And in order for a river to flow, it needs a source and a mouth.
How about a waterfall on the top of the map from chunk of forest in front of tribrush via a natural spring on the top.
And that way the river could naturally flow downwards. Instead of a patch of grass and earth splitting two pools of water in the mid lane, we could have mid lane be a bridge over water. A stone bridge, old. Ancient really. With overgrown vegetation and vines making it look ancient, and keeping the ground palette for the bridge matching the rest of the ground elsewhere on the map.
And at the bottom chunk of forest in front of tri brush we could have a cave where the water flows into. And when baron spawns the river becomes angry and bats fly out of the cave. And maybe the baron could bleed in the water, and carry it down the map slowly, to make "sneaking" unwarded baron even riskier.