## 1. Introduction

[2] Passive tracer transport through heterogeneous media has been found to be non-Fickian or “anomalous” by many researchers [e.g., *Cushman*, 1990; *Neuman*, 1993; *Haggerty et al.*, 2000; *Kennedy and Lennox*, 2001; *Dentz et al.*, 2004; *Berkowitz et al.*, 2006]. Transport parameters, such as the dispersion coefficient (*D*), are either spatially averaged or assumed to be constant in modeling anomalous diffusion (or dispersion) using recently developed nonlocal transport models, such as the space fractional advection-dispersion equation (fADE) [*Benson et al.*, 2000]. The convolutional fractional derivative used in the space fADE describes the spatial nonlocal dependency of solute transport through fractal media, and thus the dispersion coefficient was assumed to be scale-independent [*Benson et al.*, 2001]. Recent applications of the space fADE model [*Lu et al.*, 2002; *Huang et al.*, 2006], however, revealed that the dispersion coefficient should be variable to simulate realistic, superdiffusive plumes with space-dependent spreading rates. The space fADE model with variable transport parameters, which is the scaling limit of continuous time random walks (CTRWs) with heavy tailed jumps, was then derived by *Zhang et al.* [2006a] using the generalized mass balance law proposed by *Meerschaert et al.* [2006]. *Zhang et al.* [2007a] further argued that the space nonlocal model, such as the space fADE, should not overshadow the possible local variation of dispersion strength.

[3] Natural media support a nonlocal transport model conditioned on local aquifer properties. First, discontinuities in effective transport properties arising at abrupt contacts between geologic materials may interrupt the anomalous diffusion and therefore may not be simplified if an accurate prediction of contaminant transport is desirable. Some, if not most, natural media are well known to contain discontinuities at multiple scales that may significantly affect the fate and transport of contaminants [*Fogg et al.*, 2000; *LaBolle and Fogg*, 2001]. Second, the heterogeneity evolution in a nonstationary field causes the fluctuation of nonlocal dependency of solute transport, where the nonlocal spreading strength can be defined as a function of the local-scale heterogeneity [*Zhang et al.*, 2007a]. Anomalous diffusion, including superdiffusion and subdiffusion, therefore, is not limited to be homogeneous. The corresponding nonlocal models, such as the space and time fADEs, may contain space-dependent (discontinuous or continuous) transport parameters. A similar argument is made by *Berkowitz et al.* [2002]. Development of an effective approach to simulate such a process, therefore, has significant practical importance.

[4] Anomalous dispersion with space-averaged or space-dependent parameters has been studied by *Berkowitz* and his colleagues. As reviewed by *Berkowitz et al.* [2006], they first applied the generalized Master equation (GME) approach, by taking the ensemble average of the Master equation over certain scales [see, e.g., *Berkowitz and Scher*, 1995, equation (6); *Berkowitz et al.*, 2000, equation (3); *Berkowitz and Scher*, 2001, equation (2); *Kosakowski et al.*, 2001, equation (2); *Berkowitz et al.*, 2002, equation (15); *Cortis et al.*, 2004a, equation (4); *Dentz et al.*, 2004, equation (2)], to derive their continuous time random walk model with constant parameters. The Master equation (ME) with Taylor series expansions was also used by *Berkowitz et al.* [2002, equation (1)] to derive the Fokker-Planck equation (FPE) with space-dependent velocity and dispersion coefficient [see *Berkowitz et al.*, 2002, equation (12)]. *Cortis et al.* [2004b] further represented the evolution of a contaminant in a two-scale system by combining the small-scale GME terms (microscopic heterogeneity) with the large-scale ME (macroscopic heterogeneity) into a Fokker-Planck with memory equation (FPME). The FPME was solved numerically using the inverse Laplace transform (described by *Berkowitz et al.* [2006, p. 27]), and the approximation of the interface zone between different macroscopically homogeneous domains was obtained by linearizing the spatial derivative of the dispersivity over a small region surrounding the interface. The representative scale of the ME was also defined by *Scher et al.* [2002]. *Hornung et al.* [2005] applied the CTRW framework to describe morphogen gradient formation in a complex environment. The effect of an abrupt change in the transition time distribution (due to the variation of extracellular environment) on the concentration profiles was analyzed, where the semianalytical solution was provided. These studies reveal the importance of the nonlocal transport process conditioning on local aquifer properties, which is one of the motivations of this study. We will compare the model and numerical solvers developed in this study to *Berkowitz et al.*'s [2006] CTRW framework (see sections 3 and 5.2), with the main focus on the work of *Cortis et al.* [2004b] and *Hornung et al.* [2005].

[5] This study develops random walk based Monte Carlo (MC) methods to simulate and analyze anomalous transport, including superdiffusion, subdiffusion, and their combination which are described by fractional derivative models, through macroscopically heterogeneous media with discontinuous or continuous transport parameters. It is a logical extension of the work of *Zhang et al.* [2007a], who extended the homogeneous superdiffusion to more general cases with continuous dispersion coefficient. Subdiffusion is considered in this study since quite a few tracer transport processes observed in natural media show retention behaviors [see, e.g., *Haggerty et al.*, 2000; *Berkowitz et al.*, 2006].

[6] Anomalous dispersion has been characterized by various transport theories and models, see for example the non-Markovian models reviewed by *Bouchaud and Georges* [1990] and most recently, the nonlocal transport theories for non-Fickian transport in heterogeneous media reviewed by *Neuman and Tartakovsky* [2009]. The main nonlocal transport models used at present include the stochastic average of local advection-dispersion equation (SA-ADE), the CTRW framework, the multiple-rate mass transfer (MRMT) model, and the fADE model (see also the review by *Zhang et al.* [2009] and the extensive references cited therein). This study is focused on the fADE model, which shows particular promise for describing the anomalous dispersion observed in many natural systems, as reviewed extensively by *Metzler and Klafter* [2000, 2004]. The solute particle jump size and waiting times underlying fractional-order derivative models are independent and identically distributed. The correlated CTRW and/or waiting times, as focused for example by *Matheron and de Marsily* [1980], *Dentz et al.* [2008], and *Meerschaert et al.* [2009], is beyond the topic of this study and we will leave it to a future work.

[7] The lack of analytical solutions for most fractional-order partial differential equations (fPDE) motivates the development of numerical approximations. The fully Lagrangian solver is selected in this study owing to its encouraging advantages, including sub-flow-grid-scale resolution of concentrations and the construction of a sample path for the underlying stochastic process. Although the Lagrangian approach can be applied using the same flow field as that used by an Eulerian approach, the output concentration (which is related to the particle number density, or the number of particles per unit volume of the medium) can be obtained at almost any desirable spatial resolution above the support scale of the governing equations (as shown by equation (36), the spatial resolution *x* can be small enough to provide a fine resolution of concentration). In contrast, the Eulerian approach typically computes average concentrations only at the grid scale. The Lagrangian approach was first applied to tracer transport in saturated alluvial formations [*Ahlstrom et al.*, 1977; *Prickett et al.*, 1981], and then porous media of all types [*Uffink*, 1985; *Kinzelbach*, 1988; *Cordes et al.*, 1991; *Mahinthakumar and Valocchi*, 1992; *Tompson*, 1993; *Semra et al.*, 1993; *LaBolle et al.*, 1996, 1998, 2000; *LaBolle and Fogg*, 2001; *Weissmann et al.*, 2002; *Maxwell et al.*, 2007] and extended to fractured rocks with matrix diffusion [*Yamashita and Kimura*, 1990; *Wels et al.*, 1997; *Liu et al.*, 2000; *Tsang and Tsang*, 2001], among many other applications in hydrology. The successful applications are focus on Fickian diffusion. The Lagrangian solver was extended to non-Fickian diffusion described by the fPDEs by various researchers in the computational physics community (due to the increasing use of fractional dynamics and a lack of analytical solutions), as reviewed extensively by *Zhang et al.* [2008]. The pioneering work of *Fogedby* [1994a, 1994b] provided a Langevin equation for continuous time Lévy flight [see also *Chechkin et al.*, 2004]. The fractional PDEs considered in subsequent studies are typically limited to the constant dispersion coefficient; see, for example, *Magdziarz and Weron* [2007, equation (1)] (a time fPDE), *Heinsalu et al.* [2006, equation (1)] (a time fPDE), *Gorenflo et al.* [2002, equation (9)] (a time fPDE), *Gorenflo et al.* [2004, equation (2)] (a space and time fPDE with the Riesz space fractional derivative [see also *Gorenflo et al.*, 2007]), and *Fulger et al.* [2008, equation (6)] (a spatiotemporal fPDE with the Riesz-Feller space fractional derivative). The time fPDE used in these studies, however, is different from the fractal mobile/immobile (MIM) model (19) considered in this study, and it cannot distinguish the status (mobile or immobile) for solute particles [*Zhang et al.*, 2009]. *Marseguerra and Zoia* [2006] considered an interface problem, but the target governing equation [see *Marseguerra and Zoia*, 2006, equation (37)] is the time fPDE used by the above studies. In addition, the Fokker-Planck flux used by *Marseguerra and Zoia* [2006] is quite different from the Fickian flux embedded in the fractal MIM model (19), and these two equations describe different particle dynamics. In summary, previous studies have been focused on the random walk approximations of fPDEs that are different from the fADEs considered in this study, and the Lagrangian solution for the fADEs considered in this study remains unknown.

[8] The rest of the paper is organized as follows. In section 2, we develop Monte Carlo simulations of continuous time random walk to approximate the stochastic solution of the spatial fractional-order transport equation with discontinuous dispersion coefficient *D* or effective porosity *n*. Here the “effective porosity” denotes the porosity through which flow can occur [see *Fetter*, 1999, p. 50], so it is relatively smaller than the total volume fraction of pores. The subdiffusive process across a discontinuity and its Monte Carlo solution, which are quite different from superdiffusion, are explored subsequently in section 3. In section 4, we consider the mixed superdiffusion and subdiffusion with continuous dispersion coefficient and effective porosity, and build the corresponding Lagrangian framework to track particle dynamics. A field application is illustrated in section 5 and our conclusions comprise section 6. We also develop the Eulerian solution for anomalous diffusion with space-dependent *D* and *n* in Appendix A. Although the Eulerian solution is used extensively for the development and cross verification of the Lagrangian approximation in this study, it should not replace the Lagrangian solver in applications. The Lagrangian solver is the only known algorithm for many nonlocal processes that capture the leading edge and delayed tail of tracer breakthrough [*Benson and Meerschaert*, 2009], and it may be the only applicable tool for some other cases, such as the tracking of fine-scale details in large systems with strong local variations of chemical heterogeneity.