Page 111 - Percolation Models for Transport in Porous Media With

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

s
=~--------------~--------··--··!

... ~.:
:
1 .:
... ··-··········; ··············!
·- ' '
2 I
: f
0.1~.:1,-~ .... !!"'"": ...... !0!!""'."l':, ... :!"'""'l_,l!:' .. "'":l.
log C
Figure 37: The dependence of the phase saturation on the capillary number for
the distribution function /{r) = Afr 2

2
1
x 1 ~ L, for the rate of growth of the trunk we have [30) V.:(r) = tl.Pr (J.L2L)- •
Consequently lr,(r) = a(r)V.:{r). In order to determine the radius of the chains
forming the traps let us again require the minimality of time of their joining,
dtb/dr = 0. Having determined from this condition r;(tl.P), we can calculate
the dependence S(tl.P} from (5.16}. Furthermore, by taking into consideration
Darcy's law, we may obtain from (5.18) the dependence C(tl.P) and determine
the correlation S(C). The results of calculations for the above-mentioned model
function /(r) are represented by curve 1 in Fig.37, where curve 2 corresponds to
the data of the numerical experiment [18).
Thus the model proposed in this chapter allows not only to explain qualita-
tively the results of laboratory experiments on the investigation of non-steady state
displacement for immiscible fluids in porous media, but also to calculate quanti-
tatively the main parameters of the process, namely the saturation of the medium
with each phase and the conductivities of the formed IC's. To perform theoretical
calculations within the framework of the suggested "forest growth" model, it is
necessary to know the capillary function and the critical percolation indices. The
results of the calculations performed for the model function f(r) and the typical
value of the correlation radius index v for an IC demonstrates a good quantita-
tive agreement with experimental data [27, 29) and the results of direct numerical
modeling in two-dimensional networks (18).