In the paper anomalous diffusion appealing in a porous medium composed of two porous components of considerably different diffusion characteristics is examined. The differences in diffusivities are supposed to result either from two medium types being present or from variations in pore size (double porosity media). The ＇long-tail＇ effect is predicted using the homogenization approach based on the application of multiple scale asymptotic developments. It is shown that, if the ratio of effective diffusion coefficients of two porous media is of the order of magnitude smaller or equal O(epsilon(2)), where epsilon is a homogenization parameter, then the macroscopic behaviour of the composite may be affected by the presence of ＇tail-effect＇. The results of the theoretical analysis were applied to a problem of diffusion in a bilaminate composite. Analytical calculations were performed to show the presence of the ＇long-tail＇ effect in two particular cases.