* U.S. crude inventories fell by 1.1 million barrels - API
* Climbing U.S. production looms over the market
* The U.S. shale revolution: http://tmsnrt.rs/2EtJgen (Updates with comment, refreshes prices, adds graphic)
LONDON, Feb 7 (Reuters) - Oil prices were broadly steady on Wednesday, as the boost from a report showing a drop in U.S. crude inventories last week was offset by evidence of soaring U.S. output.
Brent crude futures were down 2 cents at $66.84 a barrel by 1135 GMT, while U.S. West Texas Intermediate (WTI) crude futures eased 18 cents to $63.21 a barrel.
Global equities recovered modestly after their largest one-day fall in nearly two years on Monday, when volatility surged and investors ditched stocks and bonds.
The oil price has fallen by 3.2 percent in the last week, but has still performed better than shares on Wall Street, which have lost more than 4 percent.
"The risk-off move impacted oil, but that impact has been limited because commodities are consumption or real assets, as opposed to equities or bonds, which are investment assets," BNP Paribas head of commodity strategy Harry Tchilinguirian said.
"The curve pays you for being long oil," he said, adding the main issue affecting oil prices was "the efforts of supply restraint by producers that in the second half of 2017 started to bear fruit."
The Organization of the Petroleum Exporting Countries and other producers, including Russia, have cut production since January 2017 to force down global inventories.
Crude inventories in the United States <C-STK-T-EIA> have fallen by 20 percent since hitting record highs in April 2017.
The futures curve shows prompt prices for oil are above those for future delivery, suggesting investors are counting on demand outpacing supply.
Data on Tuesday showed U.S. crude inventories fell by 1.1 million barrels in the week to Feb. 2 to 418.4 million barrels, helping support the oil price.
"Evidence points to a global inventory market that has arguably already balanced with days of forward cover in the low single digits or possibly even lower - which should support the spot price going forward," said Richard Robinson, manager of the Ashburton Global Energy fund.
But rising U.S. oil production <C-OUT-T-EIA> has been looming over the market. Output has risen by 1 million barrels per day (bpd) in the last year to about 10 million bpd.
The U.S. Energy Information Administration (EIA) expects U.S. output to reach an average of 10.59 million bpd in 2018 and 11.18 million bpd by 2019, potentially overtaking Russia as the world's largest producer.
"The strong growth that is expected in U.S. production supports our more bearish outlook for the oil market," Hamza Khan, head of ING commodities strategy, said in a note.
(Additional reporting by Henning Gloystein in Singapore; Editing by Edmund Blair)