I played around with this a bit last night after reading about irrigated/non-irrigated farms.
Once you've discovered biology you can build a farm on any flat plains or grassland. But unless this is connected via other farms to a fresh water source (river/lake) it will be a non-irrigated farm.
If you have a chain of farms stretching out from a lake, and you then break the chain by removing a farm (replacing it with another improvement), the now unconnected farms will all loose 1 food production. In some cases this can lead to cities starving.