If it's currently depleting at 10-18M barrels a week and has 370M barrels, the math here suggests it's not going to be literally depleted for 20-37 weeks, that's in a worst case scenario.
It tends to get depleted when there's profits to be taken (i.e. when world prices are high), and refilled when they're low. Frankly, this is really isn't any sort of actual emergency just yet. And given that US domestic production is very close to equal to US daily consumption, there's not any serious danger of 'running out'.
However there's a lot of price arbitrage that happens because while the US exports about 4.5M barrels a day, it also imports around 5M barrels a day. "We" sell off the stuff "we" produce (specifically, the fracked stuff, mostly) that's more valuable to other countries (i.e. they have refineries better geared for it than our own), and import from still other countries (Mexico, Canada, VZ for example) that produce oil that is both 1) cheaper on the world market than what "we" sell, and 2) fundamentally more similar to what the US was producing prior to the fracking boom (and hence what most of "our" refineries are tooled to run, given most of "ours" are 40+ years old).
If the US stocks of the (cheaper) sour stuff that local refineries prefer runs low, and therefore refineries have to switch to burning the sweet stuff that's normally sold off at a profit, it's going to ramp up petrol prices to consumers at the pump. Because of course the oil refiners aren't going to just suck up the loss of profit on their own, they'll make us all feel the pain for the increase in the prices of their feedstocks.
The US is in a financial and geopolitical position such that the worst case scenario is must more 'high prices' than 'literally out' because we can always buy from other countries if we're willing to pay.
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