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5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
FIGURE 5.22. The exclusive subdomain (Ri.,") of node
(i,rn).
number of the node in the horizontal direction, and m the number in the
vertical direction. The three-dimensional exclusive subdomain of anode (i, m)
is defined as a column whose bottom is the two-dimensional exclusive subdomain ofthe node, as shown in Figure 5.10, and whose height is the distance
L\z between two layers, as shown in Figure 5.22.
The integral form of the hydrodynamic dispersion equation, Eq. (2.4.31),
can be written for the exclusive subdomain of anode as folIows:
ff r (~C + M)dR = {f r + fr + fr } (DVC - VC)·nds,
JRi• m ) t
Js.) Jsu) JSI)
(5.3.28)
where M is the source and sink term, and (S.), (Su), (SI) represents the side,
upper base, and lower base of the exclusive subdomain (Ri,m) of node (i, m),
respectively.
The integral on the left-hand side of Eq. (5.3.28) can be calculated approximately by
ff r (~C + M)dR ~ (d~i.m + Mi,m)PiL\Z,
J(R i •m ) t
t
(5.3.29)
where Pi is the base area of (Ri,m)'
For the first surface integral on the right-hand side of Eq. (5.3.28), the
following approximate expression can be used:
f r [. ] dS ~ [f r [. ] dt] L\z
J(s.)
J(L)
= [~(AfiCi,m + AijCj,m + Af.kCk,m)]L\Z, (5.3.30)
where (L) is the periphery of (S.). Other symbols have the same meaning as
those in Section 5.2.2, and Arl (l = i,j, k) are given by Eq. (5.2.22).