Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results

WATER RESOURCES RESEARCH, VOL. 39, NO. 9, 1270, doi:10.1029/2002wr001721, 2003 Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results I. Janković Department of Civil, Structural and Environmental Engineering, State University of New York at Buffalo, Buffalo, New York, USA A. Fiori Dipartimento di Scienza dell Ingegneria Civile, Università di Roma Tre, Rome, Italy G. Dagan Department of Fluid Mechanics and Heat Transfer, Tel Aviv University, Tel Aviv, Israel Received 16 September 2002; revised 3 July 2003; accepted 16 July 2003; published 27 September 2003. [1] In parts 1 [Dagan et al., 2003] and 2 [Fiori et al., 2003] a multi-indicator model of heterogeneous formations is devised in order to solve flow and transport in highly heterogeneous formations. The isotropic medium is made up from circular (2-D) or spherical (3-D) inclusions of different conductivities K, submerged in a matrix of effective conductivity. This structure is different from the multi-gaussian one, even for equal log conductivity distribution and integral scale. A snapshot of a two-dimensional plume in a highly heterogeneous medium of lognormal conductivity distribution shows that the model leads to a complex transport picture. The present study was limited, however, to investigating the statistical moments of ergodic plumes. Two approximate semianalytical solutions, based on a self-consistent model (SC) and on a first-order perturbation in the log conductivity variance (FO), are used in parts 1 and 2 in order to compute the statistical moments of flow and transport variables for a lognormal conductivity pdf. In this paper an efficient and accurate numerical procedure, based on the analytic-element method [Strack, 1989], is used in order to validate the approximate results. The solution satisfies exactly the continuity equation and at high-accuracy the continuity of heads at inclusion boundaries. The dimensionless dependent variables depend on two parameters: the volume fraction n of inclusions in the medium and the log conductivity variance s Y 2. For inclusions of uniform radius, the largest n was 0.9 (2-D) and 0.7 (3-D), whereas the largest s Y 2 was equal to 10. The SC approximation underestimates the longitudinal Eulerian velocity variance for increasing n and increasing s Y 2 in 2-D and, to a lesser extent, in 3-D, as compared to numerical results. The FO approximation overestimates these variances, and these effects are larger in the transverse direction. The longitudinal velocity pdf is highly skewed and negative velocities are present at high s Y 2, especially in 2-D. The main results are in the comparison of the macrodispersivities, computed with the aid of the Lagrangian velocity covariances, as functions of travel time. For the longitudinal macrodispersivity, the SC approximation yields results close to the numerical ones in 2-D for n = 0.4 but underestimates them for n = 0.9. The asymptotic, large travel time values of macrodispersivities in the SC and FO approximations are close for low to moderate s Y 2, as shown and explained in part 1. However, while the slow tendency to Fickian behavior is well reproduced by SC, it is much quicker in the FO approximation. In 3-D the SC approximation is closer to numerical one for the highest n = 0.7 and the different s Y 2 =2, 4, 8, and the comparison improves if molecular diffusion is taken into account. Transverse macrodispersivity for small travel times is underestimated by SC in 2-D and is closer to numerical results in 3-D, whereas FO overestimates them. Transverse macrodispersivity asymptotically tends to zero in 2-D for large travel times. In 3-D the numerical simulations lead to a small but persistent transverse macrodispersivity for large travel times, whereas it tends to zero in the approximate solutions. The results suggest that the self-consistent semianalytical approximation provides a valuable tool to model transport in highly heterogeneous isotropic formations of a 3-D structure in terms of trajectories statistical Copyright 2003 by the American Geophysical Union. 0043-1397/03/2002WR001721$09.00 SBH 16-1

SBH 16-2 JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 moments. It captures effects like slow transition to Fickian behavior and to Gaussian trajectory distribution, which are neglected by the first-order approximation. INDEX TERMS: 1829 Hydrology: Groundwater hydrology; 1831 Hydrology: Groundwater quality; 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic processes; KEYWORDS: transport, self-consistent, inclusion, dispersion, analytic element, supercomputer Citation: Janković, I., A. Fiori, and G. Dagan, Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results, Water Resour. Res., 39(9), 1270, doi:10.1029/2002wr001721, 2003. 1. Introduction and Background [2] This paper concludes the sequence of articles (Dagan et al. [2003], part 1, and Fiori et al. [2003], part 2) dealing with the solution of flow and transport through highly heterogeneous porous formations of lognormal conductivity distribution. Its main objectives are to report numerical experiments, to compare the results with the approximate semianalytical ones of the previous parts and to conclude the series. For ease of reading and for the sake of completeness we recall here briefly the essence of the method, as exposed in the preceding parts. [3] The model of the heterogeneous structure we adopt, coined as a multi-indicator one, is a collection of M blocks of constant conductivity K ( j) ( j =1,..., M) embedded in a matrix of conductivity K 0, within the flow domain (part 1, Figures 1 and 2). The blocks of a prescribed shape do not overlap and their centroids are set at random, with no correlation between their conductivities. To achieve simple semianalytical solutions and accurate numerical ones, we selected circular (2-D) and spherical (3-D) inclusions of constant radius R, to represent isotropic formations. The inclusions are confined in, a circle of diameter L (2-D) or a spheroid of axis L (3-D) with L R. The medium of conductivity K 0 extends to infinity, where uniform flow of velocity U prevails. [4] The conductivities obey a lognormal distribution, i.e., Y =lnk is normal and defined by the mean hyi =lnk G (the geometric mean) and the variance s Y 2. The conductivity of the matrix K 0 is taken equal to the effective conductivity K ef. The latter is equal to K G in 2-D and to the selfconsistent approximation (given by equation (21), part 1) in 3-D. Under this choice the mean velocity within is equal to U and the flow outside is not disturbed by the presence of the heterogeneous medium. The spatial distribution of the conductivity K/K G is therefore completely determined by the parameters R, n and s Y 2, where n is the volume fraction of inclusions in. We are interested in values of n up to unity; however, the circular and spherical shapes of inclusions of uniform radii limit the value to about 0.9 and 0.7, respectively, as described in the sequel. [5] The two point autocorrelation function of K or Y, can be determined analytically and are given by equation (13) of part 1. In particular, the integral scales I Y are given by (8/3p)R and (3/4)R in 2-D and 3-D, respectively. [6] Two approximate solutions of flow and transport are developed in parts 1 and 2. The first one, of a semianalytical nature, is coined as the self-consistent approximation herein. It consists of deriving the velocity field by superimposing the disturbances to the uniform flow associated with each inclusion. The disturbance velocity potentials are derived analytically by regarding each inclusion as an isolated one surrounded by the matrix of conductivity K 0. This embedding matrix model [Dagan, 1979] assumes that the impact of the surrounding inclusions is taken into account by the choice K 0 = K ef. Transport is solved subsequently in a Lagrangian framework by particle tracking, leading to trajectories moments and values of macrodispersivities in terms of a few quadratures. Various results are presented in part 2 and in particular the dependence of the longitudinal and transverse dispersivities on travel time. Asymptotically, the trajectories pdf become normal and transport is completely characterized in the mean by the constant, asymptotic, dispersivities. The latter are proportional to I Y and n, but depend in a nonlinear fashion upon s 2 Y. [7] The second type of solution is the first-order approximation in s 2 Y (linearized solution). It depends only on the two-point covariance C Y and both flow and transport problem can be solved analytically [see, e.g., Dagan, 1984]. Thus one of the simplest results is for the asymptotic longitudinal dispersivity a L! s 2 Y I Y. It was shown in part 2 that the self-consistent model, valid for any s 2 Y, leads to the first-order approximation under a power expansion and retaining terms O(s 2 Y ). [8] In the present article precise and efficient numerical solutions are obtained for the multi-indicator conductivity structure by using a novel numerical technique. They are devised as numerical experiments to validate the approximate solutions. [9] The first part of the paper contains the description of the numerical flow solution. It is based on the analyticelement method introduced by Strack [1989], and it can be used to precisely simulate the flow for any value of the variance of log conductivity (values up to s 2 Y = 10 are considered here). At the same time, flow domains more than 1800 (2-D) and 250 (3-D) times the integral scale of log conductivity long were simulated. [10] The essence of the flow solution is the principle of superposition: disturbance velocity potential due to each inclusion is expressed separately and the solution is obtained by adding these velocity potentials. The expressions do not require any discretization of the flow domain: flow solution (head and velocity) are available at any location without interpolation. The flow solution of the self-consistent model is a special case, in which nonlinear interactions between inclusions are not accounted in a direct manner. [11] The present method is different from previous numerical solutions of flow in heterogeneous formations [e.g., Tompson and Gelhar, 1990; Bellin et al., 1992; Burr et al., 1994; Chin, 1997; Salandin and Fiorotto, 1998] that used finite differences or finite elements methods that limited the range of s 2 Y values and the domain size, especially in 3-D simulations. [12] The remaining part of the paper contains the details of the experimental setup, of the implementation of the numerical method and presentation and discussion of

JANKOVIC ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 SBH 16-3 analytic element method of Strack [1989], expressions for the disturbance velocity potential due to individual inclusions are superimposed on the velocity potential due to uniform flow to yield the total potential. Expressions for the velocity potential for each inclusion, expressed in spherical coordinates (r, q, y), are constructed to meet Laplace equation exactly. Here r is referred to the center of the particular inclusion. The disturbance velocity potential in the interior of a generic inclusion is given (up to an additive constant) by fðinþ ¼ 1 X k X rk Tm:k ðcos qþ½am:k cosðmyþ þ bm:k sinðmyþ ð1þ k¼1 m¼0 and, in the exterior by fðexþ ¼ 1 X k X ck r ðkþ1þ Tm:k ðcos qþ½am:k cosðmyþ þ bm:k sinðmyþ k¼1 m¼0 ð2þ Figure 1. Geometry and dimensions of the flow domain (square and rectangle) in terms of domain length (L) for 2-D numerical simulations (top) and 3-D numerical simulations (bottom) with streamlines shown for 2-D flow. results. Unlike the self-consistent model, the numerical simulations take a considerable computer time. It is found important to carry out a few numerical experiments, of an exploratory nature, to examine the performance of the more efficient self-consistent model, for prediction purposes. where Tm:k is Ferrer s function [e.g., Abramowitz and Stegun, 1965], and am:k, bm:k and ck are unknown coefficients, that have to be determined by velocity and head matching conditions at inclusions boundaries. The earliest reference to the functional form of (1) and (2) is reportedly included by Laplace in Memoires des Savants Etranges (1785). A more recent reference is given by Byerly [1893] and Hobson [1931]. [16] The coefficients ck are computed exactly by requiring continuity of the normal (r) component of the velocity [Fitts, 1991]. This gives: k R2kþ1 ck ¼ ðkþ1þ ¼ kþ1 dr dr R drk dr ð3þ where R is the radius of the inclusion. Expression (3) guarantees continuity of the normal component of the 2. Numerical Flow Solution [13] The following is an overview of analytic-based solution for three-dimensional potential flow with inclusions in hydraulic conductivity that are shaped as spheres. Equivalent solution for circular inclusions is presented by Barnes and Jankovic [1999] and is included in part 1. Solution for inclusions shaped as rotational ellipsoids are given by Jankovic and Barnes [1999]. Detailed derivation of all solutions is presented by Jankovic [1997]. [14] Each inclusion can have a unique size, conductivity, ratio of the long and the short axis (rotational ellipsoids only) and may be arbitrarily placed as long as it does not intersect or touch any other inclusion. Solutions are valid for general flow conditions including far-field uniform flow (present study), point and line sinks. [15] The governing equation for three-dimensional flow is the Laplace equation, r2f = 0 where the velocity potential is given by f = KH/q, H is head, K is hydraulic conductivity, and q is a constant porosity. Following the Figure 2. Snapshot of a plume made of 40,000 particles in input zone (rectangle) of 30R 8R of 2-D flow with sy2 = 4 and n = 0.9 at tu/iy 15.

SBH 16-4 JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 velocity regardless of the values of coefficients a m:k and b m:k. It is emphasized the conservation of mass and continuity of normal velocity component on inclusion boundaries are satisfied exactly if the series in (1) and (2) are truncated at a finite k, provided the same number of terms are kept in both interior and exterior potentials. This is a distinctive feature of the solution as compared with the ones based on conventional numerical methods. [17] The coefficients a m:k, b m:k are determined using a two-step process. The first step is to expand the velocity potential due to uniform flow and all inclusions other than the examined one, f 0, on the boundary of the inclusion itself. Following Byerly [1893] and MacRobert [1967], the orthogonal uniformly convergent expansion is (up to an additive constant): f 0 ¼ X1 X k k¼1 m¼0 T m:k ðcos q Þ½a m:k cosðmyþþb m:k sinðmyþš ð4þ Coefficients a m:k and b m:k may be computed by integrating the products of f 0 and basis functions, or, more efficiently, using over-specification [Janković and Barnes, 1999]. If a single inclusion is placed in uniform flow, only a 0:1, a 1:1 and b 1:1 have nonzero values; all other coefficients are zero. In the general case the coefficients a m:k and b m:k depend linearly on the unknown a m:k, b m:k pertaining to all other inclusions. [18] Coefficients a m:k and b m:k are obtained, following continuity of head H at r = R, by requiring f/k to be continuous across the boundary of the inclusion. Using (1), (2) and (4), this continuity condition may be written as: a m:k ¼ d k a m:k ; b m:k ¼ d k b m:k ; d k ¼ R k 1 K int K 0 K int k ð5þ K 0 k þ 1 þ 1 where K 0 is the background hydraulic conductivity and K int is the conductivity of the inclusion. Because of the aforementioned dependence of a m:k and b m:k upon a m:k, b m:k of the other inclusions, (5) constitutes a linear system of equations. Once it is solved, substituting (5) and (3) into (1) and (2) gives exactly f ðinþ ¼ X1 X k 1 K int K 0 r ktm:k ðcos qþ K k¼1 m¼0 int k K 0 k þ 1 þ 1 R ½a m:k cosðmyþþb m:k sinðmyþš ð6þ and: f ðexþ ¼ X1 X k k¼1 m¼0 1 K int K 0 K int þ k þ 1 K 0 k r ðkþ1 R ÞTm:k ðcos qþ ½a m:k cosðmyþþb m:k sinðmyþš ð7þ 3. Experimental Setup [19] Two-dimensional simulations were carried out in a circular domain of diameter L (referred to as the simulation domain) as shown on Figure 1. Far-field velocity U was uniform and oriented along x 1 axis. The conglomerate of inclusions behaves in the mean as an equivalent inclusion of radius L/2 of an equivalent conductivity. Since the analytic solution for flow past a circular inclusion of constant conductivity yields uniform flow inside the inclusion, the circular geometry has thus ensured that the mean velocity inside the simulation domain was uniform. Three-dimensional simulations were carried out using a domain shaped as a prolate ellipsoid (Figure 1). The mean flow inside the simulation domain was again uniform, following the solution for a prolate inclusion in uniform flow at infinity. Furthermore, by selecting the matrix and exterior conductivity to be equal to the effective one, the mean flow inside domains was equal to U, as discussed in the sequel. [20] The aim of the simulations was to derive the statistical moments of flow and transport variables pertaining to stationary fields, as assumed in the approximate solutions of parts 1 and 2. However, the detailed results (see Figure 2) permit one to investigate additional features, e.g., the mean concentration. These and other topics are not examined here. [21] To eliminate boundary effects along the border of the simulation domain, flow and transport were examined in a domain (referred to as the flow domain) which is smaller than the simulation domain. The boundary effects, due to the transition from the exterior matrix to the heterogeneous medium, were assessed by computing the velocity variance along various horizontal and vertical lines along the procedure of Bellin et al. [1992]. The experiments show that the boundary effects were confined to the zone which is about 5 10 inclusions wide. The flow domain was shaped as a square in 2-D and as a parallelepiped in 3-D, as shown on Figure 1, for ease of computation of various flow and transport statistics. The constancy of the velocity variance determined by sampling along lines, is regarded as a diagnostic of the given realization to represent a stationary Eulerian velocity field. [22] Hydraulic conductivity of inclusions, K, was generated using lognormal distribution in a dimensionless form by normalizing by K G, which was set equal to unity. Twodimensional simulations were carried out with variance of log conductivity (s Y 2 ), of 0.1, 0.5, 1, 2, 4, 6, 8 and 10. Threedimensional simulations were limited to s Y 2 = 2, 4 and 8 due to significantly larger computational effort. Background conductivity (K 0 from equation (5)) was set to the effective conductivity of the conglomerate: K G for 2-D flow and effective conductivity computed using self-consistent model for 3-D flow (numerical solution of equation (23), part 1): 1.291K G, 1.558K G and 2.095K G for s Y 2 =2,s Y 2 = 4 and s Y 2 = 8 respectively. [23] In the discussion of part 1 (as shown in Figure 2 there) the representation of the medium is one of inclusions of radius R, embedded in a matrix of inclusions of much smaller radii, which was replaced by a matrix of conductivity K ef. The degree of freedom introduced by the parameter n, volume fraction of inclusions, was of no consequence for the semianalytical and first-order solutions of parts 1 and 2: all the results of interest were scaled by n. This is not the case for the numerical solutions and to check the effect of varying n we adopted the values 0.4, 0.6, 0.8 and 0.9 in 2-D whereas 3-D simulations were limited to n

JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 SBH 16-5 equal to 0.4, 0.6 and 0.7. However, our main interest resides with the largest n, for which the approximate solution is subjected to the most severe test. [24] In previous numerical works Monte Carlo simulations were used primarily in 2-D flows. Furthermore, due to computer time limitations, the dimension of the flow domain was quite limited. In the present simulations we wished to use a large flow domain, to be able to analyze transport for sufficiently large travel times. This requirement and the large computer time (detailed below) restricted the number of simulations, especially for large s 2 Y and in 3-D. Hence we have used mostly single realizations and tried to identify statistical moments, by assuming stationarity and ergodicity, from space distributions. This was considered as sufficient for the main aim of this exploratory study, namely, validation of the approximate solutions. Thus each 2-D and 3-D simulation was carried out with 50,000 equally spaced inclusions. The radius of inclusions, R, was selected to yield desired volume fraction: it was set to L/707, L/577, L/500 and L/471 in 2-D simulations of n = 0.4, n = 0.6, n = 0.8 and n = 0.9 respectively, and to L/159, L/139 and L/132 in 3-D simulations of n = 0.4, n = 0.6 and n = 0.7. [25] Placement of inclusions was uniformly random for the low volume fraction n = 0.4 in 2-D, i.e., the center of each additional inclusion was set at random in the space between existing inclusions. Such a setting is in complete agreement with the pdf of centroids forwarded in part 1. For higher values of n and in 3-D simulations the inclusions centers were placed in a periodic fashion to achieve the high volume fractions adopted in the study. Thus hexagonal packing of inclusions was used for 2-D simulations and face-centered cubic lattice (also referred to as cubic closest packing) for 3-D. The maximum volume p ffiffifraction that may be obtained p ffiffi using these schemes is p/2/ 3 = 0.9069 in 2-D and p/3/ 2 = 0.7405 in 3-D. As explained in part 1, due to the random generation of conductivities, the pdf of a centroid in this setup has a discrete representation in terms of a sum of Dirac functions, rather than the uniform one assumed in the various theoretical derivations. For the high n values considered here, this setting leads to same conductivity statistics as the uniform one, as shown in the sequel. [26] As mentioned above, most of the cases were run as a single realization due to large computational expense of each simulation. Nevertheless, a limited number of cases was run as two realizations to check the assumed ergodic behavior. Results of two realizations were very similar. We therefore assumed that single realization of each case was sufficient to obtain reliable estimates of flow and transport statistics. [27] The inclusions were subject to uniform flow in x 1 direction. Subsequent to the flow solution, the conductivity and velocity statistics of 2-D simulations were computed using a 2000 2000 grid placed over the flow domain. This grid was not used during the solution process and does not influence the solution precision; it was used only for computing flow statistics. Grid size for 3-D simulations was 100 100 100. [28] Advective transport was simulated using particle tracking. Thus 2000 equally spaced particles were released at x 1 = L/3 between x 2 = L/3 and x 2 = L/3 in 2-D simulations. Similarly, 1296 equally spaced particles (36 particles in each direction) were released at x 1 = 0.275L between x 2 = 0.1L and x 2 =0.1L, and between x 3 = 0.1L and x 3 = 0.1L in 3-D simulations. Particle locations and velocities were reported at approximately 5000 (2-D) and 500 (3-D) equally spaced time instances and used to compute various transport statistics. Particle tracking was terminated once the fastest particle reached the end of the flow domain. [29] Additional 2-D simulations with 250,000 inclusions were carried out for several n = 0.4 and n = 0.9 simulations (s Y 2 =2,s Y 2 = 4 and s Y 2 = 8). Particles were distributed in input zone 6R (n = 0.4) and 4R (n = 0.9) long, and 1050R (n = 0.4) and 700R (n = 0.9) wide. The larger size did not affect the velocity variance; but transport statistics for s Y 2 = 4 and s Y 2 =8n = 0.9 simulations were more than 1% larger than those obtained for domain with 50,000 inclusions. Results from other simulations were virtually identical. All 2-D transport statistics presented in this paper are based on simulations with 250,000 inclusions. [30] Additional 3-D simulations with 100,000 inclusions and 4,900 particles (70 rows and columns) distributed in input zone 40R (n = 0.4), 35R (n = 0.6) and 33R (n = 0.7) wide (in both transverse directions) were carried for s Y 2 =4 3-D simulations examined in this paper (n = 0.4, n = 0.6 and n = 0.7). All transport statistics computed using these new simulations were within 1% of transport statistics for original simulations (with 50,000 inclusions and 1296 particles). 4. Implementation of Numerical Method [31] Implementation of infinite series (6) and (7) (and corresponding series for circular inclusions) requires truncation that may results in head discontinuities across the boundary of inclusions, while continuity of velocity is satisfied exactly. High truncations levels were selected to eliminate head discontinuities. The truncation level, N max, for 2-D simulations with 50,000 inclusions was 40 for n = 0.4 and 0.6, 70 for n = 0.8 and 100 for n = 0.9. All 3-D simulations with 50,000 inclusions were carried out with N max = 14 (truncation level for the outer sum of (6) and (7)). Number of degrees of freedom for each circular inclusion is 2 N max + 1 (e.g., 201 for n = 0.9), and (N max +1) 2 for each spherical inclusion (225 for all n). All 2-D simulations with 250,000 inclusions were carried out with N max = 35; 3-D simulations with 100,000 inclusions were carried out with N max = 16 yielding 2.89 10 7 degrees of freedom. [32] Flow problems were solved using an iterative superblock algorithm of Strack et al. [1999]. Iterations were terminated once the largest relative change in coefficients (between two subsequent iterations) over all coefficients of all inclusions was under 10 8. The relative change in a coefficient of an inclusion was computed as the absolute change in the coefficient divided by the sum of absolute values of all coefficients of the inclusion. [33] Particle tracking was carried out using a fourth-order Runge-Kutta method with a combination of a variable time step (used everywhere except across the boundary of inclusions) and a constant space-step (used when a particle crosses inclusion boundary). The velocity, required by the particle tracking procedure, was obtained analytically following Fitts [1990], by differentiating (6) and (7). The time

SBH 16-6 JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 step size was adjusted for each particle during the simulation to limit the velocity change over each time step to 2% of the velocity magnitude. The space-step was set to R/100. Few simulations were also run with an order of magnitude tighter tolerances; the results were identical to those presented later. Similarly, a simulation in which flow was reversed resulted in the plume returning practically to its initial position, indicating lack of numerical dispersion. [34] Simulations were carried out on three Linux-based distributed-memory clusters of 1 GHZ Pentium III and 2.4 GHZ Pentium IV dual-processor nodes at the Center for Computational Research, University at Buffalo. Most 2-D simulations with 50,000 inclusions were conducted on a single processor, without parallel processing. The required CPU effort has depended on the volume fraction and variance of log conductivity. Simulation have lasted up to 42 CPU days on Pentium III nodes. [35] Each 3-D simulation with 50,000 inclusions was run in parallel on 40 Pentium III processors (at 20 nodes). Each processor was responsible for a number of super-blocks (during solve phase) and a number of particles (during tracking phase). Coefficients were exchanged between processors at the end of each iteration (approximately every 6 hours) during the solve phase. The speed-up was linear with the number of processors for both the solve and the tracking phase. The simulations ran up to 12 CPU days; this corresponds to 480 CPU days on a single processor. [36] Simulations with 250,000 inclusions (2-D) and 100,000 inclusions (3-D) were run on 128 Pentium IV nodes (256 processors). Two-dimensional simulations have lasted up to 1.5 days (corresponding to 384 days on a single processor); 3-D simulations have lasted up to 7 days (corresponding to 1792 days on a single processor). [37] In Figure 2 we have represented a snapshot of a plume in 2-D flow for s Y 2 = 4, an input zone of 30R 8R and a travel time tu/r = 13 i.e., tu/i Y 15. [38] Flow nets from a 2-D simulation with 50,000 inclusions, n = 0.9 and s Y 2 = 8 are presented for illustration in Figure 3. The largest scale on Figure 3 corresponds to the flow domain. It is seen that the gap between inclusions was extremely small and they practically touch. Flow nets were visually perfect for all 2-D simulations on all investigated scales: from 10 3 R to 10 3 R. 5. Derivation of Flow and Transport Statistics From Numerical Simulations [39] This section presents expressions that were used to compute various flow and transport statistics for 3-D simulations. Expressions for 2-D simulations can be obtained in a similar manner. 5.1. Conductivity and Eulerian Velocity Statistics [40] Velocity variance, s ii 2, was computed in the longitudinal (i = 1) and transverse (i = 2 and i = 3) direction using the grid placed over the flow domain as follows s 2 ii ¼ 1 N 3 X N V i ¼ 1 N 3 X N X N X N l¼1 m¼1 n¼1 X N X N l¼1 m¼1 n¼1 ½V i ðl; m; nþ V i Š 2 V i ðl; m; nþ ð8þ with V i (l, m, n) theith component of the velocity at grid node (l, m, n) and N the number of grid nodes in each direction (100 for all simulations). [41] Autocorrelation, r ii, of the ith component of the velocity in j direction was computed by N kd 1 1 X j1 N kd X j2 N kd X j3 r ii k x j ¼ s 2 ii N 2 ½V i ðl; m; nþ V i Š ðn kþ l¼1 m¼1 n¼1 V i l þ kd j1 ; m þ kd j2 ; n þ kd j3 V i ð9þ where k =0,..., N 1, d is Kronecker delta, and x j is the flow domain size in j direction divided by the number of divisions (N 1). Autocorrelation of hydraulic conductivity or log conductivity was obtained by replacing V i in (9) by K or Y, respectively. 5.2. Transport Variable Statistics [42] The numerical experiments described here were used primarily to determine the time dependent longitudinal and transverse macrodispersivities a L and a T in advective transport and their comparison with those based on the selfconsistent model (part 2). [43] As a first step, we determined the trajectory of the centroid of the swarm of particles in each simulation x pi ðþ¼ t 1 X Np N p j¼1 x pi ðj; tþ ð10þ where x pi ( j, t) is the x i coordinate of particle j at time instance t and N p is the number of particles. It was found that the mean trajectory is linear at high accuracy and the slope was very close to the mean velocity U. [44] A few options were considered in order to determine the macrodispersivities in each realization. [45] The first one is to determine the spatial moments S ii (i = 1, 2, 3) of the swarm of particles that represent the plume, S ii ðþ¼ t 1 X Np N p j¼1 2; ðj; tþ x pi ðþ t ð11þ x pi and subsequently to derive a L = (1/2) ds 11 /dt and a T = (1/2) ds 22 /dt or a T = (1/2) ds 33 /dt. This procedure was found to lead to oscillatory behavior for a T that suggests that the plume was not large enough to ensure a smooth, ergodic, behavior. [46] A second procedure was to determine the trajectories second moment X ii (t) by computing the fluctuation of the trajectory of each particle with respect to its mean value and averaging for all particles, where X ii ðþ¼ t 1 X Np N p j¼1 2; ðj; tþ x pi ðj; 0Þ V pi t ð12þ x pi 1 V pi ¼ N p ðn t þ 1Þ X Np X Nt V pi j¼1 m¼0 ðj; m tþ ð13þ In (13)V pi ( j, t) istheith velocity component of particle j at time instance t, t is the time increment used in reporting particle positions and velocities and N t is the number of time increments. This achieved considerable smoother results,

JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 SBH 16-7 Figure 3. Flow nets and three stream tubes at four spatial scales with particle positions at a time instance (largest scale only) for 2-D flow with s 2 Y = 8 and n = 0.9 and 50,000 inclusions; solid lines are streamlines, and dashed lines are lines of constant head. but the differentiation of X ii in order to determine transverse macrodispersivity still resulted in some small oscillations. [47] The third procedure, that led to the most stable results, that were considered representative for the ensemble, was to determine first the Lagrangian velocity covariances w ii (t) (i = 1, 2, 3). They are given by w ii ðk t X Np 1 XN t k Þ ¼ V pi ðj; m tþ V pi N p ðn t þ 1 kþ j¼1 m¼0 V pi ðj; m t þ k tþ V pi ð14þ where k =0,..., N t 1. Subsequently, the well known Taylor formula a i ðþ¼ t 1 Z t V p1 0 w ii ðtþdtða L ¼ a 1 ; a T ¼ a 2 or a T ¼ a 3 Þ ð15þ was employed in order to determine macrodispersivities. The numerical experiments show that two last approaches yield close results, typical difference being under 5%. Since the dispersivity computed by (15) was smoother for both a L and a T, it was selected in the following presentations. 6. Presentation and Discussion of Results 6.1. Conductivity Statistics [48] The autocorrelation of hydraulic conductivity, r K, was quite independent of volume fraction, variance of log conductivity, setting and direction, in all simulations. Hence hydraulic conductivity was isotropic and the results are consistent with theoretical ones obtained in part 1, that are based on independence of coordinates of inclusion centers and on continuous variation of the distance between two

SBH 16-8 JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 Figure 4. Autocorrelation of hydraulic conductivity (r K ) in terms of separation (r/r) for a 2-D (n = 0.9) and a 3-D (n = 0.7) numerical simulation with s Y 2 =2. inclusions. We have represented in Figure 4 the theoretical conductivity autocorrelations (equation (13) of part 1) with results from two simulations with highest volume fractions. It is seen that the agreement is very good and the same is true for the autocorrelation of log conductivity r Y, which was the same as r K. [49] This check strengthens the confidence in both theoretical model of the structure and in the setup adopted in the numerical simulations. The simple analytical expressions of r Y (equation (13) part 1) can be used to derive directly the first-order approximation of macrodispersivities. However, we shall use in the sequel the results of part 2. 6.2. Eulerian Velocity Field Statistics 6.2.1. Average Velocity [50] The average transverse velocity (V 2 for 2-D, and V 2 and V 3 for 3-D simulations) was zero for all simulations. The average longitudinal velocity (V 1 ), was within 2.2% of far-field uniform velocity U in all 2-D and 3-D simulations. The velocity field outside the domain of heterogeneity was not disturbed and the choice of the self-consistent K ef for the embedding matrix proved to be correct. The issue of effective conductivity is discussed in detail in a separate paper [Janković etal., 2003]. 6.2.2. Velocity Variance [51] Variance of longitudinal (s 2 11 ) and transverse (s 2 22 ) velocity component, made dimensionless with respect to nu 2, are presented on Figures 5 and 6. Results for the selfconsistent approximation (SC) and first-order approximation (FO), derived in part 2, are included for comparison. In both cases, due to linear dependence on n, these are unique curves. Variance of transverse velocity component for 3-D simulations was obtained as an average of variance in x 2 and x 3 direction which were only a few percent different. The same averaging was used for all other flow and transport transverse statistics of 3-D simulations. [52] The flow solution based on the self-consistent approximation (part 2) has the same functional form as the first-order (k = 0 and k = 1) terms of (6) and (7). The differences between numerical simulations and self-consistent model observed on Figures 5 and 6 are due to highorder terms of (6) and (7), that are necessary to ensure continuity of head, and a different method of computing Figure 5. Variance of the longitudinal component of the velocity (s 2 11 ) as a function of log conductivity variance (s 2 Y ) and volume fraction (n) for numerical simulations (NS), self-consistent model (SC), and first-order approximation (FO). first-order coefficients (a 0:1, a 1:1, b 1:1 ): the first-order coefficients in the self-consistent model are function of inclusion conductivity only, while in the numerical solution they also depend on local flow conditions. [53] Comparison of the numerical solution (NS) and the SC approximation in 2-D for the variance of the longitudinal velocity (Figure 5) shows a consistent trend: they are close for s Y 2 < 1; they are close for any s Y 2 for n 0.6 and the NS is larger than the SC for n > 0.6. The latter difference is discussed in the next paragraph. [54] The 3-D results show a similar trend, though the SC approximation is in better agreement with NS for a larger range of s Y 2 and n. This is understandable in view of the more localized effect of neighboring inclusions. In contrast, the FO approximation is overestimating largely the NS for say s Y 2 > 0.25. [55] The results for the variance of the transverse velocity (Figure 6) display the same trends, but the difference between the NS and the SC values is larger than for the longitudinal component. This difference is attributed to the Figure 6. Variance of the transverse component of the velocity (s 2 22 ) as a function of s 2 Y and n; for notation, see Figure 5.

JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 SBH 16-9 nonlinear effects resulting from interaction between neighboring inclusions that are present only in the NS. For instance, in the SC approximation the interior velocities are parallel to the mean flow, whereas in the NS they can be slanted. 6.2.3. Probability Density Function of Velocity [56] For purpose of illustration, the probability density function of longitudinal velocity component (V 1 /U) fora 2-D (n = 0.9) and a 3-D (n = 0.7) numerical simulation with s 2 Y = 4 is shown on Figure 7. These pdf are highly skewed, in contrast with the Gaussian distribution of the FO approximation. This large difference is manifesting in the higherorder statistical moments of V 1, that are not presented here. It is interesting to note the presence of a small fraction of negative velocities. Negative velocities were present in 3-D only at highest volume fraction (n = 0.7) while in 2-D they were also present for lower volume fractions due to more constrained flow patterns observed in 2-D flow. [57] Another interesting feature is the presence of velocities larger than those predicted by the SC approximation SC (V 1,max =2Uin 2-D and 3U in 3-D, respectively). This effect is attributed to the nonlinear interaction between neighboring inclusions that is neglected in the SC approximation. The existence of the tails of the pdf for V 1 > V 1,max SC explains the differences between velocity variances shown in Figure 5. The tail for 3-D flow is much smaller than the one for 2-D flow, resulting in better agreement with the SC approximation. This is understandable in view of the stronger interactions existing in the 2-D configurations. [58] It is interesting to compare these results with those obtained by Bellin et al. [1992] and extended by Salandin and Fiorotto [1998] for 2-D flow in media of multi- Gaussian log conductivity structure. A qualitative comparison between the pdf in Figure 7 with Figure 9a of Salandin and Fiorotto [1998] for their largest s 2 Y = 4 reveals a high degree of similarity. Nevertheless, the velocity variance (their Figure 3a) is in excess of the FO approximation whereas the NS results (Figure 5) are below. Our main interest resides in the 3-D flow, for which the effect of the tail of the pdf is largely reduced. 6.2.4. Velocity Autocorrelation [59] Autocorrelations (r ii (k x j )) of the longitudinal (i =1) velocity component in longitudinal ( j = 1) and transverse ( j Figure 7. Probability density function (PDF) of Eulerian longitudinal velocity component (V 1 ) for a 2-D (n = 0.9) and a 3-D (n = 0.7) numerical simulation with s 2 Y =4. Figure 8. Autocorrelation (r ii (r j )) of the longitudinal (i =1) and transverse (i = 2) velocity component in longitudinal ( j = 1) and transverse (j = 2) direction as a function of separation r j /R for a 3-D s Y 2 =4n = 0.7 simulation; for notation, see Figure 5. = 2) direction, and of transverse (i = 2) velocity component in longitudinal ( j = 1) direction for a 3-D simulation with n = 0.7, s 2 Y = 4 are shown for illustration in Figure 8. The agreement with the SC values is very good for the longitudinal velocity in both longitudinal and transverse direction. Similar trends, but larger differences are present for the transverse velocity in the longitudinal direction. Two-dimensional simulations display the same features. It seems therefore that the SC approximate model captures correctly the range of lags for which velocities are correlated (approximately 6R) and the screening effect of the medium surrounding each inclusion. 6.3. Transport Statistics 6.3.1. Qualitative Discussion [60] Before embarking on a detailed analysis of the statistical moments of trajectories, it is worthwhile to examine the illustrative example of the plume displayed in Figure 2. It is seen that the multi-indicator model leads to the complex behavior found in previous works [e.g., Wen and Gomez-Hernandez, 1998]. Although the topic of mean concentration distribution is the object of a future study, it is worthwhile to point out to a few important features. First, it is observed that channels of high velocity are created by strings of high conductivity inclusions, leading to fingers of solute that move forward (the presence of transverse pore-scale dispersion, neglected here, may cause some mixing). Secondly, the retention of a large portion of the plume in the zones of low conductivity, in agreement with the discussion of part 1 on the asymmetry of the transport, is also evident. Last but not least, the extreme delay in zones of low conductivity is illustrated by the blobs of the plume retained in the injection zone. This effect reminds, at least in a qualitative manner, the findings at the MADE field test [Boggs et al., 1992]. We limit the present study to the first statistical moments of ergodic plumes, much larger than the one of Figure 2, that characterize their global behavior. 6.3.2. Average Lagrangian Velocity [61] The average Lagrangian velocity determined directly as the rate of change of x pi (t) was constant in time and different by less than 1% from the Eulerian mean V 1.

SBH 16-10 JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 Figure 9. Lagrangian longitudinal velocity covariance (w 11 ) as a function of time for a few 2-D and 3-D s 2 Y =4 simulations with infinite and finite Peclet numbers; for notation, see Figure 5. Figure 10. Longitudinal dispersivity (a L ) as a function of time for 2-D n = 0.4 simulations with s 2 Y =2,4,8;for notation, see Figure 5. 6.3.3. Lagrangian Longitudinal Velocity Covariance [62] As mentioned above, we have used the numerically determined Lagrangian velocity covariances (14) to compute macrodispersivities, the variable of major interest of this study. However, for the sake of conciseness and for illustration we display only one case of such a covariance in Figure 9. The case we chose is the 3-D flow with n = 0.7, s Y 2 =4. [63] The first-order approximation of the Lagrangian covariance is identical with the Eulerian one, after replacing the spatial lag r 1 in the mean flow direction by Ut [see, e.g., Dagan, 1989]. Therefore the difference between the FO and NS graphs in Figure 9 reflect the differences between the Eulerian and Lagrangian covariances. It is seen that the variance of FO is much larger, which was already found and illustrated in Figure 5. In contrast, the Lagrangian covariance is persistent over a much larger time, due to the large residence time of fluid particles in zones of low velocity, as discussed at length in parts 1 and 2. [64] Since the longitudinal asymptotic dispersivity results from the integration of the covariance, the compensatory effect analyzed in parts 1 and 2 is evident in Figure 9 as well. The FO approximation overestimates the effect of large conductivity zones, which determined the shape at small travel time, and underestimates that of low conductivity ones. The effect on time dependent macrodispersivity will be illustrated in the following figures. [65] The SC approximation, though underestimating the NS at small and moderate travel times, captures altogether the trend of NS. The tailing of both curves suggests that Fickian behavior may be reached only after a considerable travel time, as revealed in part 2. 6.3.4. Longitudinal Dispersivity [66] Dimensionless longitudinal dispersivity a L /R for n = 0.4 and n = 0.9 2-D simulations for three values of s Y 2 each is shown on Figures 10 and 11, respectively. Results from the self-consistent model and the first-order approximation are included. The dispersivity was computed by integration of Lagrangian longitudinal velocity covariance (equation (15) for i = 1). Because of the low volume fraction, n = 0.4 simulations could be carried out for large dimensionless travel time. [67] The numerical (NS) and the self-consistent model (SC) results are overall in excellent agreement, even for s Y 2 = 8. This is understandable in view of the diminishing nonlinear interaction between inclusions as n decreases. [68] Results for the more interesting case of n = 0.9 are displayed in Figure 11. Both the NS and SC models exhibit growth of dispersivity beyond tu/r = 150 for s Y 2 = 4 and s Y 2 = 8, while dispersivity reaches a constant value at about tu/r = 10 in the first-order solution regardless of the variance of log conductivity. At tu/r = 150, the selfconsistent model and the first-order approximation yield similar dispersivity for s Y 2 = 2. This is because overestimation of velocity variance (by the first-order approximation) is compensated by shorter integral scale of Lagrangian longitudinal velocity as discussed already. [69] Numerical simulations produce larger dispersivity than both self-consistent model and first-order approximation. The overall agreement between numerical simulations and self-consistent model is better for n = 0.4 than for n =0.9 due to lesser influence of high-order terms. For example, for s Y 2 =4attU/R = 150, the self-consistent model predicts Figure 11. Longitudinal dispersivity (a L ) as a function of time for 2-D n = 0.9 simulations with s 2 Y =2,4,8;for notation, see Figure 5.

JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3 SBH 16-11 Figure 12. Longitudinal dispersivity (a L ) as a function of time for 3-D n = 0.7 simulations with s 2 Y =2,4,8;for notation, see Figure 5. dispersivity which is 5% larger than that predicted by numerical simulation for n = 0.4 and 30% smaller for n = 0.9. Salandin and Fiorotto [1998, Figure 11a] arrived at similar results for the their NS relative to the FO approximation. However, their Monte Carlo simulations were carried out for 2-D, multi-gaussian structure and travel time limited to tu/i Y = 20. [70] Results for four 3-D simulations with n = 0.7 are shown on Figure 12. The overall agreement of numerical simulations and self-consistent model is better than in 2-D, despite the proximity of inclusions (volume fraction of n = 0.7 is close to the maximum volume fraction of about n = 0.74). For example, for s Y 2 =4attU/R = 40, the selfconsistent model predicts dispersivity which is 10% smaller than that predicted by numerical simulation. [71] As already found in part 2, the main discrepancy between the first-order and nonlinear results manifest in the transient behavior of the longitudinal macrodispersivity: the FO predicts a much quicker tendency to Fickian behavior. [72] In parts 1 and 2 we have discussed in a semiqualitative manner the impact of molecular diffusion, which is important in reducing the residence time in zones of very low conductivity. Although this is a topic of future investigations, we have tried to assess in an exploratory manner the effect of finite Pe = UR/D m number. Toward this aim a few 3-D simulations were carried out with small amounts of molecular diffusion. This has been achieved by supplementing advective displacements by random walk steps. The effect of molecular diffusion, displayed in Figure 12, makes the comparison between NS and SC even more favorable, especially at highest log conductivity variance (s Y 2 = 8) where the number of low-conductive inclusions that trap particles is the largest. Molecular diffusion has smaller impact for s Y 2 = 2 and s Y 2 = 4. Molecular diffusion of magnitude examined here (Pe = 1000) does not affect selfconsistent model results for timescales shown on Figure 12, as shown in part 2. While this was not a comprehensive study of the impact of molecular diffusion, the results show that it plays a significant role in formations of large log conductivity variance. 6.3.5. Transverse Dispersivity [73] Dimensionless transverse dispersivity, obtained by integration of Lagrangian transverse velocity covariance, is Figure 13. Transverse dispersivity (a T ) as a function of time for 2-D s 2 Y = 4 simulations; for notation, see Figure 5. presented on Figures 13 and 14 for several 2-D and 3-D simulations of varying n and constant s Y 2 = 4. The general shapes of the NS are reproduced by the approximate methods, though at different degree of quantitative agreement. [74] The results concerning the NS are similar to those of Salandin and Fiorotto [1998, Figure 11b] for s Y 2 = 4 and tu/i Y < 20, as far as the relation between the NS and the FO approximation is concerned. The underestimation of the NS results by the SC approximation was explained above: due to nonlinear interaction between neighboring inclusions the interior flow is slanted with respect to the X1 axis. Although these deviations are small, they make a relatively large accumulated contribution. Still, the transverse dispersivity is very small compared to the longitudinal one. [75] The asymptotic (large time) transverse dispersivity tends to zero for all 2-D simulations regardless of volume fraction and log conductivity variance, as demonstrated on Figure 13. The transverse dispersivity was proportional to 1/t (corresponding to logarithmic grow of X 22 ). The behavior was hence in agreement with that of the self-consistent model (part 2). [76] The asymptotic transverse dispersivity was nonzero in all 3-D simulations in the travel time interval achieved in this study. The dispersivity has diminished with a decrease in n, but in all cases a small nonzero limit is present. Figure 14. Transverse dispersivity (a T ) as a function of time for 3-D s 2 Y = 4 simulations; for notation, see Figure 5.