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6. Dispersivity under Partially-Saturated Conditions
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∆t ≤ min {T ij , T ij,k , T CU,i , T i } (6.17)
where T α denotes the residence time pertaining to the elements α within the
pore network.
After obtaining the solution for concnetrations, at any given time, BTCs at
a given longitudinal position were found by averaging the concentrations of
pores that possess the same longitudinal coordinate. In calculating BTCs,
the concentrations of pore bodies were weighted by their volumetric flow rate;
resulting in a flux-averaged concentration. That is, the normalized average
concentration, c(x, t), is given by
c(x, t) =
[ ∑N x
t
i
∑ N x
t
]
c i (x, t)Q i
i Q i
1
c 0
i = 1, 2, 3, . . . , N t (6.18)
where c 0 is inlet solute concentration, and Nt
x denotes the total number of
pore body elements that are centered at the longitudinal coordinate x. The
longitudinal coordinate could be written as multiples an interval of an l, i.e.
x = 1l, 2l, . . . , L. where l is the horizontal distance between centers of two
adjacent pore bodies. The breakthrough curve at the outlet is obtained by
plotting c(x = L, t). Figure (6.5) shows an example BTC at the outlet of
the network. We use these results to calculate (macroscopic) dispersivity as
described in the next section.
Figure 6.5: Example of resulting breakthrough curve of average concentration
computed from the network (shown by symbols). The solid line is the solution
of 1D advection-dispersion equation.
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