Science, 2012 Apr 27, p441 speaks of clock accuracies of 10−18. The Schwarzschild metric for the Earth is
2 = (1 − rs/r)dt2 + …
dτ = (1 − rs/r)½dt ≃ (1 − rs/(2r))dt + …
∂τ/∂t = (1 − rs/(2r)) = (the time dilation due to gravity well),
rs = 2GM⦿/c2 = 2 ∙ 6.7 ∙ 10−11 m3kg−1s−2 ∙ 6 ∙ 1024kg / (3 ∙ 108 m/s)2 = 0.009 m ≃ 1 cm
rs is about 1 cm for the mass of the Earth.
r = 7 ∙ 103 km = 7 ∙ 106 m (Earth radius)
∂(∂τ/∂t)/∂r = ∂2τ/∂t∂r = ∂(1 − rs/(2r))/∂r = rs/(2r2)
Near the Earth’s surface the radial gradient of the time dilation is ∂2τ/∂t∂r = 0.01 m/(1.4 ∙ 1013 m2) = 7 ∙ 10−16 m−1
Thus when r is changed by 1cm = 10−2 m, (∂r = 10−2 m), the time dilation is changed by 7 ∙ 10−18 and such a clock, one cm above another, should see the difference.