Air gap between patch substrate and ground.

How to calculate resonant patch length for aperture coupled patch?
For certain Er value there are formulas to obtain patch sizes. If substrate is separated from ground plane by air gap, how Er is affected? I guess Er becomes smaller, and patch size must be increased to compensate for this effect. Is there any numerical solution?

I usually start with a a patch length of wavelength/2 and then refine the dimensions for the particular application. You are on track with the observation that if there is dielectric under the patch the size is reduced. My view is that it helps to have one of the high end simulators to get a patch working.