Page 116 - Percolation Models for Transport in Porous Media With

6.1 MERCURY INJECTION 109

relationship is the initial condition for the equation (6.8). However in the experi-
ment, at the end of the first stage the value of saturation is Sc > 0. The obtained
"discrepancy" in data at the transition from the first to the second stage is due
to the fact that the model {6.8) does not take into account the volume of the
super-critical pores filled with mercury and adjacent to the entrance cross-section.
To take this volume into consideration, we shall use the formula (6.5) to calculate
X(S) in the interval 0 < S ~ S' and the solution of the Cauchy problem (6.8) to
calculate it for S > S'. It is natural to choose the point S' at the intersection of
the plots of these two relations in the (S, X)-plane (see fig. 38).
At the instant when the displaced phase stops coming out of the core, the
infinite cluster of the gas-containing capillaries breaks up. The value X(S~) equal
to the percolation threshold for the gas cluster X~ = 1- Xc corresponds to the
value of saturation S~ measured at the said instant. The expression
X(S~) = 1- Xc (6.9)

gives the correlation between the constants z and l of the network.
The final stage of displacement. At the third stage of the experiment,
the non-wetting phase still occupies the infinite cluster of the super-critical pores.
Therefore the equation (6.8) remains valid. Nevertheless at this stage the relation
rk(S) has to be found anew, since the pressure P2 in the formula (6.1) becomes the
pressure in the trapped compressing clusters, and is therefore unknown. Suppose
that trapping of air takes place simultaneously throughout the whole core at the
instant when the cluster of the gas-containing capillaries breaks up. Consider the
gas ideal and let all finite clusters filled with the trapped gas compress according
to the law
dVk/Vk = -dS/(1- S), (6.10)
where vk is the volume of the cluster.
Due to the low heat conductance of an actual porous medium it is possible
to consider compression to be isentropic. When heat conductance is high a poly-
tropic process is considered similarly. After differentiating the law of isentropic
compression P2 Vk =const and using the relationship (6.10), we obtain

dS
dP2 = "Yo 1 _ S P2, (6.11)

where "Yo is the isentropic exponent.
The quantity P2 can be expressed in terms of the known value of pressure Pl
from the formula (6.1) for the capillary pressure as follows, P2 = Pt- 2xcos0/rk.
After substituting this expression for P2 as well as its differential dP2 in (6.11)
and some algebraic transformations, we obtain

drk(S) = r~(S) ("Yo Pl(S)- 2xcos0/rk _ dp1(S)) (6.12)
dS 2xcos0 1- S dS