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At least according to theory, not sure what experiments have been done. Earth radius is about 6370km, a quick search gave TauTona mine as one of the deepest at 3.9km, a fractional decrease of only 0.9994! My bathroom scale won't look any better.

So I would expect to weigh a few more pounds in a deep well.

That is assuming a uniform density Earth. That's a nice simplifying assumption for a freshman physics kind of problem, but not so good assumption of the real Earth.This assumption results in gravitational acceleration dropping linearly with increasing depth. This is not what happens inside the Earth. The D" layer (the core/mantle boundary) is 2890 km below the surface. Per this simplifying assumption, gravitational acceleration at that depth would be a bit over half of surface gravity. That is off by a factor of two! Gravitational acceleration is 10.6823 m/s^2 at that depth.A better model is that gravitational acceleration is constant at 10 m/s^2 from the surface down to halfway to the center of the Earth and then drops linearly from that point inward. ...

..That is assuming a constant density, and the Earth is anything but constant density. Gravity increases with depth down to the bottom transition zone (g=10.0143 m/s^2), then decreases to a local min of 9.9314 m/s^2 in the middle of the lower mantle, increases again to a global max of 10.6823 m/s^2 at the D" layer (core/mantle boundary), and then finally drops toward zero at the center of the Earth....Thus gravity increases with decreasing r (increasing depth) if the local density is less than 2/3 of the mean density . Gravity decreases with depth only if the local density is 2/3 of the mean density or more.Now to get back to the numbers I posted at the start of this post. The crust and upper mantle are of relatively low density material. The local density is less than 2/3 of the mean density, so gravitational acceleration initially increases with depth. The rock undergoes a phase transition to a different, more dense crystalline form in the transition zone. Local density now exceeds 2/3 mean density, so gravitational acceleration starts falling with increasing depth in the top of the lower mantle. The Earth's core is much denser than the mantle rock. Mean density increases much faster than does local density in the lower mantle. Local density becomes less than 2/3 of the mean density about 1600 km below the surface (4800 km from the center), so gravity once again starts rising. Gravity is at its greatest right at the core/mantle boundary, about 2890 km below the surface. That's almost halfway to the center of the Earth!