  We analyze the dynamics of liquid filling in a thin, slightly inflated
rectangular channel driven by capillary forces. We show that although the
amount of liquid $m$ in the channel increases in time following the classical
Lucas-Washburn law, $m \propto t^{1/2}$, the prefactor is very sensitive to the
deformation of the channel because the filling takes place by the growth of two
parts, the bulk part (where the cross-section is completely filled by the
liquid), and the finger part (where the cross-section is partially filled). We
calculate the time dependence of $m$ accounting for the coupling between the
two parts and show that the prefactor for the filling can be reduced
significantly by a slight deformation of the rectangular channel, e.g., the
prefactor is reduced 50% for a strain of 0.1%. This offers an explanation for
the large deviation in the value of the prefactor reported previously.
