I got 4 desert tiles in my capital city and a great engineer.
Assuming none of these are also Flood Plains, you will get
up to an additional 4 hammers and 4 food per turn from building petra, in addition to the extra culture, the gold (or hammers or food) from the extra trade route, and the super extra culture after you discover Archaeology. So, is that worth the Great Engineer to you?
I say
up to 4 additional hammers and food because if your city is not working any of these tiles, then there's no benefit from the surrounding desert terrain tiles related to constructing Petra. If the city does not work any of these desert tiles for quite some time, was the expenditure of the Great Engineer worth it to you? Only you can answer, but you do need to make that part of your decision as to whether to pull the trigger on Petra.
Petra also gives you 1 Great Engineer point per turn in the city, so at least theoretically you will 'get back' that Great Engineer, but there is no guarantee this will be the case in practical terms. Without other modifications to great people generation, 1 GPP equates to 200 turns on standard speed will be required to 'earn back' that great engineer if you have no other wonders or buildings adding more great engineer points in the city.
I like Petra, myself, but I only really try to go for it (or rush it with a Great Engineer) if I have a city (capital or otherwise) that has substantially more qualifying desert tiles than the four you mention. Otherwise, I skip it for something else, Wonders-Wise. I'll save the Great Engineer, myself, for something more juicy, like Sistine Chapel, if I don't have enough desert tiles around one of my cities to make it worthwhile by the way
I calculate "worthwhile".
However: (a) everyone's definition of "situationally worthwhile" is going to be different, and (b) my views on these things are often not very much an overlap to the usual forum conventional wisdom.
If you are playing G&K, you don't get a trade-route or Caravan unit, you get a free Amphitheater Building and an additional Gold on every qualifying desert tile. I'm not convinced the extra 3 Culture you got from having the free Amphitheater Building wasn't better in the long-term than the extra Caravan you get in BNW.