## 1. Introduction

[2] Over the last 30 years considerable progress has been made to incorporate spatial variability of hydraulic conductivity fields and the effects of this variability such as the variability of the groundwater velocity field into stochastic models of solute transport (among many others: *Dagan* [1982, 1984, 1988, 1990]; *Gelhar and Axness* [1983]; *Neuman and Zhang* [1990]; *Rubin* [1990]; *Burr et al.* [1994]). In all these approaches, the spatial variability of the hydraulic conductivity field has been modeled with a Gaussian dependence structure. Among others, *Gómez-Hernández and Wen* [1998]pointed out the importance of non-Gaussian spatial structures for evaluating the above mentioned effects of spatial variability, in their case on groundwater travel times. Approaches exist for modeling non-Gaussian structures, for example via training images [*Strebelle*, 2002] or transition probabilities [*Carle and Fogg*, 1996]. *Zinn and Harvey* [2003]were able to model one type of non-Gaussian spatial dependence using a Chi-Square transformation, not a full copula model. This paper presents an approach where the dependence model, a multidimensional spatial copula, is fitted to the spatial dependence structure of real world data.

[3] Spatial copulas as introduced by *Bárdossy* [2006]provide full stochastic models to represent spatial dependence structures independently of marginal distributions. The distribution that fits best to the observed data can be used, and it can be nonsymmetric. Log-transformations, commonly used to ensure a symmetric marginal distribution of hydraulic conductivity (K) in variogram-based geostatistics are not necessary. In this work, two copula models are fitted to a dependence structure based on a real world K data set from a well characterized field site located at the C.F.B. Borden, Canada [*Sudicky*, 1986]: a Gaussian and a non-Gaussian copula. Both are designed such that they are not distinguishable by second-order moments (their spatial covariance functions) and they have identical marginal distributions. The impacts of these two types of spatial dependence structures of K on solute transport behavior will be tested in a series of detailed numerical tracer tests, evaluated using a Monte Carlo approach.

[4] A numerical tracer test is a tool used to evaluate the migration of solutes in heterogeneous aquifers [*Frind et al.*, 1987; *Smith and Schwartz*, 1980; *Naff et al.*, 1998]. A slug of conservative tracer is instantaneously injected and transported within a steady state flow field. The resulting concentration field is recorded and its spatial moments analyzed. By classical theory [*Bear*, 1972; *Gelhar and Axness*, 1983], the slope at the linear proportion of the second central spatial moment of a concentration field can be related to the dispersion coefficient. The advective spreading of a solute migrating in the subsurface is influenced by the groundwater velocity field, which in turn is directly impacted by the spatial distribution of K. Due to this spreading, portions of the plume advance more rapidly than the average velocity in some zones, while in other zones migration rates are slower than the average velocity. This spreading phenomenon is commonly referred to as macrodispersion. Here, solute spreading will be examined in both two- and three-dimensional high-resolution K fields.

[5] The evaluation of the solute migration is performed in a Monte Carlo style, thus enabling a statistical and hence more conclusive evaluation of the spreading process. Many realizations of K fields were generated for each type of spatial dependence of K, and for each K field, a numerical tracer experiment was conducted. For the purpose of this paper, local dispersion is reduced as much as numerically possible and the modeling domain is finely discretized in order to avoid numerical dispersion and oscillations.

[6] In the Borden aquifer, one of the most extensively studied aquifers of the world, hydraulic conductivity measurements were performed in great detail [*Sudicky*, 1986]. The Borden aquifer is considered to be relatively homogeneous with a multi-Gaussian spatial dependence structure. Traditional geostatistical analyses have been conducted on this data set, most notably by*Sudicky* [1986] and by *Woodbury and Sudicky* [1991]. *Sudicky* [1986] examined 32 cores spaced 1m apart along two orthogonally intersecting lines AA' and BB', and subsampled each 1.75 m long core in 0.05 m intervals. A permeameter test was conducted on each subsample resulting in 1152 measurements of K.

[7] The idea behind this paper is that if a non-Gaussian spatial dependence structure of K, as modeled by a fitted copula, leads to a different solute transport behavior than when it does based on Gaussian K or ln(K) assumptions, then this will have implications for solute transport in most other aquifers. Furthermore, if such deviations are found, other hydrogeological parameters that depend on the spatial structure of K could be impacted.

[8] The objective of this paper is two fold: (1) to use copulas as a non-Gaussian stochastic model to simulate spatially correlated random fields of K based on real world data, and (2) to analyze the effects of non-Gaussian dependence solute plume evolution within such K fields.

[9] The organization of this paper is as follows. As a first step, the Borden data are analyzed geostatistically with copula-based methods and compared with traditional methods. A theoretical copula function is fitted to the data (section 2). This theoretical copula model of spatial dependence is used for the simulation of K fields, which are subsequently used as input for a steady state flow and transient solute transport model (section 3). The characteristics of the solute transport plume are analyzed (section 4) and conclusions are made (section 5).