## 1. Introduction

[2] Solute transport in porous formations is ruled by the spatial distribution of the hydraulic properties, the most relevant being the hydraulic conductivity *K*. The last four decades have seen a tremendous effort by the scientific community in order to adequately capture the relevant structural features which govern solute transport in complex subsurface environments (see e.g., the monographs by *Dagan* [1989] and *Rubin* [2003]). A common assumption is to model *K* as a space random function, following experimental evidence, *Y* = ln*K* is modeled as a second-order stationary normal random variable, with geometric mean *K _{G}*, variance , and two-point covariance

*C*

_{Y}_{.}

[3] A successful and widely employed approach has been the first-order one, which is formally valid for small/moderate heterogeneity, i.e., . Despite the latter limitation, the first-order method has led to useful relations between the statistical moments of the concentration field *C* and the main flow/structure parameters, e.g., the mean velocity, the statistical moments of *K*, and the injection conditions (for a comprehensive review on the subject the reader is addressed to the previously cited monographs). The derivation of simple relations is generally not possible by numerical methods, which although are virtually free from the assumption often require a huge computational burden. Instead, the development of analytical or semianalytical solutions is important for applications, especially for a quick estimate of the concentration field, for their simplicity in the input parameters, easy software implementation and computational effort (see e.g., the popular codes BIOSCREEN [*Newell et al*., 1996] and BIOCHLOR [*Aziz et al*., 2000]).

[4] This paper focuses on approximate, semianalytical, and analytical solutions for 3-D steady transport of conservative solutes under mean uniform flow. Steady plumes, i.e., in which the concentration *C* does not depend on time, may occur for instance from continuous releases of a contaminant from a source area. Field investigations on contaminated sites, where the plume derives from old contamination events, often show that the plume is at steady state. Models for the prediction of the local concentration of steady plumes are important for many applications, e.g., those related to biodegradation of organic compounds [e.g., *Liedl et al*., 2005; *Maier and Grathwohl*, 2006; *Cirpka and Valocchi*, 2007; *Gutierrez-Neri et al*., 2009; *Cirpka et al*., 2011]; also, models for conservative solute constitute the fundamental prerequisite for building more elaborated models which embed different complex biochemical processes.

[5] Analytical or semianalytical solutions are available for the simple case of homogeneous formations [e.g., *Domenico*, 1987; *Wexler*, 1992]. Our interest here is in transport through heterogeneous porous formations, with spatially variable and random *K*. As a consequence, the concentration *C* is also random, and our main target is the calculation of its mean and variance as function of the relevant flow/structure parameters. Models of transport which explicitly consider heterogeneity are important for a few reasons. First, they provide a measure of both the mean of *C* and its uncertainty, measured by *σ _{C}*; the uncertainty analysis is quite useful in applications, and even more so in engineering applications which require a description of

*C*in probabilistic terms (e.g., risk assessment). Second, the analysis of transport in heterogeneous formations helps in elucidating fundamental issues, like e.g., the interplay between large-scale advection and local-scale dispersion/diffusion, their impact on the local dilution features (which are important in presence of additional biochemical processes), and allows distinguishing between macrodispersion and the actual decay of concentration. We are not aware of analytical models specifically designed for steady 3-D transport in heterogeneous formations; conversely, different stochastic approaches (e.g., Eulerian or Lagrangian) have been adopted in the last decades for the development of models for transient transport after instantaneous solute injection.

[6] In the present work, we shall derive the first two statistical moments of concentration for steady 3-D transport by adopting the first-order Lagrangian approach [see e.g., *Dagan*, 1989; *Fiori and Dagan*, 2000]. The plan of the paper is as follows. The general mathematical framework is introduced first; then, the first-order analysis is formally adopted and introduced in the principal expressions. Semianalytical and analytical expressions are then explicitly derived, with different levels of approximation. A set of illustration examples serves as vehicle for discussion. Then, the conclusions section ends the paper.