I have decided to change this to increasing by 1d6-3 each non-visited month. This, while averaging out to the same increases over time, should generate more volatility and less predictability.
(Note -- click on map for larger image; or click twice for much larger image.)
I have also decided to allow players to spend a month "traveling the river" instead of visiting a district. This will allow them to move from one district bordering on the river to any other district bordering on the river. Such a journey would take a month. Note that such travel may only take place on the "blue" portions of rivers -- not the white "headwater" sections.