We investigate three types of Internal Diffusion Limited Aggregation (IDLA) models. These models are based on simple random walks on Z 2 with infinitely many sources that are the points of the vertical axis I (∞) = {0} × Z. Various properties are provided, such as stationarity, mixing, stabilization and shape theorems. Our results allow us to define a new directed (w.r.t. the horizontal direction) random forest spanning Z 2 , based on an IDLA protocol, which is invariant in distribution w.r.t. vertical translations.