UPDATE 6-Oil rises on stronger demand, supply restrictions

* Global oil demand estimates rising

* U.S. Gulf oil refineries slowly returning after storms

* Output pact likely to remain in place beyond March - analysts

* BP sees crude oil prices staying in $50-$60 range (Updates detail, comment, prices; paragraphs 1-2, 10-11)

LONDON, Sept 14 (Reuters) - Oil prices rose on Thursday, taking U.S. crude above $50 a barrel for the first time in a month after the International Energy Agency (IEA) forecast the market would continue to tighten as fuel demand increased.

Brent was up 45 cents at $55.61 a barrel by 1345 GMT. U.S. light crude rose 77 cents to a high of $50.07 before easing to trade around $49.95, up 65 cents. The U.S. benchmark last traded above $50 on Aug. 10.

Brent has now climbed more than $10 a barrel over the past three months and is close to where it was at the beginning of the year, trading between about $55 and $57.

The IEA on Wednesday raised its estimate of 2017 world oil demand growth to 1.6 million barrels per day (bpd) from 1.5 million bpd.

The agency said a global oil surplus was shrinking thanks to strong European and U.S. demand as well as production declines in OPEC and non-OPEC countries.

"Stronger demand and supply restrictions from OPEC and Russia are the main reasons for the oil price upsurge," said Forex.com analyst Fawad Razaqzada.

Barclays Research said in a note that the supply side of the equation was beginning to look bullish:

"Unrest in Iraq and Venezuela should keep output there in check, regional crude oil contangos have dissipated and stocks are gradually declining," it said.

The Organization of the Petroleum Exporting Countries and other producers, including Russia, have agreed to reduce crude output by about 1.8 million bpd until next March in an attempt to support prices.

BP Chief Executive Bob Dudley told Reuters in an interview on Thursday that oil prices were likely to stay between $50 and $60 as major producers kept output restricted.

"We're all trying to make our way in this world of between $50 and $60 and I would expect that to continue."

This week's gains have come despite data showing a big build in U.S. crude inventories after Hurricane Harvey.

Data from the Energy Information Administration shows a build in U.S. crude inventories last week of 5.9 million barrels, exceeding expectations.

U.S. gasoline stocks slumped by 8.4 million barrels, the largest weekly decline since the data was first recorded in 1990. U.S. distillate stocks fell by 3.2 million barrels.

(Additional reporting by Aaron; Sheldrick and Osamu Tsukimori in Tokyo; Editing by Dale Hudson and David Goodman)