Can't seem to gain any kind of health bonus from rivers. Doesn't matter if the city is directly next to the river or within its radius. Doesnt matter what techs the civ has or what buildings exist in the city. Doesnt matter if there are roads or no roads. Doesn't matter if a citizen is working the tile or not. I tested this on a custom map using the world builder to change all these settings, so there is a possibility that there is a problem with the world builder, not the rivers themselves. See my next thread for a related issue.

Its a world builder problem afterall. When a city is built, the game makes a one time check for fresh water. If it is present, the city has the fresh water property. If no fresh water is present, then no fresh water bonus. After a city is built, this can't be changed from within the world builder as far as I can tell. If fresh water is added (or removed), there is no effect on this property. Hopefully, its a simple variable somewhere, and a script tied to an event can toggle this value, which could be usefull in a scenario (the dead sea for example, wasn't always "dead").

I am not so sure it is a WorldBuilder problem. Look at this screenshot here:

http://forums.civfanatics.com/showthread.php?t=140521

All floodplains, almost all with river. Yet, the rivers do NOT seem to be overcoming the .-4 health of the floodplains (note that the city is -4 health).

I am not so sure it is a WorldBuilder problem. Look at this screenshot here:

http://forums.civfanatics.com/showthread.php?t=140521

All floodplains, almost all with river. Yet, the rivers do NOT seem to be overcoming the .-4 health of the floodplains (note that the city is -4 health).

River = +2 Health (this bonus is only counted once, not per tile)
Floodplain = -0.4 Health per

There's something like 18 floodplains around that city, 18 * 0.4 = 7.2 (rounded to 7).

The fresh water bonus is a single +2 health bonus for the whole city. Whether it borders one river tile, ar is surrounded in river tiles, it is still just a +2 health bonus. On one of the difficulty levels between noble and diety (not sure which), you are granted 2 free health points. These two, plus the + 2 for fresh water, are the reason why the example you linked to has +4 health. Health penalties for floodplains however, are cumulative. That's why the city has +8 unhealthiness points, and why the city is doomed from the start to fail.

River = +2 Health (this bonus is only counted once, not per tile)
Floodplain = -0.4 Health per

There's something like 18 floodplains around that city, 18 * 0.4 = 7.2 (rounded to 7).

Ah... OK.

Was not aware that the +2 health from rivers was a 1 time thing. Makes sense now.