I think this is how flow works in this game..
The water source is infinite, but it's not all flowing into your section. For the water to flow in, the square next to the brook must not be 7/7. If it is, that water has to go somewhere first. It won't if the squares next to that are 7/7. Keep going down the line you have a big sequence of 7/7 water that's not really doing too much. Slowly you'll find 6/7's, then 5/7s, 4/7's. Each of those are filling up to their neighbors, and dumping into their lower neighbors. At some point from the source, this will reach and equilibrium.
Another way:
Initially when you tapped into the source, you had 7/7 next to 0/7, so the 0/7 quickly filled to 7/7. One square down from that you had 0/0 next to something quickly filling to 7/7, so it also filled to 7/7.. slightly more slowly. One square down from that you had 0/0 next to that slightly more slowly filling square, so it too filled completely, although slightly more slowly still. Work your way down, eventually you get squares filled so slowly that evaporation come into play, and EVENTUALLY you'll hit an equilibrium.
Again I'm not sure, but I think the flow direction of the brook can help provide the push needed.. There have been cases where the trench that I dug got its on waves, and there have been cases where the water appears stagnant, in spite of the fact that water must have been moving. I'm guessing you have the second case.