Brent crude futures gained 36 cents to $71.60 a barrel at 10:48 a.m. EDT (1448 GMT). US West Texas Intermediate (WTI) crude futures fell 2 cents to $61.76 a barrel.
US crude stocks rose unexpectedly last week, while gasoline stockpiles decreased and distillate inventories grew, the Energy Information Administration (EIA) said.
Crude stocks swelled by 5.4 million barrels, surprising analysts who had expected a decrease of 800,000 barrels.
"Although crude oil inventories built more than the market expected due to higher imports, prices remained supported due to the geopolitical dynamics of the Middle East," said Andrew Lipow, president of Lipow Oil Associates in Houston.
Weak economic data from the United States and China fed worries that global crude demand could slow.
Oil prices have drawn support since Saudi Arabia said on Tuesday that armed drones struck two oil pumping stations, two days after the sabotage of oil tankers near the United Arab Emirates.
"Given that nearly one-third of global oil production and nearly all of global spare capacity are in the Middle East, the oil market is very sensitive to any attacks on oil infrastructure in this region," Swiss bank UBS said, adding it expected Brent prices to rise toward $75 in coming weeks.
The attacks took place against a backdrop of US-Iranian tension. Washington has been trying to cut Iran's oil exports to zero while beefing up the US military presence in the Gulf.
"It just looks like they're setting up for some kind of conflict with them (Iran)," said Josh Graves, senior commodities strategist at RJO Futures in Chicago. "I think the oil traders of the world are looking at that and say, 'OK, what is the potential disruption there?'"
Washington ordered the departure of non-emergency American employees from its diplomatic missions in Iraq on Wednesday in another show of concern about threats from Iran.
The International Energy Agency said the world would require very little extra oil from OPEC this year as booming US output will offset falling exports from Iran and Venezuela.
The agency also revised its forecast for 2019, growth in global oil demand 90,000 bpd lower to 1.3 million bpd.