We study the influence of wettability on the morphology of fluid-fluid displacement through analog porous media in the limit of vanishing flow rates. We introduce an invasion-percolation model that considers cooperative pore filling and corner-flow mechanisms and captures interface motion at the pore scale for all quasistatic flow regimes between strong drainage and strong imbibition. We validate the method against recent experimental observations of wetting transition in microfluidic cells patterned with circular posts and we use it to explore the sensitivity of fluid invasion to wettability heterogeneity, post spacing, and post height. Our model therefore extends the Cieplak-Robbins description of quasistatic fluid invasion by reproducing the wetting transition in strong imbibition, a feature that requires incorporating three-dimensional effects.