There's no definite answer.
It varies with each shop.
Everyone has a different idea about how thin they want to leave the barrel wall at it's thinnest point.
The thinner the barrel to start with, the less they will cut away.
I can flute a featherweight for you.
I'll just grind 8 flats on it about 0.020" deep
How's that?
Seriously though, Krieger told me last week that they will NOT flute a barrel than is less than 0.900" in diameter between the chamber the the FSB.
Then they want it to be at least 0.750" out front before they will flute between the FSB and muzzle.
Having said that, let's take bushmaster as an example now.
They are fluting HBAR barrels.
HBAR's are 0.875" between the chamber and FSB. and 0.730" out front.
I don't know how deep their flutes are, maybe someone here can measure and post that.
If you fluted a medium profile barrel between the chamber and FSB, you would want to leave at least a 0.175" barrel wall thickness and 0.200" is more comfortable.
Doing the high math, that's 0.224" + 0.175" + 0.175" = 0.574" effective diamter at the bottom of the flute minimum and 0.624" being more comfortable.
Take those effective diameters and subtract them from whatever major diameter you happen to be working with and half of that is how deep you are likely to get the fluting to be.
Out in front of the FSB, it's another ballgame.
by now, pressures have subsided.
You can probably go to 0.125" wall thickness.
Now you have en effective barrel diameter of 0.474".
Subtract that from whatever your barrel happens to measure and half of that is probably how deep you can get it fluted.