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6.3 Unsaturated flow modeling
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pressure. The capillary pressure of a meniscus is given by the Laplace equation
[Bear, 1988]
P c = P n − P w = γ wn
( 1
r 1
+ 1 r 2
)
= 2γ wn
r ∗ (6.8)
where r ∗ is the mean radius of curvature. For a capillary tube of radius r,
we have r ∗ = r/ cos(θ) (Young-Laplace’s equation), in which θ is the contact
angle between fluid interface and the capillary wall. The invading fluid enters
and fills a pore throat only when the injection pressure equals or exceeds to or
larger than the entry capillary pressure of the pore.
We assume that the wetting phase is everywhere hydraulically connected. This
means that there will be no trapping of the wetting phase, as it can always
escape along the edges. The capillary pressure is increased incrementally so
that fluid-fluid interfaces will move only a short distance before coming to rest
in equilibrium at the opening of smaller pore throats.
6.3.2 Fluid flow within drained pores
To calculate the flow across the network, we need to calculate the flow of water
in saturated pores as well as along edges of drained pores. The conductance
of an angular drained pore depends on its degree of local saturation, which is
directly related to the radius of curvature of the meniscus formed along the
pore edges. However, Raoof and Hassanizadeh [2011a] have shown that if the
conductance is made dimensionless using the radius of curvature, then it becomes
independent of saturation. They used a numerical solution to calculate
the dimensionless conductances of drained pores with scalene triangular cross
section. This was done by numerically solving the dimensionless form of the
Navier-Stokes equations and the equation of conservation of mass. They performed
calculations for a range of corner half-angles, from 5 degree to a wide
corner half-angle of 75 degrees. Figure (6.4) shows the computed dimensionless
conductance as a function of corner half-angle.
The dimensional form of conductance, g, which is a function of capillary pressure,
is
g = g ∗ r4 c
µ
(6.9)
where r c is the radius of curvature of the interface and µ is the wetting fluid
viscosity. The radius of curvature r c depends on the capillary pressure prevaling
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