The Battle of the Oil Rig occurred during the Venezuelan War.
After attacking and stealing Solano's Villa, the Mercenary went to work for Universal Petroleum. After returning some stolen guns and bringing back The Devastator, they met with the CEO, Dr Lorraine Rubin. She asked the Mercenary to rescue her Marketing VP and guard her CFO. In exchange she provides them with the time of a secret meeting between Blanco and VZ officers.
After the Mercenary has complete those contracts, Acosta provides them with the location of the meeting. With everything they needed to track down Blanco, the Mercenary infiltrated the oil rig where the meeting was taking place (preferably by boat, because there are numerous Anti-Air emplacements).
In an office, Blanco and the VZ brass were talking with Solano via video link. Blanco assured Solano that everything would be operational before the government found out. Just as he said this, the Mercenary burst in and threw Blanco against the video screen, breaking it. Blanco expressed his surprise that they were still alive. The Mercenary retorted by saying they always got paid before demanding Solano's location. The Liberian refused to talk so the Mercenary grabbed the nearest officer and demanded the information. The officer blurted out that Solano was hiding in a bunker south of Cumana.
Furious, Blanco triggered a grenade and fled the office. The resulting explosion destroyed the office, killed the officers and blew the Mercenary off their feet. Blanco then drew his Hunting Pistol and prepared to do battle. (Note that the Hunting Pistol is a very powerful weapon, so it is worth killing Blanco if desired. However it can be obtained using cheats or by saving and quitting when Blanco exits the office then continuing. This will cause a clone to appear who carries the pistol but will not count as an HVT.) After capturing/killing Blanco, it is time to destroy the rig. C4 on the support pylons or the massive fuel tanks on the deck are both excellent ways to do this, but the player can do it any way they want.
After the rig's destruction, a coalition of soldiers from the Allied Nations invaded Venezuela, apparently dedicated to removing Solano from power. People's Liberation Army responded within hours, deploying their own troops with the exact same intent, though in actuality both sides were after Venezuela's considerable oil supply.