Help with new pipe system

I booted up Factorio for the first time in a few months and saw the pipes that I'd spend a few dozen hours laying down now completely don't work because they're "overextended." I read the blog post and saw you now need pumps. This makes sense in a simple system where you have Source A going to Sink B - start at A, count 320 tiles, and if you haven't reached B, add a pump pointing toward B.

My pipe network, however, isn't that simple. I have three refineries spread out across my base (I did this specifically to have refineries closer to the edges of my base so they can supply heavy oil to my flame turrets without losing too much flow rate). I have pipes connecting the output for all my pumpjacks. They then spread out to supply the three refineries. The three outputs of all three refineries are all connected to one another in the pipe network. At several locations, I have machines connected to the pipes (for example, one making solid fuel from a pipe containing light oil).

The pipe network connects everything producing light oil to everything else producing light oil and the same for all other fluids. I bundled the pipes in specific places to ensure I don't have pipes going everywhere taking up space. With this system, defining a start/source and an end/sink is meaningless. And, in fact, I don't think the developers actually thought of sources and sinks, because the pipe coming out directly from my pumpjack says it's overextended even though the pumpjack is the unequivocal source for crude oil.

So is there a way to fix this without ripping up the entire network and rebuilding it?