  In this paper, we investigate some new local Aronson-B\'enilan type gradient
estimates for positive solutions of the porous medium equation $$ u_{t}=\Delta
u^{m}, $$ under Ricci flow.
  As application, the related Harnack inequalities are derived.
  Our results generalize known results. These results in the paper can be
regard as generalizing the gradient estimates of Lu-Ni-V\'{a}zquez-Villani and
Huang-Huang-Li to the Ricci flow.
