120
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
where
~e 1
Bu = 3· 432 (170mi + 47mj + 47mk ),
~e 1
Bij = 3· 432(47mi + 23mj + 14mk),
~e 1
Bfk = 3· 432 (47mi + 14mj + 23md,
and
(5.2.19)
in which
Furthermore, let
fl
C'W
C
Fi =
--dxdy = ~W;Pi'
(D)
n
n
(5.2.20)
where C; is the solute concentration contained in the water extracted from,
or injected into, the exclusive subdomain of node i. For extraction, C; is
unknown. However, it is the same as the concentration Ci at node i. Pi is the
area of the exclusive subdomain (D) of node i. W;P i in Eq. (5.2.20) is the rate
of extraction or injection for the subdomain.
Substituting Eqs. (5.2.9), (5.2.13), (5.2.14), and Eqs. (5.2.18) through (5.2.20)
into Eq. (5.2.3), we then have:
L (AiiCi + AijCj + AikCk) + L (BuOOCi + BijOoC j + Bfk O~k) + Fi = 0,
e,
e,
t
t
t
(5.2.21)
where
Ai, = - etft + ßi~ - Yi~ + c5i~' (I = i,j, k).
(5.2.22)
Eq. (5.2.21) can alternately be written as
I C
"
I C
I oC i "B' oC j
0
A-- . + L. A.. . + ß..- + L. .. - + F =
n I
Ni
IJ J
n ot
j#i IJ ot
I
,
(5.2.23)