re: When they start to close the valves...

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Isn't the relief well and pumping heavy fluids from the bottom up to kill it, essentially the same thing as closing it or shutting it off? Wouldn't both result in the same amount of back pressure on the well?

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No, it's the opposite. Putting heavy mud into the well will increase the hydrostatic pressure at the bottom so that there's less pressure at the top.

P(top) = P(formation) - P(hydro)

The goal is to make P(hydro) greater than P(formation) so that the flow stops. (That's a bit oversimplified, but close enough.)