al .
12
presented a higher-order study on flow and transport in strongly heterogeneous formations based on the quasi-linear
theory of Neuman and Zhang.
14
Several other researchers have explored alternative theories to investigate the fluid flow and contaminant trans-
port in strongly heterogeneous formations. For example, Christakos and others
5
have used diagrammatic analysis to resolve the complex multifold integrals that appear in the
perturbative expansion of the stochastic flow solution and the numerical performance of the approach is tested.
Serrano
18
has reformulated the flow and transport equations through the aid of the Adomains Decomposition Method.
1
A great deal of research opportunities exist in these alternative
nonlinear theories. In this paper, an attempt is made to solve the problems of
transport in strongly heterogeneous formations without using a small perturbational framework. The stochastic
transport equations are solved to obtain solutions for 1 the ensemble average concentrations and 2 the variances
of concentration. In this analysis, local-scale dispersivity and diffusion are neglected based on the consideration
that the primary cause of field dispersion is field hetero- geneity. Also, the mathematical manipulation is simpler
when local dispersivity and diffusion are neglected. We are aware that neglecting the local dispersivity will cause
a loss of information on the transverse mixing of the solute and absence of the spatial correlation of hydraulic conduc-
tivity. Including the local dispersivity effect in the nonlinear framework is a future research topic and is not discussed in
this paper. The investigation in this paper focuses on the transport of leakage from two kinds of waste disposal
schemes: 1 landfills in a formation with a uniform ground- water flow direction; and 2 deep-well injection. Four kinds
of leakage transport problem will be investigated: 1 continuous leakage; 2 temporal leakage; 3 continuous
injection; and 4 temporal injection. The work presented in this paper can help us gain more insight into the
sophisticated transport process in strongly heterogeneous formations. It is the first step towards buiding a nonlinear
framework of contaminant transport in a three-dimensional, heterogeneous formation.
2 TRANSPORT OF WASTE LEAKAGE FROM A LANDFILL SITE
2.1 Continuous waste leakage from a landfill site
A simplified physical view of leakage from a landfill into a stratified formation is shown in Fig. 1a. The dispersion of
waste leakage in the unsaturated zone is not included. Con- taminant leakage with constant concentration is released at
x ¼
0 in all layers. The stratigraphy is horizontal such that the hydraulic conductivity does not change along the x axis
but does change along the vertical direction z axis. Uni- form groundwater flow is from left to right, parallel with the
stratigraphic layers. Constant head boundary conditions are applied on the left and right sides, so that the head gradient
is constant throughout the model. We should point out that the constant head boundary is only one possible condition of
the regional groundwater flow. We impose these boundary conditions to isolate the effect of heterogeneity and to con-
duct sensitivity tests of the contaminant transport upon the heterogeneity. Because of different hydraulic conductivities
in different layers, groundwater flow velocities are different in different layers. Stationarity of log conductivity is
assumed throughout this work for both the landfill and the deep-well injection configurations, therefore, the ensem-
ble average concentration is not dependent on z since the layers are statistically identical; rather, it depends on x and
time t.
It is generally accepted that, for nonreactive solute trans- port in a region without sinkssources, the concentration
distribution is governed by the Advective-Dispersion Equa- tion ADE in a local scale:
=· D
·=C ¹ qC ¼
] nC
] t
1
where D is the local-scale dispersion coefficient tensor, which is determined by local-scale pore space geometry
grain size, shape, orientation, etc. and groundwater flow velocity; q is groundwater discharge which is related to
groundwater velocity V ¼ qn, where n is porosity, C is concentration, t is time. Considering the limited informa-
tion about hydraulic conductivity, a stochastic approach is used to treat the local-scale ADE as a stochastic partial
differential equation. In a strongly heterogeneous acquifer containing different layers with different hydraulic conduc-
tivities, the dispersive effect caused by velocity distribution will be the leading cause of dispersion. Local-scale disper-
sion will cause some transverse mixing between different layers. On the basis of this consideration, the local disper-
sive term in eqn 1 is neglected and the porosity n is assumed to be constant. This results in:
] [
V z
C x
, z
, t
] ]
x ¼
] C
] t
2 For a continuous leakage problem with a constant source
concentration, the boundary condition is C0,z,t ¼ C and the initial condition is Cx,z,0 ¼ C
for x 0 and Cðx; z;
0Þ ¼ 0 for x 0. The solution of this stochastic equa- tion is a step function:
C x
, z
, t
¼ C
S [
V z
t ¹ x ]
3 where the step function Sx ¼ 1 for x 0 and Sx ¼ 0 for
x , 0. From Darcy’s law, the velocity V is:
V ¼ ¹ K
n dh
dx 4
If constant head boundary conditions are applied at both sides x ¼ 0 and x ¼ L, we assign the variable
J ¼ ¹ 1
n dh
dx 5
so that J is constant for all layers. In this way, the randomness 160
H. Zhan
of V is determined by the randomness of K. In general, lnK is assumed to be a normal distribution with a
known mean and variance. The stochastic term lnK is split into a mean part F and a random part f, where
lnðKÞ ¼ F þ f
, and eqn 4 changes to V ¼ KJ ¼
e
F þ f
J 6
The probability density function pdf of f is the well- known normal distribution with variance j
2 f
: p
f ¼
1
2p p
j
f
exp ¹
f
2
2j
2 f
7 Now, the ensemble average of concentration C is
¯ C ¼
Z
` ¹ `
C S
Vt ¹ x p
f df
8 Considering the property of the step function S, the inte-
grand of eqn 8 is a nonzero term only when Vt ¹ x 0, which means that
Vt ¹ x ¼ e
F þ f
Jt ¹ x 9
Solving the inequality 9 we get f
ln x=Jt
¹ F
10 The inequality 10 implies that the lower limitation of
integration in 8 can be replaced by lnxJt ¹ F and eqn 8 becomes
¯ C ¼
Z
` ln
x=Jt ¹
F
C S
Vt ¹ x p
f df
¼ C
2 erfc
ln x=Jt
¹ F
2
p j
f
ð 11Þ
where the complementary error function, erfc, is defined as erfc
x ¼
2
p p
Z
` x
e
¹ y
2
dy 12
The average velocity is ¯
V ¼ KJ ¼ e
F þ f
J ¼ e
F
exp j
2 f
= 2
ÿ J
13
Fig. 1. a Schematic view of leakage from a landfill site assuming that leakage originates from the central part of the site to the
underlying stratified formations not to scale; b schematic view of waste disposal through a deep injection well not to scale.
Transport of waste leakage in stratified formations 161
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
