[1] In parts 1 and 2 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 l...

[1] Flow of uniform mean velocity U takes place in a formation of spatially variable, random conductivity K(x). Advective transport of a plume of an inert solute is investigated by the Lagrangean approach. The aim of the study is to determine the spatial moments of the plume, i.e., of fluid particle trajectories, for highly heterogeneous aquifers, for which s Y > 1, where Y = ln K. A multi-indicator model of the permeability structure, which is different from the common multi-Gaussian one, is proposed: the formation is modeled as a collection of N blocks of different K ( j) . The structure is defined by the distribution of K ( j) , the blocks' shape, and the coordinates of their centroids. The following simplifications are adopted: the blocks are inclusions of a regular shape (circles, spheres for isotropic media investigated here) defined by the radius A, and the inclusions are not overlapping, and their centroids are distributed uniformly and independently in space. At the continuous limit the model is characterized by the joint pdf f(Y, A). The model is shown to be quite general and to comprise binary, bimodal, indicator variograms and unimodal distributions of Y as particular cases. The study is focused on the latter case, with Y normal N [hY i, s Y 2 ] and stationary covariance of given integral scale I Y ; these are the parameters commonly estimated for sedimentary formations. This leaves still freedom in selecting the pdf f(A). The simple model selected for semianalytical and numerical analysis is that of inclusions of radius R and volume fraction n, submerged in a matrix of effective conductivity K ef . The latter represents the effect of inclusions of much smaller radius, which appear as a nugget in the log conductivity two-point covariance. An approximate analytical solution of the flow is obtained by using a self-consistent approximation, while a fully numerical one is derived in part 3 [Janković et al., 2003a]. Transport is solved by particle tracking, and the time-dependent spatial moments (trajectories variance, skewness, kurtosis) are presented in part 2 [Fiori et al., 2003]. In the self-consistent approximation the asymptotic longitudinal macrodispersivity a L , which is a function of Y, shows strong nonlinear effects: inclusions of large positive Y lead to a finite a L , whereas a L grows unbounded for those of negative Y. This effect is not captured by the common first-order approximation in s Y , which is symmetrical and overestimates a L for Y > 0 and underestimates it for Y < 0. As a result, the second spatial moment is predicted accurately by the first-order approximation, by cancellation of errors, provided that f(Y) is symmetrical. However, the transient regime and higher-order moments are not captured 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, first-order C...

Flow and transport take place in a heterogeneous medium of lognormal distribution of the conductivity K. Flow is uniform in the mean, and the system is defined by U (mean velocity), σY2 (log conductivity variance), and integral scale I. Transport is analyzed in terms of the breakthrough curve of the solute, identical to the traveltime distribution, at control planes at distance x from the source. The “self‐consistent” approximation is used, where the traveltime τ is approximated by the sum of τ pertinent to the different separate inclusions, and the neglected interaction between inclusions is accounted for by using the effective conductivity. The pdf f(τ, x), where x is the control plane distance, is derived by a simple convolution. It is found that f(τ, x) has an early arrival time portion that captures most of the mass and a long tail, which is related to the slow solute particles that are trapped in blocks of low K. The macrodispersivity is very large and is independent of x. The tail f(τ, x) is highly skewed, and only for extremely large x/I, depending on σY2, the plume becomes Gaussian. Comparison with numerical simulations shows very good agreement in spite of the absence of parameter fitting. It is found that finite plumes are not ergodic, and a cutoff of f(τ, x) is needed in order to fit the mass flux of a finite plume, depending on σY2 and x/I. The bulk of f(τ, x) can be approximated by a Gaussian shape, with fitted equivalent parameters. The issue of anomalous behavior is examined with the aid of the model.

[1] The transport experiment at the MADE site (a highly heterogeneous aquifer) was investigated extensively in the last 25 years. The longitudinal mass distribution m(x,t) of the observed solute plume differed from the Gaussian shape and displayed strong asymmetry. This is in variance with the prediction of stochastic models of flow and transport in weakly heterogeneous aquifers. In the last decade, we have forwarded a model coined as MIM (multi-indicator), in which the heterogeneous structure consists of blocks of different of different and independent random lognormal K. Thus, the structure is completely characterized by K G (the geometric mean), 2 Y (the logconducitvity variance) and the integral scale I. Flow (uniform in the mean) and advective transport were solved by the semianalytical SCA (self-consistent approximation). The SCA models the travel time of a solute parcel from an injection to a control plane as a sequence of independent time steps, each resulting from the simple solution for isolated blocks surrounded by a uniform matrix. The aim of the article is to determine whether the model could predict the observed mass distribution of MADE ( 2 Y ' 7 based on the most recent direct-push injection logger data), by using the recently collected detailed K data and the observed mean head gradient. It was found that the agreement with the measured plume is quite satisfactory, differences related to incomplete mass recovery, injection condition and ergodicity notwithstanding. It is concluded that the physical mechanism of advection, modeled by the local ADE, and the heterogeneity of K, are able to explain the MADE plume behavior and the stochastic model could predict it.Citation: Fiori, A., G. Dagan, I. Jankovic, and A. Zarlenga (2013), The plume spreading in the MADE transport experiment: Could it be predicted by stochastic models? Water Resour.

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Uniform flow of mean velocity U takes place in a highly heterogeneous, isotropic, aquifer of lognormal conductivity distribution (variance σY2, integral scale I). A conservative solute is injected instantaneously over an area A0 at x = 0, normal to the mean flow, in a flux‐proportional mode. Longitudinal spreading is caused by advection by the fluid velocity and is quantified with the aid of the mass flux μ(t, x) through fixed control planes at x. An equivalent macrodispersivity is defined in terms of the traveltime variance. The flow and transport are solved numerically in three realizations of the conductivity field with σY2 = 2, 4, 8, respectively. The medium is modeled by a collection of a large number N = 100,000 of spherical inclusions whose conductivities are drawn at random. Transport is simulated by tracking 40,000 particles originating at a large injection area (A0 ≃ 2000 I2) and for travel distance x ≤ 121 I. It is found that the mass flux has a highly skewed time distribution because of the late arrival of solute particles that are moving through low‐conductivity blocks. The tail leads to large values of the equivalent macrodispersivity, which is highly dependent on cutoffs corresponding to the arrival of even 0.999 or 0.995 of the total mass. Furthermore, the tail is nonergodic, as it depends on the plume size. Transport appears to be anomalous in the considered interval, although by the central limit theorem it has to tend asymptotically to Fickianity and Gaussianity.

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[1] Three-dimensional advective transport of passive solutes through isotropic porous formations of stationary non-Gaussian log conductivity distributions is investigated by using an approximate semianalytical model, which is compared with accurate numerical simulations. The study is a continuation of our previous works in which formation heterogeneity is modeled using spherical nonoverlapping inclusions and an approximate analytical model was developed. Flow is solved for average uniform velocity, and transport of an ergodic plume is quantified by mass flux (traveltime distribution) at a control plane. The analytical model uses a self-consistent argument, and it is based on the solution for an isolated inclusion submerged in homogeneous background matrix of effective conductivity. As demonstrated in the past, this analytical model accurately predicted the entire distributions of traveltimes in formations of Gaussian log conductivity distributions, as validated by numerical simulations. The present study (1) extends the results to formations of non-Gaussian log conductivity structures (the subordination model), (2) extends the approximate analytical model to cubical blocks that tessellate the entire domain, (3) identifies a condition in conductivity distribution, at the tail of low values, that renders transport anomalous with macrodispersivity growing without bounds, and (4) provides links of our work to continuous time random walk (CTRW) methodology, as applied to subsurface transport. It is found that a class of CTRW solutions proposed in the past cannot be based on solution of flow in formations with conductivity distribution of finite integral scale.Citation: Fiori, A., I. Janković, G. Dagan, and V. Cvetković (2007), Ergodic transport through aquifers of non-Gaussian log conductivity distribution and occurrence of anomalous behavior, Water Resour. Res., 43, W09407,