larger range; thus it is harder to see the flat top of the concentration distribution. That is why the bimodal
distribution often seen for small j
f
is not shown for large j
f
. The bimodal shape of the variance of concen- tration has been studied before within a linear theory
framework, for instance, by Kapoor and Gelhar
15
.
3 TRANSPORT OF WASTE DISPOSAL THROUGH DEEP INJECTION WELL
3.1 Continuous injection of waste
A simplified schematic diagram of waste disposal through a deep injection well is shown in Fig. 1b. Since the flow field
generated by the high injecting pressure is much stronger than regional growndwater flow, the regional groundwater
flow will be neglected with the focus on the radial flow pattern caused by injection. The aquifer setting is exactly
the same as used in Section 2: a stratified aquifer, where hydraulic conductivity varies in the vertical direction but
does not change in the horizontal direction. Groundwater flow and solute transport are in symmetric radial patterns.
The steady-state groundwater flow is governed by equation =
2
h ¼ 29
The flow is radial so that head h is a function of r distance from the injection well. In this case, eqn 29 becomes
=
2
h ¼ 1
r d
dr r
dh dr
¼ 30
With constant boundary conditions, the solution of eqn 30 is h ¼ ¹ a
ln r=r
w
ÿ þ b
31 where a and b are constants determined by pressures at r ¼
r
w
and r ¼ R respectively, where r
w
is the radius of the injection well and R is the influence radius of the injection
well. Explicitly, a and b are: a ¼
h ¹
h
R
ln R=r
w
ÿ ,
b ¼ h 32
where h and h
R
are heads ar r ¼ r
w
and r ¼ R, re- spectively. Eqn 32 shows that if waste is injected at
high pressure h q
h
R
, ‘a’ could be very large. The radial flow velocity is then:
V z
¼ ¹ K=h=n ¼
a=n r
K z
¼ a
r K
z 33
where a ¼
a n. The longitudinal macrodispersion in a
strongly heterogeneous aquifer is dominated by field het- erogeneity which causes velocity contrasts in the different
layers. Local-scale dispersion is neglected. In a cylindrical coordinate system, the stochastic solute transport equation is:
=· CV
þ ]
C ]
t ¼
1 r
] ]
r rCV
þ ]
C ]
t ¼
a K
z r
] C
] r
þ ]
C ]
t ¼
34 If the concentration at the injection well is a constant C
, the solution to eqn 34 also is a step function:
C ¼ C S t ¹
r 2V
h i
¼ C
S t ¹ r
2
2a K
35 eqn 35 is different from the solution in Section 2 eqn
3. That is because the velocity in the radial flow is no longer a constant as was the case for uniform flow. Here,
velocity is inversely proportional to r see eqn 33. The ensemble average of concentration is obtained through an
equation of the same form as 8. Similar to the develop- ments in Section 2, the limit of integration is changed
because the step function is nonzero only when
t ¹ r
2
2a K
¼ t ¹
r
2
2a e
¹ F ¹ f
36 thus
f ln r
2
= 2a
t ÿ
¹ F 37
Changing the limit and performing the integration results in:
¯ C ¼
C 2
erfc ln r
2
= 2a
t ÿ
¹ F
2 p
j
f
38 The average flow velocity is:
¯ V ¼
a r
¯ K ¼
a r
e
F
· e
j
2 f
= 2
39 The average travel distance can be obtained by solving
dr dt
¼ ¯ V ¼
a r
e
F
· e
j
2 f
= 2
40 Thus the solution is
¯r ¼ 2a
e
F
t· e
j
2 f
= 2
h i
1=2
41 Therefore, eqn 38 changes to:
¯ C ¼
C 2
erfc
2 p
ln r=¯r ð
Þ j
f
þ j
f
2
2 p
42 The variance of concentration is calculated by using similar
steps to those adopted in Section 2. j
2 c
¼ C
2
2 erfc
2
p ln r=¯r
ð Þ
j
f
þ j
f
2
2 p
¹ C
2
4 erfc
2
2
p ln r=¯r
ð Þ
j
f
þ j
f
2
2 p
ð 43Þ
At a given time a fixed ¯r, the breakthrough curve of ¯ C
is shown in Fig. 4a. The normalized variances are exhibited
in Fig. 4b. Fig. 4aFig. 4b show that the concentrations and var-
iances of concentration distributions are similar to their counterparts discussed in Section 2, except that the flow is
radial in this section. Transport of waste leakage in stratified formations
165
3.2 Temporal injection of waste
