  We aim to elucidate cosmic ray (CR) propagation in the weakly ionized
environments of supernova remnants (SNRs) basing our analysis on the cosmic ray
cloud (CRC) model developed by \citet{2013ApJ...768...73M} and
\citet{2016MNRAS.461.3552N}. We solved two transport equations simultaneously:
one for the CR pressure and one for the Alfv\'en wave energy density where CRs
are initially confined in the SNR shock. Cosmic rays trigger a streaming
instability and produce slab-type resonant Alfv\'en modes. The self-generated
turbulence is damped by ion-neutral collisions and by noncorrelated interaction
with Alfv\'en modes generated at large scales. We show that CRs leaking in cold
dense phases such as those found in cold neutral medium (CNM) and diffuse
molecular medium (DiM) can still be confined over distances of a few tens of
parsecs from the CRC center for a few thousand years. At 10 TeV, CR diffusion
can be suppressed by two or three orders of magnitude. This effect results from
a reduced ion-neutral collision damping in the decoupled regime. We calculate
the grammage of CRs in these environments. We find that in both single and
multi-phase setups at 10 GeV, CNM and DiM media can produce grammage in the
range 10-20 $\rm{g/cm^2}$ in the CNM and DiM phases. At 10 TeV, because of
nonlinear propagation the grammage increases to values in the range 0.5-20
$\rm{g/cm^2}$ in these two phases. We also present preliminary calculations in
inhomogeneous interstellar medium combining two or three different phases where
we obtain the same trends.
