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Oil prices rise on surprise U.S. crude inventory draw

* Brent crude oil futures near $70 per barrel

* Ongoing OPEC-led supply restraint has been supporting oil

* Weak dollar also supports oil prices

* Soaring U.S. production tempers bullish mood somewhat

SINGAPORE, March 22 (Reuters) - Oil prices rose on Thursday, lifted by a surprise draw on U.S. crude inventories as well as ongoing dollar weakness which makes oil cheaper in global markets and potentially spurs demand.

U.S. West Texas Intermediate (WTI) crude futures were at $65.39 a barrel at 0021 GMT, up 22 cents, or 0.3 percent, from their previous close.

Brent crude futures were at $69.65 per barrel, up 18 cents, or 0.3 percent.

Both benchmarks are hovering just below their highest levels since early February, having risen around 10 percent from March lows.

Some support for crude futures came from currency markets, where the dollar fell as Federal Reserve officials stuck to their view of three rate increases for 2018, even as they delivered an expected quarter point rate hike.

In oil markets, U.S. crude inventories <C-STK-T-EIA> fell 2.6 million barrels in the week to March 16, to 428.31 million barrels, the Energy Information Administration (EIA) said late on Wednesday.

"Oil ... had a big session overnight although this wasn't just a function of the interest rate move. Inventory data for last week showed a surprise crude draw as well as significant drawdowns in both gasoline and distillates inventories," said William O'Loughlin, investment analyst at Australia's Rivkin Securities.

Further supporting oil prices has been supply restraint led by the Organization of the Petroleum Exporting Countries (OPEC) and Russia, which started in 2017 and is scheduled to go on for the rest of 2018.

The overall bullish mood is being somewhat tempered by U.S. crude production <C-OUT-T-EIA>, which climbed to a fresh record of 10.4 million barrels per day (bpd) last week, putting the United States ahead of top exporter Saudi Arabia and within reach of Russia's 11 million bpd.

(Reporting by Henning Gloystein; editing by Richard Pullin)