Scaling theory for spontaneous imbibition in random networks of elongated pores

We present a scaling theory for the long time behavior of spontaneous imbibition in porous media consisting of interconnected pores with a large length-to-width ratio. At pore junctions, the meniscus propagation in one or more branches can come to a halt when the Laplace pressure of the meniscus exceeds the hydrostatic pressure within the junction. We derive the scaling relations for the emerging arrest time distribution and show that the average front width is proportional to the height, yielding a roughness exponent of exactly β = 1/2 and explaining recent experimental results for nanoporous Vycor glass. Extensive simulations of a pore network model confirm these predictions.