Page 104 - Percolation Models for Transport in Porous Media With

5.1 IMMISCIBLE DISPLACEMENT 97

0 UIO 200 :r.
Figure 32: Characteristic dimension of the stagnation zones of the displaced phase
as a function of the coordinate x

Srf
~! --~-----
42_ ~
:
0 4 8 12 i
Figure 33: The dependence of the residual saturation on the form of the proba-
bility density function

consistent with the results of laboratory experiments [18]. This value substantially
exceeds that of residual saturation of water corresponding to stable displacement,
when the displacing phase flow is possible up to the breakdown of the IC, i.e., up
to the values of the order of 0.2-0.3. The diagram in Fig.31 shows that S(x, x f)
tends to the asymptotic value at xfd > 100. This means that numerical simulation
of non-steady state fluid flow presents great technical difficulties, since in order to
obtain stable reliable results it is necessary to use in the two-dimensional case a
calculation grid (capillary network simulating the pore space) of the size not less
than 200x200. It is clear that in the three-dimensional case, for calculation one
should use a network with the number of elements around 10 7 •
In the given model it is also possible to estimate relaxation times r for different
distances behind the displacement front. They will correspond to characteristic
closure times of the cells with sizeD"' R(Vo), i.e. r"' D(x)/Vo(x), where Vo(x)
is determined from (5.6} - (5.8), and D(x), from the relationship (5.14}. For the
model function ¢(V) = Vn v- 2 1J(V- Vn) in dimensionless units we obtain r ~ 1.
It is of great interest to analyze the influence of the form of the parametric
curve f(r) and the corresponding ¢(V) on the quantity S0 • Calculations of the
residual saturation dependence on the exponent n in the probability density func-
tion ¢(V) "' v-n are represented in Fig.33. As can be seen from the diagram,
the quantity S0 for dynamic displacement may be less than in the case of steady
state two- phase flow, and in the limit as n -+ oo asymptotically tends to zero.