## 1. Introduction

[2] In the turbulent diffusion field of research, many investigators were focused on analyzing the concentration probability density function (pdf) and its related moments. Those findings range from the exponential [*Lewis and Chatwin*, 1996; *Yee and Chan*, 1997b] distribution to the Clipped-Normal [*Yee and Chan*, 1997a, 1997b], generalized Pareto [*Lewis and Chatwin*, 1996; *Schopflocher and Sullivan*, 2002], Gamma [*Lewis and Chatwin*, 1996; *Klein and Young*; 2011], Clipped-Gamma [*Klein and Young*, 2011; *Yee and Chan*, 1997a; *Yee*, 2008], and α-β [*Mole*, 1995; *Yee and Chan*, 1997b] distribution. It is important to realize that investigators of the atmospheric turbulent diffusion problems had not only numerical tools at their disposal but also extensive data sets obtained through detailed concentration fluctuation experiments in open terrain [*Lewis and Chatwin*, 1996; *Klein and Young*, 2011; *Yee and Chan*, 1997a], wind tunnels [*Schopflocher and Sullivan*, 2005; *Yee et al*., 2006], and water channels [*Yee and Wilson*, 2000; *Yee et al*., 2006; *Yee*, 2008, 2009]. One of the general findings in these studies is the fact that higher-order concentration moments do provide valuable additional information for the shape of the concentration pdf.

[3] Furthermore, the studies conducted by *Yee* [2008, 2009] and *Yee and Chan* [1997a, 1997b], using a data set of large concentration fluctuations collected under different conditions, showed a remarkable collapse of data on a single curve when comparing various higher-order normalized concentration moments (*n* = 3, 4, 5, 6, 7, and 8) versus the second-order normalized concentration moment . With *c* we denote the concentration variable while *C* presents the concentration ensemble mean.

[4] Concentration moments of solute released through aquifers, atmosphere, and water channels have been of great interest in the last few decades. The analyses of concentration moments in heterogeneous aquifers have been considered in a few studies [*Andricevic*, 1998; *Bellin et al*., 1994; *Caroni and Fiorotto*, 2005; *Dagan and Fiori*, 1997; *Fiori and Dagan*, 2000; *Fiori*, 2003; *Fiorotto and Caroni*, 2002; *Kapoor and Kitanidis*, 1998]. In an early attempt to characterize groundwater contaminant transport, *Dagan* [1982] showed that in the absence of local diffusion and for point sampling, the concentration pdf is a two-state process of initial concentration, *C*_{0}, or zero. The same feature is valid even in the atmosphere and water channel flow. *Bellin et al*. [1994] analyzed the cumulative distribution function (cdf) and the first two concentration moments as a function of sampling size. In later works, local diffusion is considered and its influence is analyzed through the first two moments, concentration mean and variance [*Andricevic*, 1998; *Caroni and Fiorotto*, 2005; *Dagan and Fiori*, 1997; *Fiori and Dagan*, 1999, 2000; *Fiori*, 2003; *Fiorotto and Caroni*, 2002; *Tonina and Bellin*, 2008], in both Eulerian and Lagrangian frameworks. The calculation of concentration moments of an order higher than two is commonly constrained by a number of physical (number of measurements and its scale) and numerical reasons (number of realizations, extensive computational resources, and numerical accuracy). In order to avoid some of the above-mentioned facts, one needs to increase the number of realizations. This procedure is time consuming and reveals the necessity for finding effective tools that can determine higher-order moment values based on information about lower order ones.

[5] Knowledge of the statistical properties of concentration fluctuations in a moving plume is important in assessing the risk from adverse effects of certain highly toxic chemicals, ranging from industrial chemicals to even certain chemical warfare agents. Risk assessment studies require the knowledge of concentration fluctuations [*Andricevic et al*., 2012; *Tartakovsky*, 2007], which are described by one-point concentration pdf. At any point in space and time, the concentration pdf contains all the information about concentration fluctuations and embodies all the higher-order concentration moments.

[6] Following the first suggestion of a Beta distribution for the subsurface concentration made by *Fiori* [2001], *Caroni and Fiorotto* [2005] showed the applicability of Beta distribution and a good fit for Monte Carlo (MC) results conducted in 2-D heterogeneous aquifer. *Bellin and Tonina* [2007] confirmed results by *Caroni and Fiorotto* [2005] showing Ito Stochastic Differential Equation leads to Beta pdf. Also, in comparison with Gaussian and Log-normal distribution, Beta showed better features in order to capture the ensemble fluctuations. *Schwede et al*. [2008] reached very good agreement with Beta by semianalytical method even in 3-D, but for unity and multi-Gaussian (MG) field. Joint velocity-concentration pdf method has been developed by *Meyer et al*. [2010] and compared with results by *Caroni and Fiorotto* [2005] up to . The accuracy of this method is shown to decrease for *Pe* ≤ 100 when compared with standard MC. All the above-mentioned results indicate the asymmetric shape of the concentration pdf [*Cirpka et al*., 2011b], for a wide range of parameters. Although the Beta distribution has shown relatively good features in capturing concentration fluctuations phenomena, some limitations have been noted [*Bellin and Tonina*, 2007; *Caroni and Fiorotto*, 2005]. For the sometimes complex structure of the concentration field, the mean and variance may not be sufficient and higher-order statistical moments are required to accurately define the concentration pdf shape. Furthermore, most of the concentration findings in groundwater transport research to date have been focused on a lower range of aquifer heterogeneity and a common MG log-conductivity field.

[7] The objective of this paper is to investigate the features of the statistical properties of a plume spreading in a heterogeneous aquifer as manifested through the higher-order concentration moments (third and fourth), and their relationship to second-order normalized ones. In view of this, the question we pose is as follows: Does the concentration in a groundwater plume transported through aquifers of high heterogeneity, different *Pe* values and different log-conductivity fields exhibit a collapse of the higher-order concentration moments on the second-order concentration moment? If so, we can use this collapse feature to obtain moments of a higher order than the second one, solely from information about the first two moments.

[8] To address above questions, we will employ the two-dimensional Monte Carlo numerical experiments. Using our recently presented simulation methodology, Adaptive Fup Monte Carlo Method (AFMCM) [*Gotovac et al*., 2007, 2009a]; this methodology supports the Eulerian-Lagrangian formulation and separates the flow from the transport problem. A heterogeneous aquifer is modeled with a log-conductivity variance ranging from 1 to 8 (e.g., describing highly heterogeneous cases), including three fundamentally different log-conductivity fields [*Zinn and Harvey*, 2003], and *Pe* value ranging from 100 to 10,000.