Inductance of planar (spiral) coil near solid copper surface
You are correct that the inductance would ideally be zero at zero distance. 

You can try to model this from first principles using the Biot-Savart law and a simplified geometry, but I think an FEA approach would be faster and probably more accurate. The parasitics (resistance and distributed capacitance) may turn out to be important, depending on what frequency is used. We're measuring displacement using this method, but in our case resistance is zero and the frequency is DC so it's easier.