[61.1.4.1] The porosity

[61.2.1.1] It is often suggested to use a “pore-size distribution” as a possible geometric characterization for porous media, and mercury injection is suggested to measure it. [61.2.1.2] Already Scheidegger,[24] however, pointed out that the pore-size distribution is mathematically ill defined. [61.2.1.3] It depends on an arbitrary identification of cylindrical pores and their diameters.[25] [61.2.1.4] It is now well appreciated[26] that the result of a mercury injection measurement cannot be interpreted without having already a faithful geometrical model of the pore space, which itself requires knowledge of the pore-size distribution. [61.2.1.5] Without such a model, no reliable geometrical information such as the pore-size distribution can be extracted from the measurement.

[61.2.2.1] This paper suggests a different characterization of porous media.
[61.2.2.2] Purely geometric quantities will be introduced
which are well defined and readily
accessible to direct measurement.
[61.2.2.3] The characterization is based on viewing
the local geometry on a mesoscopic scale
as the fundamental random quantity.
[61.2.2.4] To define “local geometries”,
consider a porous medium with a homogeneously
and isotropically disordered pore space.
[61.2.2.5] The points of a Bravais lattice (in practice a simple cubic lattice)
are superimposed on the porous medium
and an arbitrary (in parctice cubic) primitive cell is chosen.
[61.2.2.6] The local geometry around the lattice point

[62.1.1.1] The local geometry inside the measurement cell
will become increasingly complex as the length scale
of resolution

[62.1.2.1] Consider first the local (or cell) porosity.
[62.1.2.2] To define it,
the characteristic functions (indicator functions)
of an arbitrary set

(2.1) |

[62.1.2.3] The local (or cell) porosity

(2.2) |

where

(2.3) |

independent of

[62.2.1.1] The local porosity distribution

(2.4) |

[62.2.1.9] The porosity autocorrelation function at scale

(2.5) |

and the porosity correlation length

(2.6) |

[62.0.1.10] In the following the “local porosity distribution”
is defined as

[62.0.2.1] The most important aspect of

(2.7) |

where

[62.1.1.1] The result of the measuring procedure described in the preceding paragraph will in general lead to a local porosity distribution of the form

(2.8) |

[page 63, §0]
[63.1.0.1] Its bulk (average) porosity

(2.9) |

is expected to result.
[63.1.0.5] The geometric information in this case
is reduced to

(2.10) |

Again, the geometrical information in

[63.1.1.1] The local porosity distribution

[63.1.2.1] The second geometric propery to characterize
local geometries is whether the pore space percolates or not.
[63.1.2.2] For cubic cells each cell is classified
as percolating or nonpercolating
according to whether or not there exists at least one face
of the cube which can be connected
to any of the other faces via a path
contained completely inside the pore space.
[63.1.2.3] For noncubic cells the classification
has to be modified appropriately.
[63.1.2.4] Let

[63.1.3.1] Twe two function