An -component porous medium in
dimensions is defined
as a compact and singly connected subset
of
which
contains
closed subsets
such that
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(2.1) | |
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(2.2) |
for all .
The set
is called the sample space and
it may represent e.g. a piece of porous rock.
The subsets
represent
different phases or components such as different
minerals or fluid phases contained in a rock.
The symbol
denotes the
-dimensional volume
of a set
and it is defined as
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(2.3) |
where is a
-dimensional vector, and
is
the
-dimensional Lebesgue measure.
Thus
denotes an area, and
is a length.
When there is no danger of confusion
will be used below.
The characteristic (or indicator) function of a
set
is defined as
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(2.4) |
and it indicates when a point is inside or outside of .
The sets
are the phase boundaries
separating the different components.
The boundary operator
is defined on a
general set
as the difference
1 (This is a footnote:) 1
The settheoretic difference operation explains the
use of the differential symbol
which may
also be motivated by the fact that the derivative
operator
applied to the
characteristic function
of a closed set
yields
the Dirac distribution concentrated on the set
.
.
The set
is called the interior of
and it is defined as the union of all open sets contained
in
.
The set
is called the closure of
and
it is defined as the intersection of all closed sets
containing
.
The condition (2.2) excludes fractal boundaries
or cases in which a boundary set is dense.
By replacing the Lebesgue measure in (2.3) with
Hausdorff measures some of these restrictions may be
relaxed [58, 59, 60, 61].
Frequently the different phases or components may be classified
into solid phases and fluid phases. An example is a porous rock.
In this case it is convenient to consider the two-component
medium in which all the solid phases are collectively denoted
as matrix space , and the fluid phases are denoted
collectively as pore space
.
The union of the pore and matrix space
gives the full porous sample space
and
the intersection of
and
defines the boundary set
.
It will usually be assumed that the boundary
of the pore or matrix space is a surface in
.
This implies that
in agreement with (2.2).
As an example consider a clean quartz sandstone filled with
water. The sets and
can be defined using the
density contrast between the density of water
, and that of quartz
.
[62]
Let
denote the total density
in a small closed ball
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(2.5) |
of radius around
.
If
exists then the matrix
and pore space are defined as
and
.
Here
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(2.6) | |
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(2.7) |
are the regions occupied by quartz and water, and
are the uncertainties in the values
of their densities
and
.
In practice it may happen that
and even
the case
is conceivable.
An example would be a muscovite overgrowth with a density
of
surrounding the quartz grains of the sandstone.
A generalization of the definition above is necessary
if the pore space is filled with two immiscible fluids
(see chapter VI).
In this case the fluid-fluid interface is mobile, and thus
all the sets above may in general become time dependent.
The same applies when the matrix is not rigid and
represents a deformable medium such as a gel.
Finally it is of interest to estimate the amount of
information contained in a complete specification of
a porous geometry according to the definitions above.
This will depend on the spatial resolution and on
the size
of the system. Assuming that the
resolution is limited by
m and that
m
the configuration of an
-component medium in
dimensions is completely specified by roughly
numbers.
For
these are
numbers.
The irregular geometry of porous media frequently appears to be random or to have random features. This observation suggests the use of probabilistic methods. The idealization underlying the use of statistical methods is that the irregular geometry is a realization drawn at random from an ensemble of possible geometries. It must be emphasized that an idealization is involved in discussing an ensemble rather than individual geometries. It assumes that there exists some form of recognizable statistical regularity in the irregular fluctuations and heterogeneities of the microstructure. This idealization is modeled after statistical mechanics where the microstructure corresponds to a full specification of the positions and momenta of all particles in a fluid while the recognizable regularities are contained in its macroscopic equation of state or thermodynamic potentials. The statistical idealization assumes that the recognizable regularities of porous media can be described by a suitable probability distribution on the space of all possible geometries. Such a description may not always be the most obvious or most advantageous [56], and in fact the merit of the stochastic description does not lie in its improved practicability. The merit of the stochastic description lies in the fact that it provides the necessary framework to define typical or average properties of porous media. The typical or average properties, it is hoped, will provide a more practical geometric characterization of porous media.
Before embarking on the definition of stochastic porous media I wish to emphasize a recent development in the foundations of statistical mechanics [63, 64, 65, 66, 67, 68, 69] which concerns the concept of stationarity or homogeneity. Stationarity is often invoked in the statistical characterization of porous media although many media are known to be heterogeneous on all scales. Heterogeneity on all scales means that the geometrical or physical properties of the medium never approach a large scale limit but continue to fluctuate as the length scale is increased from the microscopic resolution to some macroscopic length scale. Homogeneity or stationarity assumes the absence of macroscopic fluctuations, and postulates the existence of some intermediate length scale beyond which fluctuations decrease [5]. Recent developments in statistical mechanics [63, 64, 65, 66, 67, 68, 69] indicate that the traditional concept of stationarity is too narrow, and that there exists a generalization which describes stationary but heterogeneous macroscopic behaviour. Although these new concepts are still under development they have already been applied in the context of local porosity theory discussed in section III.A.5.
Consider a porous sample (e.g. cubically shaped) of extension
or sidelength , and let
be the microscopic resolution.
For concreteness let
m as before.
Then there are
volume elements inside the
sample space which are conveniently addressed by their
position vectors
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(2.8) |
with integers .
Here
is a shorthand notation for
.
A random configuration or random geometry
of an
-component medium is then given as an
-tuple
where the random
variables
defined as
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(2.9) |
indicate the presence of phase for the volume element
as identified from its density value
.
The set
is a set of indicators, here the densities, which are used
to label the phases.
Of course the density could be replaced by other quantities
characterizing or labeling the components.
The discretization is always chosen such that
for all
and
.
An -component stochastic porous medium is defined
as a discrete probability density on the set of geometries through
![]() |
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(2.10) | |
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where .
Expectation values of functions
of the random geometry are defined as
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(2.11) |
where the sum is over all configurations of the geometry.
Note the analogy between (2.11) and expectation values
in statistical mechanics.
The analogy becomes an equivalence if is
a finite dimensional normalized Boltzmann-Gibbs measure.
A stochastic porous medium is called stationary and
homogeneous (in the traditional sense) if its distribution
is translation invariant, i.e
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(2.12) |
for all .
This traditional definition of stationarity is a special
case of the more general concept of fractional stationarity
[63, 64, 65, 66, 67, 68, 69]
which is currently being developed to describe macroscopic
heterogeneity.
A stochastic porous medium is called isotropic if its distribution is invariant under all rigid euclidean motions, i.e.
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(2.13) |
for all where
denotes a combination of rotation
and translation.
The set of possible geometries contains elements.
For a two component porous cube of sidelength
cm
there are
possible configurations at
the chosen resolution
m.
Thus the complete specification of a stochastic porous
medium through
is even less practical
than specifiying all the volume elements of a particular
sample.
This does not diminish the theoretical importance of the
microscopic geometry distribution
.
In fact it is even useful to generalize it to continuous
space where the required amount of data to specify the
distribution becomes infinite.
Instead of discretizing the space it is possible to work directly with the notion of random sets in continuous space. The mathematical literature about random sets [70, 10, 71] is based on pioneering work by Choquet [72].
To define random sets recall first the concepts of a probability
space and a random variable [73, 74, 75].
An event is a subset of a set
representing all
possible outcomes of some experiment.
The probability
of an event is a set function
obeying the fundamental rules of probability
,
and
if
for
.
Formally the probability Pr is a function on a class
of subsets of a set
, called the sample space.
If the collection of sets
for which the probability
is defined is closed unter countable unions, complements
and intersections then the triple
is called
a probability space. The family of sets
is called
a
-algebra.
The conditional probability of an event
given
the event
is defined as
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(2.14) |
A random variable is a real valued function on a probability space.
Random sets are generalizations of random variables.
The mathematical theory of random sets is based on the
“hit-or-miss” idea that a complete
characterization of a set can be obtained by intersecting
it sufficiently often with an arbitrary compact test set
and recording whether the intersection is empty or not
[70, 10].
Suppose that denotes the family of all closed sets
in
including the empty set
.
Let
denote the set of all compact sets.
The smallest
-algebra
of subsets of
which contains all the hitting sets is then defined as
where
is a compact test set.
An event in this context is the statement whether or not
a random set hits a particular countable family of compact
subsets.
A random set (more precisely a random closed set)
is defined as a measurable map from a probability space
to
[10].
This allows to assign probabilities to countable union
and intersections of the sets
which are the
elements of
.
For example
![]() |
(2.15) |
is the probability that the intersection
is not empty.
This probability plays an important role in the geometric
characterizations of porous media based on capacity
functionals [72, 10] discussed below.
Note that there exists no simple mathematical analogue of
the expecation value defined for the discrete case in
(2.11).
Its definition, which will not be needed here, requires
the introduction of functional integrals
on an infinitely dimensional space, or the study of
random measures associated with the random set
[10].
While the expectation value is not readily carried over to the
continuous case the concepts of stationarity and isotropy are
straightforwardly generalized.
A random set is called stationary if
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(2.16) |
for all vectors and all compact sets
.
The notation
denotes the translated set defined as
![]() |
(2.17) |
for and
.
Using the analogous notation
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(2.18) |
for a rigid euclidean motion allows to define
a random set to be isotropic if
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(2.19) |
for all rigid motions and compact sets
.
For later reference the notation
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(2.20) |
is introduced to denote the multiplication of sets by real numbers. The traditional definition of stationarity presented in (2.16) is restricted to macroscopically homogeneous porous media. It is a special case of the more general concept of fractional stationarity which describes macroscopic heterogeneity, and which is currently under development [63, 64, 65, 66, 67, 68, 69].
The mathematical definition of random sets in continuous
space is even less manageable from a practical perspective
than its definition for a discretized space.
A complete specification of a random set would require
the specification of “all” compact or “all”
closed subsets of which is in practice impossible.
Nevertheless the definition is important to clarify the concept
of a random set.