Page 202 - Percolation Models for Transport in Porous Media With

200 CHAPTER 12. ACOUSTIC WAVES AND PERMEABILITY

4. Dissipation of the acoustic field energy due to the "flawless" movement of
the fluid in the pore channels, when the total displacement of the fluid in
any physically small volume vanishes. Temperature increase causes the same
effects as those stated in 1.
Analyze the probabilities and the nature of realization of each of the mentioned
mechanisms in the process of acoustic treatment of the medium in reservoir setting,
for the parameters of an ultrasonic wave source.

12.1 Dissipation of Energy in Viscous Poiseuille
Flow

If a total flow Q develops due to the displacement of the fluid with respect to the
solid skeleton, Poiseuille flow directed along the gradient of the external pressure
Vp, takes place in the chains of the IC skeleton.
However the existence of such flow is possible only for the frequencies lower
than a certain value, called the characteristic frequency:

(12.1)

The first to obtain this result was M. A. Biot (1956) [97], who got it studying
the solutions to the equation for the non-steady state fluid flow in porous media

Ptdvfdt = -mVp + F- (~t/ K)v

in the interval of relatively low frequencies (vo < 10 5 Hz). For frequencies higher
than Voc, the relative displacement of the fluid and the skeleton of the rock does
not take place, i.e., the solid and the liquid phases move in phase. Estimate the
value of Voc· Take IL "' w- 3 Pa·s, P! "' w- 3 kg·m- 3 and the averaged radius of
the capillary chain to be < r >.
To calculate < r >,use the model function (4.9) which reflects the qualitative
behavior of actual f(r) well. In this case< r >= a*a*(a*- a*)- 1 ln(a* fa*). We
assume, as usual, that a*>> a* and set a* fa*"' 10 3 + 10 "'e 7 +e 10 and therefore
4
< r >"' lOa*. Since a* "' w- 6 m can be taken as the minimal radius of the
conducting capillaries where Poiseuille flow is still realized [88], we have

< r >"' 10- m (12.2)
5
After substituting the above-mentioned values of /L, PJ, and< r >in (12.1), we
obtain Voc "' 10 Hz. It can be thus inferred that for Poiseuille flow to take place
4
in the medium and for the "first" mechanism of the acoustic energy transfer to be
realized, the frequency of the treatment must be lower than v0 "' 10 4 Hz. Since