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Oil supply running ahead of demand hasn't just pressured prices, it's also filling up storage space, potentially pushing crude toward another leg down.

"We're going to see pretty fast inventory builds over the next few weeks," Francisco Blanch, head of commodity research at Bank of America-Merrill Lynch, told CNBC Wednesday, noting that global supply is running around 1.4 million barrels a day above demand.

"If you run out of space, prices tend to react a lot more violently to adjust that supply and demand imbalance and that's what we expect over the next few weeks," he said, forecasting both WTI and Brent will fall toward $30 a barrel. Prices settled at $50.99 and $61.97, respectively, on Wednesday.

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He cited fresh American Petroleum Institute (API) data which showed U.S. crude inventories climbed by a larger-than-expected 8.9 million barrels in the week ended Feb. 20, for a total of around 437 million barrels squirreled away. Around 50 million to 100 million barrels of crude oil may be gathering dust in floating storage by the end of the second quarter, compared with around 110 million barrels in April 2009, during the global financial crisis, he estimated.

The supply build isn't helped by an oil market that's in contango, or when the "spot" price is lower than the price of the future contract. That makes it more profitable for traders to stick their oil in storage to sell at a higher price later.

As much as 80 percent of the commercially available storage in the U.S. may already be utilized, Premasish Das, downstream analyst at IHS Energy Insight, told CNBC last week.

Oil tanks and pump jacks are seen in an oil field near Bakersfield, Calif.
Lucy Nicholson | Reuters

"As the oversupply increases again in the second quarter, the contango structure will widen. This will further incentivize crude storage," Das said.

Others are also concerned about how quickly space could run out. "Within around two months, [onshore storage will] be completely exhausted," Ivan Szapakowski, a commodity strategist at Citigroup, told CNBC last week. "The only remaining storage globally will then be floating storage, tankers."

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Citigroup is forecasting oil prices to fall toward $20 a barrel before recovering.

Oil is already getting stashed offshore. Tanker prices and lease rates have doubled over the past 18 months, Gaurav Sodhi, a resources analyst at Intelligent Investor, told CNBC last week.

"Clearly, someone is going out there and leasing a lot of offshore storage and those lease rates are much higher. That suggests to me that there's a massive supply of oil sitting on the ocean," Sodhi said, noting each tanker can hold around 1 million to 2 million barrels each. "That sort of storage is going to take a while to unwind."

Stashing oil offshore can also create other headaches. It's harder to move the oil to where the buyer may want it and there can be issues with pollution, potentially making it more costly, IHS' Das noted.

But despite the difficulties, the oil hoard may only grow in the near term. Das noted that Asia is heading toward its refinery maintenance shutdown period in the second quarter.

Additionally, a strike among refinery workers in the U.S. is now into its fourth week, affecting plants responsible for around 20 percent of the country's production. Media reports said no talks between unions and management are scheduled for this week. If demand from the affected refineries declines further, it could be more bad news for inventory growth and prices.

—By CNBC.Com's Leslie Shaffer; Follow her on Twitter @LeslieShaffer1