Page 72 - Percolation Models for Transport in Porous Media With

64 CHAPTER 4. MULTIPHASE FLUID FLOW

MPa, g 0 = 10 3 , P,p = 1/3; r0/a* = 0.2 ·10 2 • Calculations were performed for three
values of pressure p = u 0 : the curves 1 - 3 correspond to the values 1, 7, and 10 2
MPa, respectively. It can be seen from fig.18 that the dependence k2(S1) is more
sensitive to the pressure changes than is kt(St).
In this model, the phase permeabilities are completely determined by the prob-
ability density function for conducting capillaries with respect to values of intrin-
sic conductivities (the effective radius) and the value of the percolation threshold
which characterizes the structure of the medium (the network type). Unfortu-
nately today we do not possess reliable experimental data regarding the form of
the function f(r), and therefore it is impossible to calculate the phase permeabil-
ities and compare the experimental and the theoretical dependencies numerically.
However the results of the calculations made for the model probability density
function demonstrate qualitative agreement of the experimental and the theoreti-
cal dependencies of the phase permeabilities. It should be pointed out that within
the considered percolation model, dynamic effects [22) are all but not taken into
account. At the same time, these effects can cause notable deviation of the flow
from the quasi-stationary regime when the flow rate is large enough. In this case
an important part can be played by the nature of the pressure distribution in the
medium at the micro-level, a property which depends on the flow rate. Special
research on this topic will be carried out further.
In conclusion, we want to note that the developed approach demonstrates the
presence of the "endpoint effect" very clearly. This effect expresses itself in the
trapping of the displacing phase near the surface of the rock until the saturation
of the medium by the displaced fluid reaches a value close to the limiting one,
St o ,.., ec· Therefore the rock is impermeable for the second fluid practically up
to the values of S1 !:::::! 0.2 + 0.4, and only after the saturation drops below the
mentioned value of S1 ~ S1 0 , the breakthrough of the displacing fluid occurs.

4.2 Effect of Plastic Properties of Fluids on Phase
Permeabilities

Determination of the parameters of plastic flow in porous media is based on two
principal assumptions. The first one concerns the description of the flow at the
micro-level, i.e., through an elementary capillary (pore channel). According to
this assumption, a certain analytical dependence of the shear rate :Ys on tangent
stress Ts is taken as the friction law of the fluid flow in capillaries. The flow of a
viscous incompressible fluid through a capillary in this case can be described by