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5. Pore-Network Modeling of Two-Phase Flow
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where A and P are the area and the perimeter of the cross section, respectively.
The shape factor replaces the irregular and complicated shape of a pore throat
by an equivalent irregular, but simpler, shape. The value of shape factors
for pore throats are chosen from a truncated lognormal distribution (Equation
5.1), with a minimum shape factor value of zero corresponding to a slit, and
a maximum value of 0.08 corresponding to a circular cross section. Values
between zero and 0.048 correspond to triangular cross sections (with maximum
value of 0.048 for an equilateral triangle), and values between zero and 0.062
correspond to rectangular cross section (with the maximum value corresponding
to a square).
For a triangular cross section, there is a relationship between shape factor and
corner angles [Patzek, 2001], given by
G = A P 2 = 1
∑
4 3 cot(Angle i )
i=1
= 1 tan α tan β cot(α + β) (5.3)
4
where α and β are the two corner half-angles subtended by the two longest
sides of the triangle. It is clear from the above equation that for a single value
of the shape factor, a range of corner half angles are possible. We follow the
procedure employed by Patzek [2001] to select a nonunique solution for corner
half-angles. We start by selecting the upper and lower limits of the corner
half-angle, β, assuming β < α. These two limits are
[ ( ( √ ) )]
2 arccos −12 3G
β min = arctan √ cos
+ 4π 3 3
3
[ ( ( √ ))]
2 arccos −12 3G
β max = arctan √ cos 3 3
(5.4)
(5.5)
We randomly pick a value, β = β min + (β max − β min ) .ϕ, between the two
limits. Where ϕ is a uniformly distributed random number, between 0 and 1.
We calculate the corresponding value of α by inverting Equation (5.3)
α = − 1 2 β + 1 2 arcsin ( tan β + 4G
tan β − 4G sin β )
(5.6)
The third and smallest corner half-angle, γ, is then obtained from γ = π 2 −
(α + β). After determining the corner angles, finding the triangle size requires
specifying a length scale. We have chosen the radius of the equivalent circle
of a pore cross section as the length scale. One way to select this radius is
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