In experimental samples of carbon nanotube transistors (CNT-FETs) the electrical contact and current inflow occur along relatively long portion of a CNT embedded into a metal. Only very few theoretical studies were done with geometries and materials close to realistic ones. The most common simplified approaches are using the models of point-like or very slightly embedded contacts.We perform large-scale modeling of extended metal-CNT contacts by density functional theory accompanied by Green function method in order to elucidate electrical properties of realistic metal-CNT contacts. We have obtained smooth shrinking of the band gap inside embedded portion of a semiconductor nanotube and induced by the metal doping of the embedded and free-standing part of a CNT. It causes geometry and material dependent behavior of the transmission coefficient and density of states along a CNT. We also analyze the electrostatic potential and charge redistribution and formulate an ab initio based effective transport model to calculate the current-voltage characteristics of large scale CNT-FETs.