Page 109 - Percolation Models for Transport in Porous Media With

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

completely displaced from the ri- chain it will become part of the skeleton ofiC 1.
The other chains with rk < r < Ti will remain for the moment dead ends without
any contribution to the conductivity of IC 1. Therefore, for the conductivity of
IC 1 at the given instant, taking into account [26], we have

1
k1 = (l n(r) I(r) dr) (l n( r) I(r) dr) - , I(r) = l f(rl) dr1

x [ [ f(rl)r>'drr (5.19)

and for its mass, taking account of the dead ends, we have

2
S = (I - S0 ) [ l n(r) r dr + l n( r)cr dr]
2

x [l n(r)r dr +So (5.20)
2

Relations (5.19), (5.20) represent the non-steady state phase permeability of
phase 1 in a parametric form through the parameter Ti (rk < ri < r.).
Consider the conductivity of IC 2. At the closure of traps, some part of fluid
2 will be retained in them. At the same time, there is IC 2 in the medium, which
consists of the capillaries with r < rk. Along this IC, relaxation of the trapped
phase to the equilibrium value of its saturation in the medium takes place. As new
channels for moving of fluid 2 are not formed at that moment, the conductivity of
the IC does not change in the course of relaxation and is equal to

r~o
1
!;, = (l n(r)I(r)dr) (l n(r)I(r)dr) - J f(r)dr =So (5.21)
r~

As relaxation takes place, the current value ri approaches the equilibrium value
rk, while the phase permeabilities approach theirs. Time of relaxation to the
equilibrium value depends on the values of parameters C and M. The non-steady
state phase permeabilities represented in Fig.36 are calculated using the mentioned
model function /(r), for the case lnM = 0 and InC= -1.1. It is clear from the
above-mentioned data that there are two characteristic flow regions.
1. S(r.) < S < S(rk)· In this region, traps are formed, fluid flow is essentially
non-equilibrium, and therefore for the calculation of k1 (S), it is necessary to