Page 209 - Percolation Models for Transport in Porous Media With

12.3 CAVITATION 207

After assuming that x"' 7 · w- newtons per meter, we obtain
2
(12.16}

where ao is measured in meters. For the expression (12.16} to have physical sense,
i.e., for a 0 to be greater than zero, it is necessary that a 0 < w- 11 m, as it can
be seen from the right side of (12.16}. In this case we obtain a0 < w- 10 + w- 9
m. That is, if we choose the minimal value of a 0 of the mentioned, we obtain
amin "' w-u m, or w- 2 nm, a value that is less than the characteristic size of the
atom by an order of magnitude.
The maximal size of a cavitating bubble can be estimated using the relation
for the resonance frequency of small oscillations

1
Vr =- {12.17}
ao
where 'Yp is the isentropic exponent for vapor.
Since the germs with a less than the resonance radius a0 oscillate with a bigger
amplitude, which causes the rupture of the fluid, these germs cavitate. After
representing {12.17} in the form

and transforming it into the equation

we obtain that for Po "' 10 MPa, 'Yp = 1.4, x = 7 · 10- newtons per meter,
2
PI = 10 kg/m , Vr = 2 · 10 s- , amax "' 10- m.
1
2
4
3
3
Since all germs of the initial size amin < a < amax must cavitate, it follows
that for the given negative pressure, germs of virtually any size can cavitate in
capillaries of all radii. However, due to the fact that the presented investigation
is valid only for r < w- 7 m, where there is no Poiseuille flow, and therefore for
those capillaries which do not contribute to the permeability of the medium, the
cavitation effect cannot appear in any significant quantities. If we also take into
account the fact that we considered a rather improbable case of capillary junction,
when rfr 0 "'10 2 and the boundary between them is very sharp (see fig. 73, AA'-
plane), whereas the cross-sectional views of actual reservoir rocks demonstrate a
more gradual change of radii of the capillary chains (see fig. 73, dotted line), and
the fact that the cavitational destruction in any case takes place only in the closest
vicinity of the plane of contact of the capillaries, we should admit that for the given
parameters of the process, i.e., p0 "' 10 MPa, Pa "'0.1 MPa, the cavitational effect
may be neglected.