* Brent hits five month high, U.S. WTI rises over $50 a barrel
* Global oil demand estimates rising
* U.S. Gulf oil refineries returning after storms
* Producer countries seen extending output cuts -analysts
* BP sees crude oil prices staying in $50-$60 range (New throughout, updates prices, market activity and comments)
NEW YORK, Sept 14 (Reuters) - Oil prices rose on Thursday, with Brent briefly touching a five-month high, a day after the International Energy Agency (IEA) forecast the market would continue to tighten as fuel demand increased.
Brent crude futures were up 56 cents, or 1 percent, at $55.72 a barrel by 12:16 p.m. EDT (1616 GMT). The session high for the global benchmark was $55.99, its highest since April 13. The contract remained in technically overbought territory for a second day in a row.
U.S. West Texas Intermediate, meanwhile, was up 84 cents, or 1.7 percent, at $50.14 per barrel, its first time over $50 since August. WTI touched $50.50, its highest since May 25, and surpassed its 200-day moving average.
Brent has climbed more than $10 a barrel in three months and is close to where it began the year.
On Wednesday, the IEA 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 glut was shrinking thanks to strong European and U.S. demand, as well as production declines in OPEC and non-OPEC countries.
"The IEA revising up its 2017 global oil demand growth forecast, together with persistent weakness in the U.S. dollar index, has prompted bullish sentiment in the oil market. Anticipation is growing that this could quicken the pace of oil market rebalancing," said Abhishek Kumar, Senior Energy Analyst at Interfax Energy's Global Gas Analytics in London.
The U.S. dollar index was down 0.2 percent against a basket of currencies, making oil cheaper for holders of other currencies. Last week, the dollar index fell to its lowest level since the start of 2015.
"Unrest in Iraq and Venezuela should keep output there in check, regional crude oil contangos have dissipated and stocks are gradually declining," Barclays Research said in a note.
The Organization of the Petroleum Exporting Countries and other producers including Russia have agreed to reduce crude output 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 showed 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 Christopher Johnson in London and Aaron; Sheldrick and Osamu Tsukimori in Tokyo; Editing by Dale Hudson and David Gregorio)