Page 214 - Percolation Models for Transport in Porous Media With

212 CHAPTER 12. ACOUSTIC WAVES AND PERMEABILITY

up to p ""' u* ""' 10 MPa.
We have already mentioned that the correlation between temperature and pres-
sure is established by the expression (8.17). For water we have aT :::::: 0.2 · 10- 3
K- 1 , /h :::::: 5 Pa- 1 , and for oil, aT :::::: 10- 3 K- 1 , f3T :::::: 10 Pa- 1 . The ratio of
proportionality factors aT/ f3T equals 2.5 in both cases. Therefore the lengths of
the critical temperature intervals t:l.T., for which the pressure increase is t:l.p""' u*,
coincide in order of magnitude: t:l.T. ""' 10 K. In this case it follows from (12.23)
that the minimal duration r. of the thermo-acoustic action on the capillaries with
2
r ""' 10- 6 m for Vo =20kHz, Ib = 2 watt per cm ' and Qp = 10- 1 is approximately
equal to 5 · 10 3 seconds.
We will use the formula (11.14) to find r* after estimating the temperature
gradient "VT appearing at the micro level. The following points will be taken into
consideration. Although the thick capillaries with r > 10- 4 m are the first to get
heat, and therefore, the first to colmatate, they make a very small contribution
to permeability. Therefore cutting the function f(r) in the interval of large r
virtually does not affect the change of K, while the principal interest is with the
range (10- 6 < r < 10- 5 ) m. Since we have supposed that the capillary length
l » r, we will take l ""' 10- 3 m as the characteristic length. To analyze the
temperature regimes in adjacent capillaries of different radii, return to the model
representation of a capillary junction (see fig. 73).
Let the ratio of radii of the joining capillaries be maximal, i.e., of the order
10, and consequently, the radius of the larger capillary be r ""' 10- 5 m. The
characteristic longitudinal size in the chain is""' l (capillary length); therefore, the
time of the temperature equalization after its difference occurs between different
points in the capillary chain is
(12.25)
For example, ~~:~o) "' 0.4 · w- 6 m 2 fs for oil, and from (12.25) we have r~ of
the order of several seconds. The temperature difference between the capillaries
of different radii develops during the specified period of establishment (12.25),
since for larger periods, equalization of the temperatures will take place caused by
both the heat transfer in the fluid and by the backflow of heat into the skeleton
of the rock. Note that both of the mentioned processes are identical in terms of
the characteristic time intervals, since the coefficients of temperature conductivity
for oil and sandstone differ only by a factor of two, i.e., have the same order of
magnitude (~~:~o) ~ 0.4 ·10- 6 m 2 fs, ~~:~•) ~ 0.8 ·10- 6 m 2 fs), while the characteristic
size of a grain in the rock and characteristic capillary length in a grained medium
obviously coincide (l""' 10- 3 m).
In this case we have

4 3 3
0
t:l.T = T(r , r') - T(r, r') = - 11' Qp r [1 -(ro) ] v5Ibr'
4 J.tCTCm Ct r