  Recently anomalous scaling properties of front broadening during spontaneous
imbibition of water in Vycor glass, a nano-porous medium, were reported: the
mean height and the width of the propagating front increase with time $t$ both
proportional to $t^{1/2}$. Here we propose a simple lattice imbibition model
and elucidate quantitatively how the correlation range of the hydrostatic
pressure and the disorder strength of the pore radii affect the scaling
properties of the imbibition front. We introduce an effective tension of liquid
across neighboring pores, which depends on the aspect ratio of each pore, and
show that it leads to a dynamical crossover: both the mean height and the
roughness grow faster in the presence of tension in the intermediate-time
regime but eventually saturate in the long-time regime. The universality class
of the long-time behavior is discussed by examining the associated scaling
exponents and their relation to directed percolation.
