140
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
integrals can be easily calculated. For surface (SI)' we have
f I S
1) ~~ dy dz = ~~~ [biG + bj!;2 + bkQD + 3(bi!;: + bj!;; + bk!;6)]
'(bj - b;),
f LI) ~~ dy dz = ~~~ [CiG + Cj!;2 + Ck!;3) + 3(Ci!;: + cj !;; + Ck!;6)J
. (C j - c;),
f r aa~ dy dz = -2 1 [ - 5!;~ - 5!;2 - 2!;~ + 5G + 5!;; + 2!;6J
JSI) z
. (bicj - cibj)(cj - Ci),
where, !;: (r = 1,2, ... ,6) represents the concentrations of six nodes in element (~), and bi, bj, bk, Ci' Cj' ck are determined by Eq. (5.1.33). Similar results
can be obtained for (S2)' For the surface integral over (S3), we have
f I S
3) ~~ dx dy = 11 2 (bi!;~ + bj!;2 + bk!;~ + bi!;: + bj!;; + bk!;6)'
f I S
3) ~~ dx dy = 1~ (ciG + Cj!;2 + CkG + ciG + Cj!;; + Ck!;6),
f I S
3) ~~ dx dy = 108~AZ) (- 22G - 7!;2 - 7!;3 + 22!;: + 7!;; + 7!;6)'
Adding the above surface integrals together, we obtain
f r D.p aac n. dS = t Ir!;:,
J~e)
xp
r=1
(5.3.39)
where (~e) is the part of the boundary surface of exclusive subdomain in (~),
i.e., (Sd, (S2), (S3)' In Eq. (5.3.39),
Il = 9~A {[Dxxb;(AZ) + Dxyci(Az) - ~DxAbiCj - CibJ}bj - b;)
+ [DxA(AZ) + Dyyci(Az) - ~Dyz(biCj - Cibj)}Cj - c;)
+ [Dxxb;(AZ) + Dxyci(Az) - ~Dx.(biCk - Cibk)}bk - bi)
+ [DxA(AZ) + Dyyci(Az) - ~DyAbiCk - CA)}Ck - Ci)}
1
1
11 A
- 12 Dxzbi - 12 DyA + 54 (Az) Dzz·
There are similar expressions for other 1',.'s.