  The propagation and roughening of a liquid-gas interface moving through a
disordered medium under the influence of capillary forces is considered. The
system is described by a phase-field model with conserved dynamics and spatial
disorder is introduced through a quenched random field. Liquid conservation
leads to slowing down of the average interface position H and imposes an
intrinsic correlation length \xi_\times \sim H^{1/2} on the spatial
fluctuations of the interface. The interface is statistically self affine in
space, with global roughness exponent \chi \simeq 1.25 and exhibits anomalous
scaling.
