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* U.S. oil stocks rise by 8.6 mln barrels - API
* Chinese industrial output up by 5.4% in April
* Graphic: U.S. oil output, inventories: https://tmsnrt.rs/2WhrAut (Adds comment, graphic, charts; updates prices)
TOKYO, May 15 (Reuters) - Oil prices fell on Wednesday after data showed a surprise rise in U.S. crude stockpiles and Chinese industrial output for April grew less than expected, but prices were supported by mounting tensions in the Middle East.
Brent crude futures were at $71.04 a barrel at 0358 GMT, down 20 cents, or 0.3%, from their last close. Brent closed 1.4% higher on Tuesday.
U.S. West Texas Intermediate (WTI) crude futures were at $61.38 per barrel, down 40 cents, or 0.7%, from their previous settlement. WTI closed up 1.2% in the previous session.
U.S. crude stockpiles unexpectedly rose last week, while gasoline and distillate inventories increased, data from industry group the American Petroleum Institute showed on Tuesday.
Crude inventories rose by 8.6 million barrels in the week to May 10 to 477.8 million, compared with analysts' expectations of a decrease of 800,000 barrels.
Crude stocks at the Cushing, Oklahoma, delivery hub rose by 2.1 million barrels, API said.
The U.S. Energy Department's Energy Information Administration (EIA) reports official numbers later today.
"If the EIA report confirms a strong build we could see that weigh on oil prices...but too many geopolitical risks remain that should keep prices supported," Edward Moya, senior market analyst at OANDA told Reuters by email.
Oil prices have drawn support after Saudi Arabia on Tuesday said armed drones struck two of its oil pumping stations, two days after the sabotage of oil tankers near the United Arab Emirates, while the U.S. military said it was braced for "possibly imminent threats to U.S. forces in Iraq" from Iran-backed forces.
The attacks took place against a backdrop of U.S.-Iranian tension following Washington's decision this month to try to cut Iran's oil exports to zero and to beef up its military presence in the Gulf in response to what it said were Iranian threats.
Meanwhile, the Organization of the Petroleum Exporting Countries (OPEC) on Tuesday said that world demand for its oil would be higher than expected this year as supply growth from rivals including U.S. shale producers slows. That points to a tighter market if the exporter group refrains from raising output.
Elsewhere, growth in China's industrial output slowed more than expected to 5.4% in April from a 4-1/2 year high in March, reinforcing views that Beijing will have to roll out more stimulus measures as a trade war with the United States intensifies.
(Additional reporting by Colin Packham in SYDNEY; Editing by Joseph Radford)