138
5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
where {1A{x,y)} are two-dimensional basis funetions. Using the Galerkin
finite element for x-y direetions and the finite differenee for the z direetion,
they derived a system of equations similar to Eq. (5.3.33), but with nondiagonal matriees [A 2 ]m and [A 3 ]m.
In Huyakorn et al. (1986), trial solution (5.3.34) is used for x-y direetions,
but for the z direetion, they used the following:
(5.3.35)
where ~m(z) is the one-dimensional basis funetion. This type of FEM is
espeeially suitable for layered aquifers.
5.3.4 The Multiple Cell Balance Method
Sun et al. (1984) extended the MCB method from the two-dimensional ease
to the three-dimensional ease. The 3-D MCB method also uses the triangular
prism element and basis funetions whieh are defined in Eq. (5.3.25) and
introduced in Seetion 5.3.2. In an arbitrary element (e) shown in Figure 5.20,
the unknown eoneentration C(x, y, z, t) may be approximately expressed as
6
C(x, y, z, t) = L C:(t)~r(x, y, z), (x, y, z) E (e),
(5.3.36)
r=1
where C:(t) (r = 1,2, ... ,6) are eoncentrations at the six nodes of element (e).
Approximate expressions for partial derivatives oC/ox, oC/oy, oC/oz and
aC/at in this element ean be obtained through partial differentiation, as
follows
oC = t C:(t) O~r,
OX r=1
OX
oC = t C:(t) o~r,
oy r=1
oy
oC = t C:(t) o~r,
OZ r=1
OZ
(5.3.37)
oC = t dC: ot r=1 dt r·
Using the basis funetions defined in Eq. (5.3.25), we have
O~1
1 _
O~2
1 _
O~3
1_
fiX = 2.1 Z bi , OX = 2.1 z bi , fiX = 2.1 z b,.,
O~4
1 +
O~5
1 +
O~6
1 +
fiX = 2.1 Z bi, OX = 2.1 z bi , fiX = 2.1 Z bk•
Similar expressions ean be obtained for o~r/oy (r = 1,2, ... ,6) by ehanging bi ,
b i , bk to Ci' Ci' Ck·