V.A.1 Definition
A large number of transport and relaxation processes in
porous media are governed by the diordered Laplace equation
(4.2) with variable coefficients Cr
for a scalar field Pr
within the sample region 𝕊=ℙ∪𝕄.
This “equation of motion” for P must be
supplemented with suitable boundary conditions
on the sample boundary ∂𝕊, and, if
Cr is discontinuous across ∂ℙ,
also on the internal boundary ∂ℙ .
Introducing the vector field vr the
equation (5.1) may be rewritten as
vr  =  Cr∇Pr 
 (5.2) 
∇T⋅vr  =  0. 

These equations can be used as the microscopic starting
point although, as shown below in section V.C.3
for the case of fluid flow, they may hold only in a macroscopic limit
starting from a different underlying microscopic description.
Equations (5.1) or (5.2) appear
in many transport and relaxation problems for porous and
heterogeneous media.
For Darcy flow in porous media P is the pressure,
C=K/η is the quotient of absolute hydraulic
permeability and fluid viscosity, and v is the
fluid velocity field.
For dielectric relaxation P becomes the electrostatic
potential, v becomes the dielectric displacement and
C becomes the dielectric permittivity tensor.
In diffusion or dispersion problems P is the
concentration field, v corresponds to the
diffusion flux and C becomes the diffusivity.
Table III summarizes the translation of
P,v and C into various problems.
Table III: Quantities corresponding to P,v and C in 5.2
for different transport and relaxation problems in porous media.
Problem Type 
P 
v 
C 

fluid flow 
pressure 
velocity 
permabilityviscosity 

electrical conduction 
voltage 
current 
conductivity 

dielectric relaxation 
potential 
displacement 
dielectric permittivity 

diffusion (dispersion) 
concentration 
particle flux 
diffusion constant 

For a homogeneous and isotropic medium the transport coefficients
Cr=C1, where 1 denotes the identity,
are independent of r, and (5.1) reduces to a Laplace
equation for the field P.
For a random medium the transport coefficients are random functions
of r and the solutions Pr and vr depend on the
realization of Cr.
The averaged solutions Pr and vr
are therefore of primary interest.
The tensor of effective transport coefficients is
C¯ defined as
and it provides a relation between the average fields.
The ensemble averages fr in the definition can
be replaced with spatial averages defined by
f¯=1V𝕊∫frχ𝕊rd3r 
 (5.4) 
where f stands for P or v.
Both the ensemble and the spatial average depend on the averaging
region 𝕊, and a residual variation of f¯ or f
is possible on scales larger than the size of 𝕊.
In the following it will always be assumed that
f¯=f if 𝕊 is sufficiently large.
The ensemble average notation will be preferred because
it is notationally more convenient.
The purpose of introducing effective macroscopic transport
coefficients is to replace the heterogeneous medium described
by Cr with an equivalent homogeneous medium described
by C¯.
If C¯ is known then all the knowledge accumulated
for the homogeneous problem can be utilized immediately,
and e.g. the average field v can be obtained simply
from solving a Laplace equation for P.
V.A.2 Discretization and Networks
If the function Cr is known then equation (5.1)
can be solved to any desired accuracy using standard finite difference
approximation schemes.
To this end the sample space 𝕊 of linear extension
ℒ is partitioned into cubes 𝕂j.
The cubes are centered on the sites ri of a simple
cubic lattice with lattice spacing L.
Other lattices may also be employed.
The lengths L and ℒ obey L≪ℒ.
The total numer of cubes is N=ℒ/Ld.
For a stationary and isotropic medium with
Cr=cr1 the discretization of equation
(5.1) gives a system of
linear equations for the pressure variables at the
cube centers Pi=Pri
for cubes ri not located at the sample boundary.
The boundary conditions at the sample boundary give
rise to a nonvanishing right hand side of the linear
system if ri is the center of a cube located
close to ∂𝕊.
The local transport coefficients cij are
given as
if ri and rj are nearest neighbours.
If ri and rj are not nearest neighbours
the local coefficient vanishes, cij=0.
Because the location of the cube centers ri
depends on the resolution L the coefficients
cij in the network equations depend on L
and on the shape of the measurement cells 𝕂i.
The numerical solution of the discretized equations
(5.5) can be obtained by many methods
including relaxation, successive overrelaxation or
conjugate gradient schemes, transfer matrix calculations,
series expansions or recursion methods
[263, 264, 265, 266, 248, 267, 40].
If the function cr is known then the solution
to (5.1) is recovered in the limit
L→0 to any desired accuracy.
Within a certain class of lattices the limit is known
to be independent of the choice of the approximating
discrete lattice.
To actually perform this limit, however, the function cr
must be known to arbitrary accuracy.
In most experimental and practical problems the function
cr is either completely unknown or not known to
arbitrary accuracy.
Therefore it is necessary to have a theory for the
local transport coefficients cijL
as a function of the resolution L of the discretization.
At present the only resolution dependent theories
seem to be local porosity theory
[168, 169, 170, 171, 172, 173, 174, 175]
and homogenization theory [268, 269, 270, 38, 271]
which will be discussed in more detail below.
The basic idea of local porosity theory is to use the local geometry
distributions defined in section III.A.5
and to express the local transport coefficients in terms
of the geometrical quantities characterizing the local
geometry.
The basic idea of homogenization theory is a double
scale asymptotic expansion in the small parameter L/ℒ.
The discretized equations (5.5) are
network equations.
This explains the great importance and popularity of
network models.
In the more conventional network models
[220, 221, 222, 223, 225, 187, 226, 227, 228, 229, 230, 155, 157, 231, 232, 233]
the resolution dependence is neglected altogether.
Instead one assumes a specific model for the local transport
coefficients cij such that the global geometric
characteristics (porosity etc.) are reproduced by the model.
Three immediate problems arise from this assumption:

The connection with the underlying local geometry is lost,
although the local value of the transport property depends on it.

In the absence of an independent measurement of the local
transport coefficients they become free fit parameters.
Popular stochastic network models assume lognormal or
binary distributions for the local transport coefficients.

Without a model for the local geometry an independent
experimental or calculational determination of the local
transport coefficients for one transport problem
(say fluid flow) cannot be used for another transport
problem (say diffusion) although the equations of motion
(5.1) have the same mathematical form for
both cases.
All of these problems are alleviated in local porosity theory
or homogenization theory which attempt to keep the connection
with the underlying local geometry.
V.A.3 Simple Expressions for Effective Transport Coefficients
While a numerical solution of the network equations
(5.5) is of great practical interest, its
value for a scientific understanding of heterogeneous media
is limited.
Analytical expressions, be they exact or approximate, are better
suited for developing the theory because they allow to extract
the general modelindependent aspects.
Unfortunately only very few exact analytical results are available
[272, 273, 274, 275].
The one dimensional case can be solved exactly by a change
of variable. The exact result is the harmonic average
where the average denotes either an average with respect to wc,
the probability density of local transport coefficients, or a spatial
average as defined in (5.4).
In two dimensions the geometric average
c¯=explogc=c1/2c11/2 
 (5.8) 
has been obtained exactly using duality in harmonic function theory
[272] if the microsctructure is homogeneous, isotropic and symmetric.
It was later rederived under less stringent conditions [273]
and generalized to isomorphisms between associated microstructures
[274].
Most analytical expressions for effective transport properties
are approximate.
In general dimensions approximation formulae such as [276, 277, 275]
or [278, 279]
have been suggested which reduce to the exact results for
d=1 and d=∞.
Various mean field theories also provide approximate estimates
for the effective permeabilities.
The simplest mean field theory
is obtained from equations (5.9) or (5.10) by letting d→∞.
Another very important approximation is the selfconsistent
effective medium approximation which reads
for a ddimensional hypercubic lattice.
For other regular lattices the factor d1 in the denominator
has to be replaced with z/21 where z is the coordination
number of the lattice.
Note that for d=1 and d→∞ the effective medium approximation
reproduces the exact result.
To distinguish the quality of these approximations it is
instructive to consider a probability density wc
of local transport coefficients which has a finite
fraction p=1limε→0∫0εwcdc
of blocking bonds.
In dimension d>1 this implies the existence of a
percolation threshold 0<pc<1 below which c¯=0
vanishes identically (see Table II
for values of pc).
Among the expressions (5.7) through (5.12)
only the effective medium approximation (5.12)
is able to predict the existence of a transition.
The predicted critical value pc=1/d, however,
is not exact as seen by comparison with Table
II.
Another method for calculating the effective or transport
coefficient c¯ will be discussed in homogenization
theory in section V.C.4.
The resulting expression appears in equation (5.87)
if one sets K=c1.
It is given as as a correction to the simplest mean field
expression (5.11).
The correction involves the fundamental solution
of the local transport problem (5.88).
In practice the use of (5.87) is restricted
to simple periodic microstructures [268, 280].
If the microsctructure is periodic it suffices to obtain
the fundamental solutions within the basic period, and to
extend the average in (5.87) over that period.
If the microstructure is not periodic then the solution
of (5.88) and averaging in (5.87)
quickly become as impractical as solving the original
problem, because cr is then unknown.