Two–dimensional depth–integrated model equations describing nonlinear water waves propagating over a permeable bed are developed. These model equations are expressed in terms of the free–surface displacement and representative horizontal velocity components, which could be the velocity components evaluated at certain elevation or the depth–averaged velocity components. Each set of model equations invokes different approximations and therefore imposes different limitations on the vorticity field. The frequency–dispersion properties of the linearized model equations are investigated and are compared with those of small amplitude waves over a permeable bed. Based on these comparisons, an optimal model is suggested. To check the validity of the model equations, a laboratory experiment of a wave train propagating over a submerged triangular permeable bar is performed. Numerical results of the model equations show very good agreement with experimental data. The effects of the submerged permeable bar on the wave evolution are also discussed.