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5.3 Modeling flow in the network
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1.0
0.8
g*
0.6
n=3
n=4
n=5
0.4
0.0 0.2 0.4 0.6 0.8 1.0
Figure 5.8: Relation between dimensionless conductance g ∗ and ϕ for
different number of vertices,n.
by edge elements, and also connected to corner elements of neighboring pore
bodies, j, through pore throats ij. The pore throat ij itself may have water
flowing along its corners (Nij
corner ). Therefore, the volume balance for a pore
body corner i may be written
Ni∑
Edge
n=1
Ni∑
T ube
Q in +
j=1
Q ij = 0 (5.17)
where N Edges
i is the number of edges through which corner i is connected
to other corner elements, n, within the same pore body, and Q in is the flow
through edge in (i.e., between corner i and corner n). For a cubic pore body,
N Edges
i
= 3. Q ij is the total flux through the pore throat ij connecting corner
element i and corner element j of a neighboring pore body. For drained pore
throats with flow along its edges, Q ij is the summation of fluxes through all
edges. Ni
T ube is the number of pore throats connected to the corner element
i. The combination of Equations (5.12) and (5.17), written for all nodes of
pore bodies of the network result in a set of linear equations, whose solution
gives the flow field and fluxes in all network elements. Following solution, the
overall water flux, Q t , through the pore network is calculated. Subsequently,
the relative permeability of the network to water at a given saturation and
capillary pressure is calculated from Darcy’s law
k rw =
µ wQ t
k A ∆P/L
(5.18)
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