The 2001 movie The Others did something like this–Nicole Kidman’s character was walking along in the woods and they slowly became more and more foggy. I believe the DVD extras talked about the process, but as I recall, you would basically have to isolate all the objects in the shot that are at different distances from the camera and then apply different opacities of fog to them. If you don’t need any kind of turbulence in the fog, just layers of white would probably do.
This would be a very painstaking process though, because you’d have to go through and rotoscope pretty much every tree, section of ground, etc., in addition to the character, in order to treat them as different depths. It’s a similar process to how 3D is added in post to movies that were shot 2D.
Here’s the sequence from The Others:
https://www.youtube.com/watch?v=zd0FCDiFapo
And here’s an article on adding 3D after-the-fact to Jurassic Park, which would be a similar process:
https://flavorwire.com/382554/jurassic-park-how-a-2d-movie-becomes-a-3d-movie
In the “old days,” some films would fake fog by putting a fog filter on the camera, but the problems was that the entire shot had the same level of fog, rather than more distant objects being more obscured, so it always looked fake.
If you want to do this, and you want it to look good, you’ve got a lot of work ahead of you!
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