The economic spacing of horizontal wells in a "shale" is a practical max problem. [First semester calculus] You want to get the most profit. Closer than about 2000 feet and the wells start to interfere (cannibalize or steal from each other). However, each new well bore does break (literally, that is what hydraulic fracturing is) new rock. So it is not 100% cannibalization. But they cost money. Based on the price of oil/gas, and the cost of drilling wells, there is an optimal density. You will always "leave some behind" and always "have some cannibalization". In addition, the cost function does not scale exactly. Each new well bore (drilling) is essentially a scaled cost. But the costs of the "pad" (road, permit, etc.) are essentially constant (thus divided over more wells). Also, fracturing and even drilling have some cost advantages when done in a campaign (less equipment moves).
After solving a base case, you can do some interesting questions. Like "is it possible to drill 6.5 wells (no) in a unit. So then they have to pick 6 or 7 and see which is better. Or you can discuss unitizing sections (combining two units, with the right leases/permits) to allow drilling "13 in a double wide". Plus you can do useful variances from the base case (what is the optimal density if oil price is $150/bb? How about $50?)
Personally, I think the easy thing is to just write out reasonable functions yourself, than do the analysis. But here are some papers. (Not sure they are the best, so many junky little papers!)
Maybe there is a little local connect for your kids, since you have the Utica in Ohio...although it's on the other side and further south.
P.s. In general, I'm not a fan of this sort of thing. I think research oriented teachers are too interested in the research and also don't realize that the classical examples are new/interesting to their students, even the techniques are new to them. So, it ends up being too much "push" from the teacher and a diversion. You do seem very organized in teaching your kids and getting results, so maybe you have the slack to spend time on this sort of thing. But, still..."Danger, Will Robinson!"
P.s.s. In practice companies aren't really doing calculus. They just do a spreadsheet and look at 1-20 wells per section and pick the max profit. Where added oil and diminishing returns counterbalance. I'm not sure this is even cutting edge "research" really, since companies can do it without the literature. It's just interesting FP&A. But it's definitely calculus driving the bus in terms of the math that actually affects the economics. So an econ grad student would think of it that way, write about it that way. In a sense, there are some beautiful equations and functions and calculus concepts lurking behind the matrix of the Excel. And I think it helps as an engineer/executive/consultant to have some functional insight as to what is going on, not just the black box of the Excel tables.
(Another P.S.) I understand your joy/pride in the collaboration with your sister, but I think it would be better form to lead with the question, rather than your anecdote. Looking, now at the commentary, I see a lot of discussion of the mathematics details of the problem itself. Almost as if this was more the interest than the "math ed" part at the end. So that it feels to me, more discussion of the math itself, rather than education. As a rule-breaker, of course, I applaud your (mis)use of the forum, to accomplish any objective, rather than the stated objective. All that said, I would just be very wary of this sort of "I like this hard math" that does not translate well to kids learning things (just the chain rule is new to them).
(Yet one more) If you don't like the shale industry (I love it, see capitalism in action), you can search on environmental cost benefit analysis. The Google Scholar returns lots of academic lit (if that's important to you). The regular Google search returns some case studies and the like, which may be more quickly relevant. Here is one review:
(not immediately usable for a class, but gets you into the lit, since it's a review article. Really I think a book chapter or simple case study is better than nitty gritty cutting edge research.)