## 1. Introduction

[2] The mechanism of spreading of solutes (macrodispersion) in transport through porous formations of spatially variable hydraulic conductivity *K* is dominated by the advective effect related to velocity variations. Considerable research has been carried out in the last two decades in order to investigate the relationship between heterogeneity and transport. In applications, the interest in the problem stems from its relevance to contaminant transport in aquifers. From a theoretical standpoint the topic is at the cross road between two central fields of engineering and physics: the solution of the flow problem has common ground with continuum theories of heterogeneous media while that of transport is intimately related to turbulent diffusion. It is therefore quite appropriate to include this topic in the present special issue.

[3] Because of the seemingly erratic spatial variation of *K* and scarcity of field data, it is customary to model it as a space random function and similarly for flow and transport variables. It is important to emphasize that in hydrological applications (unlike controlled laboratory experiments), there is large uncertainty in characterizing even the statistical structure of *K*. Thus understanding of the main mechanisms and development of simple, though approximate, models is highly desirable. Sophisticated and complex models, though of theoretical interest, may be of little use in practice in absence of supporting data.

[4] With these goals in mind, we consider here the simplest configuration: *K* is a three-dimensional isotropic stationary space random function, with *Y* = ln *K* of univariate normal distribution (〈*Y*〉 = ln *K*_{G}, *var*(*Y*) = σ_{Y}^{2}, *K*_{G} geometric mean) while *I* stands for the finite integral scale of *Y*; the flow is steady and uniform in the mean (pertinent to natural gradient flow); the plume of the conservative solute is of large transversal dimension as compared to *I*, to make statistical properties of transport weakly dependent on the particular realization.

[5] In the past numerical and analytical solutions were achieved primarily for weak heterogeneity (σ_{Y}^{2} < 1); the large body of literature is reviewed in a few books [e.g., *Dagan*, 1989; *Gelhar*, 1994; *Rubin*, 2003]. Numerical solutions for moderate to highly heterogeneous formations were achieved mainly for two-dimensional structures [*Bellin et al.*, 1992; *Salandin and Fiorotto*, 1998; *Wen and Gomez-Hernandez*, 1998; *Zinn and Harvey*, 2003; *Trefry et al.*, 2003].

[6] In the present study we investigate the effect of 3-D heterogeneity, as encountered in aquifer applications. Although useful information may be derived from 2-D solutions, the conclusions may not apply to actual formations [see *Janković et al.*, 2003b]. Besides, we consider high degree of heterogeneity (σ_{Y}^{2} ≤ 8), as found in some aquifers. Beyond such applications, understanding the effect of large σ_{Y}^{2} casts light on the applicability of common approaches, that were developed for weak heterogeneity. Few simulations of three-dimensional flows were carried out in the past [*Tompson and Gelhar*, 1990; *Chin*, 1997] under limiting conditions: relatively small dimension of the solute plume, effect of domain boundaries and limited accuracy of the numerical codes.

[7] Our interest resides in solving flow and transport for a given *Y* field (in a statistical sense) and we do not refer here to approaches in which transport is not directly related to the underlying permeability field.

[8] The highly accurate numerical solution is derived for a solute plume of large extent (transverse area of 2000*I*^{2}) and for a relatively large travel distance from the source (121*I*). This is achieved by using the structure model we coined as “multi-indicator” in the past: the medium is made up from a large number *N* = 10^{5} of blocks of constant *K*_{j} (*j* = 1, …, *N*) that are drawn independently from a lognormal distribution. For simplicity, they are spheres that are submerged in a homogenous matrix of conductivity equal to the effective one. While highly idealized, such a structure may represent, by selecting the radii *R*, the volume density *n* and the variance σ_{Y}^{2}, any isotropic aquifer of given σ_{Y}^{2} and *I*. Although multi-indicator model is capable of reproducing any actual covariance structure by adopting specific distributions of conductivities and radii, the present study focuses on constant R. The impacts of the variable radii on flow and transport results are expected to increase as heterogeneity levels are increased. This topic, including the inability of second-order properties of Y (〈*Y*〉, σ_{Y}^{2} and *I*) to fully capture transport, is deferred to future studies.

[9] There are alternative spatial distributions that satisfy the same constraints (e.g., the multi-Gaussian one; see discussion by *Dagan et al.* [2003]). The great advantage of the adopted structural model is that it permits one to solve flow in large domains and with high accuracy, by the use of the analytic element method [*Strack*, 1989]. We doubt this is feasible at present with the more conventional numerical methods (finite differences, finite elements).

[10] We have solved the same problem previously [*Janković et al.*, 2003a, 2003b]. The main additional features of the present study are rather than concentrating on the second spatial moment and the related macrodispersion, we analyze here the entire longitudinal mass distribution of the plume, with emphasis on its skewness; the domain and plume sizes and the travel distance are larger, permitting us to investigate ergodicity issues and anomalous behavior; the solute injection mode is different (flux proportional rather than uniform); transport is analyzed in terms of BTC (breakthrough curve) at fixed control planes rather than spatial moments. Besides, the approximate semianalytical model is extended in part 2 of the study [*Fiori et al.*, 2006] to describe the complete BTC and the tailing effect.

[11] Our main aim is to achieve a better understanding of the mechanisms affecting transport at high σ_{Y}^{2} and to capture them by simple models. Extension to anisotropic media and to more complex geometries, that resemble actual aquifers, is deferred for future studies.

[12] The plan of the paper is as follows: in section 2 we describe the modeling of the solute plume, and we summarize the definitions of ergodicity, Gaussianity, Fickianity, and anomalous behavior. Section 3 is devoted to the description of the numerical simulations; numerical results are presented in Section 4, and Section 5 concludes the paper. Part 2 of the study [*Fiori et al.*, 2006, hereinafter referred to as P2] presents an approximate semianalytical model and the summarizing discussion of the two parts.