  The propagation and roughening of a fluid-gas interface through a disordered
medium in the case of capillary driven spontaneous imbibition is considered.
The system is described by a conserved (model B) phase-field model, with the
structure of the disordered medium appearing as a quenched random field
$\alpha({\bf x})$. The flow of liquid into the medium is obtained by imposing a
non-equilibrium boundary condition on the chemical potential, which reproduces
Washburn's equation $H \sim t^{1/2}$ for the slowing down motion of the average
interface position $H$. The interface is found to be superrough, with global
roughness exponent $\chi \approx 1.25$, indicating anomalous scaling. The
spatial extent of the roughness is determined by a length scale $\xi_{\times}
\sim H^{1/2}$ arising from the conservation law. The interface advances by
avalanche motion, which causes temporal multiscaling and qualitatively
reproduces the experimental results of Horv\a'ath and Stanley [Phys. Rev. E
{\bf 52} 5166 (1995)] on the temporal scaling of the interface.
