Every other Friday off is 26 days off per year, the equivalent of 5 work weeks. Edit: oh I missed that part where you still work 80 hours in the 9 days. :(
The main problem is the culture of working through those Fridays off and many weekends. You may not be "required" to, but you're certainly expected to.
Keep in mind that I do not work in oil and gas and my knowledge is very limited, but his schedule seems to be very relaxed, more so than almost anybody else I know. I don't think I've ever seen him work on his Fridays off, and actually he frequently comes back early on the Fridays that he doesn't have off too. He also doesn't seem to work 9 hour days for the rest of the week either even though technically that's part of the 9/80 schedule.
In development and production, you're typically involved in smaller business decisions at regular intervals. In exploration, you do work to support very large (100's of millions to billions) business decisions with longer timelines, but very inflexible deadlines.
Normally, you'd have a couple of months of crunch time before a bid round/prospect assessment/etc. You work 12-16 hour days for 60 days straight, and after that there's a period where no one cares if you head out early every day for a few weeks. During the crunch time, though, there's an expectation that you're "all in".
However, I'm more of an internal contractor (I prefer the term "mercenary interpreter"). I rotate between teams that are understaffed and help out during the "crunch time". The downside to that is that I rotate off of one understaffed team and on to another with no down time in-between. It wouldn't be as bad if I were actually on a particular team long-term. (The upside is that I get to see a lot of neat geology.)
Most of this is just me griping about my current role (I'm hoping to move to R&D soon). It's not quite this bad outside of the specific teams I've been on.
However, there's a strong culture of working crazy weekends and long hours that I really wish we could change. I've worked for three of the majors. There are differences, but this part of it is the same.
Overall, I just don't understand why salary is negotiable in a job, but time off isn't.
I don't know your industry though so it would behoove you to make friends with someone else who consults for your company and figure out what arrangement they have. If the company has done it once they'll do it again.
Salary is fungible, they can give you more or less of it and have it not appreciably affect company goals. Time off often is not.