The Colorado River is a critical source of water for seven Western states, each of which gets an annual allotment according to a system that has sparked conflict and controversy for decades. But in an era of climate change, even greater difficulties loom.

The scope of those potential problems is detailed in a study being published in The Proceedings of the National Academy of Sciences. Tim P. Barnett and David W. Pierce of the Scripps Institution of Oceanography report that under various forecasts of the effects of warming temperatures on runoff into the Colorado, scheduled future water deliveries to the seven states are not sustainable.

The work builds on an earlier study by the researchers that looked at whether Lake Mead, the huge reservoir behind Hoover Dam, would eventually go dry. For the current study, they tweaked their model of river inflows and outflows and looked at the delivery shortfalls that would be needed to keep Lake Mead at the lowest functioning level. The modifications in the model “didn’t really change any of our answers,” Dr. Barnett said. “It just made the study a lot stronger.”

The study found that, with a 20 percent reduction in runoff, by 2050 nearly 9 of every 10 scheduled deliveries would be missed. But the problem may be even worse, because the allotments were determined in the 20th century, when, according to tree-ring data, the region was wetter than normal. So if drier conditions persist, delivery shortfalls will be even greater.