Hello,
Can you please explain your doubt with more detail? I am confused becasue we dont connect any extra fault model. Stuck at is the fault model. So what do you mean by connect extra fault model?
By the way ATPG tool will understand the logic of NAND gate, so it will apply patterns accordingly.
DFT ATPG(pattern generation)->Tetramax from synopsys and Encounter test from Cadence(Scan netlist,STIL are the inputs and patterns(SAF,TF) are the output)
As Maulin mentioned read some basic tutorial to understand the concepts.