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Let's say that the Yellowstone supervolcano has a (for it, quite small; some of Yellowstone's past eruptions are ~5 times greater in volume than this) VEI-7 eruption, ejecting five hundred cubic kilometers of tephra - just to provide a reference for how powerful this is.

How far away from such an eruption could you be without being immediately injured? I know that the odds are that the ash fallout and localized societal and supply chain collapses will get you eventually, but I'm not talking about that; I'm talking about "not getting carbonized by a pyroclastic flow", or "not within the range for the shockwave to cause permanent hearing loss", or "you have time to hide under a bridge before the foot-wide chunks of pumice start hitting the area".

For reference, Mount Pinatubo's eruption in 1991 pumped about ten cubic kilometers of magma above the surface. This is 50 times more powerful.

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    $\begingroup$ much closer than you think. Upwind, the longest-range immediate danger are from pyroclastic flows, and those rarely exceed 20km. But if you include any level of hearing loss.... hundreds of km. Tambora's eruption in 1815 was loud enough to interrupt indoor conversations 1000km away! $\endgroup$
    – PcMan
    Nov 8, 2021 at 19:46
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    $\begingroup$ @PcMan This is more than ten times more powerful than Tambora in terms of volume erupted, though, and, since it ejected ten times more, it stands to reason that the plume is going to be ten+ times heavier. Once the volcanic plume collapses, a pyroclastic flow occurs, and if the plume is ten+ times heavier, I imagine the pyroclastic flows contain ten+ times more matter - and that they'll therefore go further, since it takes longer for them to run out. I should edit my question; I mean "permanent hearing loss". $\endgroup$
    – KEY_ABRADE
    Nov 8, 2021 at 19:52
  • $\begingroup$ @PcMan A few sources for that claim I made: sciencedirect.com/science/article/pii/S0377027314002601 wired.com/2015/04/tambora-1815-just-big-eruption $\endgroup$
    – KEY_ABRADE
    Nov 8, 2021 at 20:03

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From 1 km to 80 km depending on what fits your story

Let's start with this alarmist article from The Express:

Zone One would extend up to 80km from the eruption, blasting the ground with sweltering hot gases and up to 3m of volcanic ash at temperatures over 400ºC. Some 70,000 people are found in this area of impact and unfortunately, the direct effects of the eruption are not survivable.

So you need to be at least 80 km (50 miles) away to stand any hope of survival. At least, that's a defensible distance if you're writing a story with maximum damage. The U.S. Geological Survey, the agency responsible for monitoring volcanic activity at Yellowstone, offers a different take.

The most likely explosive event to occur at Yellowstone is actually a hydrothermal explosion—a rock-hurling geyser eruption—or a lava flow. Hydrothermal explosions are very small; they occur in Yellowstone National Park every few years and form a crater a few meters across. Every few thousand years, a hydrothermal explosion will form a crater as much as a few hundred meters across. Though the worst-case scenario for a giant Yellowstone eruption is indeed bad and could have global implications, most past eruptions at Yellowstone were not highly explosive. Of the past 50 or so eruptions, almost all were simple lava flows. If they occurred tomorrow or next year, they would have minimal direct effect outside Yellowstone National Park.

You can scale the eruption to fit your story. Your character can be safe a click upwind or dead 75 kilometers away.

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