The permeability is the most important physical property
of a porous medium in much the same way as the porosity
is its most important geometrical property.
Some authors define porous media as media with
a nonvanishing permeability [2].
Permeability measures quantitatively the ability
of a porous medium to conduct fluid flow.
The permeability tensor

(5.54) |

where

The permeability has dimensions of an area, and it is
measured in units of Darcy (d).
If the pressure is measured in physical atmospheres
one has 1d=0.9869

The permeability does not appear in the microscopic Stokes or Navier-Stokes equations. Darcy’s law and with it the permeability concept can be derived from microscopic Stokes flow equations using homogenization techniques [268, 269, 270, 38, 321, 271] which are asymptotic expansions in the ratio of microscopic to macroscopic length scales. The derivation will be given in section V.C.3 below.

The linear Darcy law holds for flows at low Reynolds numbers in which the driving forces are small and balanced only by the viscous forces. Various nonlinear generalizations of Darcy’s law have also been derived using homogenization or volume averaging methods [268, 1, 269, 322, 321, 38, 271, 323, 324, 325]. If a nonlinear Darcy law governs the flow in a given experiment this would appear in the measurement as if the permeability becomes velocity dependent. The linear Darcy law breaks down also if the flow becomes too slow. In this case interactions between the fluid and the pore walls become important. Examples occur during the slow movement of polar liquids or electrolytes in finely porous materials with high specific internal surface.

The hydraulic radius theory or Carman-Kozeny model
is based on the geometrical models of capillary
tubes discussed above in section III.B.1.
In such capillary models the permeability can be
obtained exactly from the solution of
the Navier-Stokes equation (4.9)
in the capillary.
Consider a cylindrical capillary tube of length

(5.55) | ||||

(5.56) |

with a parabolic velocity and linear pressure profile.
The volume flow rate

(5.57) |

Consider now the capillary tube model of section III.B.1
with a cubic sample space

(5.58) |

Dimensional analysis of (5.58), (3.58)
and (3.59) shows that

(5.59) |

where the mixed moment ratio

(5.60) |

is a dimensionless number, and the angular brackets denote
as usual the average with respect to

The hydraulic radius theory or
Carman-Kozeny model is obtained from a mean field
approximation which assumes

(5.61) |

where

It must be stressed that hydraulic radius theory is not exact
even for the simple capillary tube model because in general

(5.62) |

where

Finally it is of interest to consider also the capillary slit
model of section III.B.1.
The model assumes again a cubic sample of side length

(5.63) |

which has the same form as (5.59) with a constant

The previous section has shown that Darcy’s law arises in the capillary models. This raises the question whether it can be derived more generally. The present section shows that Darcy’s law can be obtained from Stokes equation for a slow flow. It arises to lowest order in an asymptotic expansion whose small parameter is the ratio of microscopic to macroscopic length scales.

Consider the stationary and creeping (low Reynolds number)
flow of a Newtonian incompressible fluid through a porous
medium whose matrix is assumed to be rigid.
The microscopic flow through the pore space

(5.64) | ||||

(5.65) |

inside the pore space,

(5.66) |

for

The derivation of Darcy’s law assumes that the pore space

(5.67) |

and the Laplacian is replaced similarly.
The velocity and pressure are now expanded in

(5.68) | ||||

(5.69) |

where

(5.70) | ||||

(5.71) | ||||

(5.72) | ||||

(5.73) | ||||

(5.74) |

in the fast variable

(5.75) |

where the three vectors

(5.76) | ||||

(5.77) | ||||

(5.78) |

and

It is now possible to average

(5.79) |

where

(5.80) |

where the components

(5.81) |

The permeability tensor is symmetric and positive definite
[268].
Its dependence on the configuration of the pore space

(5.82) |

Equations (5.80) and (5.82)
are the macroscopic laws governing the microscopic
Stokes flow obeying (5.64)–(5.66)
to leading order in

The importance of the homogenization technique illustrated here in a simple example lies in the fact that it provides a systematic method to obtain the reference problem for an effective medium treatment.

Many of the examples for transport and relaxation in
porous media listed in chapter IV
can be homogenized using a similar technique [268].
The heterogeneous elliptic equation (4.2)
is of particular interest.
The linear Darcy flow derived in this section can be cast into
the form of (4.2) for the pressure field.
The permeability tensor may still depend on the slow variable

The permeability

If (5.80) is inserted into (5.82)
and

(5.83) |

which is identical with (4.2). The equation must be supplemented with boundary conditions which can be obtained from the requirements of mass and momentum conservation at the boundary of the region for which (5.83) was derived. If the boundary marks a transition to a region with different permeability the boundary conditions require continuity of pressure and normal component of the velocity.

Equation (5.83) holds at length scales

(5.84) |

is fulfilled.
The ratio

(5.85) |

where now

(5.86) |

where

(5.87) |

given in terms of three scalar fields

(5.88) |

analogous to (5.76)–(5.78) in the homogenization of Stokes equation.

If the assumption of strict stationarity is relaxed the
averaged permeability depends in general on the slow variable,
and the homogenized equation (5.86) has then the
same form as the original equation (5.83).
This shows that the form of the macroscopic equation does not
change under further averaging.
This highlights the importance of the averaged permeability
as a key element of every macroscopically homogeneous description.
Note however that the averaged tensor

Consider a porous medium described by equation
(5.83) for Darcy flow with a stationary
and isotropic local permeability function

To make further progress it is necessary to specify the local permeabilities. A microscopic network model of tubes results from choosing the expression

(5.89) |

for a cylindrical capillary tube of radius

Using the effective medium approximation to the network
equations the effective permeability

(5.90) |

where the restrictions on

(5.91) |

where

Consider, as in the previous section, a porous medium described
by equation (5.83) for Darcy flow with a stationary
and isotropic local permeability function

The straight line in Figure 23 corresponds to equation (5.59). The local percolation probabilities defined in section III.A.5.d complete the description. Each local geometry is characterized by its local porosity, specific internal surface and a binary random variable indicating whether the geometry is percolating or not. The selfconsistent effective medium equation now reads

(5.92) |

for the effective permeability

(5.93) |

and it gives the total fraction of percolating local geometries. If the quantity

(5.94) |

is finite then the solution to (5.92) is given approximately as

(5.95) |

for

To study the implications of (5.92) it is necessary
to supply explicit expressions for the local geometry distribution

(5.96) |

where the exponent

(5.97) |

If all local geometries are percolating, i.e. if