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6.4 Simulating flow and transport within the network
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drained pore throat ij. For a given corner unit (with concentration c CU,i and
volume V CU,i ), we can write the mass balance equation
V CU,i
d
dt (c CU,i) =
Nin∑
tube
j=1
N ij
edge
∑
k=1
c ij,k q ij,k +
N CU,i
in,edge
∑
n=1
c CU,n q i,n − Q CU,i c CU,i (6.15)
where the first term on the r.h.s. is due to the mass arriving via N ij
edge edges
of Nin
tube throats with flow towards the corner unit. The second term on the
r.h.s. accounts for the mass arriving from N CU,i
in,edge
neighboring corner units
(within the same pore body) with flow towards the corner unit. The last term
shows the mass leaving the corner unit. Q CU,i is the total water flux leaving
(or entering) the corner unit i.
We note that, for the case of saturated pores, the second term on the righthand-side
of Equation (6.15) vanishes and the value of N edge
ij in the first term
will be equal to one, since there is no edge flow present.
The mass balance equation for an edge element of a drained pore throat may
be written (assuming that corner unit j is the upstream node)
V ij,k
d
dt (c ij,k) = |q ij,k | c CU,j − |q ij,k | c ij,k (6.16)
where V ij,k , q ij,k , and c ij,k are the volume, volumetric flow rate, and concentration
of k th edge of the pore throat ij, respectively.
Combination of appropriate forms of Equations (6.15) and (6.16) results in a
linear set of equations to be solved for c ij , c ij,k , c i , and c CU,i . Since we discretize
pore bodies and pore throats on the basis of their saturation state, the number
of unknowns are different for simulations at different saturation values. For
the case of a fully saturated domain, the number of unknowns is equal to
N tube + N node (N tube is the number of pore throats and N node is the number of
pore bodies). In general, the number of pore throats is larger than the number
of pore bodies in a pore network model. To get a more efficient numerical
scheme, first, applying a fully implicit scheme, we discretized Equation (6.16)
and determined c ij,k in terms of c CU,i . This was then substituted into the
discretized form of Equation (6.15). This resulted in a set of equations for
c CU,i . In this way, we considerably reduced the number of unknowns, and thus
the computational time. The details of the method are given in Chapter 8,
Section 8.3.1. For the accuracy of the scheme, the minimum time step was
chosen on the basis of residence times [Suchomel et al., 1998c, Sun, 1996]
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