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The Aronson-B\'enilan gradient estimate for the porous medium equation hasbeen studied as a counterpart to the Li-Yau gradient estimate for the heatequation. In this paper, we give the Aronson-B\'{e}nilan gradient estimates forthe porous medium equation on weighted Riemannian manifolds under lower boundsof $N$-weighted Ricci curvature with $\varepsilon$-range for some $N < 0$. Thisis a generalization of those estimates under constant lower $N$-weighted Riccicurvature bounds with $N\in [n,\infty)$.

Aronson-B\'enilan gradient estimates for porous medium equations under lower bounds of $N$-weighted Ricci curvature with $N < 0$