The Chilean mine has a structure that is different from mines in America, where the miners walk down into the mine to get to the work area. That path has many switchbacks so that the horizontal size of the mine is limited. I see that as a characteristic which can greatly speed up the saving of the 33 trapped miners.
The conventional logic being used is to bore a large diameter borehole the entire 720 meters (nearly half a mile) down to the shelter where the 33 miners have been trapped. That process is expected to take around three months to accomplish, as enormous amounts of rock have to be bored out. Also, the miners at the bottom then have to shovel away those many tons of rock to clear away access to the new borehole.
I note that both the original collapse of the gallery and the second collapse which occurred a couple weeks later are actually relatively near the surface. So I propose either of two different approaches to a borehole, the more involved of which is shown here in the pink line I have added to the second drawing.
Note that the length of the required borehole is only around 1/3 as deep as the initial plan, around 240 meters rather than 720 meters. That should mean that completing that borehole should only take around 1/3 as long, one month instead of three. The miners might thereby be saved two months earlier than the Christmas target now being discussed. Note that the miners would then walk up the gallery for 2/3 of the exit path, to get to the bottom of the new borehole.
The second alternative might be slightly faster still. A small diameter (10 cm or 4 inches) borehole would be carefully planned to pass THROUGH the gallery shown at about the 90 meter depth, and continue down to the same gallery discussed above at the 240 meter depth. That small diameter borehole could be made in just a day. It would only be used for a passageway of a rotating shaft connected to the boring equipment on the surface.
That shaft would be inserted down to the gallery at the 90 meter depth, and a work crew would walk down the gallery from the mine entrance to that point. That work crew would then attach large diameter boring cutters to that shaft. The (surface) equipment would then bore a new borehole in the FLOOR of the gallery at the 90 meter level which would then continue to the ceiling of the gallery at the 250 meter level.
This variant would only create a large-diameter borehole which was long enough to bypass the two collapsed portions of the gallery. Since only 160 meters of rock would need to be bored out, it should only take around two weeks to accomplish. The miners would then walk up 2/3 of the way from the Shelter where they have been living, be raised up that 160 meters through the new borehole and then walk up the remaining 90 meters and out of the mine entrance.
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago