Volume 39 Number 2
  • E-ISSN: 1365-2478



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 of a constituent is an element consisting of scalar and a matrix which is 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 as → 0, where 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 as → 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 barriers can be ‘added’ and 'subtracted’ from a background homogeneous anisotropic medium commutatively and associatively, but not non‐parallel sets of channels 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.


Article metrics loading...

Loading full text...

Full text loading...


  1. Backus, G.1990. Review of a note by Hudson, J.A. and Crampin, S. submitted to Geophysics .
  2. Biot, M.A.1956. Theory of elastic waves in a fluid‐saturated porous solid. I. Low‐frequency range. Journal of the Acoustical Society of America28, 168–178.
    [Google Scholar]
  3. Crampin, S.1984. Effective anisotropic elastic constants for wave propagation through cracked solids. Geophysical Journal of the Royal Astronomical Society76, 135–145.
    [Google Scholar]
  4. Helbig, K. and Schoenberg, M.1987. Anomalous polarization of elastic waves in transversely isotropic media. Journal of the Acoustical Society of America81, 1235–1245.
    [Google Scholar]
  5. Nichols, D., Muir, F. and Schoenberg, M.1989. Elastic properties of rocks with multiple sets of fractures. 59th SEG meeting, Dallas, Expanded Abstracts, 471–474.
  6. Schoenberg, M. and Muir, F.1989. A calculus for finely layered anisotropic media. Geophysics54, 581–589.
    [Google Scholar]
  7. Schoenberg, M. and Sen, P.N.1987. Permeability and velocity in simple modes with compressible fluids. Journal of the Acoustical Society of America82, 1804–1810.
    [Google Scholar]
  • Article Type: Research Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error