and fate of contaminants in the subsurface. For well-defined, ideal aquifers, analytical solute transport models are
frequently employed. Furthermore, analytical models are often used for verifying the accuracy of numerical solutions
to complex solute transport models.
Multidimensional contaminant transport models have several advantages over one-dimensional models. For exam-
ple, multidimensional models can account for concentration gradients and contaminant transport in directions perpendicu-
lar to the groundwater flow. As indicated by Leij and Dane,
14
measuring experimentally lateral and vertical dispersion coef- ficients is not a trivial task. However, multidimensional trans-
port models can provide such parameters by direct fitting of available experimental data. In addition, multidimensional
models can easily account for a variety of boundary condi- tions, as well as contaminant source geometries.
Although several multidimensional analytical models for solute or colloidvirus transport are available in the litera-
ture,
1,3–5,8,10,14,15,20,22,23
multidimensional analytical
models that can accommodate a variety of contamination source configurations in porous media with semi-infinite or
finite thickness are nonexistent. The present study extends the collection of contaminant
transport models by presenting analytical solutions to multi- dimensional transport through saturated, homogeneous por-
ous media, accounting for first-order decay of the solute in the aqueous phase or sorbed onto the solid matrix with
different decay rates. A variety of source configurations, including continuous as well as periodic source loadings
from either point or elliptic source geometries, are consid- ered. Generalized analytical solutions applicable to solute as
well as virus transport in aquifers of semi-infinite and finite thickness are derived.
2 MODEL DEVELOPMENT
The transport of solutes in saturated, homogeneous porous media, accounting for three-dimensional hydrodynamic dis-
persion in a uniform flow field, nonequilibrium sorption, and first-order decay of liquid phase and sorbed solutes
with different decay rates, is governed by the following partial differential equation:
] C
t ,
x ,
y ,
z ]
t þ
r v
] C
p
t ,
x ,
y ,
z ]
t ¹
D
x
]
2
C t
, x
, y
, z
] x
2
¹ D
y
]
2
C t
, x
, y
, z
] y
2
¹ D
z
]
2
C t
, x
, y
, z
] z
2
þ U
] C
t ,
x ,
y ,
z ]
x þ
lC t
, x
, y
, z
þ l
p
r v
C
p
t ,
x ,
y ,
z ¼
F t
, x
, y
, z
, ð
1Þ where C is the liquid phase solute concentration; C is the
solute concentration sorbed onto the solid matrix; D
x
, D
y
and D
z
are the longitudinal, lateral and vertical hydrody- namic dispersion coefficients, respectively; U is the average
interstitial velocity; t is time; x, y and z are the spatial coor- dinates in the longitudinal, lateral and vertical directions,
respectively; r is the bulk density of the solid matrix; v is the porosity of the porous medium; l is the decay rate of
liquid phase solutes; l is the decay rate of sorbed solutes; and F is a general form of the source configuration. It should
be noted that the effective porosity, defined as percentage of interconnected pore space, may be employed instead of por-
osity in a porous medium that contains a large number of dead-end pores or in a fractured porous formation.
2,7
The accumulation of solutes onto the solid matrix is described by the following nonequilibrium expression:
r v
] C
p
t ,
x ,
y ,
z ]
t ¼
r
1
C t
, x
, y
, z
¹ r
2
r v
C
p
t ,
x ,
y ,
z ¹
l
p
r v
C
p
t ,
x ,
y ,
z ,
ð 2Þ
where r
1
and r
2
are the forward and reverse rate coefficients. Assuming that initially there are no sorbed solutes present
in the porous formation, the expression describing C is obtained by solving eqn 2 subject to the initial condition
C0,x,y,z ¼ 0 to yield
C
p
t ,
x ,
y ,
z ¼
r
1
v r
Z
t
C t
, x
, y
, z
exp ¹ r
2
þ l
p
ÿ t ¹t
dt ,
3 where t is a dummy integration variable. In view of eqns
2 and 3, the governing equation, eqn 1, can be written as
] C
t ,
x ,
y ,
z ]
t ¹
D
x
]
2
C t
, x
, y
, z
] x
2
¹ D
y
]
2
C t
, x
, y
, z
] y
2
¹ D
z
]
2
C t
, x
, y
, z
] z
2
þ U
] C
t ,
x ,
y ,
z ]
x þ A
C t
, x
, y
, z
¹ B Z
t
C t
, x
, y
, z
e
¹ H t ¹ t
dt ¼
F t
, x
, y
, z
, ð
4Þ where the following substitutions have been employed
A ¼ r
1
þ l
, 5
B ¼ r
1
r
2
, 6
H ¼ r
2
þ l
p
: 7
The derived integrodifferential equation, eqn 4, is solved analytically in the subsequent sections for the cases of
aquifers with semi-infinite and finite thickness.
2.1 Source configuration
The source configuration is represented by the following general function:
F t
, x
, y
, z
¼ G
t W
x ,
y ,
z ,
8 where Gt is the solute mass release rate per unit source
Solute transport in saturated porous media 509
area and Wx,y,z characterizes the source physical geome- try. In this work, point as well as two-dimensional source
geometries are considered. Furthermore, Gt characterizes the source loading type. Although instantaneous or contin-
uoustemporally periodic source loading types can easily be employed, the present research efforts focus only on a con-
tinuoustemporally periodic source loading.
2.1.1 Point source geometry The point source geometry is described mathematically by
the following expression:
W x
, y
, z
¼ 1
v d
x ¹ l
x
d y ¹ l
y
d z ¹ l
z
, 9
where l
x
, l
y
, l
z
represent the
x,y,z unbounded
¹ ` , l
x
, l
y
, l
z
, ` Cartesian coordinates of the point
source, respectively, and d is the Dirac delta function. It should be noted that here G represents the solute mass
release from the point source.
2.1.2 Elliptic source geometry The elliptic source geometry is described mathematically by
the following expression:
W x
, y
, z
¼ d
z ¹ l
z
v x ¹ l
x 2
a
2
þ y ¹ l
y 2
b
2
1 ,
otherwise ,
8 :
10 where l
x
, l
y
, l
z
are x,y,z Cartesian coordinates, respec- tively, of the center of the elliptic source geometry, and a
and b represent the semi-axes of the ellipse parallel to the x- and y-axes, respectively. It should be noted that here G
signifies the solute mass release rate per unit source area.
2.2 Aquifer with semi-infinite thickness
The appropriate initial and boundary conditions for the case of an aquifer with infinite longitudinal and lateral directions
and semi-infinite vertical direction thickness, as illustrated schematically in Fig. 1a, are as follows:
C ,
x ,
y ,
z ¼
, 11
C t
, 6 `
, y
, z
¼ ,
12 C
t ,
x ,
6 ` ,
z ¼
, 13
] C
t ,
x ,
y ,
] z
¼ ,
14 ]
C t
, x
, y
, `
] z
¼ ,
15 where condition 11 corresponds to the situation in which
solutes are initially absent from the three-dimensional por- ous formation, eqns 12 and 13 indicate that the aquifer
is infinite horizontally and laterally, boundary condition 14 represents a zero dispersive flux boundary and eqn
15 preserves concentration continuity for a semi-infinite vertical aquifer thickness. The vertical level z ¼ 0 defines
the location of the water table or a confining layer. Eqn 4, subject to conditions 11–15, is solved analytically. It
should be noted that z increases in the downward direction.
The analytical solution to the governing partial differen- tial equation, eqn 4, can be derived by a variety of meth-
ods, including the conventional method of separation of variables, as well as integral transform methods. Further-
more, a solution technique developed by Walker
24
invol- ving a Green function fundamental solution can also be
utilized. However, in the present study, integral transform techniques were employed because the multidimensional
models developed are an extension of our previous analy- tical work on virus transport models,
19,20
where Laplace transform techniques were employed. Similar mathematical
techniques were employed for the analytical solutions of multidimensional solute transport by Toride et al.
21
and Shan and Javandel.
18
Taking Laplace transforms with respect to time variable t and space variable z, and Fourier transforms with respect to
space variables x and y of eqn 4, and subsequently employ- ing the transformed initial and boundary conditions,
followed by inverse transformations, yields the desired ana- lytical solution for an aquifer with semi-infinite thickness
Fig. 1. Schematic illustration of point and elliptic sources of con- tamination with coordinates l
x
, l
y
, l
z
in an aquifer with semi- infinite a and finite b thickness. Note that the positive direction
for the vertical coordinate is inverted.
510 Y. Sim and C.V. Chrysikopoulos
see Appendix A: C
t ,
x ,
y ,
z ¼
1 64p
3
D
x
D
y
D
z 1=2
Z
t
Z
` ¹ `
Z
` ¹ `
Z
`
3 F t ¹ t
, q
, v
, p
L
1
t 3
Z
t
L
2
t z
3=2
L
3
z ,
x ¹ q ,
y ¹ v L
4
z ,
z þ p þ
L
4
z ,
z ¹ p ÿ
dz þ L
3
t ,
x ¹ q ,
y ¹ v t
3=2
3 L
4
t ,
zþp þ
L
4
t ,
z ¹ p dp dv dq dt
, ð
16Þ where p, q, v and z are dummy integration variables; the
following definitions were employed: L
1
t ¼
exp ¹ H t ½
ÿ ,
17a L
2
t ¼
Bz t ¹ z
1=2
I
1
2 Bz t ¹ z
ÿ
1=2
h i
, 17b
L
3
t ,
x ,
y ¼
exp Ux
2D
x
¹ 1
4t x
2
D
x
þ y
2
D
y
¹ t A ¹ H þ
U
2
4D
x
, ð
17cÞ L
4
t ,
z ¼
exp ¹
z
2
4D
z
t ,
17d and I
1
is the modified Bessel function of the first kind of first order.
2.2.1 Point source geometry Substituting eqns 8 and 9 into eqn 16 yields the analy-
tical solution for the case of point source geometry:
C t
, x
, y
, z
¼ 1
64p
3
D
x
D
y
D
z 1=2
Z
t
G t ¹ t
v L
1
t 3
Z
t
L
2
t z
3=2
L
3
z ,
x ¹ l
x
, y ¹ l
y
3 L
4
z ,
z þ l
z
þ L
4
z ,
z ¹ l
z
dz þ
L
3
t ,
x ¹ l
x
, y ¹ l
y
t
3=2
3 L
4
t ,
z þ l
z
þ L
4
t ,
z ¹ l
z
dt ,
ð 18Þ
where L
1
–L
4
are defined in eqns 17a– 17d, respec- tively, and the following property of the Dirac delta func-
tion was employed: Z
b a
f t
d t ¹ t
dt ¼ f t
, a t
b ,
19 where a and b are arbitrary constants, and f
is an arbitrary function.
2.2.2 Elliptic source geometry Substituting eqns 8 and 10 into eqn 16 leads to the
analytical solution for the case of elliptic source geometry:
C t
, x
, y
, z
¼ 1
64p
2
D
x
D
z 1=2
Z
t
Z
a
2
a
1
G t ¹ t
v L
1
t 3
Z
t
L
2
t z
L
3
z ,
x ¹ q ,
L
4
z ,
z þ l
z
þ L
4
z ,
z ¹ l
z
ÿ L
5
z dz þ
L
3
t ,
x ¹ q ,
t 3 L
4
t ,
zþl
z
þ L
4
t ,
z¹l
z
L
5
t dq dt
, ð
20Þ where L
1
–L
4
are defined in eqns 17a– 17d, respectively, L
5
t ¼
erf k
1
t ,
q ,
y ¹ erf k
2
t ,
q ,
y ,
21 a
1
¼ l
x
¹ a
, 22
a
2
¼ l
x
þ a
, 23
k
1
t ,
q ,
y ¼
y ¹ l
y
þ b
2
¹ b
2
q ¹ l
x 2
a
2 1=2
1 4D
y
t
1=2
, 24
k
2
t ,
q ,
y ¼
y ¹ l
y
¹ b
2
¹ b
2
q ¹ l
x 2
a
2 1=2
1 4D
y
t
1=2
, 25
erf[·] is the error function, and the following transformation and integral relationships were employed:
k ¼ y ¹ v
4tD
y 1=2
26a dk ¼
¹ dv
4tD
y 1=2
, 26b
Z
k
2
k
1
exp [
¹ k
2
] dk ¼
¹ p
1=2
2 erf k
1
¹ erf k
2
: 27
As noted by Chrysikopoulos,
4
solving for an elliptic source geometry is advantageous because the appropriate solution
for a circular source can easily be obtained by setting a ¼ b ¼
r in eqns 22–25, where r is the radius of the circular source.
2.3 Aquifer with finite thickness
