We propose a dynamical model for channels incised into an erodible bed by subsurface water flow. The model is validated by the time-resolved topographic measurements of channel growth in a laboratory-scale experiment. Surface heights in the experiment are measured via a novel laser-aided imaging technique. The erosion rate in the model is composed of diffusive and advective components as well as a simple driving term due to the seeping water. Steady driving conditions may exist whenever channels are incised into a flat and level erodible bed by a water table replenished via steady (on average) rainfall. Under such steady driving conditions, the model predicts an asymptotically self-similar growing shape for the channel transects. Conversely, given a transect shape that evolved under steady driving conditions and an estimate of the erosion rate at the bottom of the channel, granular transport coefficients can be inferred from the static channel shape. We report an estimate of these transport coefficients for a system of ravines incised into unconsolidated sand in the Apalachicola River basin, Florida.