## 1. Introduction

[2] Transport parameters are often obtained from interpreting the temporal evolution of concentrations at a given location or volume control section (breakthrough curves (BTCs)). The actual estimated parameters are model dependent. The conventional approach based on the advection-dispersion equation (ADE) [*Bear*, 1972] has been shown consistently to fail at completely predicting data obtained from real sites. In such cases, transport is called “anomalous” or non-Fickian. Many authors have postulated that non-Fickianity is a consequence of the presence of heterogeneity in hydraulic parameters [e.g., *Levy and Berkowitz*, 2003; *Salamon et al*., 2007; *Riva et al*., 2008].

[3] Phenomenological evidence of this effect of heterogeneity comes from observations of heavy-tailed distributions on BTCs [e.g., *Hoehn et al*., 1998; *Fernàndez-Garcia et al*., 2004; *Gouze et al*., 2008]. Sometimes, BTC tails at late times scale like power laws (PLs) of the form (long after the peak is observed). The parameter *m* is often called the “BTC slope” since PL distributions scale as straight lines in double log plots; in the literature *m* has been reported to range anywhere between one and three [e.g., *Becker and Shapiro*, 2000, 2003; *McKenna et al*., 2001].

[4] A general current goal is to find strict relationships between PL-shaped BTCs and specific spatiotemporal distributions of physical soil properties [e.g., *Dentz and Berkowitz*, 2003; *Bijeljic and Blunt*, 2006; *Willmann et al*., 2008; *Dentz and Bolster*, 2010]. In this paper, our aim is to show that heavy-tailed BTCs with PL late-time distributions can be found in finite-scale stationary hydraulic conductivity, *K*, fields, provided a third spatial dimension is accounted for and flow is convergent to a well.

[5] Convergent flow appears naturally in describing many flow configurations from real experiments. In fact, the most common form of tracer test is perhaps the convergent flow tracer test (CFTT). This is for practical reasons, such as a better control of test duration, reduced required tracer mass and large mass recoveries (compared, for example, with natural-gradient flow tracer tests).

[6] Classical examples of stationary fields include multi-Gaussian fields with log-transformed hydraulic conductivity. One of the conclusions in *Willmann et al*. [2008] is that no PL-shaped late-time behavior of BTCs could be observed from transport through 2-D stationary multi-Gaussian fields, unless an artificial modification of the conductivity field was made. Nonetheless, the presence of heavy-tailed BTCs for short travel distances in multi-Gaussian fields with log-transformed hydraulic conductivity (called *Y* field, where ) was shown by *Sanchez-Vila and Carrera* [2004] using 1-D analytical and numerical solutions.

[7] An example of BTCs that can be observed in heterogeneous media was given by *Fernàndez-Garcia et al*. [2004], who experimentally studied convergent, divergent, and uniform flow tracer tests in an intermediate-scale three-dimensional heterogeneous aquifer constructed in the laboratory with different types of sands. The resulting *K* distribution showed correlated structures, well described by an exponential variogram model. Several conservative (bromide) and sorptive (lithium) tracers were injected at different points located between one and five horizontal integral scales from the extraction well. Concentrations were recorded as depth-integrated BTCs. Four of the experimental BTCs obtained during the CFTTs using a deep-penetration source injection [*Fernàndez-Garcia*, 2003] are shown in Figure 1 and clearly display nonuniform transport behavior. We observe that (a) the BTC obtained from injecting at E1 (located two integral scales away from the pumping well) displayed a heavy-tailed distribution after the peak, well approximated by a PL with *m* = 1; (b) at the same radial distance from the pumping well, the BTCs obtained from injections at E2 and E3 were less anomalous and quite symmetric; and (c) E4, also at the same injection distance, is much more irregular and only approximately similar to a PL with .

[8] These results lead us to ask the following questions: Does scaling in BTCs with a PL with *m* = 1 occur for some specific physical reasons, or is it just a random output? More generally, are there any, and if so what are the physical mechanisms controlling BTC scaling, for large time after injection in radial convergent flow?

[9] *Willmann et al*. [2008] tried to answer to the latter question, using 2-D numerical simulations to reproduce anomalous transport under the assumptions of uniform flow conditions and finite correlated heterogeneous log conductivity fields (i.e., fields with log-normally distributed hydraulic conductivity). They concluded that classical low-order geostatistical indicators (such as variograms), usually adopted to characterize these fields, cannot be directly related to parameters associated with anomalous transport models such as those including memory functions [*Carrera et al*., 1998]. Similar results were also found by *Flach* [2012], who used dual-domain models.

[10] According to *Zinn and Harvey* [2003] and *Willmann et al*. [2008], among others, connectivity is the hydrodynamic parameter that most influences BTC late-time behavior. Here connectivity is defined as the ratio between spatially averaged and effective parameters defining flow and transport [*Knudby and Carrera*, 2006]. We refer to the recent work by *Renard and Allard* [2013] for an extensive review of connectivity concepts.

[11] In the simulations of *Willmann et al*. [2008], the BTC slopes were bounded by a minimum value of , which is not in agreement with the experimental observations reported by *Fernàndez-Garcia et al*. [2004]. It is not truly fair though to compare these works for two main reasons. First, radially converging flow is fundamentally different from uniform flow. For instance, tailing can naturally arise in BTCs, even for homogeneous fields [*Gelhar and Collins*, 1971; *Moench*, 1989; *Welty and Gelhar*, 1994]. Second, 2-D models are not suitable to reproduce this type of CFTT, where the distance between injection and controlling section is on the range of the representative heterogeneity scale such as the integral scale (*I*) in a multi-Gaussian *K* field. In this case, whenever possible, 3-D models are preferable [e.g., *Dagan*, 1989].

[12] Dimensionality is also a key factor when studying connectivity. Inclusion models [e.g., *Fiori et al*., 2006] show, for instance, that nonsymmetric BTCs naturally arise from the distribution of travel times in 3-D models under uniform flow; moreover, this method also showed that (flow) connectivity is largely enhanced by 3-D configurations relative to their 2-D counterparts [*Fiori and Jankovic*, 2012]. This is particularly true if unconditional sequential Gaussian simulations (SGSs) are used to simulate stochastic hydraulic conductivity fields [*Fernàndez-Garcia et al*., 2010]. In fact, *Willmann et al*. [2008] had to heavily condition their simulations to generate highly conductive nonstationary structures that gave rise to heavy-tailed BTCs similar to those observed in the field. It should be noted that the numerical 3-D flow and transport simulations by *Fogg* [1986] showed that one of the most influential factors controlling flow and transport is the connectivity of lenses, rather than the relative *K* value of the lenses themselves. The reliability of 2-D SGS to adequately reproduce transport connectivity patterns has been extensively debated in the past [e.g., *Sanchez-Vila et al*., 1996; *Gomez-Hernandez and Wen*, 1998]; however, 3-D models have received much less study, in large part due to the still highly computationally intensive nature of 3-D simulations.

[13] In this paper, we investigated the origin and development of heavy-tailed BTCs using 3-D numerical realizations under radially convergent flow conditions. We simulated synthetic heterogeneous fields drawn from a multi-Gaussian log-normal *K* distribution. Such a distribution is characterized by different combinations of finite-scale correlation and variances. Our aim was to provide new insights to explain how and why BTCs behave in typical CFTT field settings. We also aimed to find the key physical links to interpret the results obtained from field tracer tests in real applications.

[14] This paper is structured as follows. In section 2, we described the numerical approach we used to reproduce CFTTs in synthetic heterogeneous aquifers. Section 3 shows the results in which we highlighted how BTC tailing develops in different heterogeneous fields. This paper ends with a final discussion on the similarities between 3-D simulations and their corresponding 2-D counterparts, the role of the local dispersivity, and a possible physical explanation of PL scaling on the BTCs, in section 4, and the conclusions.