Estimating inductance change by eddy currents
'Calculating the field equations' would be one way to do it, but the geometry is far from analytically simple, so you would end up with a huge mess of terms.

If you search the literature for planar coils, you will see that most workers (mostly PhD students) mention complete analytic solutions in passing, then go on to approximate analytical solutions of simpler models like current sheets and short line segments, and FEMM based solutions. So if you take on a complete equation based solution, you are doing PhD level work.

There are numerous 'flat coil' calculators that can be found with your favourite search engine. If you get good agreement between several, you can have some faith in the answer.

Measurement might be another avenue, not least to valid answers from calculators. make a flat coil, resonate it with a capacitor, then note the change in frequency as you bring the piece of metal up to it. This gives you the mutual inductance, from which you can estimate coupling and eddy currents.