How does the gravity well change as space expands? If we assume that the Earth's gravitational field curves flat space to create a gravity well then how does the gravity well change as space expands also is the change in gravity well measurable.
For a stationary massive object like Earth or the Sun, the gravity well does not change from the expansion of space. The gravity well at one time is the same as the gravity well at future times for the same mass. Any minute force imposed by the expansion of space does not constitute a change in the gravity well itself. The well is not stretched or compressed. It remains constant in proper distance scales and not comoving distance scales.
If we assume that general relativity is the correct theory for this case, and there are currently no indications, that I am aware of, that it isn't, then the expansion of the universe adds a small modifying term to gravity wells. I doubt that it is measurable at the scale of the solar system. The current best estimate for the Hubble constant is 67.80±0.77km/s/Mpc. The size of the solar system (using the heliopause as a border) is about 0.001pc, so the total influence of the expansion would amount to approx. 1e-3pc*68km/s/Mpc=68um/s at the border of the solar system. I seriously doubt, that we can separate that from the gravitational noise of the solar system and the diverse effects of solar wind, light pressure, outgassing etc. that act on spacecraft that may be used to measure this effect.
Having said that, the effects at the edge of our Milky Way are much larger. Now we are talking about 30kpc*68km/s/Mpc=2.04km/s. While the errors bars are much larger (probably mostly due to dark matter distributions), it may be possible to find populations of objects, which show considerable effects on their orbits around the center of the Milky Way. But that's just the gut feeling of a physicist who is not an astronomer, so my intuition into the systematic errors that would undermine such a measurement could be way off.
Update: For those who would like to know what all of this means for orbits of bodies, I believe this paper may contain the required analysis:
"Analytical solution of the geodesic equation in Kerr-(anti) de Sitter space-times", Eva Hackmann, Claus L¨ammerzahl, Valeria Kagramanova and Jutta Kunz