Page 66 - Percolation Models for Transport in Porous Media With

58 CHAPTER 4. MULTIPHASE FLUID FLOW

lation with network models was first reflected in the works [22, 59, 60, 61]. This
approach does not have the drawbacks of the phenomenological and the "one-
dimensional" models but neither does it possess the necessary universality of the
obtained results, the latter being a typical advantage of analytical methods.
To obtain analytical relations which allow to calculate and analyze the behavior
of the coefficients of phase permeability, we use the approach developed in chapter
1. Based on the obtained results, consider the displacement of a wettable fluid by
a non-wettable one in a porous medium (we treat both fluids as incompressible
and viscous). For clarity, we shall use the model which looks upon the medium
as a cubic network whose sites (pores) are connected with bonds (capillaries) of
different conductivities. We shall also continue to describe the conductivities of
the capillaries by means of the probability density function f(r).
Suppose that partial displacement of the phase which saturates the core took
place in some macroscopic volume, and the IC of the displacing fluid (further
denoted by ICG) was formed. From now on, we shall mark the quantities relating
to the wettable and the non-wettable fluids with indices 1 and 2, respectively.
Assume that the fraction of capillaries filled with the wettable fluid exceeds the
percolation threshold, and also that the medium contains an infinite cluster ICD
of the capillaries containing the displaced phase. Obviously, the wettable fluid can
be displaced only from those capillaries that satisfy the following condition

PA:(r) 5 llp, PA:(r) = 2xcos9fr (4.1)

and have contact with the ICG. Here llp is the pressure difference in the fluids,
x is the coefficient of surface tension, and () is the contact angle of the surface.
In other words, displacement can take place only in those capillaries that can be
reached by the displacing fluid along the chains which belong to the ICG.
By definition, the ICG consists of those capillaries that satisfy the condition
(4.1). At the same time, the condition (4.1) can be also satisfied by some capillaries
which do not belong to the ICG and are filled with the wettable fluid. However,
as it will be shown later, the fraction of such capillaries, excluding a small domain
near the percolation threshold, is small. Moreover, such capillaries do not affect
the conductivity of the ICG at all, since they are not connected to it. From
this point of view, it does not matter whether these capillaries are filled with the
wettable or the non-wettable fluid. Thus the radius probability density for those
capillaries that are conducting for the ICG, can be represented by the function

h(r) = { f(r)/{(rA:), r ~ TA:, (4.2)
0, r < TA:

Here TA: is the minimal radius of a capillary where displacement of the wettable
fluid can take place for a given value of !1p.