LAYERED PERMEABLE SYSTEMS¹

Permeability is a second rank tensor relating flow rate to pressure gradient in a porous medium. If the permeability is a constant times the identity tensor the permeable medium is isotropic; otherwise it is anisotropic. A formalism is presented for the simple calculation of the permeability tensor of a heterogeneous layered system composed of interleaved thin layers of several permeable constituent porous media in the static limit. Corresponding to any cumulative thickness H of a constituent is an element consisting of scalar H and a matrix which is H times a hybrid matrix function of permeability. The calculation of the properties of a medium equivalent to the combination of permeable constituents may then be accomplished by simple addition of the corresponding scalar/matrix elements. Subtraction of an element removes a permeable constituent, providing the means to decompose a permeable medium into many possible sets of permeable constituents, all of which have the same flow properties. A set of layers of a constituent medium in the heterogeneous layered system with permeability of the order of 1/h as h→ 0, where h is that constituent's concentration, acts as a set of infinitely thin channels and is a model for a set of parallel cracks or fractures. Conversely, a set of layers of a given constituent with permeability of the order of h as h→ 0 acts as a set of parallel flow barriers and models a set of parallel, relatively impermeable, interfaces, such as shale stringers or some faults. Both sets of channels and sets of barriers are defined explicitly by scalar/matrix elements for which the scalar and three of the four sub‐matrices vanish. Further, non‐parallel sets of channels or barriers can be ‘added’ and 'subtracted’ from a background homogeneous anisotropic medium commutatively and associatively, but not non‐parallel sets of channels and barriers reflecting the physical reality that fractures that penetrate barriers will give a different flow behaviour from barriers that block channels. This analysis of layered media, and the representations of the phenomena that can occur as the thickness of a constituent is allowed to approach zero, are applicable directly to layered heat conductors, layered electrostatic conductors and layered dielectrics.