@ Sarithus. I think Chick is talking about the red river.
Also I took look at the map of middle earth, didn't notice the link before. If you look closely you see that the river in the north west is a confluence of multiple rivers coming from the mountains. The other river could be similar or it could be an outflow that goes to the ocean or the sea, we cant really know because it isn't shown. I bet though that it is actually a river running towards the lake. The way I think about rivers is the cardiovascular system. Basically rivers are like the capillaries of the veins they are small at first then the join together getting bigger then those join together becoming the veins.