Tailing of the breakthrough curve in aquifer contaminant transport: equivalent longitudinal macrodispersivity and occurrence of anomalous transport
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1 5 GQ7: Securing Groundwater Quality in Urban and Industrial Environments (Proc. 6th International Groundwater Quality Conference held in Fremantle, Western Australia, 2 7 December 27). IAHS Publ. no. XXX, 28. Tailing of the breakthrough curve in aquifer contaminant transport: equivalent longitudinal macrodispersivity and occurrence of anomalous transport A. FIOI 1, G. DAGAN 2, V. CVETKOVIC 3 & I. JANKOVIC 4 1 Dept. of Civil Engineering, Università di oma Tre, ome, Italy aldo@uniroma3.it 2 Dept. of Fluid Mechanics and Heat Transfer, Tel Aviv University, amat Aviv, Israel 3 Dept. of Land and Water esources Engineering, oyal Institute of Technology, Stockholm, Sweden 4 Dept. of Civil, Structural and Environmental Engineering, SUNY, Buffalo, USA Abstract We analyze the mass arrival (breakthrough curve) at control planes at x of a plume of conservative solute injected at time t= in the plane x=. The formation is of random three-dimensional stationary and isotropic conductivity K, characterized by the univariate normal distribution f(y), Y=lnK, and the integral scale I. The flow is uniform in the mean, of velocity U, and longitudinal transport is quantified by f(τ,x), the probability density function (pdf) of travel time τ at x. We characterize transport by an equivalent longitudinal macrodispersivity α L (x), which is proportional to the variance of the travel time. If α L is constant, transport is coined as Fickian, while it is anomalous if α L increases indefinitely with x. If f(τ,x) is normal (for x I), transport is coined as Gaussian and the mean concentration satisfies an ADE with constant coefficients. For the subordinate structural model transport is anomalous, in spite of the closeness of the conductivity distribution to the lognormal one. To further analyze anomalous behaviour, a relationship is established between the shape of f(k) for K and the behaviour of α L, arriving at criteria for normal or anomalous transport. The model is used in order to compare results with the recent ones presented in the literature, which are based on the Continuous Time andom Walk (CTW) approach. It is found that a class of anomalous transport cases proposed by CTW methodology cannot be supported by a conductivity structure of finite integral scale. Key words groundwater hydrology; contaminant transport; stochastic processes; random media INTODUCTION Field measurements have revealed that longitudinal dispersivity α L values, identified from field tracer tests, are much larger than the laboratory ones and furthermore, they may show scale dependence. These effects are attributed to the spatial variability of the hydraulic conductivity K(x). The identified α L were much larger, by two orders of magnitude, than the laboratory values, and they were coined as macrodispersivities. In the preceding article (Dagan et al., this issue) we have analyzed the mass arrival Copyright 28 IAHS Press
2 Tailing of the breakthrough curve in aquifer contaminant transport, Part II 51 (breakthrough curve) at control planes at x I of a plume of conservative solute injected at time t= in the plane x=. The formation is of random three-dimensional stationary and isotropic conductivity K, characterized by the univariate normal distribution f(y), Y=lnK, and the integral scale I. The flow is uniform in the mean, of velocity U, and longitudinal transport is quantified by f(τ,x), the pdf of travel time τ at x. The main topic was the presence of tailing for large τ in highly heterogeneous aquifers, result of retention of solute particles by blocks of low K. This is of significance in aquifer pollution and remediation applications. The aim of the present article is to extend the above procedure to the analysis of the longitudinal macrodispersivity α L. Of particular interest is the macrodispersivity for non-gaussian distributions of Y and the occurrence of the anomalous behaviour. The relationship with the Continuous Time random Walk (CTW) approach (Berkowitz et al., 26) is also given, with particular emphasis on asymptotic behaviour for the type of random fields considered here. SEMI-ANALYTICAL DEIVATION OF THE MACODISPESIVITY Along the simplified, semi-analytical model described in Fiori et al. (26) and the companion paper (Dagan et al., this issue), the travel time variance is easily calculated starting from the residual travel time past an isolated sphere. The final expression for the travel time variance is the following (details are given in Fiori et al., 27) = x κ K M dκ κ = 2 + κ K ef 2 ( x) a τ ( κ ) f ( κ ) 2 σ τ where K ef is the effective conductivity (Fiori et al., 26) and τ M is the residual of the travel time past a sphere along the central streamline and is obtained by the formula U M ( κ ) τ 4 1 = 3κ c 2 + κ ( ) c = 2 κ ( ln( 1+ c ) + i 4 ln( 1+ i 4 / c 1 ) i 2 ln( 1 i 2 / c 1 ) where =4/3I is the radius of the spherical inclusions and i the imaginary unit. The geometrical coefficient a has the expressions a=9n/8.79 for spheres at densest packing and a=3/2=1.5 for cubes. The linearity of σ τ ²(x) with x stems from the requirement x>, i.e. departure of the control plane from the zone adjacent to the injection plane. Since τ M is given analytically, equation (1) allows determining σ τ ²(x) by a simple quadrature for any given conductivity distribution f(k). If the travel time variance is finite, transport is Fickian. However, it may become Gaussian only after a considerable distance x/, in particular for large logconductivity variance. Nevertheless, an equivalent longitudinal macrodispersivity can be defined by (1) (2)
3 52 A. Fiori et al U στ U a = = κ 2 α L τ M ( κ ) f ( κ ) dκ (3) 2x κ The behaviour of α L for a normal f(y) was analyzed in detail in our previous studies (Fiori et al., 26). ANOMALOUS TANSPOT AND ITS OCCUENCE Anomalous behaviour is defined as the one for which the macrodispersivity α L (3) is unbounded for x>i. If a particle trajectory is represented as a sum of independent space or time steps, precisely like equation (1) of Dagan et al. (this issue), there are two reasons for the occurrence of anomalous behaviour: (i) long range correlations and (ii) broad distributions (Bouchaud and Georges, 199). The first case was considered already in the hydrological literature (e.g. Glimm & Sharp, 1991; Dagan, 1994; ajaram & Gelhar, 1993; Di Federico & Neuman, 1998; Fiori, 21) in relation with the existence of log-conductivity values correlated over increasing scales. However, the limited size of the domain and/or of the solute plume may provide an upper bound to the spatial variability encountered by the plume, thus limiting the "anomalous" features of transport in such formations. Here we examine anomalous behaviour for isotropic aquifers of a 3D structure and therefore concentrate on the second mechanism, of "broad distributions" or "fat tails", while the integral scale I is finite. The basic question we pose is: what type of conductivity pdf, which is controlling the transport mechanism, leads to anomalous behaviour? This question can be answered in a straightforward manner from expression (3) for α L. For the log-normal f(k), α L is finite and transport is Fickian, the reason being the quick drop of f(k) with K, for K. Indeed, it is the tail of τ, caused by the long residence times in blocks of low K, that may render the integral divergent. With an f K K for K and taking into account that assumed behaviour ( ) τ MU 4 3κ for K we find out that α L K 1 f 1 ( K ) dk K dk K Hence, anomalous behaviour occurs if <. Furthermore, is bounded from below, >-1, since otherwise the basic requirement of a pdf ( K ) dk = 1 (4) (5) f cannot be satisfied. Summarizing, we have arrived at the conclusion that for isotropic formations of finite integral scale I, anomalous transport can occur if the conductivity pdf behaves like K for K, with -1<<, while transport is normal for >. In a similar manner, we can calculate the tail of f(τ ) for large τ, which is responsible for possible anomalous behaviour. Details of the calculation are given in Fiori et al. (27), the final result for a power-law f(k) being
4 Tailing of the breakthrough curve in aquifer contaminant transport, Part II / ( 3τ ) ( 3+ ) f ( τ ) τ K dk = τ (6) According to the classic developments of the statistical physics literature, a powerlaw distribution of the kind of (6) leads to anomalous transport when < (see, e.g., Bouchaud & Georges, 199, Sect ); for the same reasons outlined in the previous Section, is bounded from below. Hence, transport is anomalous when -1<<, the condition being identical with the criterion based on unboundedness of α L given above. ELATION WITH THE CONTINUOUS-TIME ANDOM WALK (CTW) A modelling approach which is gaining popularity in the hydrology literature is the Continuous Time andom Walk (CTW) (Berkowitz et al., 26). We shall note here a few interesting implications of the obtained results specifically for the CTW approach. In the present work, we arrived at a transport model similar to random walk, for a spatially variable K field with a finite integral scale, described by a non-gaussian distribution, and studied asymptotic properties of transport. In the CTW framework, transport is also modeled as a random walk on a lattice where a particle waits for a time t on each step before performing the next jump l (Montroll & Weiss, 1965; Montroll & Scher, 1973). The waiting time and the length of the jump are random variables characterized by a joint pdf ψ(t,l); as a rule the random variables are assumed "separable". i.e., independent, leading to ψ(t,l)=ψ(t)p(l). A number of quantities which characterize transport, including the moments and the distribution of the total displacements, can be calculated exactly using generating function methods (see, e.g., Haus & Kehr, 1987). The function ψ(t) lies at the heart of the CTW formulation and its structure is crucial for transport predictions. Our density f(τ ) is directly related to ψ(t) by noting that the actual time for each hop is T+ τ where T is the mean waiting time for the hops. Thus we have () t = f ( t T ) Ψ (7) with f(t-t)= f(τ ) for τ =t-t. As noted earlier, we are here focused on conductivity fields with an integrable density, i.e. f ( K ) dk = 1. Physically, we are saying that there is fluid flow in the entire domain, irrespective of how low the velocity is at any given point. This integrability condition implies that we are considering the so-called "non-defective" random walks. In view of our simplifying assumption that all particles take the same number of steps to arrive at x, the first-passage time density of a particle starting at x= can be expressed by the following formula (Cvetkovic & Haggerty, 22) N ( s, t) = Ψ ˆ fˆ (8) where the hat denotes Laplace transform. It is seen that the above expression is
5 54 A. Fiori et al. formally similar to equation (2) of Dagan et al. (this issue). Thus, the present approach is formally similar to CTW, provided that the keyfunction ψ is given by f(τ ), which in turn is related to the conductivity distribution f(k). Thus, the major conclusions drawn in the previous Sections can be transferred to the CTW, and the results for asymptotic transport are discussed in the sequel. ASYMPTOTIC TANSPOT In the current literature on CTW, the extended tailing of ψ is very often quantified as a power law (Berkowitz et al., 26) 1 β () t Ψ t (9) The parameter β is calibrated against experimental results on a given scale. The range of inferred β values is quite broad, between.3 and 1.8 (Berkowitz et al., 26). We compare the results of the previous section with the distribution (9) and check what kind of conductivity distribution f(k) is able to support the model (9). Comparison between (9) and (6) leads to the identity =β-2, and the asymptotic 2 structure f ( K ) K β for K. Comparing this result with the analysis of the previous two Sections, it is seen that transport is anomalous when 1<β<2, while values of the parameter β<1 are impossible because of the basic requirement of the "nondefective random walk". Hence, it appears that the values β<1 discussed in the literature either on generic grounds or based on calibration against experimental transport data, cannot be related to a broad distribution of K with a finite integral scale. If field tracer tests nevertheless indicate β<1 then a physical interpretation presumably requires including other effects, for instance, mass transfer by diffusion or nonergodic behaviour related to the finite size of the plume. Such effects, which are not discussed here, would require introduction of additional parameters, e.g. the ratio between the initial plume size and the integral scale to account for finite size. It is doubtful that these additional factors can be encapsulated by a unique value of the parameter β which applies to the entire history of the plume. Indeed, the variation of β with distance has been revealed in an empirical manner by Trefry et al. (23). CONCLUSIONS On the basis of our simplified transport model (Fiori et al., 26; Dagan et al., this issue) we determined that transport is anomalous when the conductivity distribution f ( K ) K for K, with -1<<. Our basic assumption is that f ( K ) dk = 1, i.e., there is flow at all points no matter how small is the velocity ("non-defective" random walk). We found that the asymptotic tailing of the hopping time density f(τ ) for a ( 3+ ) "non-defective" random walk is power law f ( τ ) τ (τ ). Thus, a clear connection is established between the nature of transport and the underlying conductivity distribution of the heterogeneous formation.
6 Tailing of the breakthrough curve in aquifer contaminant transport, Part II 55 The present approach is shown to be similar to the Continuous Time andom Walk (CTW), the key function ψ of CTW being equivalent to f(τ ). With the 1 β assumption Ψ t (Berkowitz et al., 26), the identity =β-2 holds and according to our findings transport is anomalous when 1<β<2. Hence, it appears that the values β<1 sometimes attributed to experimental data cannot be related to a broad distribution of K with a finite integral scale. If field tracer tests nevertheless indicate β<1 then a physical interpretation presumably requires including additional effects, e.g. mass transfer by diffusion or non-ergodic behaviour due to finite size of the plume. EFEENCES Berkowitz, B., Cortis, A., Dentz, M. & Scher, H. (26) Modeling non-fickian transport in geological formations as a continuous time random walk. ev. Geophys. 44, G23, doi:1.129/25g178. Bouchaud, J. P. & Georges, A. (199) Anomalous diffusion in disordered media-statistical mechanisms, models and physical applications. Phys. ep. 195 (4-5), Cvetkovic, V. D. & Haggerty,. (22), Transport with exchange in disordered media. Phys. ev. E 65,5138, 1-9. Dagan, G. (1994) The significance of heterogeneity of evolving scales to transport in porous formations. Water esour. es. 3(12), Dagan, G., Fiori, A., Jankovic, I. & Cvetkovic, V. (27) Tailing of the breakthrough curve in aquifer contaminant transport: The impact of permeability spatial variability. this issue. Di Federico, V. & Neuman, S. P. (1998) Transport in multiscale log conductivity fields with truncated power variograms. Water esour. es. 34 (5), Fiori, A. (21) On the influence of local dispersion in solute transport through formations with evolving scales of heterogeneity. Water esour. es. 37(2), Fiori, A., Jankovic, I. & Dagan, G. (26) Modeling flow and transport in highly heterogeneous three-dimensional aquifers: Ergodicity, Gaussianity and anomalous behaviour-2. Approximate semianalytical solutions. Water esour. es. 42, Art. No. WO6D13. Fiori, A., Jankovic, I., Dagan, G. & Cvetkovic, V. (27) Ergodic transport through aquifers of non-gaussian log conductivity distribution and occurrence of anomalous behaviour, Water esour. es. 43, W947, doi:1.129/27w5976. Glimm, J. & Sharp, D. H. (1991) A random field model for anomalous diffusion in heterogeneous porous media. J. Stat. Phys. 62, Haus, J. W. & Kehr, K. W. (1987) Diffusion in regular and disordered lattices. Phys. ep. 15, 263. Montroll, E. W. & Weiss, G. H. (1965) andom walk on lattices. J. Math. Phys. 6 (2), 167. Montroll, E. W. & Scher, H. (1973) andom walks on lattices. IV. Continuous time random walks and influence of absorbing boundaries. J. Stat. Phys. 9(2), ajaram, H. & Gelhar, L. W. (1993) Plume scale-dependent dispersion in heterogeneous aquifers, 2, Eulerian analysis and three-dimensional aquifers. Water esour. es. 29, Trefry, M. G., uan, F. P. & McLaughlin, D. (23) Numerical simulations of preasymptotic transport in heterogeneous porous media: Departures from the Gaussian limit. Water esour. es. 39(3), Art. no 163.
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