Use of solute flux approach to contaminant transport in aquifers
Utilisation du flux du polluant lors du transport de pollution dans les nappes
aquifères
Mensur Mulabdić
Civil Engineering Faculty, University in Osijek, Croatia, leta@zg.hinet.hr
Roko Andričević
Civil Engineering Faculty, University in Split, Croatia
Keywords : solute, flux, transport, dispersion, aquifers
ABSTRACT: Analitical sollution for solute flux in dipsersive transport in aquifers is presented in
basic terms. The stochastic analysis and solute flux approach represent the efficient way to handle the
spatial variability in the properties of geologic media and to reflect its effect on the prediction of the
groundwater velocities and concentrations of contaminants
1 INTRODUCTION
One of the potential areas of greatest current interest for the application of contaminant transport
modeling lies in siting and design of new waste management facilities, and the design of remedial
activities at facilities where ground-water contamination incidents have already occurred. The threedimensional reality through which a contaminant migrates in the subsurface environment is
complicated by geologic heterogeneity and tortuous connected flowpaths. In the face of incomplete
data and insufficient resources, the three-dimensional reality has to be simplified and conceptualized
for a particular environmental engineering project. The stochastic analysis and solute flux approach
represent the efficient way to handle the spatial variability in the properties of geologic media and to
reflect its effect on the prediction of the groundwater velocities and concentrations of contaminants
(Andricevic and Cvetkovic, 1998). This paper describes the method in its basic terms.
2 DESCRIPTION OF THE METHOD
A useful representation of transport in aquifers is by the solute flux, defined as mass per unit time and
unit area. It is required to statistically describe the concentration, or the mass flux, as a function of
time and space, and not only as a mean description of the transport. In practical use this implies
quantification of mean and variance of the solute flux, since the scale of the ground velocity
variations, when larger than the plume size, can influence the solute concentration or mass flux, and in
that case its mean value and variance is of importance.
Relative and apsolute dispersion is regarded as the solute mass flux, defined as mass per unit time
and unit area through the control plane. It depends on transverse displacement and on solute travel
time evaluated at a fixed control plane. The Lagrangian formulation is used for derivation of the mean
solute flux and standard deviation, so that origin of coordinates is at the centre of mass of solute
plume, and kinematics of a particle pairs is used to evaluate the statistics for the first two moments
(mean and variance) of the solute flux.
Incompressible groundwater flow is considered in a heterogeneous aquifer, having spatialy variable
hydraulic konductivity described by K(x), x (x,y,z) beeing a Cartesian coordinate vector, and flow
velocity V(x) that satisfies the continuity equation   ( nV )  0 (n stands for effective porosity,
and velocity is defined through hydraulic head  as V  ( K / n) ). Stationary flow is considered
and the mean and covariance function function of V(Vx,Vy,Vz) are considered known (Rubin and
Dagan 1992).
Fig.1. shows the basic approach in the method. At time t=0 solute of a total mass M is reliesed into
the flow over the injection area A0. At the time t0 solute plume is formed and advected downstream
towards a plane, often called controll plane (CP), where solute flux can be measured or predicted. Any
portion a of the area A0, having its own solute mass per area 0a (0 is constant over area A0) can be
traced in transport process by watching its trajectory X=X(t,a) in the stream direction, and its relative
position in the plane A0 by transverse displacement vector (,). Transport process is described as a
time-space process (,), where  denotes travel time from the start position at t=0 to the CP, and 
quantifies transverse displacement, for the distance x of the CP from the origin.
Fig.1. Problem description (a) and definition of particle displacement: (c) fixed frame of reference and (b)
relative coordinates, for 0=const. (after Andričević & Cvetković 1998)
The travel time  and components of the transverse vector can be evaluated as:
d
V ( , ,  )
0 x
x
 ( x; a)  
 ( x; a)  X y  x; a ; a ,
 ( x; a)  X z  x; a ; a 
(1)
(2)
The mean travel time and mean transverse displacement for the entire plume crossing the CP is
defined by (3) and (4),
T ( x; A0 ) 
1
 ( x; a)da
A0 A0
(3)
Y ( x; A0 ) 
1
 ( x; a)da
A0 A0
(4)
At distance x from the point of solute injection the solute mass flux component orthogonal to CP is
calculated as
q(t; y; x, A) 
1
 0 (a)(t , ) ( y' )dy ' da
A A0 A
(5)
where
t
(t , )   (t  t ' ) (t ' , )dt '
(6)
0
in which (t) T-1 is the injection rate, and (t,) is the time retention function (available in analytical
forms for linear solute mass transfer processes, Cvetkovic and Dagan, 1984)).
From the solute mass flux, the solute discharge over the sampling area A can be defined as
Q(t , y; x, A)  q(t , y; x, A) A , where Q(t,y;x,A) quantifies the solute mass crossing A centered at y at
time t.
3 PRACTICAL USE OF THE MODEL
The presented solute flux approach to contaminant transport follows the USEPA guidelines and
incorporates real physical phenomena, such as instantaneous and/or slow release from the source,
advection, dispersion, sorption, mass transfer, and possible uncertainties in the model parameters. The
output is the expected concentration profile as a function of time (e.g., concentration breakthrough
curves) at the compliance point down gradient from the source as well as the uncertainty around the
expected concentration resulting from the natural geologic heterogeneity in general and from the
spatially variable groundwater velocity in particular.
The computational issues and comparison with numerical solutions were obtained and summarized
in Hassan et. Al (2001). The results demonstrated the robustness and accuracy of the solute flux
approach presented with the semi-analytical solutions.
The solute flux method evaluates the movement of the solute from the source (e.g., bottom of the
landfill) to a plane perpendicular to the direction of mean flow. Aquifer heterogeneity is included and
represented by the variance of log-hydraulic conductivity, its variance and the hydraulic conductivity
integral scale. The combination of the spatial variability of aquifer properties and the uncertainty in the
estimates of these properties causes the solute flux to be a random function described by the
probability density function. The mean and variance of the solute flux are converted to the flux
averaged concentration by dividing with the groundwater flux, Q. This method is particularly well
suited for the risk assessment and risk analysis that should follow such modeling exercise. The first
two moments of the flux averaged concentration are important in determining the total risk level. The
larger the magnitude of variance in the flux averaged concentrations, the larger the maximum potential
risks.
This approach was used in analysis of the solute flux for old Jakusevec landfill site, near Zagreb,
Croatia.
4 REFERENCES
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