In this paper, the time-dependent equation for diffusion in an aperiodic porous medium with weakly penetrable inclusions is considered. The asymptotic behavior of its solutions is described by an averaged equation with memory. Effective parameters are calculated with the help of solutions to local problems. The general model obtained is shown to describe three qualitatively distinguished cases differing by the degree of inhomogeneity of a medium. 1. Introduction. We consider here transport processes in a porous medium representing the spatial alternation of two types of rocks with contrast transport properties. In this case, we cannot ignore the existence of a weakly conducting subsystem since, even being virtually impenetrable, it can contain a considerable quantity of a fluid. Objects of this type are widely met in hydrology or oil engineering, where natural porous strata are composed of rocks with sharply different petrographic properties or are dissected by nets of cracks. For the description of such objects, a model of media with double porosity is used. In the framework of this model, a medium is represented as a unified highly penetrable system (matrix) Ω e having weakly penetrable inclusions (blocks) F e . The porosity of blocks is considered to be not lower than that of the matrix. If the scale of an inhomogeneity e is small, the averaged behavior of the system is of interest. This behavior is complicated due to the fact that the medium involves the second substantial parameter ω ! 1 equal to the ratio between the conductivity of blocks and that of the matrix, so that the asymptotic degeneracy of coefficients of the equations occurs on the inclusions as ω 0 . The classical model of flow through a medium with double porosity [1] was based on the hypothesis of a quasisteady exchange process between the blocks and the matrix. In the general case, the exchange process is unsteady, and, by virtue of the Duhamel principle, the appearance of operators of the time-convolution type can be expected. A similar model with memory was proposed in [2, 3] for a particular relation between the medium parameters when ω ~ e 2 ( e 2 -model). In the more general case, media with double porosity correspond to all situations when ω ~ e 2 . Four classes of such media [4] can be distinguished. For ω ! e 2 , weakly penetrable blocks can be ignored; for ω ~ e 2 , the transport is described by the above model with a long-term memory; for ω ~ e 2 , the medium has a short-term memory; and, for e ! ω ! 1 , the medium behavior is moderately inhomogeneous without memory.