266
8. Applications of Groundwater Quality Models
Let the elevation of aquifer bottom be zo, the elevation of the interface be
z l' the elevation of the fresh water table be z 2 (if the aquifer is confined, z 2 will
be the elevation of the aquifer roof). All of which are functions of x, y and t,
as shown in Figure 8.6.
Integrating (8.2.1a) along the z direction in the seawater region, we obtain
1
z1
(
Oh)
Zo V . qs + Ss 0/ dz = 0.
(8.2.11)
Using the Leibniz formula, this equation can be transformed into
+ ss[:t 1:
1 hsdz - hslzl 0;t
1 ] = 0,
(8.2.12)
where v' is a gradient operator on the xy plane and q~ is the projection
of qs on plane xy, i.e.,
V ,
0.
0.
== - I + - j ,
OX
oy
(8.2.13a)
(8.2.13b)
where qsx' qsy' and qsz are the components of qs on the three co ordinate axes.
The means of h s and q~ along the z direction are defined as
-
1 1
z1
hs = b
hsdz,
s Zo
(8.2.14a)
1 1
z1
- I
,
qs = b
qsdz,
s Zo
(8.2. 14b)
where b s = (z 1 - zo) is the thickness of the seawater layer. Suppose the
change of h s along the vertical direction is very smalI, then we have
(8.2.15)
Consequently, the term in the square brackets on the left-hand side of
Eq. (8.2.12) can be translated into
o 1
Z1
OZ 1
0 -
- obs
ohs
:l
hsdz - hslz ~ = :l(bshs) - hs~ = bs~'
ut Zo
1 ut
ut
ut
ut
(8.2.16)
Substituting this equation into Eq. (8.2.12), we obtain
V"(bsq~) + qSZlzl - qszlzo - q~IZI'V'Zl + q~IZO'V'ZO + S.bs°o~s = 0. (8.2.17a)
Using the same method, Eq. (8.2.1b) may be averaged along the vertical
direction to yield
q~bs + K.bs· V'hs = 0,
(8.2.17b)