Ergodic transport through aquifers of non-gaussian log conductivity distribution and occurrence of anomalous behavior

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1 WATER RESOURCES RESEARCH, VOL. 43,, doi:1.129/27wr5976, 27 Ergodic transport through aquifers of non-gaussian log conductivity distribution and occurrence of anomalous behavior Aldo Fiori, 1 Igor Janković, 2 Gedeon Dagan, 3 and Vladimir Cvetković 4 Received 16 February 27; revised 27 April 27; accepted 7 May 27; published 12 September 27. [1] Three-dimensional advective transport of passive solutes through isotropic porous formations of stationary non-gaussian log conductivity distributions is investigated by using an approximate semianalytical model, which is compared with accurate numerical simulations. The study is a continuation of our previous works in which formation heterogeneity is modeled using spherical nonoverlapping inclusions and an approximate analytical model was developed. Flow is solved for average uniform velocity, and transport of an ergodic plume is quantified by mass flux (traveltime distribution) at a control plane. The analytical model uses a self-consistent argument, and it is based on the solution for an isolated inclusion submerged in homogeneous background matrix of effective conductivity. As demonstrated in the past, this analytical model accurately predicted the entire distributions of traveltimes in formations of Gaussian log conductivity distributions, as validated by numerical simulations. The present study (1) extends the results to formations of non-gaussian log conductivity structures (the subordination model), (2) extends the approximate analytical model to cubical blocks that tessellate the entire domain, (3) identifies a condition in conductivity distribution, at the tail of low values, that renders transport anomalous with macrodispersivity growing without bounds, and (4) provides links of our work to continuous time random walk (CTRW) methodology, as applied to subsurface transport. It is found that a class of CTRW solutions proposed in the past cannot be based on solution of flow in formations with conductivity distribution of finite integral scale. Citation: Fiori, A., I. Janković, G. Dagan, and V. Cvetković (27), Ergodic transport through aquifers of non-gaussian log conductivity distribution and occurrence of anomalous behavior, Water Resour. Res., 43,, doi:1.129/27wr Introduction [2] Field measurements have revealed that longitudinal dispersivity a 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). A few controlled and elaborate field experiments for weakly heterogeneous aquifers [Leblanc et al., 1991; Mackay et al., 1986], in which both concentrations and conductivities were monitored by a large number of wells, showed that while local concentration varied widely in space, the spatial moments approximately reflect a Gaussian and Fickian behavior of the distributed longitudinal mass. However, the identified a L were much larger, by 2 orders of magnitude, than the laboratory values, and they were coined as macrodispersivities. A transport field experiment in a 1 Dipartimento di Scienza dell Ingegneria Civile, Universitá di Roma Tre, Rome, Italy. 2 Department of Civil, Structural and Environmental Engineering, State University of New York at Buffalo, Buffalo, New York, USA. 3 Faculty of Engineering, Tel Aviv University, Tel Aviv, Israel. 4 Department of Land and Water Resources Engineering, Royal Institute of Technology, Stockholm, Sweden. Copyright 27 by the American Geophysical Union /7/27WR5976 strongly heterogeneous aquifer [Boggs et al., 1992] displayed a skewed spatial distribution, which is poorly characterized by any a L. [3] A vast body of literature of the last three decades was devoted to modeling field-scale dispersion (e.g., the monographs by Dagan [1989], Gelhar [1993], Zhang [22], and Rubin [23]). In the common stochastic approach the conductivity K or the log conductivity Y =lnk, are regarded as random space functions, and the same is true for the dependent variables V (fluid velocity) and the concentration C. We follow this approach here and among the many possible configurations and types of solute behavior, we restrict the analysis to conservative solutes, to three-dimensional formations of stationary and isotropic Y structure (finite variance s Y 2 and integral scale I), to steady flow of mean uniform velocity hvi = U = const and to longitudinal advective spread. [4] A thin plume of total mass M is injected along the plane x = at time t = and it is subsequently advected through the heterogeneous formation. The plume spread is quantified by m(t, x), the relative solute flux through a control plane at x (the breakthrough curve). Denoting by M(t, x) the mass of solute which has moved past x at t, m is defined ). [5] The plume is assumed to be very large, namely of initial transverse extent much larger that the heterogeneity 1of13

2 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS scale I, such that the variables of interest can be assumed to be ergodic, e.g., hmi m. Although this is a global characterization of the plume, it may be of practical interest as it represents the mass intercepted by a battery of wells or by a water body in which the aquifer is discharging. Under these ergodic conditions we also have m (t, x)=f(t, x), where f(t, x) is the pdf of traveltime t of a solute particle injected at x = at t = [Dagan, 1989]. Hence transport is characterized here by f(t, x) and the associated temporal moments, and in particular the variance s 2 t. Transport is coined as Fickian if s 2 t grows linearly with x and as Gaussian if the traveltime distribution is inverse Gaussian for sufficiently large x relative to the integral scale. Transport is defined as anomalous if s 2 t /x is unbounded. [6] The present study focuses on advective transport in hydraulic conductivity fields with a realistic broad distribution, by solving first for the flow problem. The article is a direct continuation of our recent ones [Dagan et al., 23; Fiori et al., 23, Janković etal., 23b] which dealt with transport in highly heterogeneous formations, whose structure was represented by a dense ensemble of spherical inclusions of lognormal conductivity distribution. By using highly accurate numerical simulations and a semianalytical approximation, the issues of ergodicity, Fickianity and Gaussianity were examined subsequently [Janković etal., 26; Fiori et al., 26]. [7] The aim of the present article is to extend the analysis to cover the following issues: (1) application of the semianalytical approach to a cubical blocks structure; the cubical elements are closer to the common finite difference or finite elements blocks; (2) examination of the impact of a log conductivity distribution different from the Gaussian one (the one selected here is the subordination model); (3) occurrence of the anomalous behavior of transport as related to general non-gaussian, but stationary, log conductivity distributions, and (4) discussion on the relationship with the CTRW approach [Berkowitz et al., 26] with particular emphasis on asymptotic behavior for the type of random fields considered here. The plan of the paper follows these topics. 2. Review of Conductivity Structure Model and Semianalytical Approximation of Traveltime Distribution [8] The random log conductivity stationary field is of mean hyi =lnk G (the geometric mean), variance s 2 Y and isotropic two-point covariance C Y (x 1, x 2 )=s 2 Y r(x 1 x 2 ). The R autocorrelation r has a finite linear integral scale I = 1 r(x,,)dx, which is assumed to be much smaller than the characteristic length scale of the flow domain and of the plume. [9] We consider the representation of Y which is often adopted in the numerical models, i.e., a fine scale one in which the aquifer is divided into voxels, i.e., small cubes of side Dx I (Figure 1a). A numerical solution (or an analytical one at the limit Dx/I! ) of the flow and transport equations is regarded as exact. The complete statistical characterization of the medium, which is also the information needed to generate the Y field numerically, comprises the joint multipoint pdf (probability density function) of Y j ( j = 1,.., P), the values of Y(x) at the centers of an arbitrary number of voxels P. Generally, data are not detailed enough to provide such an exhaustive characterization and some simplifying assumptions are usually adopted. Thus the univariate f(y) was found to be approximately normal for many aquifers [e.g., Freeze, 1975] and encapsulated by the mean hyi and the variance s Y 2. Identification of the spatial correlation was limited to the twopoint r, which for a given analytical form (exponential, Gaussian) depends only on the integral scale I. In absence of more detailed information, it is common to model Y as multi-gaussian, i.e., the joint multipoint pdf is multivariate normal, and then the structure is completely characterized by hyi, s Y 2 and I (for a given r). The validity of this model was subject of debate and indeed one of the aims of the present study is to assess the impact of departure from multi-gaussianity. [1] Since modeling the medium by the fine-scale structure (Figure 1a) is not practical from a computational point of view, nor is the input information generally available, we simplify the model as follows: [11] 1. The voxels (Figure 1a) are replaced by large cubes of side 2R (Figure 1b) or spheres of uniform radius R (Figure 1c), of independent conductivities K. The spheres are assumed to be packed in the densest touching configuration of face-centered cubic (FCC) lattice, such that their volume density is n.7. The rest of the volume between them is supposed to be made up from small spheres of same K distribution, which we replace by a homogeneous matrix of effective conductivity (the matrix is represented by a nugget of the covariance C Y, and it has a negligible impact on dispersion since it is of vanishing integral scale). [12] This leads to considerable simplifications, since the structure is completely defined now by R and by the univariate pdf f(k) or f(y). It is easy to determine the two point correlation and integral scales for these macroscopically isotropic structures. Thus, for cubes r(r x,,)=1 (r x /2R)forr x <2R and r = for r x >2R, where r x = jx 1 x 2 j is the lag in the x direction and similarly for y, z; hence I = R. For the spheres of radius R, r =1 (3r/4R) +r 3 /(16R 3 ) for r <2R, where r = jx 1 x 2 j and correspondingly I =3R/4 [Dagan, 1989]. While in principle it is possible to determine the expression of any other multipoint correlation, the task becomes difficult [Binglin and Torquato, 199]. It is worth to mention that if the cubical or spherical blocks are divided into classes of Y intervals of equal relative volume, the integral scale of the indicator variogram of each class is the same. [13] While these simplified structures are idealized, they can model any isotropic random medium of given univariate f(y) and integral scale I up to the second-order statistics [Dagan et al., 23]. Hence, in absence of additional data they are as legitimate as any other model, e.g., the multi- Gaussian one. The model definitely serves our purpose of gaining understanding of the flow and transport processes. We have run accurate numerical simulations in the past for the spherical (or circular in two dimensions) inclusions model and we plan to extend the numerical approach to the cubical lattice in the future. Still, such simulations are computationally demanding and the model is further simplified as follows. [14] 2. Accurate flow and transport statistical moments are determined by solving separately for each block, sub- 2of13

3 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS Figure 1. (a) Heterogeneous aquifer structure, (b) conceptual model of cubical, and (c) spherical blocks of size 2R; solution (streamlines and lines of constant head in a plane that passes though the center) for (d) isolated cube and (e) sphere of conductivity K submerged in background of conductivity K ef. merged in a homogeneous matrix of conductivity equal to the effective one, K = K ef (Figures 1d and 1e) and adding up subsequently the perturbation fields of all inclusions. The intuitive justification of this approximation [Dagan, 1981] is that each block averages the effect of the neighboring inclusions and a self-consistent approach can be used to represent this effect. It has led to an exact result for K ef of circles [Fiori et al., 25] and a very accurate one for spheres of lognormal f(k) [Janković et al., 23a; Fiori et al., 26] as well as for transport in the latter case. We shall show in the sequel that this is true for a non-gaussian f(y)as well. Furthermore, these studies have cast light on the reasons for the success of this approximation. Indeed, the total velocity field in the heterogeneous medium can be represented as the sum of the perturbation velocities associated with each block. In turn, the latter can be expanded in an infinite series and the leading term of the velocity field within each inclusion of Figures 1b and 1c is the mean interior flux (constant velocity for circles or spheres), while the exterior one is of a doublet. In contrast, the following higher harmonic polynomials inside and multipoles outside lead to a net zero flux. Furthermore, the leading term is averaging the impact of neighboring blocks and is captured accurately by approximating the surrounding inclusions by an effective medium, as demonstrated by numerical simulations [see Janković etal., 23a, Figure 5]. The variability of the conductivity of the neighboring blocks influences mainly the higher-order terms of the velocity field. It turns out that transport is also dominated by the leading term of the disturbance velocity. [15] It is worthwhile to mention that this scheme leads exactly to the first-order approximation in s Y 2, applying to weak heterogeneity, as it was demonstrated for spherical inclusions [Dagan et al., 23]. [16] Along these lines and by following the approach of Fiori et al. [26], the traveltime of a particle moving from x =tox is given by t ¼ T þ Xj¼N t Rj j¼1 ð1þ 3of13

4 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS where T = x/u is the mean and the residuals t Rj are independent random variables pertaining to the N > 1 blocks encountered by the particle. In general, N nx/h i where n is the volume fraction and h i is the average length of a segment of the streamline intercepting a block. The expressions of N are given in Appendix A for both spherical and cubical inclusions at densest packing. [17] The time step residual t R associated with a block (Figures 1d and 1e) is a random variable which depends on the random conductivity K. Itspdff(t R ) was determined by Fiori et al. [26] by a semianalytical approximation for spheres. For the sake of completeness and easiness of reference the derivation is briefly recapitulated in Appendix A, where it is extended to cubical blocks as well. The final result for f(t R ), in terms of f(k) is given in an analytical form by equations (A1) and (A11). [18] In our previous work [Fiori et al., 26] we have effectively determined f(t R ) pertaining to a normal f(y) (its behavior is illustrated in Figure 4 in section 3, where it is compared with the case of non-gaussian log conductivity distributions. The main finding of Fiori et al. [26] was that f(t R ) is highly skewed, with a thin and prolonged tail for large t R. The tail is associated with blocks of low conductivity, for which t R U/I 4/(3k)!1for K!, k = K/K ef!. The solute mass pertaining to the tail is a very small portion of the total mass because low K inclusions have a thin capture zone, on one hand, and their volume fraction is very small, depending on f(k), on the other. [19] The traveltime t is defined by equation (1) as a sum of N independent time steps. Hence its pdf is given by a convolution as follows: f ðt; xþ ¼ FT 1 e ftr N ; e ftr ¼ FT ½f tr ðtþš ð2þ where FT stands for Fourier transform and FT 1 for its inverse. Since f(t R ) is given in an analytical form by (A1) and (A11), in terms of the pdf of K, f(t, x) can be determined by using fast Fourier Transform (FFT) for any given conductivity distribution. This was done precisely for the lognormal case [Fiori et al., 26] and will be generalized for the subordination model in section 3. A few breakthrough curves f(t, x) for normal Y, for an ensemble of spheres at highest density packing n =.7 and for s Y 2 = 2 are shown for illustration in Figure 4 in section 3. The cumulative effect of the tailing of f(t R ) is clearly displayed. [2] Finally, the traveltime variance, is found in Appendix A to have the simple expression s 2 t ðþ¼n x s2 t R ¼ s 2 nx t R h i k x ¼ a ½ 2 þ k t M ðkþš 2 f ðkþdk R ; k ¼ K K ef where t M is the residual of the traveltime past a sphere along the central streamline and is given in an analytical form in equations (A3) (A5). The geometrical coefficient a has the expressions a =9n/8.79 for spheres at densest packing and a = 3/2 = 1.5 for cubes. It is seen that in this simple representation the dependence on f(k) is the same for ð3þ both shapes, whereas spreading caused by the cubes ensemble (Figure 1b) is approximately twice than the one pertaining to spheres (Figure 1c). [21] The linearity of s 2 t (x) with x stems from the requirement x > R, i.e., departure of the control plane from the zone adjacent to the injection plane. Since t M is given analytically, equation (3) allows determining s 2 t (x) by a simple quadrature for any given f(k). If s 2 tr is finite, transport is Fickian. However, it may become Gaussian only after a considerable distance x/r, in particular for large s 2 Y >1. Nevertheless, an equivalent longitudinal macrodispersivity can be defined with the aid of (3) by a Leq ¼ U 2 s 2 t 2x ¼ s 2 nu 2 t R 2h i ¼ U 2 Z a 1 2R k ½ 2 þ k t M ðkþš 2 f ðkþdk [22] The behavior of a Leq for a normal f(y) was analyzed in detail in our previous studies [Fiori et al., 26] and is recalled in section 3, dealing with non-gaussian log conductivity distributions. 3. Transport in Formations of Non-Gaussian Log Conductivity Distribution (the Subordination Model) 3.1. Log Conductivity Distribution [23] In view of the relatively large influence of the tail of the distribution of the log conductivity f(y), we considered examining the impact of non-gaussian distributions of blocks conductivity. The idea is that in applications conductivity data may fit a lognormal distribution in the central portion of f(y), while identification of the tail pertaining to low Y is imprecise. Consequently, we have adopted the subordination model, a generalization of the lognormal one. The application of the subordination model in subsurface hydrology was proposed by Painter [21], and it was obtained by regarding the variance of the Gaussian distribution as variable. In particular, the model suggested by Painter [21] for the pdf of Y is a mixture of Gaussian distributions with different variances. The mathematical expression for the subordination model is f sb ðyþ ¼ ð4þ fðy; hyi; sþg sb ðs; n; I s Þds ð5þ where f (Y; hy i, s) is the Gaussian distribution of Y, with mean hyi and standard deviation (SD) s. In turn, g sb is a lognormal distribution of s, with mean n and SD I s. The resulting distribution, that depends on the three parameters hyi, n and I s, is a flexible one, which converges to the Gaussian one for I s =, and displays significant and increasing tailing as I s grows, similar to the Levy distribution. For given total variance s 2 Y, we have n =lns 2 Y /2 I 2 s and the model (5) depends only on the parameters hyi and I s. Analysis of hydrogeological data suggests values I s ].7 [Painter, 21]. [24] While the original model considered the distribution of incremental values of Y, we shall use it as a model for the Y univariate distribution. The latter was also suggested by Painter [21] in his conclusions. This way, the long-range 4of13

5 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS Figure 2. Log conductivity probability density function for normal distribution of Y and for the subordination model f sb (Y) with zero mean, s 2 Y = 2 and shape parameter I s equal to.3 and.6. correlation of Y which stems from the Y increments is filtered out from the model, consistent with our assumption of finite integral scale I. [25] In the past, analysis of solute transport aiming at the examination of the effect of departure from a multi- Gaussian structure were made by keeping the univariate f(y) normal, while changing the multipoint autocorrelation, in order to assess the impact of connectivity of low and high K. These studies were of a numerical nature and limited to two-dimensional (2-D) flows [e.g., Wen and Gomez-Hernandez, 1998; Zinn and Harvey, 23]. Here we examine the impact of the subordination model by both semianalytical method and numerical simulations for the 3-D structure of Figure 1c. Illustration is carried out for s 2 Y = 2, a value characterizing highly heterogeneous formations, beyond the domain of application of the first-order approximation. [26] The pdf f sb (Y) is represented in Figure 2 for fixed s 2 Y =2,hYi = and for a few values of I s = (normal), I s =.3 and I s =.6. While the distributions are close to the normal one for say jyj < 4, they differ in the tail regions, for values of f sb (Y) smaller by 3 orders of magnitude than f sb (). As mentioned above in this section, our aim is to examine the impact of the tail portion of the pdf that is subjected to imprecise identification in practice Solution of Transport by the Semianalytical Method and Comparison With Numerical Simulations [27] The derivation of f(t R ), the pdf of the residual of traveltime past an isolated block, is easily carried out by using f sb (Y) (5) to represent f(k) in (A1) and (A11), where k = K/K ef = e Y /K ef. The effective conductivity K ef was derived by the self-consistent approach as advanced a long time ago in the general literature on heterogeneous media (see, e.g., Janković et al. [23a] on adaptation to the Figure 3. Probability density function of traveltime residuals f(t R ) for structure made of spherical inclusions with normally distributed Y (I s = ) and subordination model (I s =.3, I s =.6). 5of13

6 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS Figure 4. Relative mass flux mu/r = f(t, x) as function of time (tu/r) at control planes at x/r = 2.75, 53.94, predicted by the semianalytical model for the Gaussian Y (I s = ) and subordination model with I s =.6. present problem) i.e., by solving numerically the integral equation R 1 [(K ef K)/(2K ef + K)] f(k) dk = (the differences for varying I s were small, as shown in the sequel). We represent f(t R )fors Y 2 = 2 for the 3 values of I s in Figure 3, after carrying out a numerical quadrature in (A1), for spheres. Obviously, for I s = the result reduces to the Gaussian distribution. It is seen that the influence of I s > on the bulk of the pdf is not large: early arrival times t Rmin U/R 2.5 are the same and the peaks are close. Differences are significant only for the tail at traveltimes t R U/R 6 for which f(t R ) ] max f(t R ). [28] The impact of f sb (Y) is further illustrated in Figure 4 by the relative mass flux at a few control planes x/r 2.75,53.94, It has been obtained for spheres at the densest packing and by inverting numerically the Fourier transform of (2). The comparison between m(t, x)=f(t, x)for the two values I s = (normal Y) and I s =.6 reveal the same features: m(t, x) are quite close most of the time and differences show up only for large arrival times and minute mass flux. Nevertheless, these thin tails have a significant impact on the variance (3) and the associated a Leq (4) as well as the skewness coefficients s t = ht 3 i/s 3 t. This effect is demonstrated by Figure 5 in which we represent a Leq (equation (4)) for s 2 Y = 2, but with the lower limit of integration k l in (4) rather than zero. It is seen that for lognormal K, a Leq converges to the asymptotic value 1.8R for k 1 1. In contrast, for the subordination model, for I s =.3 and I s =.6, a Leq grows continuously to extremely large values, for very small values of k l. Hence, in spite of the similarity of the travel time distributions of Figure 4, the difference in the tails pertaining to very large arrival times and minute mass flux have an enormous impact on s 2 t. Whether the impact of these very low K values, or conversely very large t, is felt for finite plumes is an ergodic Figure 5. Macrodispersivity a Leq /R as function of lower integration limit k l for normally distributed Y (I s = ) and subordination model with I s =.3,.6 for structure made of spherical inclusions. 6of13

7 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS Figure 6. Relative mass flux mu/r = f(t, x) as function of time (tu/r) at control planes x/r = 2.75,53.94,87.13 predicted by the semianalytical model for (a) Gaussian Y (I s = ) and (b) subordination model with I s =.6 and comparison with numerical simulations for densely packed spheres. problem. Coping with it in applications, e.g., by adopting approximations suggested in our previous work [Janković et al., 26] or similar ones, is not considered in the present article. [29] We have compared the results for m obtained with the aid of the semianalytical approximation with those derived by accurate numerical simulations, for spherical inclusions at densest packing and for fixed s Y 2 =2,hYi =.Itis reminded that for lognormal K, i.e., for I s =, the agreement was found to be excellent in the past. The numerical methodology here was the same as the one employed for lognormal K, as explained in detail by Janković et al. [26]. The numerical simulations are hence performed with 1, densely packed spherical inclusions with volume fraction equal to.7 and 4, equally spaced particles. The background conductivity was computed using self-consistent method; it equals K G and K G for I s =.3 and.6 respectively (it was K G for I s = case). Numerical simulations were conducted on a massively parallel supercomputer at Center for Computational Research, University at Buffalo. The average Eulerean velocity inside the flow domain was very close to applied uniform flow: it equals 1.3U and 1.2U for l =.3 and.6, respectively. We present here the final results that compare the mass arrival m(t, x) fori s = and I s =.6 in Figure 6, at the same control planes as in Figure 4. Again, the agreement is very good for the bulk of the mass arrival distribution, confirming again the accuracy of the semi- 7of13

8 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS analytical method and the validity of the various approximations (embedding matrix, simplified f(t R ) and f(t)). The large time tail pose the same problem as in the work by Janković et al.[26] due to the nonergodic behavior of finite plumes. 4. Occurrence of Anomalous Transport [3] In the present context anomalous behavior is defined as the one for which either s 2 tr (3) or the related a Leq (equation (4)) are unbounded for x > I. If a particle trajectory is represented as a sum of independent space or time steps, precisely like t = P t R here, there are two reasons for the occurrence of anomalous behavior: (1) longrange correlations and (2) broad distributions [Bouchaud and Georges, 199]. [31] The first case was considered already in the hydrological literature [e.g., Glimm and Sharp, 1991; Dagan, 1994; Rajaram and Gelhar, 1993; Bellin et al., 1996; Di Federico and Neuman, 1998; Fiori, 21] in relation with the existence of log conductivity values correlated over increasing scales. Since aquifers are of finite depth, this applies possibly to axisymmetric heterogeneity, of finite vertical integral scale I v, but increasing horizontal one I h. The simple and extreme case is of a stratified aquifer, for which I h = 1, and flow is parallel to the bedding. For advective transport a Leq grows like x, whereas incorporation of transverse diffusion [Matheron and de Marsily, 198] leads to growth like x 1/2. This mechanism, where spreading is faster than Fickian dispersion, is often coined as superdiffusion in the physics literature [see, e.g., Metzler and Klafter, 2]. 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. [32] Here we examine anomalous behavior for isotropic aquifers of a 3-D structure and therefore concentrate on the second mechanism, of broad distributions or fat tails, while the integral scale I is finite. While anomalous behavior of this type was discussed previously by using phenomenological approaches, our model permits us to relate it to the conductivity heterogeneous structure. Hence the basic question we pose is: What type of conductivity pdf, which is controlling the transport mechanism, leads to anomalous behavior for the present configuration? [33] This question can be answered in a straightforward manner from the expressions of s 2 tr (3) or a Leq (equation (4)) which, irrespective of the blocks shape or density, are proportional to the integral R 1 (k/(2 + k)) [t M (k)] 2 f(k) dk which in turn is proportional to R 1 K/(2K ef + K) [t M (K)/ K ef )] 2 f(k) dk. We have seen already that for the lognormal f(k), a Leq is finite and transport is Fickian, the reason being the quick drop of f(k) with K,forK!. Indeed, it is the tail of t R, caused by the long residence times in blocks of low K, that may render the integral divergent. With an assumed behavior f(k) K d for K! and taking into account that t M U/R 4/(3k) fork! in (A4), we find out that Z a Leq Z K 1 fðkþdk K d 1 dk K d for K! ð6þ [34] Hence anomalous behavior occurs if d <. Furthermore, d is bounded from below, d > 1, since otherwise the basic requirement of a pdf R 1 f(k) dk = 1 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 d for K!, with 1 <d <, while transport is normal for d >. Obviously, in the lognormal case the expression R K 1 f(k) dk R 1 exp [ Y 2 /(2s Y 2 )] dy in (6) is integrable and transport is normal. [35] The next task is to examine the traveltime distribution f(t, x) in the anomalous case, topic that has been investigated extensively in the statistical physics literature Behavior of the Traveltime pdf f(t, x) for Anomalous Transport [36] We focus on the tail of f(t R ) for large t R, which is responsible for possible anomalous behavior. The limit t R!1corresponds to K!, and by (A3) t M t M in! 4/ (3K), h 4/(3t R ),which substituted in the first of (A1) and after expansion of k/(2 + k) fork 1 it leads to f ðt R Z 4= ð 3tR Þ Þ 6t R ¼ 27t R 16 K 2 Z 4= ð 3tR Þ 4 2 fðkþdk 3K K 3 fðkþdk t R ð7þ [37] Our interest here is the distribution of K which has a power law tail for small K, i.e., f K K d for K!. The above limit then becomes f ðt R Þ Z 4= ð 3tR Þ t R K 3þd dk ¼ 3 1 d 4 2þd 4 þ d t ð 3þd Þ R [38] According to the classic developments of the statistical physics literature, a power law distribution of the kind of (8) leads to anomalous transport when d < [see, e.g., Bouchaud and Georges, 199, section 1.2.1]; for the same reasons outlined in section 3, d is bounded from below. Hence transport is anomalous when 1 < d <, the condition being identical with the criterion based on unboundedness of a Leq of section 3. Nevertheless, the traveltime distribution f(t, x) can be determined in the anomalous case by the convolution procedure, as it was shown for the subordination model. 5. Relation With Transport Modeling by CTRW 5.1. The y Function and Its Relation With the Conductivity Structure [39] Different conceptual approaches have been proposed to deal with the complexity and uncertainty of subsurface transport, such as the trajectory approach and its many extensions, e.g., the nonlocal transport equation and the fractional diffusion equation. Over the past decade, a modeling approach vigorously promoted in the hydrology literature is the continuous time random walk (CTRW) [Berkowitz et al., 26]. It was recently shown that CTRW can be related to other methods such as the fractional diffusion equation [Metzler and Klafter, 2; Benson et al., 2] and nonlocal transport formulations [e.g., Cushman ð8þ 8of13

9 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS and Ginn, 1993; Cushman et al., 1994]. Although a detailed discussion and comparison of the present approach with existing conceptual approaches for subsurface transport lies beyond the scope of this work, we shall note here a few interesting implications of the obtained results specifically for the CTRW approach. [4] 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 (subordination model), and studied asymptotic properties of transport. In the CTRW 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 and Weiss, 1965; Montroll and Scher, 1973]. The waiting time and the length of the jump are random variables characterized by a joint pdf y(t, l); as a rule the random variables are assumed separable, i.e., independent, leading to y(t, l) = y(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 and Kehr, 1987]. [41] The function y(t) lies at the heart of the CTRW formulation and its structure is crucial for transport predictions. Our density f(t R ) (A1) is directly related to y(t) by noting that the actual time for each hop is T + t R where T = h i/u is the mean waiting time for the hops. Thus we have yðþ¼ t fðt T Þ ð9þ with f(t T )=f(t R )fort R = t T. As noted earlier, we are here focused on conductivity fields with an integrable density, i.e., R 1 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 Rthe so-called R nondefective random walks and that 1 y(t)dt = 1 1 f (t R ) dt R =1. [42] Given y (equation (9)), the first-passage time density of a particle starting at x =,f(t, x), is in the general case defined by [Hughes, 1995, section 5.2.1] fðt; xþ ¼ X1 F m ðþy x m ðþ t m¼1 ð1þ with recursion y m = y*y m 1, and the asterisk denotes convolution; F m (x) is the probability that a particle arrives at x in m steps. In view of our simplifying assumption that all particles take the same N nx/h i steps to arrive at x, we have F m (x) =d mn ; (1) then yields the traveltime density in terms of y in the Laplace domain [Cvetkovic and Haggerty, 22] as ^fðs; xþ ¼ ^yn ð11þ where hat denotes Laplace transform. It is seen that (11) is formally similar to (2). [43] Summarizing, the present approach is formally similar to CTRW, provided that the key function y is given by (A7), which in turn is related to the conductivity distribution f(k). Thus the major conclusions drawn in the previously can be transferred to the CTRW, and the results for asymptotic transport are discussed in the sequel. It is emphasized that in spite of the formal similarity of the outcome, the present approach and current implementations of CTRW to subsurface transport are fundamentally different. We arrive at f(t R ) by solving the flow problem for a given conductivity structure, whereas y is typically postulated as applying to generic transport with parameters to be fitted by comparison with experiments or simulations [Berkowitz et al., 26] Asymptotic Transport [44] In the current literature on CTRW, emphasis has been given to anomalous asymptotic behavior in subsurface formations, such as fractured media, as well as granular porous media on various scales. The extended tailing of y is quantified as a power law [Berkowitz et al., 26] yðþt t 1 b ðt!1þ ð12þ [45] Equation (12) is often used in applications, also because the time for which y tends to (12) can be rather short, of the order of 1 transitions [Berkowitz and Scher, 1998]. Although a truncated form of y has been proposed [Berkowitz et al., 26], CTRW has been often applied in the past using expression (12), where parameter b is calibrated against experimental results on a given scale. The range of inferred b values is quite broad, between.3 and 1.8 [Berkowitz et al., 26]. For example, for the MADE experiment the estimated value was b =.5, implying anomalous transport. [46] The density y clearly depends on the physics of the flow field which in turn is a function of the structure of the heterogeneous conductivity field; if y is not related to the flow structure, then it is in effect treated as a black box where for instance b is calibrated by using transport experiments. Our current analysis, provides for the first time a structural interpretation of y and the possibility of investigating the asymptotic transport behavior, for a realistic broad distribution of K. [47] Specifically, we compare the results of section 4 with the distribution (12) and check what kind of conductivity distribution f(k) is able to support the model y t 1 b. Comparison between (12) and (8) leads to the identity d = b 2, and the asymptotic structure f(k) K b 2 for K!. Comparing this result with the analysis of sections 3 and 4, it is seen that transport is anomalous when 1 < b < 2, while values of the parameter b < 1 are impossible, since the basic requirement of the nondefective random walk, i.e., R 1 f(k) dk = 1 would not be satisfied. Hence it appears that the values b < 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 b < 1 then a physical interpretation presumably requires including other effects, for instance, mass transfer by diffusion or nonergodic behavior 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 9of13

10 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS factors can be encapsulated by a unique value of the parameter b which applies to the entire history of the plume. Indeed, the variation of b with distance has been revealed in an empirical manner by Trefry et al. [23]. 6. Summary and Conclusions [48] The present study on transport in heterogeneous formations of non-gaussian log conductivity structure and its conclusions are summarized as follows: [49] 1. The relative solute flux m(t, x) (BTC), which is equal to the pdf of traveltime f(t, x) for ergodic plumes, is determined by an approximate, semianalytical approach. It was derived in the past for a plume advected through a heterogeneous medium consisting of an ensemble of densely packed spherical inclusions of uniform diameter and of independent random K; the latter had a univariate lognormal distribution f(k) [Fiori et al., 26]. The traveltime residual was determined as the sum of t R,the perturbation associated with the traveltime past each inclusion crossed by the particle trajectory. The derivation is extended here for the first time to cubical, space filling, elements. The accuracy of the model for an isolated element has been tested against accurate numerical simulations. Hence the cubical (or spherical) inclusions can be used for fundamental studies of flow and advective transport in heterogeneous porous media with finite integral scale. [5] The key function for determining m(t; x) is the pdf f(t R ) the distribution of the particle residence (or hoping) time related to a generic inclusion. This distribution and particularly the variance s 2 tr are derived analytically in terms 2 of f(k). If s tr is finite, transport is Fickian and an equivalent macrodispersivity a Leq can be conveniently defined. [51] 2. We apply the theoretical analysis of advective transport to non-gaussian broad K fields, namely the subordination model, deduced from comprehensive conductivity data sets. The distribution differs from the Gaussian one mainly in the tail region of very low K; such a choice reflects the difficulty in identifying the tail from data. After comparison with highly accurate numerical solutions, we find that the BTC can be predicted quite well by the traveltime density f(t, x) (2) obtained by convoluting a density of hopping times f(t R ). [52] While the early arrival and the bulk of the BTC for the subordination and Gaussian distributions f(y) are close, the difference manifests in the very thin tail of large traveltime. This difference has a large impact on the variance s 2 tr and the related a Leq : while the latter tend to constant values (Fickian transport) for the Gaussian distribution, they are unbounded for the subordination model. [53] 3. We determined that transport is anomalous when the conductivity distribution f(k) K d for K!, with 1 <d <. Our basic assumption is that R 1 f(k)dk = 1, i.e., there is flow at all points no matter how small is the velocity; consequently, the random walk in this medium is nondefective, with R 1 1 f(t R )dt R = 1. We found that the asymptotic tailing of the hoping time density f(t R ) for a nondefective random walk is power law f(t R )! t 3 d R (t R!1). Thus a clear connection is established between the nature of transport and the underlying conductivity distribution of the heterogeneous formation. [54] 4. The present approach is shown to be similar to the continuous time random walk (CTRW), the key function y of CTRW being equivalent to f(t R ). With the assumption y t 1 b [Berkowitz et al., 26], the identity d = b 2 holds and according to our findings transport is anomalous when 1 < b < 2. Hence it appears that the values b <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 b < 1 then a physical interpretation presumably requires including additional effects, e.g., mass transfer by diffusion or nonergodic behavior due to finite size of the plume. The fundamental difference between current implementations of CTRW in the subsurface and the present approach is that we arrive at f(t R ) by solving the flow problem for a given conductivity structure, whereas y is typically postulated as applying to generic transport with parameters to be inferred from experiments or simulations. [55] 5. The present work is a first step toward the investigation of transport in heterogeneous aquifers of given non-gaussian log conductivity distribution and of finite integral scale, based on a particular structural model. To render the analysis more realistic, additional features shall be taken into account, e.g., ergodic issues related to the finite size of the plume, anisotropy and molecular and porescale diffusion. Our preliminary work on the impact of diffusion indicates that it may modify the anomalous behavior of very low conductivity zones, depending on an additional parameter (the Peclet number). The present results may be regarded as limiting ones for Pe tending to infinity. Appendix A: Derivation of Traveltime Distribution of Particles Moving Past an Solated Inclusion [56] We consider uniform flow of velocity U at infinity, past a sphere of radius R and conductivity K, submerged in a medium of conductivity K ef (Figure 1e). The velocity field is given exactly by u ex ¼rf ex ; f ex ¼ U 21 ð kþ 2 þ k u in ¼rf in ¼ U 21 ð kþ 2 þ k R 3 ðx xþ 2jx xj 3 ðjx xj > RÞ ðjx xj < RÞ ða1þ where V = U + u is the total velocity, u is the perturbation, x (x, y, z) is the coordinate vector of the center and k = K/K ef. [57] Remember that by the self-consistent argument [Janković et al., 23a] the following relationship applies: K ef /K G = hv in i/u, where the statistical averaging is carried out over the pdf f(k). [58] The traveltime residual of a particle moving on a streamline from sufficiently far upstream x = L to downstream at x = L is given by t R ðb; kþ ¼ lim L!1 Z L L dx V x ½x; r t ðþ; x kš 2L U ðl RÞ ða2þ where r t (x, b) is the equation of a streamline originating at r t =[(y y) 2 +(z z) 2 ] 1/2 = b for x = L. In words t R is the delay or advance of the traveltime of a particle with 1 of 13

11 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS Figure A1. Distribution of dimensionless traveltime residuals t R in the wake of a cubical inclusion determined numerically (solid line) and by the constant t M pertaining to a sphere (dashed line). respect to the mean time and it becomes independent of L for L larger than a few radii R. [59] While the computation of t R requires a quadrature, it can be determined analytically along the central streamline b = as follows U t in M ðkþ ¼ R t M ðkþ ¼ t in M ðkþþtexð Þ Z R R M k dx=r ð Þ 41 1 þ ðu in 2 ¼ ð k Þ =UÞ 3k 8 9 U t ex M ðkþ < ZL ¼ lim R 2 dx=r ð Þ L!1 1 þ ½u ex ðx; ; Þ=UŠ 2 L = : R 1 ; R ¼ 41 ð kþ dx 2 þ k 1 x 3 þ 2ðk 1Þ= ð2 þ kþ ( 2 h ¼< ln 1 þ c 1=3 þ i 4=3 ln 1 þ i 4=3 =c 1=3 3c 1=3 ða3þ ða4þ i 2=3 ln 1 i 2=3 =c 1=3 ða5þ i ) with t M in and t M ex pertaining to the part of the streamline within and outside the sphere, respectively, i =( 1) 1/2 and c =(2+k)/[2(k 1)]. [6] In Figure A2 we have represented the dependence of t M in, t M ex and t M upon k. It is worthwhile to recall a few properties of t M that are underlain Figure A2: (1) it is highly skewed due to the different behavior of the velocity (A1) for k!1and k! ; (2) t M is dominated by t M in! 4/(3k) for k 1, and (3) t M = 2k under an expansion at first order in k = k 1; that is, it is antisymmetrical, with same behavior for negative and positive k. [61] Examination of the behavior of t R for different streamlines [Fiori et al., 23] has shown that the contribution of streamlines that do not cross the sphere is negligible. Furthermore, the variation of t R for streamlines that cross the sphere can be approximated by t R ¼ t M ðkþ ðrþ 2R ð < r RÞ ; t R ¼ ðr > RÞ ða6þ where = 2(R 2 r 2 ) 1/2 is the segment of the sphere intercepted by the streamline and < r < R is the radial coordinate in the center plane (Figure 1e). In this simplified representation the entire profile of traveltime residuals is given in terms of the analytical expressions (A5) and (A6)). [62] A similar analysis was carried out for isolated cubes (Figure 1d). In this case, however, no simple expressions are available for the velocity field and we have solved the flow problem numerically, by using the finite difference method. Traveltime residuals were computed numerically by tracing particles between head specified boundaries at x = L = 6R and x = L =6R. The distance between impermeable lateral boundaries was also 12R. High-resolution numerical solution was obtained by setting the size of finite difference cells to.4r. For illustration we display in Figure A1 the distribution of t R as function of distance d from the center of the cube, along a vertical axis, for a few values of k. A few striking findings facilitate our further analysis: (1) t M for the central streamline at y = z = is quite close to that pertaining to a sphere (A3). This is further illustrated in Figure A2 displaying the dependence of t M U/R upon k, (2) t R is approximately constant in the wake of the cube, and (3) the ratio between the wake, a square of area A L and the cross section A =4R 2, can be approximated by the ratio applying to the sphere A L /A = V in /U =3k/(2 + k) (equation (A1)). The step function approximating t R in the wake is shown in Figure A1. [63] In view of these findings, we shall treat the traveltime distributions behind a sphere or a cube by the same 11 of 13

12 FIORI ET AL.: ANOMALOUS TRANSPORT IN NON-GAUSSIAN FIELDS Figure A2. Total dimensionless traveltime residual t M, the interior t in M and the exterior t ex M,onthe center streamline for a spherical inclusion by the analytical solution (A3), the first-order approximation and the numerically determined one for a cubical inclusion. formulae (A3) and (A6), the only difference being that in (A6) =2R is now constant. [64] We discuss next the pdf f(t R ) of the traveltime of a fluid particle that moves past a generic isolated inclusion. We start by recapitulating briefly the case of the sphere [Fiori et al., 26], to be generalized for the cube. [65] We consider a particle that starts far upstream in the plane x = L. Under our simplified representation, the traveltime residual t R at x = L is a random variable for two reasons: (1) due to the randomness of the conductivity ratio k = K/K ef, the traveltime residual on the central streamline t M (A3) as well as the magnitude of the wake area A L,are random and (2) since we assume that the particle starts with uniform random location within A L, it crosses the sphere at a random segment (A6). The two random variables and K are independent. As a result, we can write f ðt R Þdt R ¼ FðkÞf ðþdk d ; FðkÞ ¼ A L A f ðkþ ¼ 3k 2 þ k f ðkþ ða7þ where f(k) is the pdf of k = K/K ef. For spheres, the geometric function f( ) has the simple expression f s ( ) = /(2R 2 )for< <2R [Fiori et al., 26, equation (9)], which results from the uniform distribution of the particle location in the circle A ( < r t < R). [66] In view of the results obtained for the cubical block (Figure 3), equation (A7) is valid for it as well provided that we substitute in (A7) f c ( ) =d ( 2R). From (A6) and (A7) it is easy to obtain any moment of t R as follows: hðt R Þ m i¼ 1 ð2rþ m ¼ cðmþ Z 2R m f ðþd ½t M ðkþš m FðkÞdk ½t M ðkþš m FðkÞdk ða8þ [67] Integration over leads to c s = [1/(2R) m ] R 2R m f s ( ) d =2/(m +2),m = 1,2,.. for spheres and c c =[1/(2R) m ] R 2R m f c ( ) d = 1 for cubes. Thus the moments of t R have same dependence on K for both shapes, since t M (equation (A3)) is the same, whereas the difference manifests through the coefficient 2/(m + 2) or 1, for spheres or cubes, respectively. In particular the variance is given by s 2 t R ¼ 3 cð2þ k ½ 2 þ k t M ðkþš 2 f ðkþdk ða9þ with c s = 1/2 and c c =1. [68] It is easy to derive the complete analytical expression of f(t R ) by using (A5), (A6), and (A7). Thus the result for spheres [Fiori et al., 26, equation (11)] is f ðt R Z htr ð Þ Þ ¼ 6t R f ðt R Þ ¼ 6t R ht ð R Þ k 1 2 þ k t 2 M ðkþf ðkþdk t R > k 2 þ k 1 t 2 M ðkþf ðkþdk t R < ða1þ where k = h (t M ) is obtained by inverting the relationship t M (k) (equation (A3)) with hkh1. [69] For cubes the even simpler result is f ðt R Þ ¼ fðht ð R ÞÞ dt 1 M dk k¼h ð tr Þ ða11þ where f(h(t R )) is f(k) calculated at h(t R ). [7] The function f(t R ) (equations (A1) and (A11)) plays a central role in determining the traveltime pdf f(t, x) and in analyzing transport behavior, as shown previously by Fiori et al. [26] and in the present study. [71] The total traveltime of a particle results from summation of t R pertinent to the N > 1 inclusions crossed by the trajectory. In general, N nx/h i where n is the volume fraction and h i is the average length of a segment of the 12 of 13