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II.B Definition of Porous Media

II.B.1 Deterministic Geometries

An n-component porous medium in d dimensions is defined as a compact and singly connected subset \mathbb{S} of \mathbb{R}^{d} which contains n closed subsets \mathbb{P}_{i}\subset\mathbb{S} such that

\displaystyle\mathbb{S} \displaystyle= \displaystyle\mathbb{P}_{1}\cup...\cup\mathbb{P}_{n} (2.1)
\displaystyle 0 \displaystyle= \displaystyle V_{d}(\partial\mathbb{P}_{i}) (2.2)

for all 1\leq i\leq n . The set \mathbb{S} is called the sample space and it may represent e.g. a piece of porous rock. The subsets \mathbb{P}_{i},(i=1,..n) represent n different phases or components such as different minerals or fluid phases contained in a rock. The symbol V_{d}(\mathbb{G}) denotes the d-dimensional volume of a set \mathbb{G}\subset\mathbb{R}^{d} and it is defined as

V_{d}(\mathbb{G})=\int\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{G}}}({\bf r})d^{d}{\bf r} (2.3)

where {\bf r} is a d-dimensional vector, and d^{d}{\bf r} is the d-dimensional Lebesgue measure. Thus V_{2} denotes an area, and V_{1} is a length. When there is no danger of confusion V_{3}(\mathbb{G})=V(\mathbb{G}) will be used below. The characteristic (or indicator) function of a set \mathbb{G} is defined as

\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{G}}}({\bf r})=\left\{\begin{array}[]{r@{\quad:\quad}l}1\quad:&\mbox{for}\quad{\bf r}\in\mathbb{G}\\
0\quad:&\mbox{for}\quad{\bf r}\notin\mathbb{G}\end{array}\right. (2.4)

and it indicates when a point is inside or outside of \mathbb{G}. The sets \partial\mathbb{P}_{i} are the phase boundaries separating the different components. The boundary operator \partial is defined on a general set \mathbb{G} as the difference 1 (This is a footnote:) 1 The settheoretic difference operation explains the use of the differential symbol \partial which may also be motivated by the fact that the derivative operator \nabla\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\bf r}) applied to the characteristic function \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\bf r}) of a closed set \mathbb{P} yields the Dirac distribution concentrated on the set \partial\mathbb{P}. \partial\mathbb{G}=\stackrel{\circ}{\mathbb{G}}\setminus\stackrel{\bullet}{\mathbb{G}}. The set \stackrel{\circ}{\mathbb{G}} is called the interior of \mathbb{G} and it is defined as the union of all open sets contained in \mathbb{G}. The set \stackrel{\bullet}{\mathbb{G}} is called the closure of \mathbb{G} and it is defined as the intersection of all closed sets containing \mathbb{G}. 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 \mathbb{M}, and the fluid phases are denoted collectively as pore space \mathbb{P}. The union of the pore and matrix space \mathbb{S}=\mathbb{P}\cup\mathbb{M} gives the full porous sample space \mathbb{S} and the intersection of \mathbb{P} and \mathbb{M} defines the boundary set \mathbb{P}\cap\mathbb{M}=\partial\mathbb{P}=\partial\mathbb{M}. It will usually be assumed that the boundary \partial\mathbb{P} of the pore or matrix space is a surface in \mathbb{R}^{3}. This implies that V_{3}(\partial\mathbb{P})=0 in agreement with (2.2).

As an example consider a clean quartz sandstone filled with water. The sets \mathbb{P} and \mathbb{M} can be defined using the density contrast between the density of water \rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{W}}\approx 1\mbox{g/cm}^{3}, and that of quartz \rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{Q}}\approx 2.65\mbox{g/cm}^{3}. [62] Let \rho({\bf r},\varepsilon) denote the total density in a small closed ball

\mathbb{B}({\bf r},\varepsilon)=\{{\bf q}\in\mathbb{R}^{3}:|{\bf q}-{\bf r}|\leq\varepsilon\} (2.5)

of radius \varepsilon around {\bf r}. If \lim _{{\varepsilon\rightarrow 0}}\rho({\bf r},\varepsilon) exists then the matrix and pore space are defined as \mathbb{M}=\stackrel{\bullet}{\mathbb{Q}} and \mathbb{P}=\stackrel{\bullet}{\mathbb{W}}. Here

\displaystyle\mathbb{Q} \displaystyle= \displaystyle\{{\bf r}\in\mathbb{S}:|\lim _{{\varepsilon\rightarrow 0}}\rho({\bf r},\varepsilon)-\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{Q}}|<\Delta\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{Q}}\} (2.6)
\displaystyle\mathbb{W} \displaystyle= \displaystyle\{{\bf r}\in\mathbb{S}:|\lim _{{\varepsilon\rightarrow 0}}\rho({\bf r},\varepsilon)-\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{W}}|<\Delta\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{W}}\} (2.7)

are the regions occupied by quartz and water, and \Delta\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{Q}},\Delta\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{W}} are the uncertainties in the values of their densities \rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{Q}} and \rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{W}}. In practice it may happen that \mathbb{P}\cup\mathbb{M}\neq\mathbb{S} and even the case \mathbb{P}\cap\mathbb{M}=\emptyset is conceivable. An example would be a muscovite overgrowth with a density of 2.82\mbox{g/cm}^{3} 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 \mathbb{M} 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 a and on the size L of the system. Assuming that the resolution is limited by a\approx 10^{{-10}}m and that L\approx 10^{{-2}}m the configuration of an n-component medium in d dimensions is completely specified by roughly (L/a)^{d}\approx 10^{{8d}} numbers. For d=3 these are O(10^{{24}}) numbers.

II.B.2 Stochastic Geometries

II.B.2.a Discrete Space

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 L, and let a be the microscopic resolution. For concreteness let a=10^{{-10}}m as before. Then there are N=(L/a)^{d} volume elements inside the sample space which are conveniently addressed by their position vectors

{\bf r}_{i}={\bf r}_{{i_{1}...i_{d}}}=(ai_{1},...,ai_{d}) (2.8)

with integers 1\leq i_{1},...,i_{d}\leq L/a. Here {\bf r}_{i} is a shorthand notation for {\bf r}_{{i_{1}...i_{d}}}. A random configuration or random geometry G of an n-component medium is then given as an N-tuple G=(X_{1},...,X_{N})=(X({\bf r}_{1}),...,X({\bf r}_{N})) where the random variables X_{i}\in\mathbb{I}_{n}=\{\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{1}}},...,\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{n}}}\} defined as

X_{i}=X({\bf r}_{i})=\sum _{{j=1}}^{{n}}\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{j}}}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{j}}}({\bf r}_{i}) (2.9)

indicate the presence of phase \mathbb{P}_{i} for the volume element {\bf r}_{i} as identified from its density value \rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{i}}}. The set \mathbb{I}_{n}=\{\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{1}}},...,\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{n}}}\} 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 {\bf r}_{i}\notin\partial\mathbb{P}_{j} for all 1\leq i\leq N and 1\leq j\leq n.

An n-component stochastic porous medium is defined as a discrete probability density on the set of geometries through

\displaystyle\mu(x_{1},...,x_{N}) \displaystyle= \displaystyle\mbox{Prob}\{ G=(x_{1},...,x_{N})\} (2.10)
\displaystyle= \displaystyle\mbox{Prob}\{(X_{1}=x_{1})\wedge...\wedge(X_{N}=x_{N})\}

where x_{i}\in\mathbb{I}_{n}=\{\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{1}}},...,\rho\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}_{n}}}\}. Expectation values of functions f(G)=f(x_{1},...,x_{N}) of the random geometry are defined as

\left\langle f(G)\right\rangle=\left\langle f(x_{1},...,x_{N})\right\rangle=\sum _{{x_{1}\in\mathbb{I}_{n}}}....\sum _{{x_{N}\in\mathbb{I}_{n}}}f(x_{1},...,x_{N})\mu(x_{1},...,x_{N}) (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 \mu is a finite dimensional normalized Boltzmann-Gibbs measure.

A stochastic porous medium is called stationary and homogeneous (in the traditional sense) if its distribution \mu(x_{1},...,x_{N})=\mu(x({\bf r}_{1}),...,x({\bf r}_{N})) is translation invariant, i.e

\mu(x({\bf r}_{1}),...,x({\bf r}_{N}))=\mu(x({\bf r}_{1}+{\bf q}),...,x({\bf r}_{N}+{\bf q})) (2.12)

for all N\in\mathbb{N},{\bf q}\in\mathbb{R}^{d}. 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.

\mu(x({\bf r}_{1}),...,x({\bf r}_{N}))=\mu(x(\bf R{\bf r}_{1}),...,x(\bf R{\bf r}_{N})) (2.13)

for all N\in\mathbb{N} where \bf R denotes a combination of rotation and translation.

The set of possible geometries contains n^{N} elements. For a two component porous cube of sidelength L=1cm there are 2^{{10^{{24}}}} possible configurations at the chosen resolution a=10^{{-10}}m. Thus the complete specification of a stochastic porous medium through \mu(x_{1},...,x_{N}) 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 \mu(x_{1},...,x_{N}). In fact it is even useful to generalize it to continuous space where the required amount of data to specify the distribution becomes infinite.

II.B.2.b Continuous Space

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 \mathbb{E} is a subset of a set \mathbb{O} representing all possible outcomes of some experiment. The probability \mbox{Pr}(\mathbb{E}) of an event is a set function obeying the fundamental rules of probability \mbox{Pr}(\mathbb{O})=1, \mbox{Pr}(\mathbb{E})\geq 0 and \mbox{Pr}\left(\bigcup _{{i=1}}^{\infty}\mathbb{E}_{i}\right)=\sum _{{i=1}}^{\infty}\mbox{Pr}(\mathbb{E}_{i}) if \mathbb{E}_{i}\cap\mathbb{E}_{j}=\emptyset for i\neq j. Formally the probability Pr is a function on a class \mathfrak{O} of subsets of a set \mathbb{O}, called the sample space. If the collection of sets \mathfrak{O} for which the probability is defined is closed unter countable unions, complements and intersections then the triple (\mathbb{O},\mathfrak{O},\mbox{Pr}) is called a probability space. The family of sets \mathfrak{O} is called a \sigma-algebra. The conditional probability of an event \mathbb{E} given the event \mathbb{G} is defined as

\mbox{Pr}(\mathbb{E}|\mathbb{G})=\frac{\mbox{Pr}\{\mathbb{E}\cap\mathbb{G}\}}{\mbox{Pr}\{\mathbb{G}\}}\;\;,\;\;\mbox{Pr}\{\mathbb{G}\}\neq 0. (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 \mathcal{F} denotes the family of all closed sets in \mathbb{R}^{d} including the empty set \emptyset. Let \mathcal{K} denote the set of all compact sets. The smallest \sigma-algebra \mathfrak{F} of subsets of \mathcal{F} which contains all the hitting sets is then defined as \mathcal{F}_{\mathbb{K}}=\{\mathbb{F}\in\mathcal{F}:\mathbb{F}\cap\mathbb{K}\neq\emptyset\} where \mathbb{K} 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 \mathbb{X} (more precisely a random closed set) is defined as a measurable map from a probability space (\mathbb{O},\mathfrak{O},\mu) to (\mathcal{F},\mathfrak{F}) [10]. This allows to assign probabilities to countable union and intersections of the sets \mathcal{F}_{\mathbb{K}} which are the elements of \mathfrak{F}. For example

\mbox{Pr}(\mathcal{F}_{\mathbb{K}})=\mu(\mathbb{X}^{{-1}}(\mathcal{F}_{\mathbb{K}})) (2.15)

is the probability that the intersection \mathbb{X}\cap\mathbb{K} 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 \mathbb{X} [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 \mathbb{X} is called stationary if

\mbox{Pr}\{\mathbb{X}\cap\mathbb{K}\neq\emptyset\}=\mbox{Pr}\{(\mathbb{X}+{\bf r})\cap\mathbb{K}\neq\emptyset\} (2.16)

for all vectors {\bf r}\in\mathbb{R}^{d} and all compact sets \mathbb{K}. The notation \mathbb{G}+{\bf r} denotes the translated set defined as

\mathbb{G}+{\bf r}=\{{\bf q}+{\bf r}:{\bf q}\in\mathbb{G}\} (2.17)

for {\bf r}\in\mathbb{R}^{d} and \mathbb{G}\subset\mathbb{R}^{d}. Using the analogous notation

\bf R\mathbb{G}=\{\bf R{\bf q}:{\bf q}\in\mathbb{G}\} (2.18)

for \bf R a rigid euclidean motion allows to define a random set to be isotropic if

\mbox{Pr}\{\mathbb{X}\cap\mathbb{K}\neq\emptyset\}=\mbox{Pr}\{(\bf R\mathbb{X})\cap\mathbb{K}\neq\emptyset\} (2.19)

for all rigid motions \bf R and compact sets \mathbb{K}. For later reference the notation

c\mathbb{G}=\{ c{\bf q}:{\bf q}\in\mathbb{G}\} (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 \mathbb{R}^{d} which is in practice impossible. Nevertheless the definition is important to clarify the concept of a random set.