  The thermal conductivity and shear viscosity of dense nuclear matter, along
with the corresponding shear viscosity timescale of canonical neutron stars
(NSs), are investigated, where the effect of Fermi surface depletion (i.e., the
$Z$-factor effect) induced by the nucleon-nucleon correlation are taken into
account. The factors which are responsible for the transport coefficients,
including the equation of state for building the stellar structure, nucleon
effective masses, in-medium cross sections, and the $Z$-factor at Fermi
surfaces, are all calculated in the framework of the Brueckner theory. The
Fermi surface depletion is found to enhance the transport coefficients by
several times at high densities, which is more favorable to damping the
gravitational-wave-driven $r$-mode instability of NSs. Yet, the onset of the
$Z$-factor-quenched neutron triplet superfluidity provides the opposite
effects, which can be much more significant than the above mentioned $Z$-factor
effect itself. Therefore, different from the previous understanding, the
nucleon shear viscosity is still smaller than the lepton one in the superfluid
NS matter at low temperatures. Accordingly, the shear viscosity cannot stablize
canonical NSs against $r$-mode oscillations even at quite low core temperatures
$10^6$ K.
