where e
f
¼ exp j
2 f
= 2
ÿ is obtained from a general property of the moment-gener-
ating function
15
e
fq
: E e
f q
¼ exp ¯fq þ j
2 f
q
2
= 2
, where E[] is the expectation sign, f is a Gaussian distribution with
mean ¯f and variance j
2 f
and q is a constant. The average travel distance is
¯x ¼ ¯ Vt ¼
e
F
J exp j
2 f
= 2
ÿ t
14 eqn 11 becomes:
¯ C ¼
C 2
erfc ln x=¯x
ð Þ
2
p j
f
þ j
f
2
2 p
15 At a fixed time ¯x is fixed, C versus x=¯x is plotted in Fig.
2a which is the breakthrough curve. The variance of concentration is j
2 c
¼ C
2
¹ ¯ C
ÿ
2
. Similar to the calculation processes from eqns 8–11, we have
C
2
¼ Z
` ¹ `
C
2
S
2
Vt ¹ x p
f df
¼ C
2
2 erfc
ln x=¯x ð
Þ
2 p
j
f
þ j
f
2
2 p
ð 16Þ
j
2 c
¼ C
2
¹ ¯ C
2
¼ C
2
2 erfc
ln x=¯x ð
Þ
2 p
j
f
þ j
f
2
2 p
¹ C
2
4 erfc
2
ln x=¯x ð
Þ
2 p
j
f
þ j
f
2
2 p
ð 17Þ
The normalized variances of concentration as functions of x=¯x
time is fixed are plotted in Fig. 2b. Several interesting results are illustrated in Fig. 2a and
Fig. 2b: 1. Fig. 2a shows that when j
f
is very small ¼ 0.1, the point with ¯
C ¼ C =
2 is located approximately at x ¼ ¯x
this point is usually called the 50 concentra- tion point or C
50
. However, when heterogeneity increases j
f
increases, and the C
50
point moves towards the source point x ¼ 0 of contamination.
The C
50
point is decided by the geometrical mean of hydraulic conductivity,
¯ K
G
¼ e
ln K
¼ e
F
see eqn 11, while the average travel distance ¯x is decided
by the arithmetical mean of hydraulic conductivity, ¯
K ¼ e
F
exp j
2 f
= 2
ÿ ¼ ¯
K
G
exp j
2 f
= 2
ÿ see eqn 14.
Thus ¯ K= ¯
K
G
¼ exp j
2 f
= 2
ÿ . 1. When j
f
increases, ¯ K
G
is smaller than ¯ K
and the difference between ¯ K
and ¯
K
G
gets larger, thus C
50
point moves further from x ¼ ¯x
to the left in Fig. 2a. 2. Fig. 2b shows that the point of maximal variance of
concentration is close to x ¼ ¯x when the medium is nearly homogeneous j
f
¼ 0.1. Therefore, using ¯
C as
the concentration estimator has maximal uncertainty at the point x ¯x when j
f
¼ 0.1. The largest concen-
tration variance is in the area where the mean concentration ¯
C changes most rapidly; such a result
has been pointed out by Gelhar
11
p. 242. When j
f
increases, the maximal variance point shifts towards the origin in Fig. 2b. This is because the point of
maximal mean concentration gradient also moves towards the origin. Fig. 2b shows that the maximal
variance is j
2 c
= C
2
¼ 0.25 or j
c
C ¼
0.5. 3. Fig. 2a and Fig. 2b reveal that heterogeneity has a
great impact on the concentration and variance of concentration distribution curves. When j
f
increases, the corresponding concentration and variance of con-
centration curves are deformed significantly from their counterparts for j
f
¼ 0.1.
2.2 Temporal waste leakage from a landfill site
In this section, contaminant transport with a temporal leak- age source is investigated. If the leakage time is T and the
other physical parameters are the same as those used in Section 2.1, the solution to the stochastic eqn 2 becomes
C ¼ C
S [
Vt ¹ x ]
; if t , T
C S
[ Vt ¹ x
] ¹
S [
V t ¹ T
¹ x
] ;
if t . T 18
Fig. 2. a Normalized concentration ¯
C=C as a function of x=¯x
breakthrough curve; b normalized variance of concentration as a function of x=¯x. ——, j
f
¼ 0.1; 3 3 3 , j
f
¼ 0.5; – – –, j
f
¼ 1.0; p p p , j
f
¼ 3.0.
162 H. Zhan
The solution for t , T is the same as that of Section 2.1, so only t . T is considered here. Following similar calcula-
tions as those utilized in Section 2.1, the ensemble average concentration is
¯ C ¼
Z
` ¹ `
C S
[ Vt ¹ x
] ¹
S [
V t ¹ T
¹ x
] p
f df
¼ C
2 erfc
ln x=¯x ð
Þ
2 p
j
f
þ j
f
2
2 p
¹ erfc
ln x= ¯x ¹ x ÿ
2
p j
f
þ j
f
2
2 p
ð 19Þ
where ¯x is the average travel distance given by eqn 14 and x
is defined as x
¼ ¯ VT ¼
e
F
J exp j
2 f
= 2
ÿ T
20 The variance of concentration is
j
2 c
¼ C
2
¹ ¯ C
2
¼ C
2
2 erfc
ln x=¯x ð
Þ
2 p
j
f
þ j
f
2
2 p
¹ erfc
ln x= ¯x ¹ x ÿ
2
p j
f
þ j
f
2
2 p
¹ C
2
4 erfc
ln x=¯x ð
Þ
2 p
j
f
þ j
f
2
2 p
¹ erfc
ln x= ¯x ¹ x ÿ
2
p j
f
þ j
f
2
2 p
2
ð 21Þ
The ¯ C=C
versus x=¯x and j
2 c
= C
2
versus x=¯x curves are plotted in Fig. 3.
An interesting problem here is to see what happens when
Fig. 3.
a1, b1 and c1 Normalized concentrations ¯ C=C
as functions of x=¯x with different Tt; a2, b2 and c2 normalized variances of concentrations as a functions of x=¯x with different Tt. – – –, j
f
¼ 0.1; 3 3 3 , j
f
¼ 0.5; ———, j
f
¼ 1.0; p p p , j
f
¼ 3.0.
Transport of waste leakage in stratified formations 163
the leakage time is very short the so-called pulse source. In this situation, T p t and x
p ¯x
, we have ln
x ¯x ¹ x
ÿ ¼ ln
x ¯x
¹ ln 1 ¹
x ¯x
ln x
¯x þ
x ¯x
22 By using the following approximation of the complemen-
tary error function: erfc
x ¹
erfc x þ Dx
2
p p
e
¹ x
2
D x
, when Dx p x
23 eqns 19 and 21 are reduced to
¯ C
C
2p p
j
f
exp ¹ ln x=¯x
ð Þ
2
p j
f
þ j
f
2
2 p
2
· T
t 24
j
2 c
C
2
2p
p j
f
exp ¹ ln x=¯x
ð Þ
2
p j
f
þ j
f
2
2 p
2
· T
t ¹
C
2
2pj
2 f
exp ¹ 2 ln x=¯x
ð Þ
2
p j
f
þ j
f
2
2 p
2
· T
t
2
ð 25Þ
The macrodispersivity in a stratified aquifer was discussed by Gelhar
11
pp. 204–207 and Dagan
7
p. 293, so it is not repeated here. Our interest is to derive directly the concen-
tration breakthrough curve, the variance of concentration, and to test the sensitivity of these results with the variance
of log conductivity. Gelhar
11
has derived the concentration for an instantaneous pulse source by recognizing that the
travel time probability density function pdf is essentially equal to the normalized concentration. Although the physi-
cal base of Gelhar’s analysis is correct, his eqn 5.1.14 is inaccurate and dimensionally inconsistent. As a matter a
fact, in Gelhar’s analysis of 5.1.12, if we do one step further to transform the probability density function f
K
to the f
ln K
, we can easily derive the correct eqn 5.1.14 and find that it is identical to our solution 24. The mathema-
tical process is shown below. Recognizing that the notation ‘J’ used by us see eqn 5
equals ‘Jn’ used by Gelhar, his eqn 5.1.11 becomes t ¼
x V
¼ x
JK 26
where t, V and K are the time, velocity, and hydraulic conductivity, respectively. By using Gelhar’s relationship
2.1.7, which is
f
z
z ¼
] h
] z
f
h
h where f
z
and f
h
are the pdfs to z and h respectively, his eqn 5.1.12 becomes
f
t
t ¼
x Jt
2
f
K
x JT
¼ x
Jt
2
· 1
K f
ln K
ln x
Jt h
i ¼
1 t
f
ln K
ln x
Jt h
i 27
If lnK obeys the normal distribution, from eqn 13 we have ¯
V ¼ e
ln K
exp j
2 f
= 2
ÿ J, and eqn 27 becomes:
c t
Z
` ¹ `
c dt
¼ f
t
t ¼
1
2p p
j
f
t exp
¹ ln
x=Jt ¹
ln K
2
2j
2 f
¼ 1
2p
p j
f
t exp
¹ j
2 f
= 2 ¹ ln ¯
Vt=x ÿ
2
2j
2 f
ð 28Þ
eqn 28 is the accurate eqn 5.1.14 in Gelhar’s analysis. Remeber that
Z
` ¹ `
c dt is the normalization factor which
equals the product of C and T used in Section 2.2,
¯ Vt ¼ ¯x
and ln ¯ Vt=x
ÿ ¼ ¹ ln x=¯x
ð Þ
, then eqn 28 is identical to our solution 24. Therefore, Gelhar’s solution eqn
5.1.14 for a pulse source is a special case of our general solution 19 for any leakage time T.
The following results are obtained in this section: 1. When Tt is small, for instance Tt ¼ 0.05 as in Fig.
3a1, the leakage can be considered as a pulse source. Therefore, the concentration distribution has
a bell shape. When Tt gets larger, the leakage time becomes longer, the scenario changes from a pulse
source towards the continuous source, and the con- centration ¯
C=C distribution changes from a bell
shape to the shape of continuous leakage Section 2.1. Fig. 3a1Fig. 3b1Fig. 3c1 exhibit the transi-
tion from a pulse-type source to a continuous-type source. In Fig. 3a1, the maximum ¯
C=C is about
0.2 when j
f
¼ 0.1. It drops rapidly when j
f
gets larger because the concentration spreads over a
wider range. The maximal ¯ C=C
increases when Tt gets larger for each of the cases of j
f
¼ 0.1, 0.5, 1.0
and 3.0, because a longer leakage time will bring more solute into the aquifer and enhance the concentration.
2. Fig. 3a2Fig. 3b2Fig. 3c2 are the variances of concentration for different leakage times. At an inter-
mediate leakage time Fig. 3b2, the variances of concentration show bidmodal two peaks shapes for
j
f
¼ 0.1, 0.5 and 1.0. For j
f
¼ 3.0, the bimodal shape
is not clearly shown in the Fig. 3b2. Fig. 3b1 shows that the concentration changes most rapidly
in the two corners of the flat top for j
f
¼ 0.1, thus
there are two maximal peaks of concentration variances corresponding to these two corners. The
rationale is still the same as addressed in point 2 of Section 2.1: the largest concentration variances
occur in the areas where concentrations change most rapidly. The region between these two corners
is a flat top where the concentration change is extre- mely small, so the concentration variance in that area
is near zero as shown in the Fig. 3b2 for j
f
¼ 0.1.
When T is larger, the flat top region is wider, thus the distance between the two peaks of concentration
variances is larger. For a strong heterogeneity, the concentration distribution spreads over a much
164 H. Zhan
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
