Page 46 - Percolation Models for Transport in Porous Media With

2.5 CONDUCTIVITY AND STRAINED STATE 37

Consider, for instance, a channel oriented parallel to the principal axis i = 1.
The change of its cross-section is determined from the values of £2 and £3, i.e.,
by those components of the strain tensor, which are perpendicular to the axis of
the channel. Assuming that the cross-section s, of a channel is proportional to
the product of its dimensions along the directions perpendicular to its axis, we
obtain that s, "'(r + 0.5le2)(r + 0.5le3). Consequently the effective channel radius
1
is reff = s{. If the condition 0.5le2,3 > reff is satisfied, then the change in the
effective radius of the channel in the first approximation is

Using (2.36) we get

3
!;,.ref/ = lf(4g ) L(02; + Oa;) log(1 + g a;/C). (2.41)
0
0
j=l
Similar dependencies take place for the channels oriented in directions i = 2, 3.
Note that if the condition !;,.reff < ref/ is satisfied, then the channels do not
close under the applied load, the value of K. does not change, and the structure of
the IC is preserved. Therefore the change of the conductivity in the medium for
this case is caused only by the change in the conductivities of the chains. If the
tortuousity of the chains is not taken into account, then the average conductivity of
a unit length of chains is described by (1.9). In this case the distribution function
f(r') is of the following form, f(r') = f(r + !;,.r), where /(r) is the distribution
function for Ui = 0 and !;,.r =!;,.ref! found from (2.41). When deformation ofthe
medium happens at a constant pressure p = (1/3)(a1 + a2 + ua) and the condition
!;,.u1 = 0.5a2 = 0.5aa is satisfied, a case which is of practical importance, the
expression (2.41) appears in a less complicated form

!;,.al)/C)}. (2.42)
Formulas (1.9), (1.11), (2.20), and (2.41) define the values of the corresponding
components of the permeability tensor. Similar relationships take place for the
specific electric conductivity of the medium.
The relationships presented allow to calculate the change of the coefficients of
permeability in the medium given the distribution function of pore channels with
respect to values of intrinsic conductivities. The problem of determining /(r) will
be studied in detail in further sections of this book. Now we shall use the most
common approximation of the porometric curves obtained from experiment (48]

(2.43)