## 1. Introduction

[2] Fractures are the main responsible of contaminant transport in low-permeability formations because they offer preferential pathways to water flow and associated advective transport in the host rock [*Painter and Cvetkovic*, 2005]. For this reason, the quantitative analyses of the physical and chemical processes affecting transport through fractured media play a fundamental role in the safety assessments of disposal facilities of hazardous wastes. Typical examples of such facilities are land sites and deep wells for chemical or low-level radioactive wastes, or geological repositories for high-level nuclear wastes [*Tang et al.*, 1981; *Berkowitz and Scher*, 1997; *Benke and Painter*, 2003; *Delay et al.*, 2005].

[3] To address these issues, in this paper we propose both a novel particle tracking scheme for solute transport in fractured media and an original two-step procedure for the identification of its parameters from measured data. Our work is motivated by the fact that, regardless the rich literature on the subject, quantitative analyses of solute transport in fractured media are still challenging tasks, due to their inherent complexities. In particular, two issues contribute most to the general difficulty of the problem:

[4] (1) It is difficult, if not impossible, to obtain complete and detailed maps of the subsurface fractures; thus, the analyses must rely on stochastic representations of the fracture network based on observations of exposed fractures [*Odling*, 1997; *Berkowitz and Scher*, 1998; *Davy et al.*, 2010]. These representations are complicated by the fact that fracture sizes range from micrometers to kilometers and fracture patterns range from relatively regular polygons to apparently random distributions [*Berkowitz and Scher*, 1998; *Bonnet et al.*, 2001].

[5] (2) In such heterogeneous domains, the mechanical mixing due to local, irregular advective displacements cannot be represented as a Fickian-like diffusion process, as assumed in most predictive models based on some form of the advection-dispersion equation (ADE), and the dispersivities are observed to grow with the solute residence time or displacement distance, as opposed to the classical ADE assumptions of constant flow fields and dispersion coefficients [*Sudicky et al.*, 1985; *Gelhar*, 1993; *Berkowitz and Scher*, 1997; *Berkowitz and Scher*, 1998; *Berkowitz et al.*, 2001; *Berkowitz and Scher*, 2001; *Berkowitz et al.*, 2002; *Dentz et al.*, 2004]. This phenomenon, generally known as “scale-dependent dispersion,” “anomalous transport,” or “non-Fickian” behavior, has been shown to be due to heterogeneities in the velocity field at every scale of observation: (i) the microscale of the pores [*Berkowitz et al.*, 2006]; (ii) the local scale of a single fracture [*Sudicky et al.*, 1985]; (iii) the large scale of the entire fracture network [*Williams*, 1992; *Berkowitz and Scher*, 1997; *Painter et al.*, 2002]. Moreover, chemical interactions, i.e., adsorption/desorption processes between the solutes and the rock matrix and/or colloidal particles, are also shown to be responsible of non-Fickian contaminant migration.

[6] Some authors addressed the problem of quantifying non-Fickian transport by resorting to stochastic generalizations of ADEs, for example by spatially varying velocity fields [*Rubin*, 1990; *Gelhar*, 1993]. The use of these methods is suitable when many site-specific geological observations are available, which is rarely true [*Gao et al.*, 2010].

[7] In many applications, the most practical choice is to resort to transport modeling approaches based on the deterministic ADE. Then, efforts have been devoted to the development of models which (i) explicitly incorporate empirical time- or space-dependent dispersivities into the ADE [*Pickens and Grisak*, 1981; *Gao et al.*, 2010] and (ii) generalize the ADE by including multicontinua/multirate concepts for both water flow and/or solute transport [*Barenblatt et al.*, 1960; *Warren and Root*, 1963], like mobile-immobile [*van Genuchten and Wierenga*, 1976], dual-porosity [*Simunek et al.*, 2003], dual-permeability [*Gerke and van Genuchten*, 1993a, 1993b], multirate [*Haggerty and Gorelick*, 1995], etc., where a common mathematical framework [*Haggerty and Gorelick*, 1995] is adopted to represent mass (water and/or solute) exchanges between different phases or, within the same phase, different velocity regions (see *Simunek and van Genuchten* [2008] for a comprehensive categorization of all these methods). It is to be noted, however, that all these models are based, explicitly or implicitly, to an underlying Fickian transport behavior at some scale [*Berkowitz et al.*, 2002].

[8] Recently, the continuous time random walk (CTRW) theory has been proposed as an alternative (i.e., not ADE-based) description of non-Fickian transport at different scales, both in fractured [*Berkowitz and Scher*, 1997; *Berkowitz and Scher*, 1998] and heterogeneous media in general [*Berkowitz et al.*, 2001; *Berkowitz and Scher*, 2001; *Berkowitz et al.*, 2002; *Dentz et al.*, 2004]. In the CTRW methods, the transport of a solute particle is modeled as a random walk in space and time, with the mass transfers being described in terms of a joint probability distribution of the displacements and the time . In practical applications, problems may arise in identifying this distribution, which should reflect the medium heterogeneities at the proper scales and account for any additional relevant influence on the transfer process. Another, not ADE-based modeling framework, alternative to the CTRW, was proposed by*Williams* [1992], where the actual motion of solute particles was intuitively described in analogy with neutron transport.

[9] In all generality, the differential equations of the model adopted have to be solved to give the contaminant concentrations in time and space. The solution of the model equations represents another difficult task.

[10] For a single, longitudinal fracture, i.e., the simplest type of heterogeneity made of a perfect stratification with continuous layers of different hydraulic conductivity [*Sudicky*, 1983], analytic approaches have been proposed which are based on some form of the ADE. For example, *Neretnieks* [1980] proposed an analytic solution for contaminant transport in a single fracture under the assumption that dispersion and diffusion along the fracture are negligible; *Tang et al.* [1981] proposed a more general analytic solution approach taking into account all dispersive and diffusive processes, but where transversal diffusion occurs in an infinite matrix; *Sudicky and Frind* [1982] extended such solution to the finite matrix case, including also sorption and radioactive decay; *Zhan et al.* [2009]also included the non-negligible transverse dispersion in the fracture.

[11] Analytic solutions have the advantage of providing an explicit physical explanation of the underlying driving processes. However, for realistic domains, often characterized by complex boundary conditions, analytic solutions are difficult to obtain, if not impossible; moreover, analytic solutions often rely on a Laplace transformation of the relevant flow and transport equations, so that complex numerical antitransformations may be required. In these cases, the adoption of numerical schemes becomes more attractive.

[12] Numerical schemes can in general be divided in two categories, i.e., Eulerian and Lagrangian methods. Eulerian approaches to solve the transport equations are based on the application of finite difference or finite element methods [*Grisak and Pickens*, 1980; *Salamon et al.*, 2006b]. A wide spectrum of alternatives were proposed in the literature and several well-established commercial tools are available. Specific numerical tools were developed to solve dual-continua and multirate models, also in variably saturated conditions (e.g., HYDRUS-2-D [*Simunek and van Genuchten*, 2008]). The major drawbacks of Eulerian methods are that (i) mass conservation is not guaranteed and (ii) they may require very fine discretization of the transport domain to reduce instabilities of the numerical solutions and numerical dispersion, thus leading to large computational expenses, in particular for advection-dominated problems [*Delay et al.*, 2005; *Salamon et al.*, 2006b].

[13] To address these issues, several Langrangian (or mixed Eulerian-Lagrangian) random walk particle tracking (RWPT) methods were developed, based on the simulation of the motion of solute particles within the flow field. In the first particle tracking methods proposed, i.e., the space-based particle tracking methods, the particles are moved by random spatial displacements at specified time steps. The most popular of these approaches is the one based on the Fokker-Planck-Kolmogorov Equation (FPKE) [*Kinzelbach*, 1987; *Uffink*, 1990; *Delay et al.*, 2005], whose first application to hydrologic problems dates back to the late 1970s and early 1980s. Further extensions of the FPKE-based particle method were developed in the following years to include multicontinua and/or multirate transport representations. For example, a particle tracking algorithm was applied by*Kinzelbach* [1987]within a mobile-immobile scheme for solute transport in single longitudinal fractures;*Michalak and Kitianidis* [2000]proposed an analogous RWPT method for handling single-rate, first order mass transfer with no mesh cell communication, whereas*Salamon et al.* [2006a]extended it to the multiple-rate mass transfer case; recently, with regards to fractured porous media,*Liu et al.* [2000], *Pan and Bodvarsson* [2002], *Hassan and Mohamed* [2003], and *Roubinet et al.* [2010]improved RWPT modeling of contaminant transport in a dual-continua representation by devising fracture-matrix transfer probabilities which account also for diffusion into the matrix blocks in irregular fracture grids.

[14] Other particle tracking approaches are based on the ordinary Langevin equation: They are less common in transport problems in natural porous media, although they are well suited for applications involving colloid transport. It can be shown that the Langevin equation reduces to the classical FPKE under certain simplifying assumptions [*Uffink*, 1990; *Delay et al.*, 2005].

[15] In very heterogeneous media, such as fractured rocks, space-based random walk calculations may become very time consuming, mainly due to the large number of jumps that have to be simulated for a particle moving in regions of very low velocities [*Delay et al.*, 2005]. More recently, time domain particle methods were shown to offer a possible solution to this problem, consisting of moving the particles by constant spatial displacements at randomly sampled times: This way the particles cross a constant-velocity segment in a single step, with an increase in the computational efficiency [*Wen and Gomez-Hernandez*, 1996; *Delay et al.*, 2005; *Painter et al.*, 2008].

[16] In this paper we present an alternative approach to the modeling of solute transport in a fractured medium based on an original extension of the Kolmogorov-Dmitriev (KD) stochastic branching model [*Kolmogorov and Dmitriev*, 1947] applied to the transport of contaminants in groundwater [*Marseguerra and Zio*, 1997]. The model aims at describing the stochastic transport of the populations of solute particles in the fractures and host matrix by means of a discrete-states, continuous time Markov process based on properly defined transition probability distribution functions. Partial differential equations, known as the forward Kolmogorov equations (FKEs), are then derived for the expected values of the different particles' populations, which are proportional to the solute concentrations. The Markovian formalism then allows a simple implementation of a particle tracking algorithm for solving the FKEs.

[17] In particular, at the scale of a single, longitudinal fracture, the FKEs take a form similar to that of the coupled differential equations of a classical dual-continua description of the migration process [*Simunek and van Genuchten*, 2008]. In principle, this analogy can allow the identification of the transition rates of the stochastic model as a function of the dual-continua model parameters, as originally proposed by*Ferrara et al.* [1999]for a standard, single-continuum problem.

[18] The Kolmogorov-Dmitriev particle tracking (KDPT) scheme is here applied to model the solute transport in a single-fracture medium, corresponding to the setting of an experimental case study of literature [*Sudicky et al.*, 1985]. The model transition rates for the solute migration within the fracture and the matrix are then identified by a two-step procedure which first exploits the analogy with the corresponding dual-continua representation and then a genetic algorithm for optimal fitting to the experimental data. The proposed modeling scheme and parameter estimation procedure are shown to effectively reproduce the anomalous transport features in the simple heterogeneous layout experimentally observed at the scale of the single fracture.

[19] The paper is organized as follows. In section 2the modeling of branching stochastic processes is specialized to a one-dimensional, dual-continua transport domain and the operative particle tracking simulation procedure for solute concentration estimation is described. Insection 3the method is applied to the experimental data from the single-fracture case study of literature. Conclusions on the capabilities and limitations of the proposed modeling scheme and parameter estimation procedure are drawn insection 4.