  Semi-inclusive transverse momentum spectra observed in proton-proton and
proton-lead nuclear collisions at LHC energies obey a geometric scaling with a
scaling variable using multiplicity-dependent saturation momentum. The
saturation momentum extracted from the experimental data is proportional to the
1/6 power of the hadron multiplicity in the final state. On the other hand, the
system's transverse size is proportional to the 1/3 power of the multiplicity,
and the saturation momentum and the transverse size of the system are strongly
correlated with the hadron multiplicity in the final state. Since the
saturation momentum is proportional to the average transverse momentum of
hadrons, one predicts average transverse momentum is also proportional to the
1/6 power of the multiplicity, which is consistent with experimental results at
the LHC energy. We found that a nuclear modification factor $R_{\rm pPb}$
calculated by the multiplicity-dependent saturation momentum decreases at
$p_{\rm T} \lesssim$ 1GeV/$c$ and that our model can partially explain the
$R_{\rm pPb}$'s behavior thought to be caused by nuclear shadowing. On the
other hand, Cronin enhancement experimentally observed at $2 \lesssim p_{\rm T}
\lesssim $ 6 GeV/$c$ is not reproduced. However, the experimental result,
including the Cronin effect, can be reproduced well by introducing $p_{\rm T}$
dependence as a 4$\sim$5\% correction to the multiplicity-dependent saturation
momentum.We also discuss a relation between the geometric scaling in the
semi-inclusive distributions and the string percolation model.
