Time-dependent transport in heterogeneous formations of bimodal structures: 2. Results

Transcription

1 WATER RESOURCES RESEARCH, VOL. 39, NO. 5, 25, doi:0.029/2002wr00398, 2003 Time-dependent transport in heterogeneous formations of bimodal structures: 2. Results A. Fiori Faculty of Engineering, Università di Roma Tre, Rome, Italy G. Dagan Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel Received 22 April 2002; accepted 5 September 2002; published 3 May [] The theoretical results of part [Dagan and Fiori, 2003] for modeling time-dependent, advective transport of a conservative solute in porous formations of bimodal structure are applied to illustrate the behavior of a few trajectory statistical moments as function of time, of the permeability contrast k, and of the inclusions volume density n. The computations are carried out for circular (2D) and spherical (3D) inclusions to represent isotropic media. Advective transport is solved by studying the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the plume is determined from the residual trajectories of solute particles that are derived numerically by a quadrature. The longitudinal macrodispersivity is defined by a L (t; n, k) =(2U) dx /dt, where X is the trajectories second moment in the mean flow direction. The general behavior of the time-dependent longitudinal dispersivity a L (t; n, k) and, in particular, its constant, large time limit are examined. The tendency of a L to the Fickian limit with time depends strongly on the conductivity contrast; in particular, for low permeable inclusions (k ) it may be extremely slow. It is shown that the first-order approximation in the conductivity contrast k is of limited validity (0.3 < k < 2). The transverse moment X 22 tends asymptotically to a constant value. The analysis of the trajectory high order moments shows that the probability density function (pdf) of the solute trajectories tends to normality at large time. Similar to the Fickian limit the normal distribution may be reached at very large time in presence of low conductivity inclusions, with the pdf characterized by significant tailing for the trailing part of the pdf. INDEX TERMS: 829 Hydrology: Groundwater hydrology; 83 Hydrology: Groundwater quality; 832 Hydrology: Groundwater transport; 869 Hydrology: Stochastic processes; KEYWORDS: groundwater flow, groundwater transport, bimodal formations, heterogeneous formations, composite medium Citation: Fiori, A., and G. Dagan, Time-dependent transport in heterogeneous formations of bimodal structures: 2. Results, Water Resour. Res., 39(5), 25, doi:0.029/2002wr00398, Introduction [2] In part [Dagan and Fiori, 2003] of the present set of papers we have set the theoretical foundations for modeling time dependent, advective transport of a conservative solute in porous formations of bimodal structure. The porous formation is represented as a collection of inclusions of assigned shape (circles, spheres) and hydraulic properties, placed at random in an uniform matrix. Advective transport is characterized with the aid of the statistical moments of the solute trajectories, that are derived in terms of a few parameters: () the inclusion/matrix permeability contrast k, (2) the inclusions volume fraction n, (3) the mean velocity U, and (4) the horizontal radius of the inclusion A. The theoretical analysis has been carried out for finite volume fractions n by means of the composite inclusions model. The latter Copyright 2003 by the American Geophysical Union /03/2002WR00398$09.00 SBH 9 - converges to the dilute limit analysis of Eames and Bush [999] and Dagan and Lessoff [200], developed under the assumption that the inclusions are sparse (i.e., when n ). [3] Advective transport is solved in terms of the distortion of a thin plume, linear (2D) or planar (3D), normal to the mean velocity U and moving through a single inclusion. The deformation of the thin plume is described by the residual trajectory X d (t; b) =X(t; b) Ut b of a solute particle, injected at x = b at t = 0. The derivation of X d (t; b) is based on the potential and stream functions, as well as the associated velocity field, for the flow exterior and interior to the single inclusion, which are given in an analytical form in part [Dagan and Fiori, 2003] (section 3.2). Once the velocity and the flow potential are known, the residual trajectory is obtained by a quadrature. The details of the procedure for determining X d as a function of time are given in Appendix A. Finally, the solutions for the trajectory statistical moments are derived in terms of integrals of

2 SBH 9-2 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 Figure. The longitudinal velocity variance s 2 u as function of the permeability contrast k for a few values of the volume density n. The results are for circular and spherical inclusions. powers of X d, that are calculated by a few numerical quadratures. The second order moments X ij (t), and the correspondent definition of dispersivity, are obtained by solving equations 4 and 5 of part ; the procedure is illustrated in Appendix A. [4] Here we apply the methodology derived in part to illustrate the behavior of a few trajectory statistical moments, both in the longitudinal and in the transverse direction, as a function of time and of the permeability contrast k. The influence of the finite density volume n is also investigated by means of the composite inclusion model. The plan of the paper is as follows: the velocity variance is examined first; then, the asymptotic, large-time dispersivity is studied as a function of n and k, and a novel approximate solution is introduced; the time-dependent dispersivity and the trajectories skewness and kurtosis are then illustrated for a few cases; the transient moments are also studied. In the last part of the paper the probability density function of the solute trajectories is briefly discussed. 2. Velocity Variance [5] The velocity variance s 2 u, which is representative of the magnitude of the velocity fluctuations in the porous medium, is given as a function of the medium structure in part [Dagan and Fiori, 2003, equation ()] in terms of a few integrations. These can be carried out exactly for the cases of composite circular (2D) and spherical (3D) inclusions, obtaining the following expressions: s 2 u 3n ¼ ð n Þð kþ2 U 2 2þ ð k n þ knþ 2 ðcircleþ s 2 u U 2 ¼ 24n ð n Þð kþ2 52þ ð k 2n þ 2knÞ 2 ðsphereþ ðþ with U = juj. The transverse velocity variance s 2 u2 is obtained in a similar manner, as follows s 2 u 2 U 2 ¼ nð nþð kþ2 2þ ð k n þ knþ 2 ðcircleþ s 2 u 2 U 2 ¼ 3n ð n Þð kþ2 52þ ð k 2n þ 2knÞ 2 ðsphereþ The dilute limit is obtained expanding the latter expressions near n = 0 and retaining the linear term in n. [6] The longitudinal variance for both cases is represented in Figure as a function of the permeability contrast k for a few values of the inclusions density n. Starting with the dilute limit, which is attained for n ffi 0.00, it is seen that the variance tends to the same asymptotic value for k! 0 and k! for circular inclusions, while for spherical inclusions the variance is larger for large permeability contrast. The picture changes when the volume fraction is not very small, say when n > 0.00, for which the velocity variance behavior depends on the sign of k. The dependence of the variance on n can be analyzed as follows. When n, the inclusions are sparse and the contribution of each inclusion is relatively modest, resulting in a low velocity variability; in particular, if n! 0, the variance tends to zero in any case. As n increases, the velocity fluctuations intensify because of the increasing number of inclusions. However, a limit is reached when the interaction between the inclusions does not allow a further increase of the velocity fluctuations, after which the velocity variance start to decrease. In the limit n!, the medium tends to be homogeneous and the velocity fluctuations vanish every- ð2þ

3 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 SBH 9-3 where. Hence a density ratio n* can be defined, for which the velocity variance displays a maximum. The same reasoning applies to any other quantity examined in this paper, as it will be shown in the sequel due to the proportionality to n( n) of the conductivity variance. It is easy to recognize that n* is larger than /2 if the dimensionless s u 2 /[U 2 n( n)] increases with n, and is smaller than /2 in the other case. Thus it is seen that n* < /2 when k >, and n* >/2ifk < ; these results are illustrated in the small box in Figure, where the variance is represented as a function of n for two particular values of k for circular inclusions. Similar conclusions can be drawn for spherical inclusions. 3. Longitudinal Moments 3.. Asymptotic Dispersivity [7] The asymptotic, large-time longitudinal dispersivity a L () for dilute suspensions has been studied in the past by Eames and Bush [999], Dagan and Lessoff [200], and Lessoff and Dagan [200]. The main results concerning the asymptotic a L, which is function of the permeability contrast %k, are reproduced in Figure 2, where the dilute limit is shown by the thick lines. For a thorough discussion on the behavior of the asymptotic a L at the dilute limit one can refer to Lessoff and Dagan [200]. A simplified expression for a L () at the dilute limit based on the centerline trajectory is forwarded in Appendix B. Its solution is depicted in Figure 2 as a dashed line. [8] Our main concern here is the behavior of a L () for finite values of the density volume ratio n, which was investigated for the first time in part by the composite inclusions formulation. Figures 2a (circular inclusions) and 2b (spherical inclusions) display the dimensionless a L () as a function of k for a few fixed n. The behavior is similar to the dilute case [Lessoff and Dagan, 200]. Thus, for very low conductivity contrast (k ) the dispersivity diverges because of the solute hold up in the inclusions, while for large conductivity contrast (k ), a L tends to a constant. The density ratio n plays a role similar to that examined in the previous section concerning the velocity variance: the asymptotic a L is equal to zero for both n! 0, n! and it has a maximum at n* that is function of the particular value for k. For the present case, n* is always smaller than /2; for small values of k, itisn* ffi /2, as it can be observed by the closeness of the different curves. The dilute limit is reached for values of the density n that depend on k. Thus, for k > the dilute limit is attained in all cases for n ffi 0 3, while for k < the convergence with n to the dilute limit is much quicker. It is observed that the approximate solution (Appendix B) predicts quite accurately the asymptotic dispersivity at the dilute limit in the entire range of conductivity contrast k, with higher accuracy for low conductivity inclusions. [9] The behavior of a L at the small conductivity limit can be analyzed by the asymptotic results illustrated in section 3.4, Paper. The asymptotic dispersivity for circular and spherical inclusions when k is given at leading order by a L =[4n( n) A/(3pk)] and a L =[n( n) A/(2k)], respectively. This is shown by the dimensionless representation of Figure 2, the asymptotic dispersivity tending to an unique hyperbola, independent of n. Similarly, the various curves tend to collapse into a single one for decreasing values of k. [0] In the small windows of Figures 2a and 2b we represent the dimensionless dispersivity a L /(na) as a function of k in a magnified region around k =. We depict a L at the dilute limit together with its first-order approximation in the conductivity contrast e = Y =lnk (see section 3.4 of Dagan and Fiori [2003]) and the approximate solution of Appendix B (equation (B4)). Figures 2a and 2b show that the first-order approximation, where a L e 2, tends to underestimate the dispersivity for low conductivity inclusions, and to overestimate a L when k > [see also Dagan and Lessoff, 200]. The first-order results seem to apply in all cases for 0.3 k 2. Conversely, the approximate solution (B4) leads to much better results for a L, both for large and small permeability values of the inclusions Time-Dependent Dispersivity [] We examine now the transient behavior of the longitudinal dispersivity, defined as a L (t) =/(2U) dx /dt. For the sake of illustration, we depict in Figure 3 the time dependent dimensionless dispersivity for a few combinations of values of k and n for the case of spherical inclusions. At the small-time limit, a L grows linearly with time and is proportional to the velocity variance (this is a general Lagrangean property). The results for the longitudinal velocity variance, illustrated in section 2, explain the different rate of growth of a L for tu/a for the different selected values of k. The behavior is similar for the circular inclusions case. [2] At large time, a L tends to its asymptotic limit, discussed in the preceding section. The dilute limit is approximately obtained for n =0 3 in all cases. It is seen that the time needed to reach the asymptotic dispersivity (i.e., the setting time ) depends strongly on the conductivity contrast k and the density ratio n. To assess the setting time of the asymptotic dispersivity, we define the quantity t 99 as the time needed for a L to reach 0.99 a L ( ). Figure 4 represents the dimensionless setting time t 99 U/A as a function of n for a few values of the conductivity contrast k for circular and spherical inclusions. At the dilute limit (see the curves at n ffi 0 3 ), it is seen that when k > the setting time is roughly constant, being t 99 U/A = O(). The picture is completely different when k <, for which the setting time grows being approximately proportional to /k. Thus the reaching of the asymptotic dispersivity when the conductivity of the inclusions is small can be very slow, much slower than that related to high conductivity contrasts. The main reason is the large travel time of the solute particle needed to cross the inclusions; this travel time is roughly proportional to the inverse of the permeability contrast, for low k. The result is similar to both circular and spherical inclusions. Thus the important result is that the preasymptotic (or non-fickian ) regime of solute transport when in presence of very small conductivity zones in the aquifer may last for a very large time. In principle, when k! 0the Fickian limit is never reached, because the solute particles remains trapped in the inclusions for a time that diverges like /k. However, molecular diffusion that was

4 SBH 9-4 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 Figure 2. The asymptotic dispersivity a L () as function of the permeability contrast k for a few values of the volume density n: (a) circular and (b) spherical inclusions. neglected here may change the picture for say k < 0.0 (see discussion in part ). [3] We explore now the dependence of the dispersivity on the inclusions volume fraction n by means of the composite inclusions model developed in part. While the impact of the density ratio on the asymptotic dispersivity was discussed in the preceding section, here we illustrate the time dependency through the setting time t 99 as shown in Figure 4. Examining first the case k <, we observe that the setting time decreases as n!. Such a result is explained by observing that the inclusions interior velocity V (in), which is the principal cause of the large travel time inside the inclusion, is strongly influenced by the density of the inclusions. In particular, the interior velocity tends to be equal to the outer uniform velocity U as the inclusions density approaches unity. At the limit n =,t 99 is identically equal to zero because the medium tends to be homogeneous. Matters are slightly different for the case k >, for which the behavior of t 99 with n is not monotonous, and the setting time may increase for increas-

5 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 SBH 9-5 Figure 3. The transient dispersivity a L (t) as function of dimensionless time tu/a, for a few values of the volume density n, and for k = 0. and k = 0. Spherical composite inclusions. ing values of n. The reason is that the setting time is determined by the slowest solute particles, and for certain values of n the velocity field may slow down some particles compared to the dilute limit, in particular the particles moving around the inclusion (see, e.g., the two cases represented in Figure 4b of part ). [4] Consistent with the large-time analysis, a L (t) is maximum for a determined value n*, which obeys the Figure 4. The dimensionless setting time t 99 U/A as a function of the density ratio n, for a few values of the conductivity contrast k. Spherical (solid lines) and circular (dashed lines) composite inclusions.

6 SBH 9-6 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 Figure 5. The longitudinal trajectories skewness as function of time, for a few values of the permeability contrast k. Spherical dilute inclusions. condition n* > /2 for small time; at large time n* < /2 for both the circular and spherical inclusions Higher-Order Moments [5] The third and fourth order moments of the longitudinal trajectories are now examined. For simplicity, we shall limit the analysis to the dilute case. The third moment is given by equation (6) of Dagan and Fiori [2003], under the limit n, while the relation that gives the 4th moment X is obtained along the lines of section 3.3 [Dagan and Fiori, 2003], as follows: Z X ðþ¼ t X 04 n ¼ w Xd 4 ðt; bþdb þ 3X 2 ðþ t Hence the 4th trajectory moment is a function of the trajectory variance X. [6] The quantities of our concern here are the skewness and the kurtosis, which are defined as sk = X /X 3/2 and ku = X /X 2 3. These quantities are of particular interest because for a normal distribution they are both equal to zero. Hence the skewness and the kurtosis provide a measure of the deviation of the trajectories distribution from the normal one. The issue of the probability density function (pdf) of the trajectories is retaken in section 5. Summarizing, the skewness and the kurtosis are given by the following expressions: Z skðþ¼ t n wx 3=2 ðþ t Z n kuðþ¼ t wx 2 ðþ t Xd 3 ðt; bþdb ð3þ ð4þ Xd 4 ðt; bþdb Asymptotic, large-time expressions for sk and ku can be obtained along the lines of section 3.3, part. It is seen that both quantities decay with time, and in particular: sk t /2 and ku t. An approximate analysis of the higher order moments is pursued in section 5. [7] For the sake of illustration we represent in Figure 5 the skewness as a function of time for a dilute system with spherical inhomogeneities for a few values of k. The results for the case of circular inclusions are similar. At the limit tu/a! 0, all the curves tend to the same quantity which is the ratio between the velocity third moment m (3) u, defined as (3) m u = n/w R u 3 (x)dx, and s 3 u. For circular and spherical inclusion the third moment can be calculated exactly, the result being: m u 3Þ ¼ nu 3 k 3 k þ ðcircleþ m u 3Þ ¼ nu k 3 35 k þ ðsphereþ sk ¼ pffiffi 2 3=2 signðk Þ ðcircle; tu=a Þ n 3 pffiffiffiffiffi sk ¼ p ffiffi n 30 7 signðk Þ ð sphere; tu=a Hence the skewness at t! 0 does not depend on the permeability contrast k, except for its sign. [8] For inclusions of conductivity higher than the matrix (k > ) the skewness is always positive, indicating that the bulk of the solute advective displacement is slower than the average velocity. The dependence on k for this case seems ð5þ

7 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 SBH 9-7 Figure 6. The longitudinal trajectories kurtosis as function of time, for a few values of the permeability contrast k. Spherical dilute inclusions. rather weak, and the decay with time is relatively fast, and it is proportional to t /2 when tu/a. Taking for example the value sk = 0. as a reference limit, it is seen that the time needed to reach the above limit is tu/a ffi 40; for circular inclusions the time is tu/a ffi 70. [9] Conversely, if inclusions of conductivity smaller that the matrix conductivity are considered, the skewness is always negative, indicating the presence of slowlymoving solute particles trailing behind the mean. The dependence on the permeability contrast k is quite strong for this case, in particular for the smallest values of k considered. In fact, when k the skewness has large maximum values, and decays at larger times. Taking for example the same limit for sk as the previous case, it is seen that the time to reach the limit is, for k = 0., equal to tu/a ffi 000, and for k = 0.0, tu/a ffi 0000; similar values are obtained for circular inclusions and even larger values are obtained for k < 0.. These results indicate that the trajectory pdf of a solute particle moving in a medium where the inhomogeneities conductivity is much smaller that the matrix conductivity, (e.g., when in presence of lenses of very small permeability) can be very skewed and differ from the normal distribution for a considerably large timescale. These results are discussed at detail in section 5. [20] The kurtosis as a function of time, for a few values of k, is represented in Figure 6, again for spherical inclusions. At the limit tu/a, the kurtosis tends to the ratio between the velocity fourth moment m (4) (4) u, defined as: m u = n/w R u 4 (x) dx, and the squared velocity variance s 4 u. For circular and spherical inclusion the velocity fourth moment can be calculated exactly, the result is: m ð u 4Þ ¼ nu 4 9 k 4 ðcircleþ 8 k þ m ð u 4Þ ¼ nu ku ¼ 2n ku ¼ 5 7n k 4 ðsphereþ k þ ðcircle; tu=a Þ ðsphere; tu=a Þ Similar to the skewness, the kurtosis does not depend on the particular value for k when tu/a. When considering positive conductivity contrasts, i.e., k >, the kurtosis, which is always positive by definition, is slightly dependent on k. The same result was found for the skewness. The time needed to reach a value of ku = 0. is equal to tu/a ffi 8 (circle) and tu/a ffi 6 (sphere). For inclusions of conductivity smaller than that of the matrix, the kurtosis exhibits an increase with time at an earlier stage, to reach a maximum, which is larger for decreasing values of the permeability contrast k. After the peak, ku decreases with time, and the decay is proportional to t. It is seen that the dependence on k is strong, and for k the kurtosis remains high for a large time. 4. Transverse Moments [2] We consider now the effect of random inhomogeneities on the plume transverse spreading by analyzing the ð6þ

8 SBH 9-8 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 Figure 7. The transient transverse trajectory moment X 22 (t) as function of dimensionless time tu/a, for a few values of the volume ratio n, and for k = 0. and k = 0. (a) Circular and (b) spherical composite inclusions. transverse trajectory second order moment X 22. As explained in part, because of the symmetry of trajectories, the large time transverse dispersivity a T (t) = /(2U)dX 22 /dt is equal to zero in all cases. Thus the time dependent moment X 22 (t) is our main concern here. Standing the isotropy of the system, X 22 = X 33 for spherical inclusions. Starting with the dilute limit, we represent in Figure 7 the transverse moment as a function of time for a few values of the permeability contrast k and the density ratio n, for circular (Figure 7a) and spherical (Figure 7b) inclusions. [22] It can be shown that X 22 grows with time at the dilute limit, having a logarithmic behavior for tu/a. Such a behavior can be observed in the curves for n =0 3. The result is consistent with the first-order analysis of transport in continuous, two-dimensional random permeability fields [Dagan, 989]. It must be noted, however, that the above behavior is attained asymptotically at the dilute limit, when n, and since X 22 is a linear function of n, the transverse moment is expected to be small in any case. The rate of growth does not seem to depend strongly on the permeability contrast, except perhaps when k approaches unity. [23] The transverse moment tends to a finite limit as time grows large for finite density ratio n. Hence the density of the inclusions controls transverse spreading. For any finite value of n, the transverse moment ceases to grow after a certain time and reaches a constant limit. However, the time to reach the limit increases with decreasing values of n, and it is approximately proportional to n /2. For example, the

9 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 SBH 9-9 Figure 8. The dimensionless trajectory pdf f(x ) 0 pffiffiffiffiffiffiffi X as function of the dimensionless longitudinal trajectory X 0 pffiffiffiffiffiffiffi Sk/ for a few values of the time t s = ntv (in) /A. Circular inclusions. X curves for n =0 3 of Figure 7a level off after a transient stage in which the growth with time is logarithmic. Similar to the case of the longitudinal spreading, we may also define here a value of the density n* for which X 22 is maximum, with n* < /2. [24] The picture is different for the three-dimensional, spherical inclusions case (Figure 7b). In particular, at the dilute limit, X 22 levels off to a constant, asymptotic value, after a transient stage (see the curves for n =0 3 ). This is again consistent with the first-order general analysis for continuous conductivity fields [Dagan, 989]. The sill depends on the particular k value, and it typically grows with increasing values of the contrast. The time to reach the asymptotic X 22 is roughly tu/a ffi 30, except for the smallest values of k, for which the time to the sill may increase dramatically. Such an increase is due to the long solute hold up in the inclusions. Hence when k the time to reach the asymptotic X 22 is similar to that for the longitudinal moment. The effect of the finiteness of the density ratio is also represented in Figure 7b, where the relatively high impact of the density n on the transverse moment is quite evident. In particular, X 22 and the time to reach its asymptotic value decrease quickly as n!. The dilute limit is reached for lower values of n than those needed for longitudinal transport. 5. Large-Time Longitudinal Trajectories Pdf [25] The determination of the longitudinal trajectories probability density function f(x ) is a difficult task, which requires the statistical moments of X at any order. Our aim here is to asses the large time limit shape of f(x ) and how fast the pdf converges to its asymptotic, normal, limit for dilute inclusions. An approximate derivation of the trajectory pdf can be carried out with the aid of the cumulant neglect approximation [Beran, 968], and the main derivations are given in Appendix C. One of the main results, which are encapsulated in equation (C6), (circular inclusions), and equation (C7), (spherical inclusions), is that f(x ) tends to a normal distribution when t!. The tendency to the Gaussian distribution, however, depends strongly on the permeability contrast k. The behavior can be observed pffiffiffiffiffiffiffi in Figure 8, where the dimensionless trajectory pdf f(x ) X is represented pffiffiffiffiffiffiffi as a function of the dimensionless trajectory X S k / X (with Sk = sign(k )) for a few values of the dimensionless time t s = ntv (in) /A and for circular inclusions. The time t s is a significant timescale, which is proportional to the number of inclusions crossed by the solute plume at time t. The small negative tail of the pdf is a consequence of the truncation of the infinite series in (C3); however, the most interesting part is the one for positive values of the dimensionless longitudinal trajectory, that shows a tailing behavior for X (the tailing is hardly visible in Figure 8 becausep of ffiffiffiffiffiffiffi the scaling effect of the dimensionless variable X Sk/ X ). The tail is more persistent when t s tends to infinity, i.e., when the interior velocity V (in) i is small; this is the case of small permeability inclusions, i.e., k. Furthermore, the tailing is larger for low values of k because in such cases the trajectory second moment X is an increasing function of /k and X is dimensionless with respect to X. [26] It is seen that the tendency to normality is much slower for inclusions characterized by low permeability. This behavior can be quantified through the dimensionless time t s. For illustration, assuming that the normal distribution is almost reached for t s = 50, for circular inclusions, of given radius and density ratio n, the time is equal to ntu/a = 27.5 (for k = 0), ntu/a = 275 (k = 0.) and ntu/a = 2525 (k = 0.0). We therefore conclude that the tendency to the normal distribution for systems with low permeability inclusions can be extremely slow and characterized by a persistent tailing of the (negative) residual displacements.

10 SBH 9-0 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 Thus the implication for solute travel times to a compliance boundary of the tailing behavior is a slow arrival of a significant part of the solute body, due to the solute hold up in the inclusions. 6. Summary and Conclusions [27] We have derived expressions for time dependent, advective trajectories of a conservative solute in porous formations of bimodal structure. The porous formation is represented as a collection of inclusions of assigned shape (circles, spheres) and hydraulic properties, placed at random in an uniform matrix. The theoretical analysis has been carried out for finite volume fractions n by means of the composite inclusions model. The various approximations and limitations of the approach are discussed in part. The method allows for arbitrary permeability contrast between the inclusions and the porous matrix. In the following we summarize and discuss the main findings. [28] The longitudinal dispersivity a L (tu/a; n, k), that is function of time, density ratio n and permeability contrast k, was analyzed first. Its temporal behavior follows closely the Lagrangean general results. In particular, the dispersivity grows linear with time when tu/a, with the rate of growth proportional to the velocity variance, and it tends to an asymptotic limit for tu/a!. The behavior of a L for weak heterogeneity, i.e., at the limit j kj!0 can be analyzed by a first-order approximation around k =;a L is proportional to (ln k) 2 and therefore symmetrical with respect to k =. The validity of first-order solution is limited to small to moderate conductivity contrasts, say when 0.3 k 2. Furthermore, the asymptotic dispersivity is no longer symmetric for large values of the conductivity contrast, as found for dilute systems by Eames and Bush [999] and Lessoff and Dagan [200]. In particular, for high conductivity contrast (k ), such as for isolated fractures or cracks, the asymptotic dispersivity converges toward a limit value. The opposite is true for k, e.g., in presence of lenses of clay, for which a L grows unboundedly as k! 0. The two different behaviors can be explained by analyzing the travel time of a solute particle crossing the inclusion. For high conductivity inclusions, the interior (in) velocity V tends to a finite limit as k!and so the residence time. Conversely, the interior velocity tends to zero as k! 0, increasing dramatically the solute travel time inside the inclusion and leading to an ever increasing dispersivity as k tends to zero. However, the increase of the solute residence time may cease for values of the conductivity contrast smaller than a certain, cutoff value because of the effects of molecular and pore-scale diffusion which make the solute particle escape from the inclusion. The effect of the inclusions volume density n on dispersivity is quite complex. Generally speaking, the dispersivity is identically zero for n! 0 and n!, because for both cases the medium tends to be homogeneous. Hence a L reaches a maximum for a particular value of the density ratio, usually for n < /2. [29] The tendency of a L (tu/a; n, k) to its asymptotic limit, i.e., the reaching of the Fickian regime, was analyzed through the setting time, which is defined as the characteristic time needed to approximately reach the asymptotic dispersivity. We have found that the setting time tends to be constant for k, while for low conductivity inclusions k the time grows with decreasing k, roughly proportional to its inverse. The above behavior can be explained by the solute residence time inside the inclusion, as done before for the asymptotic a L. Hence the Fickian regime is approximately reached at a time that depends on the conductivity contrast, and for low permeable inclusions it may be reached at very large time. [30] The transverse second moment X 22 (t) after a transient stage, reaches an asymptotic, large-time value for spherical (3D) inclusions; for the circular (2D) inclusions case, X 22 tends to a constant limit only for finite n. In particular, the setting time grows unbounded as n! 0, with the transverse moment growing with time like X 22 nlnt. The behavior of the setting time with the conductivity contrast is similar to the longitudinal moment, i.e., it tends to a constant for k and it grows like /k for k. Again, the setting time may be consistently reduced by the effects of the molecular diffusion, for the same reasons previously illustrated. Furthermore, molecular diffusion may modify the large time behavior of X 22, as it is the case for continuous distributions of permeability [see, e.g., Dagan, 989]. [3] The probability density function (pdf) of the solute trajectories, which is proportional to the local mean concentration, tends to normality when tu/a!. The tendency is illustrated through the higher order moments of trajectories, and in particular the skewness and the kurtosis which tend to zero at large time. The generic dimensionless m-order cumulant, that measures the deviation from the normal distribution, decays with time like t m/2. A setting time can also be defined for the reaching of normality which depends on the permeability contrast k, similar to the one defining the Fickian limit. However, the time required for the reaching normality seems to be larger than that related to the Fickian regime, because the convergence to suitable small values of the generic cumulant, like, e.g., the kurtosis, can be quite slow, in particular for low conductivity inclusions. A very interesting feature of the trajectory pdf is its significant tailing for negative relative displacements that occurs for k, that is a consequence of the slow mass release from the low conductivity inhomogeneities. [32] The present study and results have been obtained by means of a few approximations, and some generalizations may be considered in the future. For example, the impact of molecular diffusion on transport should be considered when dealing with very low conductivity inclusions (k ;) [Guswa and Freyberg, 2000]. A complete assessment of the trajectory pdf is also desiderable for studying the transition of the pdf to the normal distribution. The extension of the analysis to continous distributions of the permeability is also an important issue. Appendix A: Numerical Derivation of the Trajectory Moments [33] The first step to determine the statistical moments of the trajectories is the calculation of the trajectory X(t; b) of a particle, injected at x = b (with b =(b, b 2, b 3 )), moving past a single inclusion. Assuming for simplicity x = 0, with x the position of the center of the inclusion, the expression for R X given in part can be written as X (t; b) =b + 0 t V [X(t 0 ; b)]dt 0. Toward the numerical calculation of X, once the velocity field is known, it is convenient to rewrite

11 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 SBH 9 - the same expression in term of travel time t of the solute particle to a distance L, as follows Z L dx tðl; bþ ¼ b V ½x ; x 2 ðx ; yþš ðaþ where the coordinate x 2 (x ; y) depends on the stream function y and the horizontal position x. Expression (A) results from the Lagrangian definition dx /dt = V [X(t)]. The longitudinal trajectory X is found by solving the implicit relation t (X ; b) =t. The solution for X (t; b) is obtained through a discrete version of (A), switching to the coordinates q, r, and proceeding for prescribed increments of q; the inverse relation r = r (q; y) and the formulae x = x (r, q), x 2 = x 2 (r, q) are needed. The velocity V switches from the exterior to the interior field when the particle crosses the inclusion. The residual of the trajectory is calculated as X d (t; b) =X (t; b) Ut b. The transverse residual trajectory X d2 is obtained in a similar manner. [34] Once X d is obtained, the longitudinal moment is obtained by integration the formula (4) of part. The integration for the composite scheme is illustrate here, the extension to the one for the dilute system being straightforward. First, we split the domain of integration 0 of (7) in two parts: one X () for values of b inside the outer shell, and the other X (2) for b outside the outer shell. Then, we switch to polar (3D) or cylindrical (2D) coordinates and solve numerically the following integrals X ðþ t X ðþ t ðþ¼ n w ðþ¼ 2pn w Z 2p Z Ae rxd Z p Z Ae 0 0 ðt; r; qþdrdq ð circle Þ r 2 Xd 2 ðt; r; qþsin qdrdq ð sphere Þ where r, q are the polar coordinates, and A, A e are the interior and the exterior radii of the composite inclusion (A e = A/n /2 and A e = A/n /3 for circular and spherical inhomogeneities, respectively). The moment for spherical inclusions is obtained exploiting the symmetry of the system, and r, q are related to any plane crossing the sphere center, parallel to the mean flow. (2) [35] The term X is obtained by observing that the contributing particles are those located upstream the border of the outer shell, at a distance shorter than Ut; outside the outer shell, the particles move with constant velocity U. Thus the component can be obtained by solving numerically the following integrals Appendix B: Approximate Analysis of the Asymptotic Dispersivity [37] A simple, approximate formula for a L (t) = /(2U) dx /dt X /(2Ut) at large time for dilute systems can be derived observing that the main contribution to the determination of a L comes from the solute particles that cross the inclusion [see, e.g., Dagan and Fiori, 2003, Figure 4]. In particular, the largest relative displacement X M is the one relative to the centerline trajectory, which can be calculated exactly from the expressions for the velocity fields given in part ; the results are as follows X M A ¼ rffiffiffiffiffiffiffiffiffiffiffi k 2 k tanh þ k X M A ¼ 2 3 ðk Þ r ffiffiffiffiffiffiffiffiffiffiffi! k þ k 2 k þ 3 2F ð2=3; ; 5=3; ð2 kþ=ð2 þ kþþ 2 þ k ðcircleþ ðbþ ðsphereþ ðb2þ where 2 F is the hypergeometric function. [38] The approximate analysis for a L () assumes a particular shape for X d (;, b 2, b 3 ) along the projection of the inclusion wake at infinity, around on its maximum value X M. By continuity reasons, the wake at infinity has a width equal to l w = AV (in) /U for circular inclusions, and a radius l w = A(V (in) /U) /2 for spherical (in) ones, with V the interior, constant velocity of the inclusion. Assuming that the displacement X d is proportional to the distance travelled by the particle inside the inclusion, the shape of X d (, y) can be described by the following expressions: sffiffiffiffiffiffiffiffiffiffiffiffiffi X d ð; ; b 2 Þ ¼ X M b2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X d ð; ; b 2 ; b 3 Þ¼X M b2 2 þ b3 3 X d ¼ 0 l 2 w l 2 w ð ðj b 2 jb 2 j; jb 3 j l w ; circleþ j l w ; sphereþ ðotherwiseþ ðb3þ Hence, integrating equation (2) of part with the above assumptions, the large time dispersivity has the following expression: X ð2þ t ðþ¼ nu w Z 3p=2 Z t p=2 0 Xd 2 ð t t0 ; A e ; qþ A e cos qdt 0 dq ðcircleþ a L ¼ X 2Ut ¼ G 2 V ðinþ U n A X M 2 ðtu=a Þ ðb4þ X ðþ 2 t ðþ¼ 2pnU w Z p p=2 Z t 0 Xd 2 ð t t0 ; A e ; qþ A 2 e cos q sin qdt0 dq ðsphereþ ða2þ [36] The longitudinal moment X is obtained as the sum of the two components X (), X (2). The calculation of the higher order moments and the transverse ones follows the same procedure illustrated above. with G = 4/(3p) for the circle and G = 3/8 for the sphere. It can be shown that the asymptotic limits for k and k are the exact ones (see section 3.4 of part and Dagan and Lessoff [200]). The values for k are, however, approximate only, with a relative error usually less that 0%. Thus expression (B4) represents a reasonable approximation for a L for the entire range of values assumed by k.

12 SBH 9-2 FIORI AND DAGAN: TRANSPORT IN BIMODAL STRUCTURES, 2 Appendix C: Approximate Derivation of the Trajectory Pdf for Dilute Systems at Large Time [39] The characteristic function M = hexp (ikx)i 0 (i.e., the Fourier transform of the probability density function) of the trajectory residual X 0 = X Ut b can be expanded in the following manner [Beran, 968] " # Mv ðþ¼exp 2 l 2v 2 þ X l m ðivþ m m! m¼3 ðcþ where v is the wave number and l m is the generic m-order cumulant of X. The one-dimensional Gaussian distribution has the form M G (n) = exp ( /2l 2 n 2 ). The cumulants l m measure the degree from which M differs from the Gaussian form M G. The relations between the first four cumulants and the trajectory moments are the following: l = hx i 0 =0, l 2 = X, l 3 = X, l 4 = X 3X 2. The generic cumulant can be evaluated through the following expression: l m ¼ n w Z Xd m ðt; bþdb ðm 2Þ ðc2þ The trajectory pdf is calculated by the means of the inverse Fourier transform of the characteristic pffiffiffiffiffiffiffi function M. Switching to the coordinate k = v/ X we obtain for the probability density function the following expression Z f X 0 ¼ p 2p ffiffiffiffiffiffiffi X Z ¼ p 2p ffiffiffiffiffiffiffi X e ikx 0 = e ikx 0= p X pffiffiffiffiffi X p ffiffiffiffiffiffiffi M k= dk X ffiffiffiffiffi " # exp k2 2 þ X l 0 m ðikþ m dk 0 m! m¼3 ðc3þ where l 0 m = l m /l 2 m/2. It is interesting to note that l 0 3 and l 0 4 are identical to the skewness and the kurtosis, as defined in section 3.3. [40] The derivation of f(x ) requires the determination of all the cumulants l 0 m. Here we wish to derive an approximate expression which is valid for tu/a ; the aim is to assess the limit shape of the pdf as time grows. The m-order cumulant can be calculated by the means of the approximation outlined in Appendix B, i.e., by sing the simplified expressions (B3) for X d (;, b 2, b 3 ) in (C2), to obtain the following result l m ffi V ðin U Þ X M m nut A Fm ð Þ ðc4þ where V (in) the p ffiffiffiinternal velocity and F(m) = ( + m/2)/ ( (3/2 + m/2) p ) for circular inclusions, and F(m) = 3/(2(2 + m)) for spherical ones. The dimensionless cumulants are, then, equal to l 0 m ¼ l m ¼ l m=2 2 in nv ð Þ t A! m=2 Fm ð Þ F m=2 ð2þ sign X M m It can be noted that sign(x M )=Sk = sign(k ). ðm > 2Þ ðc5þ [4] It is seen that the cumulants of order higher than 2 tend asymptotically to zero as time grows large like t m/2. As a consequence, at that limit the trajectory pdf (C3) degenerate to the normal distribution, which is, Thus the limit distribution of the trajectory pdf at tu/a!. Our aim is to find out how fast is the tendency to normality as a function of time and the permeability contrast k. Tothisaim we seek an approximate solution of (C3) based on the cumulant neglect approximation [Beran, 968], where we set l m = 0 for m > N in (C3). The idea is to neglect the contribution of cumulants over a certain order, since they decay faster for tu/a. We set here N = 4, limiting our analysis up to the fourth moment. At the limit tu/a, it is l m, we perform a series expansion around l 0 3, l 0 4 =0 and solve the integrals after insertion of (C5). After a few manipulation, we finally obtain for the pdf the following simplified expressions, for circular and spherical inclusions respectively fðx* Þ! exp X * 2 ( =2 pffiffiffiffiffiffiffiffiffiffiffiffi þ ð3pþ3=2 2pX 28 ffiffiffi p X * X * 2 3 t s þ p 3 6X * 2 þ X * 4 40t s þ 27p3 X * 6 5X * 4 þ 45X * 2 5 ðc6þ 32768t s exp X *2 =2 fðx* Þ! pffiffiffiffiffiffiffiffiffiffiffiffi þ 4 p ffiffi 2 p 2pX 5 ffiffiffiffiffi X * ðx * 2 3Þ 3t s þ 2 ð8x * 2 95X * 4 þ 20X * 2 45Þ 675t s ðc7þ where X* =S k X/ 0 pffiffiffiffiffiffiffi X and ts = nv (in) t/a is a dimensionless time variable that is based on the interior velocity V (in) ; t s is a slowed down time or an accelerated one depending on whether k <ork >. References Beran, M. J., Statistical Continuum Theories, Wiley Interscience, New York, 968. Dagan, G., Flow and Transport in Porous Formations, Springer-Verlag, New York, 989. Dagan, G., and A. Fiori, Time-dependent transport in heterogeneous formations of bimodal structures:. The model, Water Resour. Res., 39, doi:0.029/2002wr00396, in press, Dagan, G., and S. C. Lessoff, Solute transport in heterogeneous formations of bimodal conductivity distribution:. Theory, Water Resour. Res., 37, , 200. Eames, I., and J. W. Bush, Longitudinal dispersion by bodies fixed in a potential flow, Proc. R. Soc. London, Ser. A, 455, , 999. Guswa, A. J., and D. L. Freyberg, Slow advection and diffusion through low permeability inclusions, J. Contam. Hydrol., 46, , Lessoff, S. C., and G. Dagan, Solute transport in heterogeneous formations of bimodal conductivity distribution: 2. Applications, Water Resour. Res., 37, , 200. G. Dagan, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel. A. Fiori, Dipartimento di Scienza dell Ingegneria Civile, Università di Roma Tre, via Volterra 62, 0046 Rome, Italy. (aldo@uniroma3.it)