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4.3 Simulating flow and transport within the network
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Figure 4.3: An example of interconnected pore bodies and pore
throats. Flow direction is from pore body j into pore body i in tube
ij. Node j is the upstream node.
We assume that each pore body and pore throat is a fully mixed domain.
Therefore, a single concentration is assigned to each pore body or pore throat
[De Jong, 1958, Li et al., 2007b]. For a given pore body, i, (e.g., in Figure 4.3)
we can write the mass balance equation
N
dc
in
i
V i
dt = ∑
q ij c ij − Q i c i (4.4)
j=1
where c i is the pore-body average mass concentration, c ij is the pore-throat
average mass concentration, Q i is the total water flux leaving the pore body,
V i is the volume of pore body i, and N in is the number of pore throats flowing
into pore body i. As the total water flux entering a pore body is equal to the
flux leaving it, we have
∑N in
Q i = q ij (4.5)
j=1
Note that, in Equation (4.4), we have neglected adsorption of solutes to the
pore body walls. Adsorption of the solutes to the walls of the pore throats is
taken into account as explained below. At the local-scale, i.e., at the wall of
the pore throats, the solute adsorption is assumed to occur as an equilibrium
process. Assuming linear equilibrium, we may write s = k d c| wall
, where s is
the adsorbed concentration at the grain surface [ML −2 ], c| wall
[ML −3 ] is the
solute concentration in the fluid phase next to the wall, and k d [L] denotes the
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