Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol21.Issue7.1998:
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Advances in Water Resources 21 (1998) 487–498
q 1998 Elsevier Science Limited
All rights reserved. Printed in Great Britain
0309-1708/98/$19.00 + 0.00
Modeling plume behavior for nonlinearly
sorbing solutes in saturated homogeneous
porous media
A. Abulaban a,*, J. L. Nieber b & D. Misra c
a
Department of Biosystems and Agricultural Engineering, University of Minnesota, St Paul, MN, USA
Department of Biosystems and Agricultural Engineering, University of Minnesota, St Paul, MN, USA
c
Army High Performance Computing Research Center, University of Minnesota, Minneapolis, MN, USA
b
(Received 15 March 1996; revised 8 January 1997; accepted 29 January 1997)
Transport of a sorbing solute in a two-dimensional steady and uniform flow field is
modeled using a particle tracking random walk method. The solute is initially
introduced from an instantaneous point source. Cases of linear and nonlinear sorption
isotherms are considered. Local pore velocity and mechanical dispersion are used to
describe the solute transport mechanisms at the local scale. The numerical simulation
of solute particle transport yields the large scale behavior of the solute plume.
Behavior of the plume is quantified in terms of the center-of-mass displacement
distance, relative velocity of the center-of-mass, mass breakthrough curves, spread
variance, and longitudinal skewness. The nonlinear sorption isotherm affects the
plume behavior in the following way relative to the linear isotherm: (1) the plume
velocity decreases exponentially with time; (2) the longitudinal variance increases
nonlinearly with time; (3) the solute front is steepened and tailing is enhanced. q 1998
Elsevier Science Limited. All rights reserved.
Key words: solute transport, Freundlich nonlinear sorption, saturated flow, particle
tracking-random walk, retardation.
and transport these equilibrium times are considerably
small. Therefore, the Local Equilibrium Assumption
(LEA) seems to be a reasonable approximation to adopt in
most instances.
A primary issue of solute transport is the definition of
an appropriate value for dispersivity. The assignment
of laboratory-scale dispersivities to field scale problems
has been found to underestimate plume spreading compared
to field observed behavior.31,45,46 One way to circumvent
this problem is to scale up the laboratory-scale dispersion
coefficients,33 but this generally results in overestimation
of dispersion close to the source.32 The problem of
defining a unique dispersivity relates to the fact that the
dispersivity commonly increases with distance travelled as
the transported mass encounters successively larger scales
of heterogeneity.14
To better represent the scale dependence of dispersivity
it is necessary to account for variability at the smallest scale
possible. In porous media, mixing at the junctions of the
1 INTRODUCTION
Sorption is probably the major factor controlling the movement of many hazardous substances through the vadose
zone and in ground water aquifers.49 The term sorption
refers to the physical/chemical attachment (adsorption)
and detachment (desorption) of some chemical to a solid
surface such as a porous medium solid. The system strives
to attain equilibrium between adsorbed and desorbed
phases. Equilibrium times vary for different chemicals and
porous media types. However, for most chemicals of concern in ground water and different porous media types,
equilibrium times are of the order of minutes to hours.
Equilibrium times have been reported for an extensive
list of chemicals in different soil environments to have a
maximum of 72 h, and to be less than 24 h for most
common chemicals.15 In the context of groundwater flow
Corresponding author.
487
488
A. Abulaban et al.
flow paths at the pore scale is the major cause of hydrodynamic dispersion.10 However, modeling at the pore-scale
is prohibitive when applying the model to large scale
effects. Hence, the minimum scale possible is the Darcyscale or continuum scale,6 where pore–scale effects are
represented by core-scale measured dispersivities. With
spatial discretizations given at the core-scale it is possible
to represent the scale-dependent dispersivity with appropriate computational models for solute transport.
In this paper we utilize the particle tracking random
walk (PTRW) method3,17,47 to examine the behavior of
solute plumes initiated from an instantaneous point source
in a uniform flow field. We examine cases of solutes which
obey linear and Freundlich sorption isotherms. The results
presented are a lead into the evaluation of plume behavior
in hydraulically and chemically heterogeneous domains.
Related studies have been reported by Tompson40 and
Bosma and van der Zee,8 who considered only a single
Freundlich isotherm and emphasized only the first two
spatial moments of the solute plume. More recently,
Bosma et al.9 considered several Freundlich isotherms,
but used Monte Carlo simulations to predict average
plume behavior in terms of the first two spatial moments.
In this paper we show the sensitivity of plume behavior
to the Freundlich isotherm exponent, in terms of spatial
moments, including plume skewness, as well as mass breakthrough curves. Also we look at the dilution of the peak
concentration, as well as the total solute mass in liquid
phase which varies due to nonlinearity in the Freundlich
isotherm.
2 THE TRANSPORT MODEL
The transport of a sorbing chemical can be described by the
advection–dispersion–sorption equation:
]C
r ]S
þ =·(Cv) ¹ =·(D·=C) þ
¼0
]t
F ]t
(1)
where C is the concentration in the liquid phase (ML ¹3); S
is the concentration in the adsorbed phase expressed as
mass of solute per mass of solid (1); v is the local velocity
vector (LT ¹1); D is the local hydrodynamic dispersion
tensor10 (LT ¹2); r is the dry bulk density of the porous
matrix (ML ¹3); F is the effective porosity (1); and t is the
time variable (T).
The hydrodynamic dispersion tensor can be written
as,5,6,30
vv
(2)
D ¼ (aT V þ Dm )I þ (aL ¹ aT )
V
where D m is the molecular diffusion coefficient (L 2T ¹1); I
is the identity matrix; V is the magnitude of the velocity
vector (LT ¹1); a L and a T are, respectively, the longitudinal
and transverse dispersivities (L); and vv is the diadic of the
velocity vector. Utilizing the identity
=2 : (DC) ¼ =·(D·=C) ¹ (=C)·(=·D)
where = 2 ; =·=, eqn (1) can be written as
]C
r ]S
þ =·½Cðv þ =·DÞÿ ¹ =2 : (DC) þ
¼0
]t
F ]t
(3)
where =·D is a velocity-like term that prevents the accumulation of solute in stagnation zones.37,38
Rearranging terms, eqn (3) can be written as
]
rS
C 1þ
þ =·½Cðv þ =·DÞÿ ¹ =2 : (DC) ¼ 0
]t
FC
(4)
r(x) S
where 1 þ F(x) C is a concentration-dependent and spacedependent retardation coefficient, R(C,x). The expression
inside the time derivative is the total (sorbed plus dissolved) solute concentration, C T ¼ R(C,x)C. Substituting
C ¼ CT =R(C, x) in eqn (4) results in the transport equation
in terms of the total solute concentration, C T:
]CT
þ =· vr (x, t, C)CT ¹ =2 : Dr (x, t, C)CT ¼ 0
]t
(5)
where vr (x, t, C) ¼ ½v þ =·[Dÿ=R(C, x)] and Dr (x, t, C) ¼
D=R(C, x) are the retarded velocity and dispersion tensor,
respectively.
Several mechanistic and phenomenological relationships have been derived to describe the equilibrium concentrations in the solid and fluid phases.48 The most commonly
used among those relationships is the Freundlich isotherm,13,15,28,35,48 expressed as
S ¼ Kf ðxÞC n
(6)
where K f(x)((L 3M ¹1) n) relates to sorption capacity, while n
relates to sorption intensity. Using the Freundlich isotherm
to express S in terms of C the retardation coefficient can be
written as
R(C, x) ¼ 1 þ aðxÞC n ¹ 1
(7)
where aðxÞ ¼ (rðxÞKf ðxÞ)=FðxÞ.
Although the linear version (n ¼ 1.0) of the Freundlich
isotherm has been widely used to model the transport of
sorbing chemicals,11,23,26 it has become increasingly apparent that this model frequently fails to provide adequate
representation of the effects of sorption processes on contaminant transport.48 Nonlinear isotherms often represent
equilibrium sorption phenomena more satisfactorily. Even
for slight nonlinearity in the isotherm Weber et al.48 showed
that plume behavior, in terms of center-of-mass retardation,
tailing of breakthrough, and front sharpening, was drastically affected for an isotherm exponent of 0.95 in contrast
with that for a linear isotherm. Tailing of breakthrough
curves is commonly perceived to be a result of kinetic processes,43 while in the case of a nonlinear isotherm it occurs
because solute retardation increases exponentially as the
concentration decreases.
It is commonly assumed that the equilibrium relationship between S and C is unique in both the adsorption and
desorption regimes. There can be a different relationship for
desorption than that for adsorption, in the case of sorption
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
Fig. 1. Behavior of the retardation coefficient function for the case
when a ¼ 1.0.
hysteresis.27 Such a non-unique relationship, however, will
make the simulation process more complicated, similar to
the increased complexities involved in modeling unsaturated flow processes with hysteresis in the water retention
function.24
It should be apparent from eqn (5) that the retardation
coefficient is a critical factor in the transport equation and
in the development of the solute plume. Therefore, it would
be of help to shed some light on its behavior.
As can be seen from eqn (7) above, the retardation coefficient for the case of a linear isotherm is
R(x) ¼ 1 þ
rðxÞKd ðxÞ
FðxÞ
(8)
which is dependent on the spatially-variable sorption capacity coefficient, dry bulk density, and porosity. In this case
the transport equation is the linear PDE for a nonreactive
solute scaled by the retardation coefficient. The parameter
K d(x) above is the special case of the sorption capacity for
the linear isotherm, and is commonly known as the distribution (or partition) coefficient.
For nonlinear isotherms it can be readily seen from eqn (7)
that: (1) the retardation coefficient is proportional to the
sorption capacity; (2) the retardation coefficient increases
nonlinearly with decreasing concentration; and (3) the effect
of the isotherm exponent on the retardation coefficient
depends on the concentration magnitude.
For the range of values of n used in this paper (0.7 # n #
1.0), Fig. 1 illustrates the behavior of the retardation coefficient for a wide range of concentrations for the special case
when a(x) ¼ 1.0, that is R(C,x) ¼ 1 þ C n ¹ 1 ¼ RðCÞ. Examination of Fig. 1 shows that for concentrations greater than
1.0 mg/l, R(C) increases with the Freundlich isotherm n. For
concentrations below 1.0 mg/l the trend is reversed; R(C)
decreases as n increases. Also the values of R(C) in this case
are much larger than in the case of large concentrations.
Note that when n ¼ 1.0, C n¹1 ¼ 1.0, and therefore R(C) is
constant and is equal to 1 þ a(x).
Due to hydrodynamic dispersion, concentrations within
489
the plume decrease towards the edges of the plume. For any
exponent less than 1.0, the retarded velocities on the edges
of the plume will be smaller than the velocity of the peak,
and hence the solute closer to the peak of the plume will be
moving faster than that at the edges. Thus as local dispersion
tends to spread the plume and dilute the concentration, the
retardation coefficient increases across the advancing front
of the plume and slows its progress. This can be construed as
a self-restraining process imposed by the plume itself, that
is, the plume tends to resist hydrodynamic dispersion. This
produces a sharpening front, because the solute within the
plume moves faster than the advancing front. On the other
hand, as the solute leaves sites along the trailing edge of
the plume the retardation coefficient increases and hence
hinders solute removal. As concentration approaches zero,
the retardation coefficient tends to infinity, meaning that at
very small concentrations solute will hardly be moving. The
result will, therefore, be an ever stretching tail of the plume.
Theoretically the tail of the plume should extend back at
least to the injection point (it may travel a short distance
along the upgradient of the injection point due to local dispersion). The implication of the Freundlich isotherm is that,
in theory, since the retardation coefficient tends to infinity
as concentration vanishes, once solute invades a site it will
never be totally recovered.
3 SOLUTION PROCEDURE
When used to solve the advection–dispersion–sorption
equation, conventional computational techniques, such as
finite difference and finite element methods, suffer from
numerical complications chief among which is numerical
dispersion. The numerical dispersion results when there
are sudden spatial and temporal changes of solute concentrations. Operator splitting methods are one means to
improve the solution accuracy.16,25 However, the computational costs involved with any of these techniques is generally high, inhibiting applications of these methods to large
flow domains.
An alternative to the conventional numerical techniques
for solute transport is the particle tracking-random walk
(PTRW) method where a solute plume is represented with
a large number of particles that move according to spatially
and/or temporally variable local velocities (deterministic
translation). A random component proportional to the
local dispersivity of the porous medium is added to
the deterministic translation of a particle to account for
the Brownian-like motion caused by heterogeneities at
scales smaller than the Darcy scale.
The use of the PTRW technique to model transport of
solutes with dispersion has been shown to be superior in
accuracy, efficiency, and computational cost compared to
more conventional finite difference and finite element
schemes.19–21,37,38 In addition to its application for modeling solute transport in porous media,3,17,21,33,34,37,38,41,42,47
the PTRW technique has been applied to transport problems
490
A. Abulaban et al.
in free water bodies, where random motions are the result
of turbulence in flow velocities. PTRW techniques have also
been used successfully for the simulation of dispersion in
turbulent boundary layers,4 effluent plumes,7,18 effluent
patches,2,18 and in the ocean.19
Like other numerical methods that approximate continuous variables with discretized ones, the PTRW method also
suffers some drawbacks. One such drawback is the wide
oscillations that occur at locations when numbers of particles are too small, which can occur at the leading and trailing edges of a solute plume. This problem can be improved
by increasing the number of particles or by using a concentration smoothing technique.39 Another way to improve the solution is by increasing particle resolution at low-concentration
sites. In the present application we used 200 000 particles,
which we found to be large enough to minimize concentration oscillations during the simulation times considered.
The mathematical foundation for the PTRW technique
lies in the similarity between the advection/dispersion transport equation6 and the Fokker–Planck equation,12,29,36 a
conservation equation for the probability distribution of
particles moving independently in a random field. Detailed
mathematical development of the PTRW technique and its
application to simulate contaminant transport in porous
media can be found in Tompson et al.37 and Abulaban.1
The motion of a particle moving in a random field can be
described by the nonlinear Langevin equation12
ÿ
ÿ
dX
¼ A X(t), t þ B X(t), t ·yðtÞ
(9)
dt
where X(t) is the position of the particle at time t. The
vector A(X,t) is a known function of space and time, and
is used to represent the deterministic driving forces acting
to change X. The second order tensor B(X,t) is also a
known function of space and time that aligns the random
forces with A(X,t). The vector y(t) represents the uncorrelated and rapidly changing random forces. Integration of
the Langevin equation over a small Dt yields the discretized
stepping equation
ÿ
ÿ
DX ¼ A X(t), t Dt þ B X(t), t ·yðtÞDt
(10)
If a large number of independently moving particles
started from the initial position X 0 at t 0, then the fraction
of particles expected to be found in a small volume around
the point X at time t, denoted f(X,t/X 0,t 0), will obey the
Fokker–Planck conservation equation12,29,44 given by
ÿ
]f
BBT
2
f ¼0
(11)
þ =· Af ¹ = :
2
]t
Comparing the Fokker–Planck equation, eqn (11), with
the transport equation, eqn (5), we can see that if v r and
D r are not dependent on C T (through C), as is the case for
conservative and for linearly sorbing solutes, then the
solute transport equation and the Fokker–Planck equation
will be equivalent if A and B are chosen such that
A ¼ vr ; BBT ¼ 2Dr
(12)
while f corresponds to C T. However, v r and D r for nonlinearly sorbing solutes are concentration-dependent, which
means that the particles will not move independently; the
movement of a particle depends on the intensity of particles
around it. This is a violation of the assumption of independence used to derive the Fokker–Planck equation. Thus it is
necessary to adjust for this dependence in the application of
the method. The issue of particle dependence has not been
addressed explicitly in any of the previous applications8,9,40
of the analogy to the solution of transport with nonlinear
sorption.
The problem of particle movement dependence can be
circumvented by an iterative solution procedure.22 The
way in which the iterative procedure circumvents the
dependence of particles can be explained by using an argument based on the existence of an analytical solution. Let
us suppose that an analytical solution to the nonlinear sorption problem is known. Then the retardation coefficient
distribution will be known exactly for all time and all
space. If one were to then use this distribution in moving
a set of particles, the particles can be moved independently
because the retardation coefficients do not depend on particle movement, but instead are known a priori. The application of the Fokker–Planck analogy for such a condition is
exact.
With the iterative solution, the appropriate spatial distribution of the retardation coefficient for a particular time
step is determined by repeated trial (iterative) movements
of the particles until a specified convergence criterion is
satisfied. Note that within any particular iteration, a trial
retardation coefficient distribution is used and the particles
are assumed to move independently of each other. Thus,
for each iteration, the application of the Fokker–Planck
analogy is exact. The distribution of the particles at the
end of the time step following such an iterative procedure
will be the same as the distribution that would be computed
with independently moving particles using the retardation
coefficient distribution had it been known a priori. We can
therefore conclude that the application of the Fokker–
Planck analogy to the case of nonlinear sorption is valid
when iteration is used to account for dependence among
particle movements.
In our application of iteration the retardation field for a
new time step is initiated using the current retardation field.
The particles are then moved and a retardation coefficient
field is calculated for the final positions of the particles. The
retardation coefficient field is then averaged between that
for the initial field and the final field. The particles are
moved again, starting from their positions at the beginning
of the time step. Iterations continue until the retardation
field does not change appreciably.
In our application of the PTRW method we used a grid of
computational cells to define the local values of the concentration, hydraulic parameters, and transport parameters and
variables. The concentration and the retardation coefficient
were assumed to be constant within a computational cell so
that all the particles within that cell move with the same
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
retarded velocity and dispersion, but with a different
random component since each particle uses a different random number. It should be noted here that, in order to avoid
random changes in the retardation coefficient field due to
random numbers, in our application a set of random numbers is generated at the beginning of each time step and
the set is used for all the iterations within that time step.
We found that the retardation coefficient field generally
converged within three or four iterations.
Experience with the numerical solution showed that
errors due to not iterating the solution depend on initial
conditions, number of particles used to represent the total
solute mass, length of the solution time step, as well as the
nonlinearity of the sorption isotherm. The error increased
for higher initial concentrations, smaller injection area, larger time steps, and stronger nonlinearity in the sorption
isotherm. Also errors grow with time into the simulation.
The quantitative effect of not iterating on the retardation
coefficient has not been reported in the literature.
Other computational aspects related to the implementation of the random walk technique for conservative solutes
are discussed in detail by Tompson et al.37 and Tompson.40
Inserting A and B back into eqn (10) yields the stepping
equation
Xm ¼ Xm ¹ 1 þ vr Dtm þ B·wm
(13)
where the subscript m indicates the time level, Dt m ¼ t m ¹
t m¹1, and w m contains the random component. In the
present application the flow field is homogeneous and
therefore the divergence of the dispersion tensor within
the v r term vanishes.
By the central limit theorem the random component w m
can take any form, provided that it possesses a zero-mean
and a variance proportional to Dt m. Therefore, it will suffice
to use a standardized random vector, Z, whose components
Z i (i ¼ x,y,z) are statistically independentpwith
a zeromean and a unit variance such that w m ¼ Z Dtm . Consequently, any of a wide range of distributions can be implemented, such as a normal N(0,1) distribution,3 or a simple
uniform distribution spread over 6 Î3 to represent the Z i
components.
4 RESULTS
Whereas it is recognized that the hydraulic properties as
well as chemical properties are usually spatially variable
in a flow field, results are presented in this paper for a
physically and chemically homogeneous flow field. This
paper is intended as background for a subsequent paper to
consider spatial heterogeneity of the hydraulic conductivity
and the sorption capacity.
The flow field illustrated in Fig. 2 is a 1000 m by 200 m
rectangle with one-dimensional flow along the longer
dimension of the domain. Parameters are assumed to be
constant with depth, so a 1.0 m slice is considered. The
dimensions were chosen to be large enough to allow us to
491
Fig. 2. Size and orientation of the flow domain. Also shown are
the boundary conditions, mean velocity direction, and location of
the solute injection point.
observe the behavior of the plume at large times where the
concentrations become very small. Since this is a nonlinear
problem, large scale behavior cannot be deduced from
Monte Carlo simulations similar to those of Bosma and
van der Zee8 and Bosma et al.9 The boundaries are treated
as absorbing boundaries where the solute particles are
allowed to cross the boundaries and leave the flow domain
forever. This treatment is similar to that of Tompson and
Gelhar38 and Bosma et al.,9 whereas Schwartz31 and Smith
and Schwartz33,34 treated the boundaries as reflecting
boundaries.
The flow domain was discretized into 1-m square cells.
Solute is injected as an instantaneous pulse in the horizontal
plane. To avoid very high initial concentrations the solute is
uniformly distributed over a square region containing 16
cells centered in the middle of the transverse direction and
10 m downgradient from the inflow boundary of the domain
to ensure that no solute would cross the inflow boundary
early in the simulation.
The mean flow velocity was 1.0 m/day in the x-direction.
The effective porosity was constant at 0.35, and the soil bulk
density was 1.75 g/cm 3. Longitudinal and transverse dispersivities were 0.5 m and 0.1 m, respectively. The sorption
capacity factor for all isotherms was 0.2 (1/g) n, which was
chosen to give a retardation coefficient of 2.0 for the linear
case. Four simulations were performed using 200 g of solute
and varying the isotherm exponents over four values: 0.7,
0.8, 0.9, and 1.0. The initial liquid concentration depends
on the value of the exponent, n. Initial liquid concentrations
for the different exponent values used are summarized in
Table 1. The number of particles used to represent the solute
was 200 000 with a constant particle mass, m p, of 1.0 mg.
The particles carry the total mass and move with the local
retarded velocities. Even though only the dissolved mass is
available to move with the flow velocity, the instantaneous
equilibrium assumption makes it appear like all the mass is
moving with a phase-averaged velocity which can be easily
shown to be the flow velocity divided by the retardation
coefficient.
Results are presented below in terms of concentration
profiles, mass breakthrough, plume velocity, spread variance, and plume skewness. For all of the simulations the
plumes are symmetric in the transverse direction due to
the one-dimensional flow, and, except for the spread
492
A. Abulaban et al.
Table 1. Initial liquid concentrations for different values of
isotherm exponent corresponding to an injected solute mass
of 200 g
Isotherm exponent
Initial liquid concentration
(mg/l)
1.0
0.9
0.8
0.7
17.85
21.49
25.16
28.42
variance, the transverse moments were unremarkable.
Results were, therefore, evaluated in terms of the longitudinal spatial moments and breakthrough curves, as well
as the transverse spread variance. Mathematical definitions
of the spatial moments can be found elsewhere.1,38 In our
simulations moments are estimated from cell concentrations
assigned to the centers of the cells as:
X1 (t) ¼
N
F Xc
x C (t)DxDy
M(t) j ¼ 1 j j
(14)
2
(t) ¼
X11
N
F Xc
(x ¹ X1 (t))2 Cj (t)DxDy
M(t) j ¼ 1 j
(15)
2
(t) ¼
X22
N
F Xc
(y ¹ X2 (t))2 Cj (t)DxDy
M(t) j ¼ 1 j
(16)
"
#
Nc
X
1
F
3
(xj ¹ X1 (t)) Cj (t)DxDy
Cˆ s, 1 ¼
(X11 )3=2 M(t) j ¼ 1
(17)
where M(t) is the total mass in liquid, calculated as:
M(t) ¼ F
Nc
X
Cj DxDy
Fig. 3. ConcenEtration plume at 500 days and 1500 days for
different isotherm exponents.
balance equation:
(18)
j¼1
In the expressions above C j is the concentration in the
computational cell, N c is the number of computational
cells in the domain, and Dx and Dy are the lengths of
the rectangular computational cells in the longitudinal
and transverse directions, respectively, and x j and y j are
the coordinates of the center of computational cell j. Also
X 1 is the longitudinal position of the center of mass, X 211 and
X 222 are the longitudinal and transverse spread variances
ˆ s,1 is the longabout the center of mass, respectively, and C
itudinal coefficient of skewness. Note that the longitudinal
displacement is X 1(t) ¹ X 1(0).
The cell liquid concentration, C j, is calculated by solving
the balance equation relating the total mass in a cell with the
dissolved phase concentration. The total mass in a cell, M j,
is the sum of the masses of all the particles in that cell, N jm p,
where N j is the number of particles in the cell. This total
mass is distributed between the dissolved phase, C j, and
adsorbed phase, Sj ¼ K fC j n, according to the following
DxDy(FCj þ rCjn ) ¼ Mj
(19)
For a linear isotherm (n ¼ 1.0) eqn (19) is linear and therefore it is solved directly to give C j as the total mass divided
by the retardation coefficient given by eqn (8). For nonlinear isotherms, however, eqn (19) is nonlinear and is
solved numerically using the Newton–Raphson method.
An example of the concentration distribution in the
plumes for the various isotherm exponents is presented in
Fig. 3. The plumes are illustrated for 500 and 1500 days. It
is seen that the plume is virtually symmetric for the linear
isotherm case, which is the expected behavior since the
transport equation is linear and the Peclet number is 2.0.
In contrast, the plumes for the nonlinear cases have sharp
fronts and long tails, and the tails increase in length as the
isotherm exponent decreases.
As has been discussed earlier the tail should extend back
to the injection point, but this does not appear to be the case
(Fig. 3). The reason is that there is a minimum concentration
that can be resolved in each case due to numerical limitations. The smallest resolvable concentration and the largest
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
Table 2. Smallest resolvable solute concentrations and the corresponding retardation coefficients when using 200 000 particles to represent a mass of 200 g of chemical, for various
values of the Freundlich isotherm exponent
Freundlich
exponent
1.0
0.9
0.8
0.7
Smallest concentration
(mg/l)
0.0143
0.0111
0.0078
0.0047
Retardation
coefficient
2.00
2.57
3.64
6.00
Fig. 4. Longitudinal velocity of the plume center of mass.
corresponding retardation coefficient depend on the value of
the Freundlich exponent. They correspond to the concentration resulting from one particle in a computational cell.
With 200 000 particles carrying 200 g of solute the smallest
concentrations and the corresponding retardation coefficients are given in Table 2.
The plume velocity is defined as the instantaneous timederivative of the plume center of mass. The plume retardation
coefficient can be defined as the inverse of the ratio of the
plume velocity to the flow velocity.40 Fig. 4 shows such a
retardation coefficient for the different isotherm exponents.
It is readily seen in this figure that the plume velocity remains
constant for the linear isotherm, which is due to the fact that
the retardation coefficient does not depend on concentration.
In contrast, the plume velocities for all the nonlinear isotherms decrease with time due to the global decrease in
concentration. It is observed that initially the plume for n ¼
0.7 is the fastest, and the speed decreases with increasing n.
Shortly after that the plume for the linear isotherm becomes
the fastest and the velocity decreases with decreasing n.
Fig. 5. Breakthrough curves at three cross-sections located at (a)
50 m, (b) 200 m, and (c) 500 m downgradient from the inflow
boundary of the flow domain.
493
494
A. Abulaban et al.
Fig. 6. Longitudinal spread variance of the solute plume.
A theoretical assessment requires that the displacement for each of the nonlinear isotherms be asymptotic
to some constant value at large time. This is due to the
increasing retardation as the concentration decreases. As
the concentration approaches zero, the retarded velocity
also approaches zero, and the plume will then become stagnant. In their analysis of large time behavior, Bosma et al.9
derived a relationship for the center of mass displacement.
Differentiating their equation with respect to time gives the
plume velocity, which tends to zero as t → `.
The longitudinal breakthrough curves at the crosssections 50 m, 200 m, and 500 m are shown in Fig. 5 in
terms of the rate of mass that crossed a cross-section
within a time step. The curves were slightly smoothed by
taking the 7-point moving average of the relative mass flux
across the corresponding cross-section. As expected, the
curves for the linear isotherm are symmetric. The curves
for the nonlinear isotherms, however, show the sharpening
fronts and the elongating tails of the plume. The slowing
speed of the plume is also clear in Fig. 5 where we see that at
the 50 m cross-section the curve for the n ¼ 0.7 exponent
appears first and the one for n ¼ 1.0 comes last, while for the
200 m and 500 m cross-sections the curve for n ¼ 1.0 comes
first and the one for n ¼ 0.7 comes last. The behavior of
the breakthrough curves also indicates that the plumes for
the nonlinear isotherms are skewed to the left, which is due
to the sharp fronts and long tails.
The longitudinal spread variance of the plume for the
different values of the Freundlich exponent used is shown
in Fig. 6. It is obvious that the variance increases rapidly for
smaller values of the exponent. It can also be seen that,
except for the linear isotherm case (n ¼ 1.0) where the
time dependence is linear, the variance grows in a nonlinear
fashion. The nonlinearity of the growth is shown to be
stronger for smaller n values. As was discussed previously,
Fig. 7. Transverse spread variance of the solute plume.
the solute plume should slow asymptotically to zero due
to increasing retardation with decreasing concentration,
and thus we should expect the plume variance for the
nonlinear cases to level off. This argument is supported
by the theoretical large time behavior of the longitudinal
spread variance as expressed by eqn (28) of Bosma et
al.9 The time derivative of this equation tends to zero as
t → `.
The transverse spread variance is shown in Fig. 7. Contrary to the behavior of the longitudinal variance, Fig. 7
shows the transverse spread variance to decrease as the
isotherm exponent decreases. This means that spreading
is hindered in the direction of zero velocity. In fact, as
was discussed earlier, the large longitudinal variances are
largely due to the elongating tails, and the same behavior
responsible for sharpening the leading front also causes
the slower transverse spreading.
Because the ratio between the solute mass in the liquid
phase and that in the adsorbed phase decreases as concentration decreases for the nonlinear cases, the total mass in
the liquid phase decreases as the plume spreads. Illustrated
in Fig. 8(a) is the ratio of total solute mass in liquid phase
to the total injected mass. This is the inverse of another
plume retardation defined as the total solute mass divided
by the total mass in the liquid phase.40 It is obvious from
Fig. 8(a) that the liquid mass for the linear isotherm remains
constant, which is due to the retardation coefficient being
independent of concentration. However, the behavior for
the nonlinear isotherms indicates that the plume retardation
as defined above increases, a behavior similar to that of the
retardation coefficient defined with the velocity of the
plume center of mass relative to the flow velocity and
depicted in Fig. 4. The relationships depicted in Fig. 8(a)
are important for clean-up efforts. Since we know that for
the nonlinear cases the more spread the solute is, the more
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
495
Fig. 9. Longitudinal skewness coefficients of the solute plume.
Fig. 8. (a) Total mass in liquid phase normalized by initial
injected mass. (b) Peak concentration normalized by initial peak
concentration.
strongly it adsorbs to the solid, clean-up will be significantly
enhanced by doing it earlier.
Another behavior related to the spreading of the plume is
that of the peak concentration. Shown in Fig. 8(b) is the
peak concentration normalized by the initial peak concentration. It should be noted here that since the total injected
solute mass is the same for the different isotherm exponents,
and because of the different partitioning behavior with the
different isotherm exponents, the initial peak concentrations
are different. In fact Fig. 8(a) shows that initially the total
mass in the liquid phase, and thus the initial concentration,
are largest for the smallest isotherm exponent and they
decrease as the exponent increases. It is interesting to see
from Fig. 8(b) that the normalized peak concentration does
not change appreciably with the isotherm exponent, especially after we saw the great increase of the longitudinal
spread variance with decreasing the isotherm exponent.
This is further evidence that the enhancement of the longitudinal spread variance is due to the elongating tail, and
that spreading in the nonlinear cases is actually increased
stretching of the tail rather than dilution of the peak.
Finally the skewness coefficient of the plume is shown in
Fig. 9. Since the plume for the linear case is symmetric, its
skewness coefficient should be zero, which is what is seen
from Fig. 9. However, for the nonlinear cases we observe
that the solute plume is skewed to the left, which is indicated
by the negative skewness coefficients shown in Fig. 9. It can
also be seen that the absolute skewness coefficient is, for
some time, larger for the smaller isotherm exponent, indicating a sharper front and longer tail relative to the size of
the plume. However, as Fig. 9 shows, the skewness coefficient tends to zero after a long time. This behavior is due to
the tendency of the concentration gradients to vanish in the
case of spatial uniformity of concentration. This behavior is
also evidenced by the behavior depicted in Fig. 8(b) which
shows the normalized peak concentration to asymptotically
approach zero.
Another measure of skewness is the distance between the
center of mass and the peak concentration, which is shown
in Fig. 10. The behavior depicted in Fig. 10 was slightly
smoothed by taking the 7-point average of the actual simulation results. It is seen from Fig. 10 that the center-of-mass
to peak distance hovers around zero for the linear isotherm
case, as expected since the plume is symmetric. For the
nonlinear isotherms, however, it increases as the isotherm
exponent decreases, attesting to the behavior that the front
gets sharper and the tail flatter. Note the linear trend of the
behavior in Fig. 10, indicating that the tail of the plume
stretches at a steady rate.
496
A. Abulaban et al.
Fig. 10. Distance between plume center of mass and peak
concentration.
5 DISCUSSION
This paper dealt with the simpler case of a hydraulically and
chemically homogeneous flow domain. Other researchers
have already presented results for more complicated flow
domains. Tompson40 considered flows in three-dimensional
homogeneous and heterogeneous domains and a solute
injected from an instantaneous point source. He evaluated
simulated results from single long-term simulations in terms
of plume center of mass velocity and plume spread variance.
More recently Bosma et al.9 used the PTRW method to
solve the transport equation for an instantaneously injected
solute pulse. They used only 2000 particles to represent the
solute plume and analyzed averaged spatial moments
obtained from Monte Carlo simulations.
The results presented in this paper closely correspond to
the results presented by Tompson40 for the case of the
homogeneous domains. The behaviors of the plume center
of mass velocity, plume spread variance, and skewness of
plume concentration field presented in this paper are
comparable to those illustrated by Tompson. In addition
to the results presented by Tompson, we have demonstrated
the sensitivity of the plume behavior to the isotherm exponent, which is in agreement with the findings of Bosma
et al.9
One problem with the current application of the PTRW
method is the limited concentration resolution associated
with using a fixed number of particles to represent the solute
mass for cases of nonlinear sorption. This limitation was
observed to influence the amount of tailing for the plumes.
These tails should extend back to the injection point, but
due to the limited concentration resolution the tails did not
extend that far. To test the effect of particle numbers we
have performed selected simulations with various numbers
of particles, including 20 000, 100 000, 200 000 and 500 000
particles. We found that the greater the number of particles
used, the greater is the amount of tailing. We also found that
at small times, the simulations with 20 000 particles are
identical to the results for 500 000 particles. However, as
time increases the number of particles needed to provide
accurate results increases. It is expected that at some sufficiently large time, even 500 000 particles will not provide
adequate accuracy. As the number of particles used directly
influences the computational demand for the simulation,
it will be necessary to improve the computational efficiency
of the PTRW method.
One improvement to the PTRW method would be to use
an adaptive procedure where particles in regions with low
particle numbers would be subdivided to increase concentration resolution. Such a procedure would allow a simulation to begin with a relatively small number of particles, and
then the number of particles would be increased to automatically adapt to conditions of either low concentration
or high concentration gradients. This should reduce the
overall computation demand of a simulation.
6 SUMMARY AND CONCLUSIONS
In this paper the Particle Tracking Random Walk method
has been implemented for the simulation of the transport
of sorbing solutes with nonlinear equilibrium isotherms in
a homogeneous flow field. Numerical experiments were
performed to study the effects of the nonlinearity of the
sorption isotherm on the behavior of the plume of a sorbing
solute. The flow field was a saturated two-dimensional
rectangle with constant mean flow velocity in one direction.
The numerical experiments consisted of releasing 200 g of
solute as a pulse and tracking their movements using
200 000 particles to represent the solute mass. The Freundlich isotherm exponents examined varied between 0.7 and
1.0. Results were presented in terms of the plume center-ofmass velocity, spread variance, skewness, and breakthrough
curves. Findings of the study can be summarized as follows:
1. As the plume spreads the velocity of its center-ofmass decreases because of the increased retardation
caused by decreasing concentrations. The decrease in
the velocity was observed to be fastest at earlier times
due to a larger initial decrease of the concentrations.
The plume was observed to move faster for smaller
values of the Freundlich isotherm exponent at earlier
times, but the order was reversed at large times.
2. Except for the symmetric case of the linear isotherm
(n ¼ 1.0), the plume was observed to develop a long
tail due to the differential retardation behavior.
Theoretically the tail should extend back to the
point of injection because, according to the Freundlich isotherm, the retardation coefficient tends to
infinity in the limit as the liquid concentration
approaches zero. The implication of this is that once
a site is invaded by a contaminant that obeys the
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
Freundlich isotherm, it would be impossible to clean
it completely.
3. The advancing front of the solute was seen to get
sharper as the plume advanced, again due to the
behavior of the retardation coefficient rapidly increasing across the front of the plume.
4. While advancement of the solute front was hindered
by the nonlinear sorption, the spread variance was
found to increase nonlinearly with time. Nonlinearity
was stronger for smaller isotherm exponents. The
increase in the spread variance is primarily due to
the stretching tail behind the plume, not to increased
spreading at the front. This is because the variance is
related to the square of the distance from the center of
mass.
5. Due to the long tail and the sharp front of the plume
the skewness was found to be negative and its absolute value increasing with time for nonlinear isotherms;
for linear isotherms the plume was symmetric. At large
times the trend of the absolute skewness was found
to reverse and tended to zero, which is due to the
flattening of the tail and vanishing of concentration
gradients. It was observed that the more nonlinear the
isotherm, the stronger the trend toward zero.
ACKNOWLEDGEMENTS
Published as Paper No. 22,585 of the scientific journal
series of the Minnesota Agricultural Experiment Station on
research conducted under Minnesota Agricultural Experiment Station Project No. 12-047. This work was partially
supported by the Army High Performance Computing
Research Center under the auspices of the Department of
the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-0003/contract number
DAAH04-95-C-0008, the content of which does not necessarily reflect the position or the policy of the government,
and no official endorsement should be inferred. Additional
support for numerical simulations was provided by the University of Minnesota Supercomputer Institute.
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Advances in Water Resources 21 (1998) 487–498
q 1998 Elsevier Science Limited
All rights reserved. Printed in Great Britain
0309-1708/98/$19.00 + 0.00
Modeling plume behavior for nonlinearly
sorbing solutes in saturated homogeneous
porous media
A. Abulaban a,*, J. L. Nieber b & D. Misra c
a
Department of Biosystems and Agricultural Engineering, University of Minnesota, St Paul, MN, USA
Department of Biosystems and Agricultural Engineering, University of Minnesota, St Paul, MN, USA
c
Army High Performance Computing Research Center, University of Minnesota, Minneapolis, MN, USA
b
(Received 15 March 1996; revised 8 January 1997; accepted 29 January 1997)
Transport of a sorbing solute in a two-dimensional steady and uniform flow field is
modeled using a particle tracking random walk method. The solute is initially
introduced from an instantaneous point source. Cases of linear and nonlinear sorption
isotherms are considered. Local pore velocity and mechanical dispersion are used to
describe the solute transport mechanisms at the local scale. The numerical simulation
of solute particle transport yields the large scale behavior of the solute plume.
Behavior of the plume is quantified in terms of the center-of-mass displacement
distance, relative velocity of the center-of-mass, mass breakthrough curves, spread
variance, and longitudinal skewness. The nonlinear sorption isotherm affects the
plume behavior in the following way relative to the linear isotherm: (1) the plume
velocity decreases exponentially with time; (2) the longitudinal variance increases
nonlinearly with time; (3) the solute front is steepened and tailing is enhanced. q 1998
Elsevier Science Limited. All rights reserved.
Key words: solute transport, Freundlich nonlinear sorption, saturated flow, particle
tracking-random walk, retardation.
and transport these equilibrium times are considerably
small. Therefore, the Local Equilibrium Assumption
(LEA) seems to be a reasonable approximation to adopt in
most instances.
A primary issue of solute transport is the definition of
an appropriate value for dispersivity. The assignment
of laboratory-scale dispersivities to field scale problems
has been found to underestimate plume spreading compared
to field observed behavior.31,45,46 One way to circumvent
this problem is to scale up the laboratory-scale dispersion
coefficients,33 but this generally results in overestimation
of dispersion close to the source.32 The problem of
defining a unique dispersivity relates to the fact that the
dispersivity commonly increases with distance travelled as
the transported mass encounters successively larger scales
of heterogeneity.14
To better represent the scale dependence of dispersivity
it is necessary to account for variability at the smallest scale
possible. In porous media, mixing at the junctions of the
1 INTRODUCTION
Sorption is probably the major factor controlling the movement of many hazardous substances through the vadose
zone and in ground water aquifers.49 The term sorption
refers to the physical/chemical attachment (adsorption)
and detachment (desorption) of some chemical to a solid
surface such as a porous medium solid. The system strives
to attain equilibrium between adsorbed and desorbed
phases. Equilibrium times vary for different chemicals and
porous media types. However, for most chemicals of concern in ground water and different porous media types,
equilibrium times are of the order of minutes to hours.
Equilibrium times have been reported for an extensive
list of chemicals in different soil environments to have a
maximum of 72 h, and to be less than 24 h for most
common chemicals.15 In the context of groundwater flow
Corresponding author.
487
488
A. Abulaban et al.
flow paths at the pore scale is the major cause of hydrodynamic dispersion.10 However, modeling at the pore-scale
is prohibitive when applying the model to large scale
effects. Hence, the minimum scale possible is the Darcyscale or continuum scale,6 where pore–scale effects are
represented by core-scale measured dispersivities. With
spatial discretizations given at the core-scale it is possible
to represent the scale-dependent dispersivity with appropriate computational models for solute transport.
In this paper we utilize the particle tracking random
walk (PTRW) method3,17,47 to examine the behavior of
solute plumes initiated from an instantaneous point source
in a uniform flow field. We examine cases of solutes which
obey linear and Freundlich sorption isotherms. The results
presented are a lead into the evaluation of plume behavior
in hydraulically and chemically heterogeneous domains.
Related studies have been reported by Tompson40 and
Bosma and van der Zee,8 who considered only a single
Freundlich isotherm and emphasized only the first two
spatial moments of the solute plume. More recently,
Bosma et al.9 considered several Freundlich isotherms,
but used Monte Carlo simulations to predict average
plume behavior in terms of the first two spatial moments.
In this paper we show the sensitivity of plume behavior
to the Freundlich isotherm exponent, in terms of spatial
moments, including plume skewness, as well as mass breakthrough curves. Also we look at the dilution of the peak
concentration, as well as the total solute mass in liquid
phase which varies due to nonlinearity in the Freundlich
isotherm.
2 THE TRANSPORT MODEL
The transport of a sorbing chemical can be described by the
advection–dispersion–sorption equation:
]C
r ]S
þ =·(Cv) ¹ =·(D·=C) þ
¼0
]t
F ]t
(1)
where C is the concentration in the liquid phase (ML ¹3); S
is the concentration in the adsorbed phase expressed as
mass of solute per mass of solid (1); v is the local velocity
vector (LT ¹1); D is the local hydrodynamic dispersion
tensor10 (LT ¹2); r is the dry bulk density of the porous
matrix (ML ¹3); F is the effective porosity (1); and t is the
time variable (T).
The hydrodynamic dispersion tensor can be written
as,5,6,30
vv
(2)
D ¼ (aT V þ Dm )I þ (aL ¹ aT )
V
where D m is the molecular diffusion coefficient (L 2T ¹1); I
is the identity matrix; V is the magnitude of the velocity
vector (LT ¹1); a L and a T are, respectively, the longitudinal
and transverse dispersivities (L); and vv is the diadic of the
velocity vector. Utilizing the identity
=2 : (DC) ¼ =·(D·=C) ¹ (=C)·(=·D)
where = 2 ; =·=, eqn (1) can be written as
]C
r ]S
þ =·½Cðv þ =·DÞÿ ¹ =2 : (DC) þ
¼0
]t
F ]t
(3)
where =·D is a velocity-like term that prevents the accumulation of solute in stagnation zones.37,38
Rearranging terms, eqn (3) can be written as
]
rS
C 1þ
þ =·½Cðv þ =·DÞÿ ¹ =2 : (DC) ¼ 0
]t
FC
(4)
r(x) S
where 1 þ F(x) C is a concentration-dependent and spacedependent retardation coefficient, R(C,x). The expression
inside the time derivative is the total (sorbed plus dissolved) solute concentration, C T ¼ R(C,x)C. Substituting
C ¼ CT =R(C, x) in eqn (4) results in the transport equation
in terms of the total solute concentration, C T:
]CT
þ =· vr (x, t, C)CT ¹ =2 : Dr (x, t, C)CT ¼ 0
]t
(5)
where vr (x, t, C) ¼ ½v þ =·[Dÿ=R(C, x)] and Dr (x, t, C) ¼
D=R(C, x) are the retarded velocity and dispersion tensor,
respectively.
Several mechanistic and phenomenological relationships have been derived to describe the equilibrium concentrations in the solid and fluid phases.48 The most commonly
used among those relationships is the Freundlich isotherm,13,15,28,35,48 expressed as
S ¼ Kf ðxÞC n
(6)
where K f(x)((L 3M ¹1) n) relates to sorption capacity, while n
relates to sorption intensity. Using the Freundlich isotherm
to express S in terms of C the retardation coefficient can be
written as
R(C, x) ¼ 1 þ aðxÞC n ¹ 1
(7)
where aðxÞ ¼ (rðxÞKf ðxÞ)=FðxÞ.
Although the linear version (n ¼ 1.0) of the Freundlich
isotherm has been widely used to model the transport of
sorbing chemicals,11,23,26 it has become increasingly apparent that this model frequently fails to provide adequate
representation of the effects of sorption processes on contaminant transport.48 Nonlinear isotherms often represent
equilibrium sorption phenomena more satisfactorily. Even
for slight nonlinearity in the isotherm Weber et al.48 showed
that plume behavior, in terms of center-of-mass retardation,
tailing of breakthrough, and front sharpening, was drastically affected for an isotherm exponent of 0.95 in contrast
with that for a linear isotherm. Tailing of breakthrough
curves is commonly perceived to be a result of kinetic processes,43 while in the case of a nonlinear isotherm it occurs
because solute retardation increases exponentially as the
concentration decreases.
It is commonly assumed that the equilibrium relationship between S and C is unique in both the adsorption and
desorption regimes. There can be a different relationship for
desorption than that for adsorption, in the case of sorption
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
Fig. 1. Behavior of the retardation coefficient function for the case
when a ¼ 1.0.
hysteresis.27 Such a non-unique relationship, however, will
make the simulation process more complicated, similar to
the increased complexities involved in modeling unsaturated flow processes with hysteresis in the water retention
function.24
It should be apparent from eqn (5) that the retardation
coefficient is a critical factor in the transport equation and
in the development of the solute plume. Therefore, it would
be of help to shed some light on its behavior.
As can be seen from eqn (7) above, the retardation coefficient for the case of a linear isotherm is
R(x) ¼ 1 þ
rðxÞKd ðxÞ
FðxÞ
(8)
which is dependent on the spatially-variable sorption capacity coefficient, dry bulk density, and porosity. In this case
the transport equation is the linear PDE for a nonreactive
solute scaled by the retardation coefficient. The parameter
K d(x) above is the special case of the sorption capacity for
the linear isotherm, and is commonly known as the distribution (or partition) coefficient.
For nonlinear isotherms it can be readily seen from eqn (7)
that: (1) the retardation coefficient is proportional to the
sorption capacity; (2) the retardation coefficient increases
nonlinearly with decreasing concentration; and (3) the effect
of the isotherm exponent on the retardation coefficient
depends on the concentration magnitude.
For the range of values of n used in this paper (0.7 # n #
1.0), Fig. 1 illustrates the behavior of the retardation coefficient for a wide range of concentrations for the special case
when a(x) ¼ 1.0, that is R(C,x) ¼ 1 þ C n ¹ 1 ¼ RðCÞ. Examination of Fig. 1 shows that for concentrations greater than
1.0 mg/l, R(C) increases with the Freundlich isotherm n. For
concentrations below 1.0 mg/l the trend is reversed; R(C)
decreases as n increases. Also the values of R(C) in this case
are much larger than in the case of large concentrations.
Note that when n ¼ 1.0, C n¹1 ¼ 1.0, and therefore R(C) is
constant and is equal to 1 þ a(x).
Due to hydrodynamic dispersion, concentrations within
489
the plume decrease towards the edges of the plume. For any
exponent less than 1.0, the retarded velocities on the edges
of the plume will be smaller than the velocity of the peak,
and hence the solute closer to the peak of the plume will be
moving faster than that at the edges. Thus as local dispersion
tends to spread the plume and dilute the concentration, the
retardation coefficient increases across the advancing front
of the plume and slows its progress. This can be construed as
a self-restraining process imposed by the plume itself, that
is, the plume tends to resist hydrodynamic dispersion. This
produces a sharpening front, because the solute within the
plume moves faster than the advancing front. On the other
hand, as the solute leaves sites along the trailing edge of
the plume the retardation coefficient increases and hence
hinders solute removal. As concentration approaches zero,
the retardation coefficient tends to infinity, meaning that at
very small concentrations solute will hardly be moving. The
result will, therefore, be an ever stretching tail of the plume.
Theoretically the tail of the plume should extend back at
least to the injection point (it may travel a short distance
along the upgradient of the injection point due to local dispersion). The implication of the Freundlich isotherm is that,
in theory, since the retardation coefficient tends to infinity
as concentration vanishes, once solute invades a site it will
never be totally recovered.
3 SOLUTION PROCEDURE
When used to solve the advection–dispersion–sorption
equation, conventional computational techniques, such as
finite difference and finite element methods, suffer from
numerical complications chief among which is numerical
dispersion. The numerical dispersion results when there
are sudden spatial and temporal changes of solute concentrations. Operator splitting methods are one means to
improve the solution accuracy.16,25 However, the computational costs involved with any of these techniques is generally high, inhibiting applications of these methods to large
flow domains.
An alternative to the conventional numerical techniques
for solute transport is the particle tracking-random walk
(PTRW) method where a solute plume is represented with
a large number of particles that move according to spatially
and/or temporally variable local velocities (deterministic
translation). A random component proportional to the
local dispersivity of the porous medium is added to
the deterministic translation of a particle to account for
the Brownian-like motion caused by heterogeneities at
scales smaller than the Darcy scale.
The use of the PTRW technique to model transport of
solutes with dispersion has been shown to be superior in
accuracy, efficiency, and computational cost compared to
more conventional finite difference and finite element
schemes.19–21,37,38 In addition to its application for modeling solute transport in porous media,3,17,21,33,34,37,38,41,42,47
the PTRW technique has been applied to transport problems
490
A. Abulaban et al.
in free water bodies, where random motions are the result
of turbulence in flow velocities. PTRW techniques have also
been used successfully for the simulation of dispersion in
turbulent boundary layers,4 effluent plumes,7,18 effluent
patches,2,18 and in the ocean.19
Like other numerical methods that approximate continuous variables with discretized ones, the PTRW method also
suffers some drawbacks. One such drawback is the wide
oscillations that occur at locations when numbers of particles are too small, which can occur at the leading and trailing edges of a solute plume. This problem can be improved
by increasing the number of particles or by using a concentration smoothing technique.39 Another way to improve the solution is by increasing particle resolution at low-concentration
sites. In the present application we used 200 000 particles,
which we found to be large enough to minimize concentration oscillations during the simulation times considered.
The mathematical foundation for the PTRW technique
lies in the similarity between the advection/dispersion transport equation6 and the Fokker–Planck equation,12,29,36 a
conservation equation for the probability distribution of
particles moving independently in a random field. Detailed
mathematical development of the PTRW technique and its
application to simulate contaminant transport in porous
media can be found in Tompson et al.37 and Abulaban.1
The motion of a particle moving in a random field can be
described by the nonlinear Langevin equation12
ÿ
ÿ
dX
¼ A X(t), t þ B X(t), t ·yðtÞ
(9)
dt
where X(t) is the position of the particle at time t. The
vector A(X,t) is a known function of space and time, and
is used to represent the deterministic driving forces acting
to change X. The second order tensor B(X,t) is also a
known function of space and time that aligns the random
forces with A(X,t). The vector y(t) represents the uncorrelated and rapidly changing random forces. Integration of
the Langevin equation over a small Dt yields the discretized
stepping equation
ÿ
ÿ
DX ¼ A X(t), t Dt þ B X(t), t ·yðtÞDt
(10)
If a large number of independently moving particles
started from the initial position X 0 at t 0, then the fraction
of particles expected to be found in a small volume around
the point X at time t, denoted f(X,t/X 0,t 0), will obey the
Fokker–Planck conservation equation12,29,44 given by
ÿ
]f
BBT
2
f ¼0
(11)
þ =· Af ¹ = :
2
]t
Comparing the Fokker–Planck equation, eqn (11), with
the transport equation, eqn (5), we can see that if v r and
D r are not dependent on C T (through C), as is the case for
conservative and for linearly sorbing solutes, then the
solute transport equation and the Fokker–Planck equation
will be equivalent if A and B are chosen such that
A ¼ vr ; BBT ¼ 2Dr
(12)
while f corresponds to C T. However, v r and D r for nonlinearly sorbing solutes are concentration-dependent, which
means that the particles will not move independently; the
movement of a particle depends on the intensity of particles
around it. This is a violation of the assumption of independence used to derive the Fokker–Planck equation. Thus it is
necessary to adjust for this dependence in the application of
the method. The issue of particle dependence has not been
addressed explicitly in any of the previous applications8,9,40
of the analogy to the solution of transport with nonlinear
sorption.
The problem of particle movement dependence can be
circumvented by an iterative solution procedure.22 The
way in which the iterative procedure circumvents the
dependence of particles can be explained by using an argument based on the existence of an analytical solution. Let
us suppose that an analytical solution to the nonlinear sorption problem is known. Then the retardation coefficient
distribution will be known exactly for all time and all
space. If one were to then use this distribution in moving
a set of particles, the particles can be moved independently
because the retardation coefficients do not depend on particle movement, but instead are known a priori. The application of the Fokker–Planck analogy for such a condition is
exact.
With the iterative solution, the appropriate spatial distribution of the retardation coefficient for a particular time
step is determined by repeated trial (iterative) movements
of the particles until a specified convergence criterion is
satisfied. Note that within any particular iteration, a trial
retardation coefficient distribution is used and the particles
are assumed to move independently of each other. Thus,
for each iteration, the application of the Fokker–Planck
analogy is exact. The distribution of the particles at the
end of the time step following such an iterative procedure
will be the same as the distribution that would be computed
with independently moving particles using the retardation
coefficient distribution had it been known a priori. We can
therefore conclude that the application of the Fokker–
Planck analogy to the case of nonlinear sorption is valid
when iteration is used to account for dependence among
particle movements.
In our application of iteration the retardation field for a
new time step is initiated using the current retardation field.
The particles are then moved and a retardation coefficient
field is calculated for the final positions of the particles. The
retardation coefficient field is then averaged between that
for the initial field and the final field. The particles are
moved again, starting from their positions at the beginning
of the time step. Iterations continue until the retardation
field does not change appreciably.
In our application of the PTRW method we used a grid of
computational cells to define the local values of the concentration, hydraulic parameters, and transport parameters and
variables. The concentration and the retardation coefficient
were assumed to be constant within a computational cell so
that all the particles within that cell move with the same
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
retarded velocity and dispersion, but with a different
random component since each particle uses a different random number. It should be noted here that, in order to avoid
random changes in the retardation coefficient field due to
random numbers, in our application a set of random numbers is generated at the beginning of each time step and
the set is used for all the iterations within that time step.
We found that the retardation coefficient field generally
converged within three or four iterations.
Experience with the numerical solution showed that
errors due to not iterating the solution depend on initial
conditions, number of particles used to represent the total
solute mass, length of the solution time step, as well as the
nonlinearity of the sorption isotherm. The error increased
for higher initial concentrations, smaller injection area, larger time steps, and stronger nonlinearity in the sorption
isotherm. Also errors grow with time into the simulation.
The quantitative effect of not iterating on the retardation
coefficient has not been reported in the literature.
Other computational aspects related to the implementation of the random walk technique for conservative solutes
are discussed in detail by Tompson et al.37 and Tompson.40
Inserting A and B back into eqn (10) yields the stepping
equation
Xm ¼ Xm ¹ 1 þ vr Dtm þ B·wm
(13)
where the subscript m indicates the time level, Dt m ¼ t m ¹
t m¹1, and w m contains the random component. In the
present application the flow field is homogeneous and
therefore the divergence of the dispersion tensor within
the v r term vanishes.
By the central limit theorem the random component w m
can take any form, provided that it possesses a zero-mean
and a variance proportional to Dt m. Therefore, it will suffice
to use a standardized random vector, Z, whose components
Z i (i ¼ x,y,z) are statistically independentpwith
a zeromean and a unit variance such that w m ¼ Z Dtm . Consequently, any of a wide range of distributions can be implemented, such as a normal N(0,1) distribution,3 or a simple
uniform distribution spread over 6 Î3 to represent the Z i
components.
4 RESULTS
Whereas it is recognized that the hydraulic properties as
well as chemical properties are usually spatially variable
in a flow field, results are presented in this paper for a
physically and chemically homogeneous flow field. This
paper is intended as background for a subsequent paper to
consider spatial heterogeneity of the hydraulic conductivity
and the sorption capacity.
The flow field illustrated in Fig. 2 is a 1000 m by 200 m
rectangle with one-dimensional flow along the longer
dimension of the domain. Parameters are assumed to be
constant with depth, so a 1.0 m slice is considered. The
dimensions were chosen to be large enough to allow us to
491
Fig. 2. Size and orientation of the flow domain. Also shown are
the boundary conditions, mean velocity direction, and location of
the solute injection point.
observe the behavior of the plume at large times where the
concentrations become very small. Since this is a nonlinear
problem, large scale behavior cannot be deduced from
Monte Carlo simulations similar to those of Bosma and
van der Zee8 and Bosma et al.9 The boundaries are treated
as absorbing boundaries where the solute particles are
allowed to cross the boundaries and leave the flow domain
forever. This treatment is similar to that of Tompson and
Gelhar38 and Bosma et al.,9 whereas Schwartz31 and Smith
and Schwartz33,34 treated the boundaries as reflecting
boundaries.
The flow domain was discretized into 1-m square cells.
Solute is injected as an instantaneous pulse in the horizontal
plane. To avoid very high initial concentrations the solute is
uniformly distributed over a square region containing 16
cells centered in the middle of the transverse direction and
10 m downgradient from the inflow boundary of the domain
to ensure that no solute would cross the inflow boundary
early in the simulation.
The mean flow velocity was 1.0 m/day in the x-direction.
The effective porosity was constant at 0.35, and the soil bulk
density was 1.75 g/cm 3. Longitudinal and transverse dispersivities were 0.5 m and 0.1 m, respectively. The sorption
capacity factor for all isotherms was 0.2 (1/g) n, which was
chosen to give a retardation coefficient of 2.0 for the linear
case. Four simulations were performed using 200 g of solute
and varying the isotherm exponents over four values: 0.7,
0.8, 0.9, and 1.0. The initial liquid concentration depends
on the value of the exponent, n. Initial liquid concentrations
for the different exponent values used are summarized in
Table 1. The number of particles used to represent the solute
was 200 000 with a constant particle mass, m p, of 1.0 mg.
The particles carry the total mass and move with the local
retarded velocities. Even though only the dissolved mass is
available to move with the flow velocity, the instantaneous
equilibrium assumption makes it appear like all the mass is
moving with a phase-averaged velocity which can be easily
shown to be the flow velocity divided by the retardation
coefficient.
Results are presented below in terms of concentration
profiles, mass breakthrough, plume velocity, spread variance, and plume skewness. For all of the simulations the
plumes are symmetric in the transverse direction due to
the one-dimensional flow, and, except for the spread
492
A. Abulaban et al.
Table 1. Initial liquid concentrations for different values of
isotherm exponent corresponding to an injected solute mass
of 200 g
Isotherm exponent
Initial liquid concentration
(mg/l)
1.0
0.9
0.8
0.7
17.85
21.49
25.16
28.42
variance, the transverse moments were unremarkable.
Results were, therefore, evaluated in terms of the longitudinal spatial moments and breakthrough curves, as well
as the transverse spread variance. Mathematical definitions
of the spatial moments can be found elsewhere.1,38 In our
simulations moments are estimated from cell concentrations
assigned to the centers of the cells as:
X1 (t) ¼
N
F Xc
x C (t)DxDy
M(t) j ¼ 1 j j
(14)
2
(t) ¼
X11
N
F Xc
(x ¹ X1 (t))2 Cj (t)DxDy
M(t) j ¼ 1 j
(15)
2
(t) ¼
X22
N
F Xc
(y ¹ X2 (t))2 Cj (t)DxDy
M(t) j ¼ 1 j
(16)
"
#
Nc
X
1
F
3
(xj ¹ X1 (t)) Cj (t)DxDy
Cˆ s, 1 ¼
(X11 )3=2 M(t) j ¼ 1
(17)
where M(t) is the total mass in liquid, calculated as:
M(t) ¼ F
Nc
X
Cj DxDy
Fig. 3. ConcenEtration plume at 500 days and 1500 days for
different isotherm exponents.
balance equation:
(18)
j¼1
In the expressions above C j is the concentration in the
computational cell, N c is the number of computational
cells in the domain, and Dx and Dy are the lengths of
the rectangular computational cells in the longitudinal
and transverse directions, respectively, and x j and y j are
the coordinates of the center of computational cell j. Also
X 1 is the longitudinal position of the center of mass, X 211 and
X 222 are the longitudinal and transverse spread variances
ˆ s,1 is the longabout the center of mass, respectively, and C
itudinal coefficient of skewness. Note that the longitudinal
displacement is X 1(t) ¹ X 1(0).
The cell liquid concentration, C j, is calculated by solving
the balance equation relating the total mass in a cell with the
dissolved phase concentration. The total mass in a cell, M j,
is the sum of the masses of all the particles in that cell, N jm p,
where N j is the number of particles in the cell. This total
mass is distributed between the dissolved phase, C j, and
adsorbed phase, Sj ¼ K fC j n, according to the following
DxDy(FCj þ rCjn ) ¼ Mj
(19)
For a linear isotherm (n ¼ 1.0) eqn (19) is linear and therefore it is solved directly to give C j as the total mass divided
by the retardation coefficient given by eqn (8). For nonlinear isotherms, however, eqn (19) is nonlinear and is
solved numerically using the Newton–Raphson method.
An example of the concentration distribution in the
plumes for the various isotherm exponents is presented in
Fig. 3. The plumes are illustrated for 500 and 1500 days. It
is seen that the plume is virtually symmetric for the linear
isotherm case, which is the expected behavior since the
transport equation is linear and the Peclet number is 2.0.
In contrast, the plumes for the nonlinear cases have sharp
fronts and long tails, and the tails increase in length as the
isotherm exponent decreases.
As has been discussed earlier the tail should extend back
to the injection point, but this does not appear to be the case
(Fig. 3). The reason is that there is a minimum concentration
that can be resolved in each case due to numerical limitations. The smallest resolvable concentration and the largest
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
Table 2. Smallest resolvable solute concentrations and the corresponding retardation coefficients when using 200 000 particles to represent a mass of 200 g of chemical, for various
values of the Freundlich isotherm exponent
Freundlich
exponent
1.0
0.9
0.8
0.7
Smallest concentration
(mg/l)
0.0143
0.0111
0.0078
0.0047
Retardation
coefficient
2.00
2.57
3.64
6.00
Fig. 4. Longitudinal velocity of the plume center of mass.
corresponding retardation coefficient depend on the value of
the Freundlich exponent. They correspond to the concentration resulting from one particle in a computational cell.
With 200 000 particles carrying 200 g of solute the smallest
concentrations and the corresponding retardation coefficients are given in Table 2.
The plume velocity is defined as the instantaneous timederivative of the plume center of mass. The plume retardation
coefficient can be defined as the inverse of the ratio of the
plume velocity to the flow velocity.40 Fig. 4 shows such a
retardation coefficient for the different isotherm exponents.
It is readily seen in this figure that the plume velocity remains
constant for the linear isotherm, which is due to the fact that
the retardation coefficient does not depend on concentration.
In contrast, the plume velocities for all the nonlinear isotherms decrease with time due to the global decrease in
concentration. It is observed that initially the plume for n ¼
0.7 is the fastest, and the speed decreases with increasing n.
Shortly after that the plume for the linear isotherm becomes
the fastest and the velocity decreases with decreasing n.
Fig. 5. Breakthrough curves at three cross-sections located at (a)
50 m, (b) 200 m, and (c) 500 m downgradient from the inflow
boundary of the flow domain.
493
494
A. Abulaban et al.
Fig. 6. Longitudinal spread variance of the solute plume.
A theoretical assessment requires that the displacement for each of the nonlinear isotherms be asymptotic
to some constant value at large time. This is due to the
increasing retardation as the concentration decreases. As
the concentration approaches zero, the retarded velocity
also approaches zero, and the plume will then become stagnant. In their analysis of large time behavior, Bosma et al.9
derived a relationship for the center of mass displacement.
Differentiating their equation with respect to time gives the
plume velocity, which tends to zero as t → `.
The longitudinal breakthrough curves at the crosssections 50 m, 200 m, and 500 m are shown in Fig. 5 in
terms of the rate of mass that crossed a cross-section
within a time step. The curves were slightly smoothed by
taking the 7-point moving average of the relative mass flux
across the corresponding cross-section. As expected, the
curves for the linear isotherm are symmetric. The curves
for the nonlinear isotherms, however, show the sharpening
fronts and the elongating tails of the plume. The slowing
speed of the plume is also clear in Fig. 5 where we see that at
the 50 m cross-section the curve for the n ¼ 0.7 exponent
appears first and the one for n ¼ 1.0 comes last, while for the
200 m and 500 m cross-sections the curve for n ¼ 1.0 comes
first and the one for n ¼ 0.7 comes last. The behavior of
the breakthrough curves also indicates that the plumes for
the nonlinear isotherms are skewed to the left, which is due
to the sharp fronts and long tails.
The longitudinal spread variance of the plume for the
different values of the Freundlich exponent used is shown
in Fig. 6. It is obvious that the variance increases rapidly for
smaller values of the exponent. It can also be seen that,
except for the linear isotherm case (n ¼ 1.0) where the
time dependence is linear, the variance grows in a nonlinear
fashion. The nonlinearity of the growth is shown to be
stronger for smaller n values. As was discussed previously,
Fig. 7. Transverse spread variance of the solute plume.
the solute plume should slow asymptotically to zero due
to increasing retardation with decreasing concentration,
and thus we should expect the plume variance for the
nonlinear cases to level off. This argument is supported
by the theoretical large time behavior of the longitudinal
spread variance as expressed by eqn (28) of Bosma et
al.9 The time derivative of this equation tends to zero as
t → `.
The transverse spread variance is shown in Fig. 7. Contrary to the behavior of the longitudinal variance, Fig. 7
shows the transverse spread variance to decrease as the
isotherm exponent decreases. This means that spreading
is hindered in the direction of zero velocity. In fact, as
was discussed earlier, the large longitudinal variances are
largely due to the elongating tails, and the same behavior
responsible for sharpening the leading front also causes
the slower transverse spreading.
Because the ratio between the solute mass in the liquid
phase and that in the adsorbed phase decreases as concentration decreases for the nonlinear cases, the total mass in
the liquid phase decreases as the plume spreads. Illustrated
in Fig. 8(a) is the ratio of total solute mass in liquid phase
to the total injected mass. This is the inverse of another
plume retardation defined as the total solute mass divided
by the total mass in the liquid phase.40 It is obvious from
Fig. 8(a) that the liquid mass for the linear isotherm remains
constant, which is due to the retardation coefficient being
independent of concentration. However, the behavior for
the nonlinear isotherms indicates that the plume retardation
as defined above increases, a behavior similar to that of the
retardation coefficient defined with the velocity of the
plume center of mass relative to the flow velocity and
depicted in Fig. 4. The relationships depicted in Fig. 8(a)
are important for clean-up efforts. Since we know that for
the nonlinear cases the more spread the solute is, the more
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
495
Fig. 9. Longitudinal skewness coefficients of the solute plume.
Fig. 8. (a) Total mass in liquid phase normalized by initial
injected mass. (b) Peak concentration normalized by initial peak
concentration.
strongly it adsorbs to the solid, clean-up will be significantly
enhanced by doing it earlier.
Another behavior related to the spreading of the plume is
that of the peak concentration. Shown in Fig. 8(b) is the
peak concentration normalized by the initial peak concentration. It should be noted here that since the total injected
solute mass is the same for the different isotherm exponents,
and because of the different partitioning behavior with the
different isotherm exponents, the initial peak concentrations
are different. In fact Fig. 8(a) shows that initially the total
mass in the liquid phase, and thus the initial concentration,
are largest for the smallest isotherm exponent and they
decrease as the exponent increases. It is interesting to see
from Fig. 8(b) that the normalized peak concentration does
not change appreciably with the isotherm exponent, especially after we saw the great increase of the longitudinal
spread variance with decreasing the isotherm exponent.
This is further evidence that the enhancement of the longitudinal spread variance is due to the elongating tail, and
that spreading in the nonlinear cases is actually increased
stretching of the tail rather than dilution of the peak.
Finally the skewness coefficient of the plume is shown in
Fig. 9. Since the plume for the linear case is symmetric, its
skewness coefficient should be zero, which is what is seen
from Fig. 9. However, for the nonlinear cases we observe
that the solute plume is skewed to the left, which is indicated
by the negative skewness coefficients shown in Fig. 9. It can
also be seen that the absolute skewness coefficient is, for
some time, larger for the smaller isotherm exponent, indicating a sharper front and longer tail relative to the size of
the plume. However, as Fig. 9 shows, the skewness coefficient tends to zero after a long time. This behavior is due to
the tendency of the concentration gradients to vanish in the
case of spatial uniformity of concentration. This behavior is
also evidenced by the behavior depicted in Fig. 8(b) which
shows the normalized peak concentration to asymptotically
approach zero.
Another measure of skewness is the distance between the
center of mass and the peak concentration, which is shown
in Fig. 10. The behavior depicted in Fig. 10 was slightly
smoothed by taking the 7-point average of the actual simulation results. It is seen from Fig. 10 that the center-of-mass
to peak distance hovers around zero for the linear isotherm
case, as expected since the plume is symmetric. For the
nonlinear isotherms, however, it increases as the isotherm
exponent decreases, attesting to the behavior that the front
gets sharper and the tail flatter. Note the linear trend of the
behavior in Fig. 10, indicating that the tail of the plume
stretches at a steady rate.
496
A. Abulaban et al.
Fig. 10. Distance between plume center of mass and peak
concentration.
5 DISCUSSION
This paper dealt with the simpler case of a hydraulically and
chemically homogeneous flow domain. Other researchers
have already presented results for more complicated flow
domains. Tompson40 considered flows in three-dimensional
homogeneous and heterogeneous domains and a solute
injected from an instantaneous point source. He evaluated
simulated results from single long-term simulations in terms
of plume center of mass velocity and plume spread variance.
More recently Bosma et al.9 used the PTRW method to
solve the transport equation for an instantaneously injected
solute pulse. They used only 2000 particles to represent the
solute plume and analyzed averaged spatial moments
obtained from Monte Carlo simulations.
The results presented in this paper closely correspond to
the results presented by Tompson40 for the case of the
homogeneous domains. The behaviors of the plume center
of mass velocity, plume spread variance, and skewness of
plume concentration field presented in this paper are
comparable to those illustrated by Tompson. In addition
to the results presented by Tompson, we have demonstrated
the sensitivity of the plume behavior to the isotherm exponent, which is in agreement with the findings of Bosma
et al.9
One problem with the current application of the PTRW
method is the limited concentration resolution associated
with using a fixed number of particles to represent the solute
mass for cases of nonlinear sorption. This limitation was
observed to influence the amount of tailing for the plumes.
These tails should extend back to the injection point, but
due to the limited concentration resolution the tails did not
extend that far. To test the effect of particle numbers we
have performed selected simulations with various numbers
of particles, including 20 000, 100 000, 200 000 and 500 000
particles. We found that the greater the number of particles
used, the greater is the amount of tailing. We also found that
at small times, the simulations with 20 000 particles are
identical to the results for 500 000 particles. However, as
time increases the number of particles needed to provide
accurate results increases. It is expected that at some sufficiently large time, even 500 000 particles will not provide
adequate accuracy. As the number of particles used directly
influences the computational demand for the simulation,
it will be necessary to improve the computational efficiency
of the PTRW method.
One improvement to the PTRW method would be to use
an adaptive procedure where particles in regions with low
particle numbers would be subdivided to increase concentration resolution. Such a procedure would allow a simulation to begin with a relatively small number of particles, and
then the number of particles would be increased to automatically adapt to conditions of either low concentration
or high concentration gradients. This should reduce the
overall computation demand of a simulation.
6 SUMMARY AND CONCLUSIONS
In this paper the Particle Tracking Random Walk method
has been implemented for the simulation of the transport
of sorbing solutes with nonlinear equilibrium isotherms in
a homogeneous flow field. Numerical experiments were
performed to study the effects of the nonlinearity of the
sorption isotherm on the behavior of the plume of a sorbing
solute. The flow field was a saturated two-dimensional
rectangle with constant mean flow velocity in one direction.
The numerical experiments consisted of releasing 200 g of
solute as a pulse and tracking their movements using
200 000 particles to represent the solute mass. The Freundlich isotherm exponents examined varied between 0.7 and
1.0. Results were presented in terms of the plume center-ofmass velocity, spread variance, skewness, and breakthrough
curves. Findings of the study can be summarized as follows:
1. As the plume spreads the velocity of its center-ofmass decreases because of the increased retardation
caused by decreasing concentrations. The decrease in
the velocity was observed to be fastest at earlier times
due to a larger initial decrease of the concentrations.
The plume was observed to move faster for smaller
values of the Freundlich isotherm exponent at earlier
times, but the order was reversed at large times.
2. Except for the symmetric case of the linear isotherm
(n ¼ 1.0), the plume was observed to develop a long
tail due to the differential retardation behavior.
Theoretically the tail should extend back to the
point of injection because, according to the Freundlich isotherm, the retardation coefficient tends to
infinity in the limit as the liquid concentration
approaches zero. The implication of this is that once
a site is invaded by a contaminant that obeys the
Modeling plume behavior for nonlinearly sorbing solutes in saturated homogeneous porous media
Freundlich isotherm, it would be impossible to clean
it completely.
3. The advancing front of the solute was seen to get
sharper as the plume advanced, again due to the
behavior of the retardation coefficient rapidly increasing across the front of the plume.
4. While advancement of the solute front was hindered
by the nonlinear sorption, the spread variance was
found to increase nonlinearly with time. Nonlinearity
was stronger for smaller isotherm exponents. The
increase in the spread variance is primarily due to
the stretching tail behind the plume, not to increased
spreading at the front. This is because the variance is
related to the square of the distance from the center of
mass.
5. Due to the long tail and the sharp front of the plume
the skewness was found to be negative and its absolute value increasing with time for nonlinear isotherms;
for linear isotherms the plume was symmetric. At large
times the trend of the absolute skewness was found
to reverse and tended to zero, which is due to the
flattening of the tail and vanishing of concentration
gradients. It was observed that the more nonlinear the
isotherm, the stronger the trend toward zero.
ACKNOWLEDGEMENTS
Published as Paper No. 22,585 of the scientific journal
series of the Minnesota Agricultural Experiment Station on
research conducted under Minnesota Agricultural Experiment Station Project No. 12-047. This work was partially
supported by the Army High Performance Computing
Research Center under the auspices of the Department of
the Army, Army Research Laboratory cooperative agreement number DAAH04-95-2-0003/contract number
DAAH04-95-C-0008, the content of which does not necessarily reflect the position or the policy of the government,
and no official endorsement should be inferred. Additional
support for numerical simulations was provided by the University of Minnesota Supercomputer Institute.
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