  We study the recently-introduced directed percolation depinning (DPD) model
for interface roughening with quenched disorder for which the interface becomes
pinned by a directed percolation (DP) cluster for $d = 1$, or a directed
surface (DS) for $d > 1$. The mapping to DP enables us to predict some of the
critical exponents of the growth process. For the case of $(1+1)$ dimensions,
the theory predicts that the roughness exponent $\alpha$ is given by $\alpha =
\nu_{\perp} / \nu_{\parallel}$, where $\nu_{\perp}$ and $\nu_{\parallel}$ are
the exponents governing the divergence of perpendicular and parallel
correlation lengths of the DP incipient infinite cluster. The theory also
predicts that the dynamical exponent $z$ equals the exponent $d_{\rm min}$
characterizing the scaling of the shortest path on a isotropic percolation
cluster. The exponent $\alpha$ decreases monotonically with $d$ but does not
seem to approach zero for any dimension calculated ($d \le 6$), suggesting that
the DPD model has no upper critical dimension for the static exponents. On the
other hand, $z$ appears to approach $2$ as $d \rightarrow 6$, as expected by
the result $z = d_{\rm min}$, suggesting that $d_c = 6$ for the dynamics. We
also perform a set of imbibition experiments, in both $(1+1)$ and $(2+1)$
dimensions, that can be used to test the DPD model. We find good agreement
between experimental, theoretical and numerical approaches. Further, we study
the properties of avalanches in the context of the DPD model. We relate the
scaling properties of the avalanches in the DPD model to the scaling properties
for the self-organized depinning (SOD) model, a variant of the DPD model. We
calculate the exponent characterizing the avalanches distribution $\tau_{\rm
aval}$ for $d = 1$ to $d = 6$, and compare our results with recent theoretical
