"We have seen healthy demand and a flattening of rig counts in the United States," Mazroui told reporters.
"This is the beginning of the third quarter and demand picks up in the third quarter and I hope the agreement will have a significant impact in the third and fourth quarter."
Brent crude oil prices remain just under the key $50/bbl mark on concerns about high supplies from the Organization of the Petroleum Exporting Countries (OPEC), despite a pledge to cut output in a bid to tighten the market.
OPEC, together with some non-members like Russia, has extended a deal to cut production by 1.8 MMbpd to March 2018.
However, OPEC's compliance slumped to 78% in June as higher-than-allowed output from Algeria, Ecuador, Gabon, Iraq, the UAE and Venezuela offset strong compliance from Saudi Arabia, Kuwait, Qatar and Angola, the International Energy Agency said last week.
"The UAE is committed to its cut," Mazroui said.
"We have seen some increase in production in some of the countries that were not part of the agreement because of their special stance."
Oil traders are looking ahead to Monday's meeting between OPEC and non-OPEC members to see if it will address rising production from Nigeria and Libya, which have been exempted from the cuts.
OPEC's Joint Ministerial Committee monitors compliance with the supply pact and will meet in St. Petersburg, Russia.
OPEC's supply cuts have also been countered by rising U.S. production, which has increased almost 12% since mid-2016 to 9.4 MMbpd. The number of rigs drilling for new U.S. oil supply has also climbed since last year though the pace has slowed in recent weeks.
