Page 119 - Percolation Models for Transport in Porous Media With

112 CHAPTER 6. PORE SIZE DISTRIBUTION

is a known exact dependence. After integrating by parts we find

a a
I <P( r) f ( r) dr = <P( 0) - c/J( a) X (a) + I <P' ( r) X ( r) dr
0 0
Therefore the integral, which is necessary for the calculation of the coefficients of
permeability, is stable towards errors of the experiment.
The obtained pore radii distribution function, together with the constants z and
l of the network, forms sufficient initial data for the calculation of the coefficients
of permeability for a porous medium.

6.2 Percolation Model for the Electric Porome-
try Method

If one of the sides of an initially non-saturated core is immersed in a container
with a wetting electrolyte (see fig. 41), a saturation distribution, decreasing with
height, is formed there. This phenomenon is due to the fact that a wetting fluid
rises in a capillary of radius r in the gravitational field up to a height

L = 2xcosOf(PJgr), (6.14)

where xis the coefficient of surface tension,() is the contact angle, PJ is the density
of the fluid, and g is the acceleration of gravity. In general, vertical capillary chains
are not isolated, but nevertheless the fraction of the saturated pores at the height
L can be considered, up to some proportionality factor, as determined by a critical
radius r(L) which correlates with L through (6.14). In this case the specific electric
conductivity in the vertical direction is a function of saturation, and therefore of
height L, and can serve as a source of information about the size distribution of
pores.
Measure the specific electric conductivity at the heights { Li} in sufficiently
thin layers tl.Li <t:: Li (see fig. 41), so that within the portion measured, the
specimen can be considered uniformly saturated. Estimate the specific electric
conductivity of an arbitrary portion of the specimen and the contributions to it
made by the subsystems of the site pores and the bond pores. Let the specific
electric conductivity of the electrolyte be ae and of the skeleton of the specimen,
a = 0. Obviously, the electric conductivity of the material in the vertical direction
is determined by the vertically-oriented chains of pores filled with the electrolyte.
Taking account of the transverse bonds between them in the considered case brings
coefficients of the order unity into the calculations and does not affect the esti-
mates. Consider a unit cube of an element of the specimen. If l is the period of the
network and "' is the fraction of the vertical chains in a unit volume filled with the