* U.S. crude prices up 9 pct this week
* U.S. crude at slight premium over Brent for most of day
* Drilling activity, crude inventories both fall
* Enterprise and Vitol plan U.S. crude exports
* Overall market conditions remain weak (Updates prices)
By Henning Gloystein
SINGAPORE, Dec 24 (Reuters) - U.S. crude prices rose for a fourth straight session on Thursday, headed for a 9 percent weekly gain in the lead-up to Christmas as the market tightened on the back of falling supplies and looming exports.
Front-month West Texas Intermediate (WTI) crude futures CLc1 were trading at $37.82 per barrel at 0749 GMT, up 32 cents on the day and set for the biggest weekly gain since early October.
With internationally traded Brent futures LCOc1 trading at $37.79 a barrel, U.S. crude defended a small premium which it regained this week for the first time in around a year CL-LCO1=R .
The strengthening U.S. market is a result of falling stocks, reduced drilling activity, and looming exports following the lifting of a 40-year old ban of most U.S. crude exports. urn:newsml:reuters.com:*:nL1N1471KH
"We continue to see the front month WTI-Brent spreads widen... It will reach $0.50," Singapore-based Phillip Futures said on Thursday
On the production front, Baker Hughes (N:BHI) reported that U.S. oil drillers cut rigs for a fifth week out of six. urn:newsml:reuters.com:*:nL3N14C12K
"The current rig count is... pointing to U.S. production declining sequentially between 2Q15 and 4Q15 by 320,000 barrels per day," Goldman Sachs (N:GS) said.
In storage, U.S crude inventories fell 5.88 million barrels to 484.78 million, still near record highs, the Energy Information Administration (EIA) said. urn:newsml:reuters.com:*:nZXN03MI00
The tightening physical market came just as U.S. energy group Enterprise and oil trader Vitol raced to exploit the end of the ban on most U.S. crude exports, loading a 600,000-barrel cargo of domestic light crude oil scheduled for the first week of January, reportedly heading for Europe, though Asian buyers might also be interested in U.S. cargoes. urn:newsml:reuters.com:*:nL3N14C3O5 urn:newsml:reuters.com:*:nL3N14D1F1
"Judging from the strong production (despite the falling rig count) as well as inventories in the U.S., it may not be surprising to see the U.S. playing a bigger role in exporting," Phillip Futures said.
Despite this week's bull-run of U.S. crude, overall market conditions remain weak due to a global overhang in production that sees between 0.5 and 2 million barrels of crude produced every day in excess of demand, and analysts said it would take time for the glut to be worked down.
"Energy prices are likely to rise slightly as production slips, which will ease the current supply versus demand gap. Demand growth should remain solid, but inventories will remain an overhang to markets for much of the year," said Rob Haworth, senior investment strategist at U.S. Bank Wealth Management.
U.S. crude prices have fallen over 10 percent since the beginning of the month and remain almost two-thirds below mid-2014 when prices began to tumble. (Editing by Christian Schmollinger and Biju Dwarakanath)