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Someone over on r/WriterResearch asked how long a character could survive after a traumatic limb amputation. I had my copy of Blood On The Page handy, so I took a shot at answering.
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The chapter on traumatic amputation rates the injury at 5/5 lethality, meaning it will usually be fatal without treatment.
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In terms of treatment attempts, a full amputation is often simpler to deal with than a partial amputation when the limb is not viable—it saves time not having to finish the amputation after the fact. This scenario would be more likely with a crushing injury than with a clean cut.
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A cross reference to Maim Your Characters also tells us that a crushing injury has an overall worse prognosis than a clean cut.
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Speed of the bleed out is obviously influenced by whether a tourniquet is applied, and applied correctly. The chapter also directs to the exact shock/blood loss chart also featured here, which is the author’s blog.
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The books didn’t give us a complete answer, so it was time for a good old-fashioned Google search. We learn from The Internet that a person can die of an arterial bleed (virtually certain in an amputation unless it was partial) in as little as two minutes, with 2-5 minutes being the usual timeline. A different page on the same site gives a well-placed tourniquet a timeline of about two hours before it needs to be removed, the concern after that being damage to the tissues below the tourniquet.
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I would say that based on the two sources together, I give this character 2-5 minutes until bleed out with 30 seconds – 2.5 Minutes of actual coherence within that if the bleeding is uncontrolled. If someone properly gets a tourniquet on it in that time, freeze the symptoms in whatever stage of shock he was at before that happened and let him linger as desired. If someone does it unsuccessfully, so that the bleeding is slowed but not stopped, I would double the time (4-10 minutes, 1-5 of coherence) but still have him die rapidly.
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If the artery was severed as the result of a crushing injury, it may also come with its own form of bleeding control. If the large rock (or whatever it may be) is still on top of him, it might apply enough pressure to slow or stop the bleeding on its own, meaning he’s trapped in that position but will die almost immediately if anyone pulls him out.