A simple model for porous media is the capillary tube
model in which the pore space is represented as an array
of cylindrical tubes.
The crucial assumption of the model is that the tubes
do not intersect each other.
Often it is also assumed that the tubes are straight or
parallel to each other.
Consider a cubic sample

(3.58) |

The specific internal surface on the other hand is given as

(3.59) |

In a stochastic model
the radii

Several special cases are of particular interest.
If the random radii

(3.60) | ||||

(3.61) |

where

(3.62) |

Moreover in this case the ratio

(3.63) |

is a characteristic length independent of the tortuosity and
sample size.
Two further special cases arise from setting all radii equal
to each other,

(3.64) |

In the case where all radii are equal,

(3.65) |

Although this relation holds only in a special case it has become the basis for defining the so called hydraulic radius

(3.66) |

as a characteristic length scale of porous media [1, 2]. The hydraulic radius concept was used in section III.A.3.d for defining pore size distributions. The capillary tube model and the hydraulic radius concept play an important role for fluid flow through porous media, and will be discussed further in section V.C.2 below.

A model which is closely related to the capillary tube model
is obtained by considering

(3.67) |

and specific internal surface area

(3.68) |

independent of the widths of the slits. Here the specific internal surface is independent of the distribution of widths. As in the capillary tube model one finds a relation

(3.69) |

similar to (3.65) also for the model of capillary slits. The model may be generalized by allowing small undulations and smooth fluctuations of the slits.

Grain models of various sorts have long been studied in optics
[193, 179], colloids [194, 195],
phase transitions [196] and disordered systems
[197].
An important class of grain models are random bead packs
[198, 199, 200, 201, 202, 203, 204] which provide a
reasonable starting point for modeling unconsolidated sediments.
In grain models either the pore or the matrix space are
represented as an array of convex grains
[205, 108, 109, 206, 207, 107, 208, 209].
The grains could be regularly shaped such as spheres,
cubes or ellipoids, or more irregularly shaped convex sets.
They may be positioned randomly or regularly in space, and
they may have equal or varying diameters.
If the grains are placed randomly their centroids are
assumed to form a stochastic point process.
For a Poisson point process the centers or centroids of
the grains are placed randomly and independently in space such
that the number

(3.70) |

The Poisson point process with constant density is stationary
and isotropic.
The contact distribution

(3.71) |

A simple class of grain models is obtained by attaching compact
sets to the points of a Poisson point process.
The compact sets are called primary grains.
An example are spheres of constant radius.
Important generalizations are obtained by randomizing the primary
grains.
An example would be spherical grains with random radii.
More generally it is possible to use as grains independent
realizations of a random compact set as defined in section
II.B.2.b.
If

(3.72) |

if the grains are interpreted as pores.
If the grains are matrix then

(3.73) |

where

(3.74) |

where

(3.75) |

and it determines the specific internal surface according to
(3.16) and (3.13).
The contact distribution

(3.76) |

where the interior

Two simple classes of grain models are obtained by randomly
placing penetrable or impenetrable spheres of radius

(3.77) |

for the porosity of fully penetrating spheres. The relation

(3.78) |

applies to hard spheres of radius

(3.79) |

and for hard spheres

(3.80) |

is obtained [108].

A basic question of stereology concerns the “unfolding” of
threedimensional information from planar sections [210].
For a stationary Poisson distributed grain model with spherical grains
of random diameter the problem of calculating the probability
density

(3.81) |

where the mean sphere diameter

(3.82) |

The solution to this equation is

(3.83) |

for

(3.84) |

with

(3.85) |

for

Even the simplest grain model with penetrable spheres of equal
diameter and Poisson distributed centers still poses unsolved problems.
At low dimensionless densities

A different parallel with statistical mechanics emerges if the grains are identified with the particles in statistical mechanics [6, 7, 8, 208]. This identification suggests generalizations of the underlying uncorrelated Poisson point process, which corresponds to an ideal gas of noninteracting particles, by adding interactions between the points. A large variety of new models such as hard sphere models or Gibbs point fields [37, 34] emerges from this generalization.

Network models represent the most important and widely used class of geometric models for porous media [220, 221, 222, 223, 224, 225, 187, 226, 227, 228, 229, 230, 155, 157, 231, 232, 233]. They are not only used in theoretical calculations but also in the form of micromodels in experimental observations [157, 234, 235, 232, 236, 237, 238]. For random bead packs a random network model has recently been derived starting from the microstructure [204, 200]. Network models arise generally and naturally from discretizing the equations of motion using finite difference schemes. As such they will be discussed in more detail in chapter V.

A network is a graph consisting of a set of vertices or sites connected by a set of bonds. The vertices or sites of the network could for example represent the grain centers of a grain model. If the grains represent pore bodies the bonds represent connections between them. The vertices can be chosen deterministically as for the sites of a regular lattice or randomly as in the realization of a Poisson or other stochastic point process. Similarly the bonds connecting different vertices may be chosen according to some deterministic or random procedure. Finally the vertices are “dressed” with convex sets such as spheres representing pore bodies, and the bonds are dressed with tubes providing a connecting path between the pore bodies. A simple ordered network model consists of a regular lattice with spheres of equal radius centered at its vertices which are connected through cylindrical tubes of equal diameter. Very often the diameters of spheres and tubes in a regular network model are chosen at random. If a finite fraction of the bond diameters is zero one obtains the percolation model.

The purpose of the present section is not to review percolation theory but to introduce the concept of a percolation transition, and to collect for later reference the values of percolation thresholds. The name “percolation” derives from fluid flow through a coffee percolator, and it has been used extensively to model various aspects of flow through porous media [239, 154, 41, 153, 156, 113, 240, 241, 242, 243, 226]. Invasion percolation has become a frequently studied model for displacement processes in porous media [244, 245, 246, 42, 246a]. Percolation theory itself is a well developed branch in the theory of disordered systems and critical phenomena, and the reader is referred to [247, 197, 248, 213] for thorough information on the subject.

Percolation as a geometrical model for porous media is closely related both to grain models and to network models. The model of spherical grains attached to the points of a Poisson process is also known as continuum percolation or the “swiss cheese” model [213]. Site percolation is an abstract version of a grain model, while bond percolation may be seen as an abstraction from network models. The distinguishing feature of percolation theory from other models is its focus on a sudden phase transition associated with the connectivity of random media.

The simplest model of percolation is bond percolation on
a lattice.
In bond percolation the bonds of a regular
(e.g. simple cubic) lattice are occupied randomly
with connecting (=conducting) elements ( e.g. tubes)
with a certain occupation probability

Similarly site percolation may be viewed as a lattice
version of a grain model.
In site percolation the sites of the underlying regular lattice
are occupied randomly with spherical pore bodies of radius at least
half the lattice constant.
Two nearest neighbour sites are called connected if they are both
occupied.
As in bond percolation there exist a critical occupation
probability

These basic bond and site percolation models may be modified in many ways. The underlying lattice may be replaced with an arbitrary regular or random graph. The radii of tubes and pores may be randomized, and the connectivity criterion may be changed. Table II shows the values of the critical occupation probabilities (thresholds) for bond and site percolation for some common two and three dimensional lattices [213]. The table lists also the coordination number of each lattice defined as the number of bonds meeting at an interior lattice site.

Lattice Type | Dimension | Coordination | ||
---|---|---|---|---|

honeycomb | 2 | 3 | 0.6962 | |

square | 2 | 4 | 1/2 | 0.592746 |

triangular | 2 | 6 | 1/2 | |

diamond | 3 | 4 | 0.3886 | 0.4299 |

simple cubic | 3 | 6 | 0.2488 | 0.3116 |

body centered cubic | 3 | 8 | 0.180 | 0.246 |

face centered cubic | 3 | 12 | 0.119 | 0.198 |

The transition between the permeable regime and the impermeable
regime becomes a phase transition in the limit of an infinitely
large lattice.
The role of the order parameter is played by the percolation
probability

Recently a new type of geometric models has appeared
[249, 250, 171, 251, 118, 252] which is
based on image processing techniques.
These models attempt to reconstruct a porous medium
with prespecified statistical characteristics such
as its porosity

The motivation for studying reconstructed porous media is to generate precisely known microgeometries whose transport properties can then be calculated numerically [249, 252]. As shown in [171] two media with the same porosity and correlation function may still show significant differencs in their geometric characteristics. More importantly, the porosity and two point correlation function are not sufficient to determine the connectivity of a medium. The connectivity controls the transport, and therefore it is unclear to what extent reconstructed porous media are useful for predicting transport or relaxation properties.

The geometric models discussed in the previous sections do not account
for the fact that the pore space configuration is often the result of a
physical process.
This suggests the use of dynamic process models which
describe the formation of the porous medium.
While such models are employed routinely for unconsolidated
bead packs [201, 253, 199, 202] they are relatively rare
for consolidated porous media.
The so called bond shrinkage model [254] was developed
for sedimentary rocks.
In this model one starts from a random resistor network on a
simple cubic lattice in

The so called grain consolidation model [255, 256]
does give rise to a percolation threshold.
In the grain consolidation model one starts from a grain model
with nonoverlapping grains

Another process model, called local porosity reduction model,
was introduced in [170].
Consider a simple regular lattice (e.g. simple cubic) with
lattice constant

(3.86) |

where

(3.87) |

and

(3.88) |

summarizes several idealized processes such as
crack compaction,