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.
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