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

Lattice Model for Spontaneous Imbibition in Porous Media: The Role of Effective Tension and Universality Class