A four-dimensional variational assimilation (4D-Var) scheme is now widely used by meteorological centres in a operational way. However, most of these applications do not take account of model error.Indeed, the classical 4D-Var with imperfect model formulation is unaffordable for current computational means. This paper presents a low-cost method for dealing with model errors in 4D variational assimilation. This method can be formally compared to a Kalman filter. This new scheme is tested on two configurations: first on a Burger equation, which allows one to calibrate the method, and second on a more relevant shallow-water equations model, both in a twin experiment framework. It is shown that, compared to classical 4D-Var results, this method provides a noticeable improvement.