Fiber reinforced composite materials often consist of fibers gathered in bundles. Thus, during manufacturing, a liquid resin impregnates a multiscale porous medium. For wetting systems the capillary pressure jump becomes much higher in the smaller pores, i.e., within the bundles, and in addition to any applied pressure gradient there will be a local driving pressure gradient between the small- and large-scale areas. Such gradients will influence mechanisms, such as void formation and particle filtration. Hence, it is of interest to clarify the mechanisms for the wetting in general and the influence from the detailed geometry of the fiber network in particular. In this article, a porous pore-doublet model is studied in order to determine if an overflow of liquid can explain a leading flow in the smaller capillaries, and at which conditions it takes place. Experiments, as well as theoretical calculations on this generic geometry show that the leading front can be in the smaller capillary, as well as in the larger one. The outcome is dependent on the actual permeability of the porous material being a parameter that determines to what extent the larger capillary feeds the smaller one.