Page 76 - Percolation Models for Transport in Porous Media With

68 CHAPTER 4. MULTIPHASE FLUID FLOW

We shall not go into realization details of the outlined plan for the calculations
here and shall present only the final result of computing the absolute permeability
for the discussed type of visco-plastic fluids

23-1/n rc [ rc 00 dr ]1/n
Kg= 3 + 1/n Ao I I f(r) dr/ I f(r) r3n+l
0 r1 r1

(4.24)
[/ /(r) dr ]" /(r1) dr1

To calculate the phase permeabilities, address the displacement of a hydrophilic
fluid by a hydrophobic one. In accordance with the analysis carried out in §4.1, we
select the parts of f(r) which characterize the distribution of each of the phases
in the pore space. This operation is valid for plastic fluids as well, if the plastic
resistance("' Tp/R 9 ) is much less than the capillary resistance ("' x/R~). This
problem was studied in [64], and it was shown that both the hydrodynamic (vis-
cous) and the plastic forces are small compared to the capillary forces in a common
domain of relatively small values of Vp, and therefore, of relatively small local ve-
locities. Since the subject of this part is only steady state flow, i.e., the case of
small flow velocities, it follows that the assumption on the nature of the distribu-
tion of the fluids in the pore space used in §4.1 may be considered valid for the
visco-plastic fluids as well. Consequently the absolute permeabilities for the phases
K;•d (i = 1,2) can be calculated using formulas (4.22) and (4.24) as follows. For
K;·d, the lower limit of integration is changed to rk, and the function h(r) acts as
f(r). For K~·d, relationships {4.22) and {4.24) do not change, only ft(r) is used
instead of f(r), and rc is found from (4.6) forK.= 1.
Note that the flowing fluids can in fact have different plastic properties, the
algorithm of calculating K;•d using (4.22) and (4.24) not being affected by this at
all. It is only necessary, when passing to the relative phase permeabilities k:•d =
K;•d / Kg•d, to take into account the fact that a specific absolute permeability Kg•d
of the medium corresponds to each of the fluids. In analyzing the qualitative trends
in the behavior of the quantities k:·d we shall confine ourselves to studying the
flow of fluids with identical plastic properties. In this case each way of calculations
is characterized by a common value of the absolute conductivity of the medium,
Kg or Kg.
The choice of a model for the calculation of S(rk), naturally, depends on the
actual pore space structure of a specific porous medium.
Calculations for a simple cubic network (z = 6) with the model function f(r) =
ror- 77(r- ro) will be presented as an illustration. In this model, the saturation
2
S2 of the medium by the displacing less wettable fluid was used as the saturation
variable S, and the relation S(rk) was established using model I.