In this paper we study the possibility to account for preferential diffusion effects in lean turbulent premixed flames in numerical predictions with reduced chemistry. We studied the situation when hydrogen is added to methane at levels of 20% and 40% by volume in the fuel, at lean combustion (φ = 0.7) with air. The base case of pure methane was used as a reference. In this case preferential diffusion effects are negligible. First the sensitivity of the mass burning rate to flame stretch was investigated, in one dimensional computations with detailed chemistry, to set reference values. Then the framework of the Flamelet Generated Manifolds (FGM) was used to construct an adequate chemical method to take preferential diffusion into account, without the need for using detailed chemistry. To that end a generalization of the method was presented in which five controlling variables are required. For this system, proper transport equations and effective Lewis numbers where derived. In practice not all five variables are necessary to include and as a first step we limited the amount in the numerical tests in this study to two controlling variables. The method was then tested in configurations in which there was an interaction of coherent vortices and turbulence with flames. It was demonstrated that a minimum of two controlling variables is needed to account for the changed mass burning rate as function of stretch and curvature. It was shown that one-dimensional FGM as well as one-step Arrhenius kinetics can not describe this relation.