ࡱ> .0-|Y .bjbjWW "H=="](((((,T$( *******lnnnnnn$  u********x**l((*lll*x l^,(( lUse of solute flux approach to contaminant transport in aquifers Utilisation du flux du polluant lors du transport de pollution dans les nappes aquifres Mensur Mulabdi Civil Engineering Faculty, University in Osijek, Croatia, leta@zg.hinet.hr Roko Andri evi 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 three-dimensional 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  EMBED Equation.3  (n stands for effective porosity, and velocity is defined through hydraulic head ( as  EMBED Equation.3 ). 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 Andri evi & Cvetkovi 1998) The travel time ( and components of the transverse vector can be evaluated as:  EMBED Equation.3  (1)  EMBED Equation.3  (2) The mean travel time and mean transverse displacement for the entire plume crossing the CP is defined by (3) and (4),  EMBED Equation.3  (3)  EMBED Equation.3  (4) At distance x from the point of solute injection the solute mass flux component orthogonal to CP is calculated as  EMBED Equation.3  (5) where  EMBED Equation.3  (6) 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  EMBED Equation.3 , 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 Ahmed E. Hassan, R. Andricevic and V. Cvetkovic, 2001. Computational issues in the detrmination of solute discharge moments and implications for comparison to analytical solutions, Advances in Water Resources, Vol. 24, pp. 607-619. Andri evi, R. 1996 Evaluation of sampling in the subsurface. Water Resour. Res. 32, 863-874. Andri evi, R. & Cvetkovi, V. 1996. Evaluation of risk from contaminants migrating by groundwater. Water Resour. Res. 32, 611-621. Andri evi, R. & Cvetkovi, V. 1998 Relative dispersion for solute flux in aquifers, J. Fluid Mech., Vol. 361, pp. 145-174. Andri evi, R. 1998 Effects of local dispersion and sampling volume on the evolution of concentration fluctuations in aquifers, Water Resour. Res. 34, 1115-1129. Cvetkovi, V. Dagan, G. & Shapiro, A. 1992. A solute flux approach to transport in heterogeneous formations, 2, Uncertainty analysis, Water Resour. Res. 28, 189-215. Cvetkovi, V. & Dagan, G. 1996 Reactive transport and immiscible flow in geological media, 1. Applications, Proc. R.Soc.Lond. A 452, 303-328. Dagan, G. & Cvetkovi, V. 1996 Reactive transport and immiscible flow in geological media, 1. General theory, Proc. R.Soc. Lond. A 452, 285-301. Purvance, D.T. & Andri evi, R. 2000 Geoelectric characterisation of the hydraulic conductivity field and its spatial structure at variable scales, Water Resour. Res. 36, 2915-2924. Rubin, Y. 1990 Stochastic modelling of macrodispersion in heterogeneous porous media. Water Resour. 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