should not need to wait.
Also, I am playing without mods.