Science, 2012 Apr 27, p441 speaks of clock accuracies of 10⁻¹⁸. The Schwarzschild metric for the Earth is
dτ² = (1 − r_s/r)dt² + …
dτ = (1 − r_s/r)^½dt ≃ (1 − r_s/(2r))dt + …
∂τ/∂t = (1 − r_s/(2r)) = (the time dilation due to gravity well),
r_s = 2GM_⦿/c² = 2 ∙ 6.7 ∙ 10⁻¹¹ m³kg⁻¹s⁻² ∙ 6 ∙ 10²⁴kg / (3 ∙ 10⁸ m/s)² = 0.009 m ≃ 1 cm
r_s 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 − r_s/(2r))/∂r = r_s/(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.