Soil wettability has been recently recognized as a factor that can dramatically influence the dissolution behavior of residual nonaqueous phase liquids (NAPL). A NAPL that wets the solid surface is trapped within the smaller pores and along the corners of pores invaded by water (the nonwetting phase). We present a two-dimensional network simulator of wetting NAPL dissolution, inspired by observations of this process in transparent glass micromodels. The network model idealizes the pore space as a network of cubic pores connected by square tubes, following respective distributions. In accordance with micromodel observations, capillary equilibrium is assumed to exist between NAPL-water interfaces along pore corners and within pores. Advection and diffusion of the organic dissolved in the aqueous phase, as well as dissolution mass transfer from residual NAPL, are explicitly accounted for in the model. Pores filled with NAPL are invaded at a rate which is controlled by mass transfer from dissolving thick NAPL films in pore corners and in order of increasing entry capillary pressure, resulting in quasi-static drainage and fingering of the aqueous phase. Loss of NAPL continuity due to rupture of thick NAPL films and heterogeneity are found to affect profoundly the dissolution behavior, resulting in concentration tailing. The network simulator reproduces qualitatively the behavior observed in column experiments with oil-wet media.