* Brent posted biggest third-quarter rise since 2004
* U.S. drillers add oil rigs for first week since mid-August
* OPEC oil output rises in September -Reuters survey
* Libya's Sharara oil field under force majeure -sources (Adds quotes, adds context, updates final prices)
By Julia Simon
NEW YORK, Oct 2 (Reuters) - Oil fell more than $1 a barrel on Monday as a rise in U.S. drilling and higher OPEC output put the brakes on a rally that helped prices notch their biggest third-quarter gain in 13 years.
Iraq announced its exports rose slightly in September while a Reuters survey showed OPEC overall boosted output.
U.S. drillers added six oil rigs in the week to Sept. 29, bringing the total count to 750, data from General Electric Co's Baker Hughes energy services firm showed on Friday.
"We’ve seen them add rigs for the first time in seven weeks, so that changes sentiment as well," said John Tjornehoj, energy market analyst at CHS Hedging.
Brent crude, the global benchmark, settled down 67 cents or 1.2 percent to $56.12 a barrel. It had notched a third-quarter gain of about 20 percent, the biggest increase for that quarter since 2004, and traded as high as $59.49 last week.
U.S. crude closed down $1.09 or 2.1 percent to $50.58. The U.S. benchmark posted its strongest quarterly gain since the second quarter of 2016.
Oil prices climbed last week on tension in Iraqi Kurdistan after the region's independence vote.
"The big short-term risk is obviously the pipeline," said James Williams, president of energy consultant WTRG Economics. "So far Turkey hasn't closed the Kurdish pipeline."
The rally had also been driven by signs that a three-year crude glut is easing, helped by a production cut deal among global producers led by the Organization of the Petroleum Exporting Countries.
But a Reuters survey on Friday found OPEC oil output rose last month, mostly because of higher production in Iraq and also Libya, an OPEC member exempt from cutting output.
However, Libya's National Oil Company in a letter on Monday declared force majeure on deliveries from Sharara, the country's largest oilfield.
Middle Eastern oil producers are concerned the price rise will stir U.S. shale producers into more drilling and push prices lower again. Key OPEC producers consider a price above $60 as encouraging too much shale output.
Hedge funds have accumulated a record bullish position in middle distillates such as diesel, heating oil and gasoil, anticipating stocks will be relatively tight this winter.
"We've seen a run up in heating oil futures, and I think that particular product has supported the rise of WTI," said Tjornehoj, while noting that distillate prices fell on Monday. "As we reverse here lower we see the recent strong correlation continuing."
Additional reporting by Alex Lawler in London and Aaron Sheldrick; editing by Susan Thomas; Editing by Edmund Blair and David Goodman