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So basically you are making either 90 or 110 petroleum gas per second.
Of course, if you are using advanced oil processing you would need to handle light and heavy oil outputs as well.
3 chemical plants making plastic would take 60 per second, so you should be making a bit of surplus over what you are consuming.
That is assuming that you are providing enough crude oil to the refineries of course but 200 per second shouldn't really be hard to get from 6 pumpjacks.
So what remains would be the piping, since pipes get worse as the "distance" grows (calculated in entities used so underground pipes are much better).
Even then, 200 per second would require a very lengthy pipe (over 1000 entities) before throughput being an issue.
You might want to include some screenshots of your setup because on paper what you described should work just fine.
Fluids aren't as intuitive as solids, both because the visual indication of a supply shortage is less obvious on pipes than it is on belts, but also because flow rate does not remain constant. Essentially, the more non-pump entities that the fluid has to pass though, the slower it will move. So if you have a very long line of pipes, especially regular pipes instead of max distance undergrounds, the flow rate through the pipes will slow to a crawl.
Also, some napkin math:
3 chem plants making plastic use ~60 petroleum per second.
Each refinery, using basic processing** produces 9 petroleum/s, meaning you need ~7 refineries just to supply those chem plants. In turn, they need 20 crude/s per refinery, or ~140 crude/s total.
Pumpjack output of crude oil per second is highly variable, so you need to look at your crude production to get an idea. Worst case scenario, 6 jumpjacks could be producing ~12 crude per second.
The other odd thing about pumpjacks/oil patches is that while it will never fully deplete, production will slow down to a trickle--often, about 2/s per pumpjack. So, much like an ore patch that has shrunk significantly, even if it is still producing raw materials, the output per second can drop below what your factory needs. And just like ore, the solution is to find another patch to supplement your supply. Though, switching to advanced oil processing ASAP helps, because even though it is more complicated you get far more resources out of the same amount of crude oil/refineries.
**I'm making the assumption that you are using basic. Please correct me if I'm wrong.
It seems like this mod[mods.factorio.com] is what you are looking for.
I doubt that 6 pumpjacks would be providing anything like that much crude. Oil wells near the spawning point (which is where I presume these are) rarely provide much over 10/s. A little further out, and you might get 15/s-20/s in the core of a deposit.
So 60/s is likely, 90/s-100/s is possible. That is almost certainly the bottleneck, and it's only going to get worse as the wells deplete.
Once I get basic oil I need 24 refineries right away just to keep up with science and expansion.
[EDIT]I have to add, though, that global bonusses add into this, in this case extra productivity for mines and to my knowledge, pumpjacks also benefit from this[/edit]
You should also check if your refineries and plants get enough water, sometimes that's the bottleneck and all you need is an aditiona pump have the way. Or maybe you do have a pump but it lost connection to power. There could be a lot of things that slow you down without even noticing, so hwat shadain said is probably the best approach: Check everything in altmode and identify the bottlenecks. You don't need 10 refineries for 6 jacks, something is definately off.
Not a huge deal since you are using punctuation properly but I know some people just directly skip big blocks of text so it's a bit detrimental when asking for help.
Example: Let's say there's an 11 tile gap between two structures containing fluid. Say, a pump pushing fluid into an assembler or chem plant. If you connect them using normal pipes, there's 11 pipe entities in between the structures, so the flow rate is ~1300 fluid per second. If you connect them with max distance underground pipes, you only need 2 pipe entities, meaning the max flow rate is ~3000 fluid per second. In this example, the undergrounds are about 3x faster, but either setup should have a high enough flow rate for the task at hand. But sometimes you need a very high flow rate, or to cross hundreds of tiles.
https://wiki.factorio.com/Fluid_system
As for why your production temporarily spiked, it's hard to say for certain, but most likely the act of switching out the pipes allowed a fluid buffer to build up, which slowly drained as it was consumed faster than it was produced, eventually leaving you with the same shortage as before.
Also, if you really aren't a fan of fluids in Factorio and want to look at other mod options, especially for future playthroughs, there's always this:
https://mods.factorio.com/mod/omnimatter_fluid
IMO, its a little better balanced than the one Fel linked, though I probably wouldn't try adding it to an existing run. It may actually be a bit harder than playing vanilla, especially at the start when you are trying to keep your boilers topped off.