  Local porosity distributions for a three-dimensional porous medium and local
porosity distributions for a two-dimensional plane-section through the medium
are generally different. However, for homogeneous and isotropic media having
finite correlation-lengths, a good degree of correspondence between the two
sets of local porosity distributions can be obtained by rescaling lengths, and
the mapping associating corresponding distributions can be found from
two-dimensional observations alone. The agreement between associated
distributions is good as long as the linear extent of the measurement cells
involved is somewhat larger than the correlation length, and it improves as the
linear extent increases. A simple application of the central limit theorem
shows that there must be a correspondence in the limit of very large
measurement cells, because the distributions from both sets approach normal
distributions. A normal distribution has two independent parameters: the mean
and the variance. If the sample is large enough, LPDs from both sets will have
the same mean. Therefore corresponding distributions are found by matching
variances of two- and three-dimensional local porosity distributions. The
variance can be independently determined from correlation functions. Equating
variances leads to a scaling relation for lengths in this limit. Three
particular systems are examined in order to show that this scaling behavior
persists at smaller length-scales.
