LONDON—It was once a behemoth, a massive North Sea oil field whose importance to the world crude market was summed up by its name atop the global benchmark price: Brent.

Today, the Brent field, northeast of Scotland’s remote Shetland Islands, is all but tapped dry. It produces about 1,000 barrels a day in a global market of 93 million daily barrels. Royal Dutch Shell PLC is awaiting approval to scrap the first of four Eiffel Tower-sized platforms that have sucked Brent-branded crude from the seabed for almost 40 years.

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