Page 107 - Percolation Models for Transport in Porous Media With

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

s
t.DD .......................... f ........................... :
... : ; 'f
.
!2
... . .
.
.
.
.
.
.
... •••••••••• .... L •••••••••-•••' .
.
.
.
... .
.
.
log M
Figure 34: The dependence of the residual saturation of the displacing phase on
the viscosity ratio (for a fixed capillary number: logC = 3.5)
x (l n(r) r 2dr) -t +So (5.16)

The nature of the relation S(M) obtained for the case of the model probability
density function f(r) = (10/9)r- 2 , 1 < r < 10 is represented in Fig.34 by line 1. It
can be seen that in the limiting cases M ~ 1 and M « 1, theoretical calculations
give the maximum (the case of piston-type displacement) and the minimum (the
case of viscous fingering) saturation and correlate well quantitatively with the
results of numerical experiments [18), represented in Fig.34 by curve 2. These
relations S(M) are obtained for the case C -+ oo (in the numerical experiment C
was set to equal 3.5).
Draw our attention to the investigation of the effect that parameter C has on
the nature of the displacement. The minimum radius of a capillary which can be
reached by the displacing fluid for the given pressure difference is determined from
the condition of the equality between the hydrodynamic and the capillary pressure
differences
{5.17}
From Darcy's law we have V P 9 = QJ.ttf(k0ki). The characteristic sizes in this
problem are pore sizes (for example, the average radius < r >) and the size L of
the specimen. It is obvious that using the hydrodynamic pressure difference on
pore size is physically unjustified since the action of mass forces is not taken into

account. Therefore in {5.17) it is necessary to use the estimate of the value A.P 9
over the size L: A.P 9 = VP 9 L. From Laplace's formula we have APe= 2ufr. In
this case, after substituting these relations in (5.17), we have

C = koktf(rL) (5.18)