Contaminant Isolation By Cutoff Walls: Reconsideration Of Mass Fluxes

Transcription

1 GeoHalifax29/GéoHalifax29 Contaminant Isolation By Cutoff Walls: Reconsideration Of Mass Fluxes Christopher. Neville S.S. Papadopulos & ssociates, Inc., Waterloo, Ontario BSTRCT Cutoff alls are used frequently to isolate contaminants at both controlled and uncontrolled hazardous aste sites. Neville and ndres (26) presented a containment criterion for contaminant isolation by a cutoff all. Their analysis yields the arcy flux required to achieve containment, based on the condition that long-term advective and diffusive mass fluxes across the all are balanced. In this paper e sho that the condition of zero net mass flux represents only one particular case. Straightforard expressions for the long-term mass fluxes across a cutoff all can also be developed from the same theory for any arcy flux. The expressions for the long-term mass fluxes may be used to estimate the mass flux to the environment in cases here it is difficult to satisfy the criterion of zero net mass flux. RÉSUMÉ es barrières souterraines imperméables sont souvent utilisées pour confiner les sources de contamination présentes sur les terrains contaminés. Neville et ndres (26) ont présenté, pour de telles barrières, un critère de confinement des sources de contamination. Le résultat de leur analyse indique quel est le flux de arcy permettant d atteindre le confinement, en se basant sur l équilibre, à long-terme, entre les flux de masse advectifs et dispersifs à travers la barrière. Cet article démontre que la condition d équilibre, où le flux de masse net de contamination traversant la barrière est nul, représente un cas particulier. es simples expressions mathématiques décrivant les fluxes de masse à travers une barrière peuvent êtres obtenues pour toutes les valeurs de flux de arcy. es expressions quantifiant les débits massiques à long-terme peuvent être utilisée pour estimer la charge de contamination relâchée dans l environnement souterrain, dans les cas où il est difficile d obtenir un parfait confinement. 1 INTROUCTION Cutoff alls are used frequently to isolate contaminants at both controlled and uncontrolled hazardous aste sites. Neville and ndres (26) presented a containment criterion for contaminant isolation by a cutoff all. Their containment criterion builds on the analysis of evlin and Parker (1996) and is developed from the assumption that containment is achieved hen the long-term advective and diffusive mass fluxes across the all are balanced so that the net mass flux is zero. The conceptual model of Neville and ndres (26) is shon schematically in Figure 1. Their analysis may be used to estimate the arcy flux required to achieve containment according to this criterion. In this paper, e sho that the assumption of zero net mass flux represents only one particular case. The same theory may be used to develop a straightforard expression for the long-term net mass flux across a cutoff all for any arcy flux. This expression may be used to estimate the net mass flux to the environment in cases here it may not be possible to satisfy the criterion of zero net mass flux. Figure 1. Conceptual model of source isolation 875

2 GeoHalifax29/GéoHalifax29 2 NEVILLE N NREWS (26) CONTINMENT CRITERION The analysis of Neville and ndres (26) assumes a constant concentration ithin the source zone and steady one-dimensional flo and dispersive diffusive transport across the cutoff all. In Figure 1, c and c denote the steady concentrations along the inside and outside faces of the all, respectively, and is the thickness of the all. The terms, diff, and mech represent the advective, diffusive, and mechanical-dispersive mass fluxes, respectively. The advective mass flux,, is thus defined: = qc [1] here c is the concentration and q is the arcy flux (positive outards from the source zone). The steadystate dispersive diffusive mass flux,, is given by: dc = θ [2] dx Given a knon concentration of the source and a target concentration of outside of the cutoff all, the Neville- ndres containment criterion yields the arcy flux that is required to achieve a long-term net mass flux of zero: θ c q = ln c 3 GENERL INTERPRETTION OF THE NEVILLE N NREWS (26) NLYSIS The analysis of Neville and ndres (26) accommodates groundater seepage in either direction, and any combination of specified concentrations over both faces of the cutoff all. Hoever, Equation 5 represents only one specific case, in hich the dispersive diffusive and advective mass fluxes are balanced. It is important to note that steadystate conditions may be attained for any arcy flux. To demonstrate this point, e consider the more general case of transient conditions. The governing equation for transient transport is: [5] here θ is the effective porosity and is the dispersion coefficient. The diffusive flux in Equation 2 is a lumped representation of the mechanical-dispersive and diffusive mass fluxes. Both process are assumed to be Fickian processes and additive; therefore, the dispersion coefficient can be interpreted as the sum of the mechanical-dispersive and diffusive fluxes (Bear, 1972): θ = q + θ t x x 2 c c c The initial and boundary conditions are: 2, x [6] c ( x, ) = [7a] α v * = L + [3] c(, t) c = [7b] here α L is the longitudinal dispersivity, v is the groundater velocity, and * is the effective diffusion coefficient. The mechanical-dispersive flux accounts for variations in the groundater velocity across the all that are beneath the scale of resolution of the analysis. For a properly constructed all, this flux should be negligible. Neville and ndres (26) derived a containment criterion for any given all design (all thickness, diffusion coefficient, and arcy flux across the all). Folloing the approach of evlin and Parker (1996), they assumed that containment is achieved hen the net mass flux on the outside face of the all is zero: c(, t) c = [7c] + = [4] 876

3 GeoHalifax29/GéoHalifax29 The general solution for transient conditions is derived by generalizing the derivation of l-niami and Rushton (1977): qx sinh 2θ (, ) = q sinh c x t c n= 1 2θ q( x ) { } 2 2π q x q t + c 2 2 2θ 4θ 1 n n π t nπ x sin n θ 2 2 n π q θ The solution for the steady-state concentration profile can be derived directly from Equation 8 and is given in Neville and ndres (26) as: [8] ifferentiating the general steady-state solution yields: qx dc ( x, ) qv θ = c dx θ q qx q θ c θ q [12] Evaluating the derivative at x = and simplifying yields: q dc(, ) q θ ( c c [13] = ) dx θ q (, ) = ( ) c x c c c qx θ q θ 4 STEY-STTE NET MSS FLUX [9] The expression for the diffusive flux is obtained by substituting Equation 13 into 11: = q c c q θ q [14] The Neville and ndre (26) containment analysis is generalized by deriving expressions for the advective and dispersive diffusive mass fluxes for any magnitude of the arcy flux across the all. The advective mass flux across the outside face of the all is: The net mass flux is defined as the sum of the advective and diffusive mass fluxes: net = + [15] = qc = qc [1] x= The steady-state dispersive diffusive mass flux across the outside face of the all is defined as: dc = θ [11] dx x= Substituting Equations 1 and 14 into Equation [15] yields the expression for the steady-state net mass flux across the outside face of the cutoff all: = qc q c c net q θ q [16] 877

4 GeoHalifax29/GéoHalifax29 5 EXMPLE CLCULTIONS To illustrate the characteristics of the analysis, transient and steady-state concentration profiles across the all are calculated for to cases. The parameters for the calculations are listed in Table 1. Table 1. Parameters for example calculations Parameter Value * m 2 /d α L. m θ.3 1. m c 1. c.1 Case 1: m/d (inard flo) q Case 2: m/d (outard flo) The results for Case 1 are shon in Figure 2. s shon in Figure 2, groundater flo directed inards across the all gives rise to a steady-state concentration profile that is concave-inards. s shon in Figure 3, groundater flo that is directed outards across the all gives rise to a steady-state concentration profile that is concave-outards. The length of time required for steady concentrations to evolve differs for the to cases, but a steady profile is eventually attained in both cases. For Case 1, the inards flo opposes the concentration gradient. The steady-state net mass flux is given by: net ( 6 1 / )(.1) ( 6 1 / )(( 1.) (.1) ) = m d m d ( 6 1 m / d )( 1. m) 2 (.3)( m / d ) ( 6 1 m / d )( 1. m) 2 (.3)( m / d ) = + = [17] In this case, the net mass flux is negative, indicating that the inard advective mass flux exceeds the outard diffusive mass flux. The results for Case 2 are shon in Figure 3 In this case, the net mass flux is directed inards across the all. For Case 2, the outard flo is in the same direction as the concentration gradient. The steady-state net mass flux is given by: net ( 6 1 / )(.1) ( 6 1 / )(( 1.) (.1) ) = m d m d ( 6 1 m / d )( 1. m) 2 (.3)( m / d ) ( 6 1 m / d )( 1. m) 2 (.3)( m / d ) 7 = + = + [18] In the second case, diffusion and advection both direct solute toards the outside of the all. The solution for Case 2 does not predict that the entire all attains a concentration of the source c. Rather, solute is dran off continuously at x = to maintain the concentration at the fixed level c. s the outards advective flux increases, the concentration across the all becomes nearly uniform at c and declines abruptly to c at the boundary. The net mass flux for a range of arcy fluxes is plotted in Figure 4. The condition for zero net mass flux is also indicated. These results reinforce the point made previously; the condition of zero net mass flux is only one state along a continuous spectrum of possibilities. If the aquifer outside of the all is extensive, it is reasonable to assume significant dilution of the solute that migrates out of the all. The orst case ith respect to the predicted outard mass flux ill arise hen the outside concentration c is assumed to be zero. In this case, there is no advective flux to counteract the outard diffusion of solute and the expression for the net mass flux reduces to: net q θ = qc q [19] Results ith Equation 2 for a range of values of c are plotted in Figure 5. For reference purposes, results are included from the previous example c /c =.1. To general observations can be made regarding the results. When the arcy flux is directed inards (negative), the net mass flux is only eakly sensitive to the outside concentration. When the arcy flux is directed outards, the net mass flux is essentially independent of the outside concentration. The assumption of an outside concentration c of zero provides an appropriate preliminary estimate of the net mass flux. The solution given by Equation 15 is straightforard in interpretation and implementation, and yields conservative estimates of the net mass flux. 878

5 GeoHalifax29/GéoHalifax Relative concentration, c/c , d.1 t = 1, d Steady-state (dashed) istance across all, x (m) Figure 2. Concentrations for Case 1, inards flo 1.9 Steady-state.8 5, d Relative concentration, c/c t = 1, d 1, d istance across all, x (m) Figure 3. Concentration for Case 2, outards flo 879

6 GeoHalifax29/GéoHalifax29.15 Net mass flux net at x =.1.5 Zero net mass flux at q = m/s arcy flux across all, q (m/d) Figure 4. Net mass flux outside of the all.15 Net mass flux net at x = c /c =.1 c /c =.5 c /c =.1 c /c = arcy flux across all, q (m/d) Figure 5. Net mass flux outside of the all: Effect of c 88

7 GeoHalifax29/GéoHalifax29 6 CONCLUSIONS The containment criterion of Neville and ndres (26) is based on the condition that the net mass flux across the cutoff all is zero. steady-state concentration profile ill develop across the all for any magnitude and direction of the arcy flux, and the condition of zero net mass flux may be unnecessarily restrictive to achieve the containment objectives at a particular site. This supplement to the analysis of Neville and ndres (26) presents an expression for the steadystate net mass flux across a cutoff all for any value of the arcy flux. The results of example calculations reveal that hen the arcy flux is directed inards toards the source zone, the net mass flux is eakly dependent upon the concentration on the outside face of the all. When the arcy flux is directed outards from the source zone, the net mass flux is essentially independent of the outside concentration. The assumption of an outside concentration equal to zero yields a simple expression for the net mass flux that is straightforard to interpret and evaluate, yielding a conservative estimate of the net mass flux across the all. 7 REFERENCES l-niami,.n.s. and Rushton, K.R nalysis of Flo gainst ispersion in Porous Media, ournal of Hydrology, 33: Bear, ynamics of Fluids in Porous Media, merican Elsevier Publishing Company, Inc. Ne York, NY, US. evlin,.f., and Parker, B.L Optimum Hydraulic Conductivity to Limit Contaminant Flux through Cutoff Walls, Ground Water, 34(4): Neville, C. and ndres, C.B. 26. Containment Criterion for Contaminant Isolation by Cutoff Walls, Ground Water, 44(5):