Page 105 - Percolation Models for Transport in Porous Media With

98 CHAPTER 5. NON-STEADY STATE TWO-PHASE FLOW

This effect is explained by the fact that as n -t oo, the function cf>(V) approaches
the6-function in form. Then the displacement velocities in all chains are equal, as
a result of which the trapping of the displacing phase turns out to be impossible.
We note that all the above-mentioned arguments are also true for the three-
dimensional case. The forms of the corresponding analytical expressions for the
three-dimensional problem are different, but the obtained relationships are quali-
tatively the same.

5.2 Effects of Viscosities and Interfacial Tension

A great impact of the flow velocity, interfacial tension, and viscosities of fluids on
the IC structure was observed [18, 27, 28]. From these quantities, one can make
up two dimensionless parameters, namely the capillary number C = (Qp,t)fu {the
ratio of the viscous forces to the capillary forces), and the viscosity ratio M =

p,tfp,2 • Here Q is the flow velocity, J.tt is the viscosity of the injected fluid, p,2 is
the viscosity of the displaced fluid, u is the coefficient of interfacial tension. When
M ~ 1 a stable interphase border is formed in the medium, and the displacement
has a piston-type nature. When M .g: 1 the displacement front turns out to be
unstable: so-called "fingers," formed by the largest pores and pore channels along
which displacing phase breaks through, develop.
It is natural to take into account the influence of the pore space structure,
where the displacement takes place, in the form of the capillary function f(r). It
is interesting to observe the effects of the above-mentioned factors on the structure
of the IC of the displacing phase during the displacement, because knowledge of
the structure allows for qualitative estimates of saturation of the porous medium
with each fluid for non-stable two-phase flow.
In §5.1 the "forest growth" model is suggested and its calculation is carried
out for the simplest limiting case M = 0, C -t oo . The main attention was
paid to the investigation of residual saturation of the medium with the displaced
phase behind the displacement front, while the difference in phase permeabilities
for steady state and non-steady state fluid flows was not analyzed at all. However,
the approach suggested in §5.1 allows to consider non-steady state two-phase flow
in the general case M -1 0, C < oo , as well as to calculate the relative phase
permeabilities ki(S) for it (i = 1, 2; 1 is the index of the displacing phase, and 2,
the index of the displaced phase, S = St).
Invoke the famous solution of the Buckley-Leverett problem [14], where for the
saturation front velocity we have
v 1 = QF(S)s-tm- 1
Here v, is the velocity of the phase interface averaged over an elementary
physical volume, F(S) = [k1 (S)/ p,1 + k2{S)/ p,2]- 1 , ~ is the porosity. From the