Sorry you could not find anything on Seagate's site. Actually, for the Expansion Drive, according to that first link that I provided, there is no software available.
Did you try to connect the drive to another USB port on the Mac Mini? Also, no matter where you connect it, can Disk Utility "see" the drive at all?
Also, in doing some searching about your drive model and High Sierra, saw this link:
https://discussions.apple.com/thread/8170082
That actually indicates that the drive would need to be erased and formatted before it can be recognized. But I suspect you have a lot of important stuff on your drive, and thus Erasing it is out of the question.
Sorry about so many links, but here is another one:
https://discussions.apple.com/thread/8170082
Note one of them, where the user states:
"Actually, the screenshot you sent is exactly the one if you have not activated Show All Devices, but only the Show Only Volumes option in the View Dropdown (there are subtle differences. For example, you wouldn't see Macintosh HD, but the drive denomination, and not the used quantity , but the smart status, etc). So, as I see it, Eau Rouge was the right path form start on.
Has High Sierra introduced as default the view volumes only option? what for, to protect a user form himself who has found its way to disk utility?"
This link shows how to set the "Select (All) Devices":
https://www.google.com/search?rlz=1...DbAhVQ72MKHTSQAVQQ1QIItgEoAA&biw=1280&bih=792
Hopefully those links will help.