We consider the effects of a hierarchical, multiple layered system of fractures on the flow of a single-phase, slightly compressible fluid through a porous medium. A microscopic flow model is first defined which describes precisely the physics of the flow and the geometry of the fracture system and porous matrix, all of which depends on a positive parameter is an element of that determines the scale of the various fracture-level thicknesses. We then show by a rigorous mathematical argument that the unique solution of this microscopic problem converges as is an element of --> 0 to the solution of a double-porosity model of the global macroscopic flow. Our techniques make use of the concept of reiterated homogenization and essentially consist of an adaptation of the methods of extension and dilation operators to the reiterated-homogenization context. We show how the porosities and permeability tensor of the porous medium are determined in a precise way by certain physical and geometric features of the microscopic fracture domain, the microscopic matrix blocks, and the interface between them.