Page 44 - Percolation Models for Transport in Porous Media With

2.5 CONDUCTIVITY AND STRAINED STATE 35

elastic fractured capillary porous medium. It is assumed in this model that the
size of a conducting channel depends linearly on e,H the component of the strain
tensor which is normal to the channel. The value of fn is calculated using the
model of a nonlinearly elastic grained medium which takes account of the contact
compressibility of the grains.
Note that generally the correlation between the size of the channels and the
strain tensor can be more complex, especially for those media which have a plastic
(clay) component. At the same time, as long as elastic deformations of grained
media are considered, the assumption of a linear dependence of the channel size on
the value of the strain tensor appears reasonable. Under this assumption, find the
change of the distribution function f(r) caused by the stress tensor O'i applied to
the medium. In the case of a uniform comprehensive contraction of the medium,
the correlation between the radius r' of a channel in the loaded medium and the
initial radius r can be described as follows

r' = r + 0.5lfn· (2.33)

Here fn = e 0 /3, where e 0 is the volumetric deformation, and l is the grain
diameter. e 0 depends not only on the stress applied, but also on Young's mod-
ulus which is a function of the stress. If the discussed medium is isotropic with
respect to elastic properties, then the correlation between the small changes of
deformations and stresses can be expressed by Hooke's law in the differential form

3 du·
dei = L fJi; E( '·), (2.34)
j=l u,
where
.. _ { -1 when i :f. j,
(} ,, - . .
JLp, when t = 3;
#Lp is Poisson's modulus.
According to experimental data (49], the dependence of Young's modulus E
of a grained medium on the principal stress O'i applied along the Xi-axis can be
written in the following form

E(ui) = A[1- exp( -Bui)] + C (i = 1, 2, 3). (2.35)

Here (A+ C) and Care the maximum and the minimum values of Young's
modulus, B is a constant. By "stress" we mean the "effective" stress, equal to the
difference between the external load and the interior pressure in the pores. After
integrating (2.34) using (2.35), we obtain the following ([49])

3
fi =(A+ c)- L fJi;{u; + B- log[E(u;)/C]}. (2.36)
1
1
i=l