I have not seen the particular rig in question, but it's quite likely a GOSP(Gas Oil Separator Plant). If the flame is bright white, then it's usually natural gas being burned off from a large, high pressure gas pocket that was hit unexpectedly. If it's a more yellow flame, it's just typical low volume/pressure gas burned off during the separation process when it's less cost effective to capture it than to burn it off.
Sometimes those unexpected gas pockets are not simply natural gas, but a much more combustible, toxic gas, that NEEDS to be burned off as it's vented to the air, so as not to create a hazard to people.
If you see the smoke cloud sink to the water level after the gas is burned, it can indicate heavier than air gases that would pose a significant risk to people if not burned.
Keep in mind, I'm just a pilot monkey, not an engineer or chemist, but these are the things I've learned from co-workers when I worked in the oil and gas industry.