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5. Pore-Network Modeling of Two-Phase Flow
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phase. Figure (5.5) shows a strong dependency of conductance on corner angle
(note the logarithmic scale of the vertical-axis in Figure 5.5). If the angle of
the edges within a pore body are larger than the corner angles of the neighboring
pore throats, then they will have much less conductance to the flow.
In particular, for cubic pore bodies, their edge half angle is 45 and thus their
conductance will be less than that of pore throats with triangular cross section.
5.3.3 Regular hyperbolic polygons
It is possible to use similar procedure to calculate conductances for different
cross sectional shapes. Considering porous media of type glass beads or well
mature sand grains with spherical grain shapes, we may want to consider hyperbolic
polygons as the cross-sectional shapes of the pores. Here, we present
calculation of dimensionless conductances for saturated pores with hyperbolic
polygons as their cross sectional shapes. A given hyperbolic polygons with ′ n ′
vertices can be generated using ′ n ′ circles with the same length and the same
radius of curvature. These circles (with radius R 1 shown in Figure 5.6) could
be tangential to each other or merge into each other to give different corner
angles (shown as ϕ in Figure 5.6). For radius of circles, R 1 , Area of polygon,
A, and shape factor, G, we have the following relations [Joekar-Niasar et al.,
2010]
sin π n
R 1 = R
cos ϕ − sin π n
(5.14)
A =
nR 2 sin 2 π [
n
( )
cos ϕ − sin
π 2
cos 2 ϕ cot π ( 1
n − π 2 − 1 )
]
+ ϕ − 0.5 sin 2ϕ
n
n
(5.15)
G = A P 2 = cos2 ϕ cot π n − π ( 1
2 − n) 1 + ϕ − 0.5 sin 2ϕ
4n ( π ( 1
2 − ) ) 1 2
(5.16)
n − ϕ
We have considered different domain with different number of vertices (n),
starting from three up to n = 5. For each choice of n, we have considered
domains with different corner angles (ϕ), ranging between zero and the maximum
possible value for each polygon, covering a range of shape factors. We
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