The Rio Grande is the border. The river in between the banks is kinda like the 'neutral zone'. The northern bank is US territory, the southern bank is Mexican territory, but right smack dab in the middle is more of an 'either-or'. It sounds like the Mexican Army wanted the pot more than the BP did, and got a bit more creative in their method of retrieving it.
Unless there is a specific treaty defining the border as the river bank(s), international law controls. Under customary international law the border follows the river's main channel with the law of erosion and accretion applying to a meandering change in the border. Under a less well accepted doctrine, the border is the constant middle of the river. Both are subject to amendment by treaty.