In this work we develop nonlinear models to study the propagation of surface water waves.The Nonlinear Schrödinger Equation (NLSE) is used to describe the evolution of the envelope of a wave packet. However, the NLSE, an equation obtained at third order in steepness in the multiscale expansion, is symmetric in time. Therefore it is unable to reproduce the asymmetric development of the water waves envelope. We then propose four models obtained at fourth order in steepness to recover asymmetry and improve the agreement with experiments. Those four models are adapted from the fourth-order NLSE found in the literature. With those models, we perform numerical simulations of the envelope propagation in space. We simulate the water waves as propagating in one direction along a wave tank of constant depth. This simulation is performed with the third-order NLSE and the four fourth-order models for different physical parameters (water depth, carrier wave frequency, steepness of the wave). We then study the impact of the modified fourth-order models on the envelope propagation and we carefully investigate the role of each term in such model equations. On the other hand we compare our simulations with real measurement in the wave flume of the University of Sydney. We then analyse the agreement between our models and the experiment.