Infiltrations with certain boundary and initial conditions show an anomalous behavior where the water saturation and pressure profile are inverted in the vertical direction. This occurs when the water at the infiltration front oversaturates the porous medium, causing drainage behind the front, and is most likely due to the front being sharp at the pore scale. Here we attempt to quantitatively model water flow at infiltration fronts by using a physically based network model that includes viscous effects. The network model includes pore and throat elements of different shapes and sizes, and a connection topology based on geologic media. Viscous effects are added quasi-statically through calculating the water pressure in each element for a particular applied flux. For media with a small initial saturation, the infiltration flux range for which saturation overshoot occurs is predicted by the network model. For initially dry media, the model predictions scale correctly with media size but require an independent parameter to predict the absolute transition flux. The predicted magnitude of the overshoot does not match particularly well with the measured magnitude for coarse media, although this may be the result of certain simplified assumptions in the model. The results suggest that pore-scale modeling can make certain quantitative predictions of saturation overshoot.