I hinted at this in another thread. I've been a fan of the anime 'Gate' where a standard LoTR type universe opens a portal to our world and charges in with their level of warfare. Doesn't work well and there is a counter invasion by modern Japan. Also a novel called the Shield where modern Israel lands in 1941 WWII.
So, for this: Physicists playing around, open a portal to the LoTR just before the Fellowships takes their stroll. Thru the gate and a base in an isolated area, recon drones, satellites, etc, map the world. Middle Earth - not the magic Eleven Disneyland across the seas as I don't want to play with the chance of some divine intervention.
Well, here's an Earth full of unspoiled energy resources for us. Lots of oil, probably. It is decided to 'take over' Middle Earth in a nice way. The nasty dictator will be overthrown. The nice folks - humans, eleves, dwarves, hobbits will get delegations, education, etc. The nasty humans in the South, they get a lesson to behave.
But how does it go down? Massive armies, air power strikes, special forces - etc. Or just trying to convince the good folks to buy some munitions and get training.