Monte Carlo simulations are developed to approximate one-dimensional superdiffusion and subdiffusion in macroscopically heterogeneous media with discontinuous or continuous transport parameters. For superdiffusion characterized by a space fractional (α-order) derivative model, one empirical reflection scheme is built to track particle trajectory across an interface with discontinuous dispersion coefficient D, where the reflection probability depends on both α and the ratio of D. Different from the superdiffusive case, anomalous diffusion described by a time fractional derivative model can be decomposed into a motion component and a hitting time process, where the discontinuity affects only the motion process, implying an efficient Monte Carlo simulation of decoupled continuous time random walks. The discontinuity of effective porosity n is also discussed, and results show the influence of the ratio of n on solute particle dynamics. In addition, for anomalous superdiffusion and subdiffusion in heterogeneous media with spatially continuous D and n, Langevin analysis reveals that the corresponding particle dynamics contain three independent stable Lévy noises scaled by D, the gradient of D, and the gradient of ln(n). A new implicit Eulerian finite difference method is also developed to solve the spatiotemporal fractional derivative models and then extensively cross verify the Lagrangian solutions. Further testing against one field example of mixed superdiffusion and subdiffusion reveals the applicability and flexibility of the novel Monte Carlo approach in simulating realistic plumes in macroscopically heterogeneous media with locally variable transport parameters.
 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. . 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.
 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. . Development of an effective approach to simulate such a process, therefore, has significant practical importance.
 Anomalous dispersion with space-averaged or space-dependent parameters has been studied by Berkowitz and his colleagues. As reviewed by Berkowitz et al. , 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. . Hornung et al.  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  CTRW framework (see sections 3 and 5.2), with the main focus on the work of Cortis et al. [2004b] and Hornung et al. .
 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].
 Anomalous dispersion has been characterized by various transport theories and models, see for example the non-Markovian models reviewed by Bouchaud and Georges  and most recently, the nonlocal transport theories for non-Fickian transport in heterogeneous media reviewed by Neuman and Tartakovsky . 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.  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 , Dentz et al. , and Meerschaert et al. , is beyond the topic of this study and we will leave it to a future work.
 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. . 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  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  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.
 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.
2. Superdiffusive Dynamics Across an Interface
 Monte Carlo realizations of particle sample path yield a stochastic solution for the scaling limit of the underlying CTRW. A specific Monte Carlo algorithm is needed to overcome the local mass conservation problem across the discontinuity. For Fickian diffusion, three methods, including the reflection method [e.g., Uffink, 1985; Cordes et al., 1991; Lim, 2006], the interpolation method [LaBolle et al., 1996; LaBolle and Zhang, 2006], and the generalized stochastic differential equation (SDE) method [LaBolle et al., 2000], have been developed to track the particle trajectory across the discontinuity with conserved local mass.
 The space fractional advection-dispersion equation with space-dependent fluid velocity (v) and dispersion coefficient (D) can be written as [Zhang et al., 2006a]
where C(x, t) denotes the solute concentration, and α (1 < α < 2) [dimensionless] is the order of the space fractional derivative. The other fADE model is also possible, as discussed by Zhang et al. [2006a], but the fADE (1) is favored on the basis of a prior estimation of equation parameters, and it also performs better in simulating field plumes [Zhang et al., 2007a]. All parameters here are time-independent, representing the steady state condition. If α = 2, (1) reduces to the classical second-order advection-dispersion equation (ADE). If v and D are constant, (1) reduces to the fADE
which has been used extensively to simulate the superdiffusive process by the hydrology community, including Benson et al. [2000, 2001], Pachepsky et al. , Zhou and Selim , Deng et al. [2004, 2006], Chang et al. , Zhang et al. , Clarke et al. , and Kim and Kavvas . These applications (i.e., the transport models) are all in one dimension (1-D) (where the medium itself is usually multidimensional), and we constrict our attention to the 1-D model too. Extension to high dimensions will be performed in a future study. Note that the spatial moments of plumes higher than the first-order diverge for a space fractional derivative model of order 1 < α < 2. However, the sample moments exist [see, e.g., Baeumer et al., 2001] and the underlying process is typically treated as superdiffusion (since the second-order sample moment σ2(t) increases faster than linear: σ2(t) ∝ t2/α), as reviewed by Metzler and Klafter [2000, 2004]. Also note that the space fractional dispersion process in (1) can incorporate the effects on transport of the fine-scale spatial variations in the velocity field (which is not limited to be 1-D) (see the review by Zhang et al. ), and thus the velocity in equation (1) can actually be an average velocity over a certain range of the heterogeneous system. Therefore, the 1-D equation (1) should not be confused with the classical ADE, which generally applies at the pore scale, and is therefore more restrictive in terms of the velocity field.
 Since a typical Markovian process holds for discontinuous advective velocities, here we only need to consider the diffusion with discontinuous dispersion coefficient. In sections 2.1 through 2.3, we consider the extensions of the three classical particle-based Monte Carlo simulations to the superdiffusion with a discontinuous D, and then test the methodologies with numerical experiments in section 2.4. The discontinuity of effective porosity is also discussed in section 2.5.
2.1. An Empirical and Generalized Reflection Method
 The traditional reflection techniques are built based on either an analytical solution to the specific problem (see for example, the analytical solution for discontinuous D given by Carslaw and Jaeger ) or the specialized numerical treatment of particle jumps to maintain mass balance at an interface between regions with constant, but different, diffusion coefficient (see the review by LaBolle et al. ). Difficulties arise for the case of anomalous diffusion (with even constant parameters) since the exact analytical solutions are either difficult to derive or contain high approximations (see, e.g., the Fox's H function introduced by Metzler and Klafter [2000, 2004]).
 For a particle crossing the interface (with the coordinate Xface) over the time step Δt, the particle's displacement can be divided heuristically into two steps with time intervals Δt1 and Δt2. The particle is assumed to start at the left region (denoted as region 1), and the right region is region 2. The governing equations where D is a step function can be written as
The conservation of mass across the interface requires
The particle moves to the interface over Δt1 first, and then displaces over Δt2 from the interface either into the other region 2 (with the diffusion coefficient D2) with a probability P1 or reflecting back to the original region 1 (with the diffusion coefficient D1) with a probability 1 − P1. The second step can be expressed numerically as
where U is a uniform [0, 1] random number, and dLα is a random noise (rescaled by time Δt2) representing the jump size [see Zhang et al., 2006c], which will also be introduced in section 4. The time interval Δt2 can be calculated by
where Xt is the particle position at time t, and dSα [dimensionless] is a standard α stable Lévy random variable (with β, σ, and μ denoting the skewness, scale, and shift, respectively). Note here we assume region 1 is located at the left of region 2, otherwise the signs in equations (5a) and (5b) are reversed.
for which we define the reflection probability P1 as
where a and b are unknown parameters to be determined and n1 and n2 denote the effective porosity at the left and right region, respectively (n1 = n2 in this section). In the case of a homogeneous medium, we have n1 = n2, D1 = D2, and P1 is equal to the probability of backward jumps (denoted as Pr, which is 1/α for a superdiffusion). So we have b = Pr. For the case of impermeable medium at region 2, we have n2 = 0, R = 1, and P1 = 1 (i.e., all particles are reflected back to region 1). Thus the parameter a = 1 − Pr. Therefore, the empirical P1 is
which is the analytical solution for the reflection probability for normal diffusion, verifying that the second-order ADE is a specific case of the nonlocal transport model (1). The number density of particles, whose trajectories are simulated using (5a), (5b), and (9), can approximate the solution of equations (3a) and (3b). We will consider the validity of formula (9) in section 2.4.
2.2. Stochastic Diffusion Equation Method
 The classical SDE approach designed for the second-order ADE divides the particle jumps artificially into two steps [LaBolle et al., 2000]:
where dY and dX denote the jump size during one time step dt, dW is a Brownian random variable with mean zero and variance dt, and parameter B2 = 2D. The first step (11a) is to calculate the jump size using the dispersion coefficient at current location X, while the second step (11b) tracks the particle jump using the dispersion strength at the renewed location X + dY. It is noteworthy that one can expand (11b) as
where (B)α = 2D. Similar to (12), Taylor expansion of B in (13b) yields
which is a nonlinear Langevin equation. However, we find, after analyzing (14) using the Langevin approach proposed by Zhang et al. [2006a], that (14) is not analogous to the space fADE
Numerous alternatives to (13a) and (13b) have been proposed (not shown here), but as yet no feasible generalization of the SDE approach to superdiffusion has been found. Inverse analysis also suggests that the space fADE (15) cannot be decomposed to any two-step jump scheme similar to (13a) and (13b).
2.3. Interpolation Method
 The interpolation method adds an artificial zone surrounding the discontinuity and then interpolates the discontinuous transport parameter. The length of the interpolation zone must be larger than the jump size for each particle during one time step, and thus each Monte Carlo sample path or individual particle dynamics can experience the influence of the interface. This method is applicable for discontinuous porosity for solutes undergoing Fickian diffusion, where the interpolation zone needs not to be extremely large due to the limited jump size and the typically small time step. In the case of superdiffusion, however, the classical interpolation method may not be feasible due to the finite probability of a large particle jump.
 One intuitive modification is to partition the motion time for particles passing the interface into two periods. During the first period, the particle enters into the interpolation zone; during the second period, the particle is driven by a linearly interpolated transport parameter and then exits the interpolation zone. Thus the interpolation zone has a limited size. This modification, however, is not computationally efficient if the particle has a relatively high probability of large jumps (corresponding to a small index α). In addition, as pointed out by LaBolle et al. [1996, 2000] and LaBolle and Zhang , interpolation of the dispersion coefficient can significantly alter the problem being solved, since interpolating D affects both the drift and the dispersion with the interpolation zone.
2.4. Numerical Experiments and Methodology Improvement
 Numerical experiments of Monte Carlo approximations confirm that both the SDE approach and the modified interpolation approach fail to track superdiffusive particle trajectories across the interface (Figures 1b and 1c). Compared to the “true” solution using an implicit Euler finite different method (Appendix A), the generalized SDE approach smears the plume distribution across the interface, although it captures well the particle snapshot at locations far away from the discontinuity. This discrepancy might be due to the backward dispersion embedded in a Lévy motion. Particularly, different from the normal diffusion, the Lévy motion for superdiffusion is not symmetric, but positively skewed. During both the first step (defined by (13a)) and the second step (defined by (13b)), the particle has a high probability to move backward in a relatively short distance, and thus it tends to remain in the starting zone, forming a delayed density peak.
 In contrast, the modified interpolation approach overestimates the shift and dispersion of particles crossing the interface (the same problem is found for Fickian diffusion; see Figure 1a). This is because the interpolation of dispersion coefficient for superdiffusion changes both the drift and the dispersion of particles, as also implied by the positive random noise expressed by equations (47), (48a), (48b), and (48c).
 The generalized reflection approach described in section 2.1, surprisingly, overestimates the backward diffusion and underestimates the plume mass at the interface (denoted as Reflection 1 in Figures 1b and 1c). Further tests reveal that the maximally positive skewness defined in the fADE (1) might be the reason for these discrepancies. In particular, the Lévy motion dLα is positively skewed with relatively large and positive jumps, describing the fast displacements in front of the mean for solute particles in natural porous media [Schumer et al., 2001] and fractured aquifers [Reeves et al., 2008]. Similar to the Fickian dispersion used in the traditional ADE model, here the fractional dispersion also accounts for the deviation of local velocity from the mean, where the mean shift due to dispersion remains zero. The backward dispersion due to either the local flow slower than the mean velocity or the backward flux, therefore, should not be as apparent as its forward counterpart [Lu et al., 2002]. The positively skewed flow velocities may be due to the heavy tailed hydraulic conductivity (K) distribution, such as the Pareto distribution of K found by Aban et al. . When a particle's dynamics is affected by the heterogeneity interface and it needs to be “reflected back,” the consequent backward jump does not require the same magnitude as the forward one. To mathematically describe the discrepancy between backward and forward jumps at the interface, we replace the random variable dLα in equation (5a) with a newly generated, negative jump
Now the reflection method can produce a better approximation of particle dynamics (denoted as Reflection 2 in Figures 1b and 1c). Here dLαR(Δt2) denotes a negative stable random variable generated for the reflected particle, and dLαE(Δt2) denotes a positive stable random variable generated for the particle passing the interface. The above numerical algorithm is also mathematically sound for the case of D1 = D2, where the sum of heavy tailed jumps approximates the same α stable, according to the generalized central limit theory [Meerschaert and Scheffler, 2001].
 In the case shown in Figure 1, the best fit reflection probability (0.88), however, is slightly larger than the probability predicted using (9), which is 0.86. This discrepancy might also be due to the nonsymmetric Lévy motion compared to the symmetric Brownian motion. We ran numerous numerical experiments by trial and error (with a few examples shown in Figure 2) to determine the relationship between P1 and the ratio D1/D2 for a series of α (varying from 1.2 to 1.8). Analysis shows that, interestingly, P1 varies almost linearly with r/(1 + r), where the parameter r = (D1/D2)1/α (Figure 3). Thus we obtain the following empirical reflection probability
which also contains the reflection probability for normal diffusion (i.e., (10)) as an end member.
 The dynamics of superdiffusive particle movement across a heterogeneity interface using the Lagrangian framework has not been previously explicitly defined. The generalized reflection approach developed above builds the possible sample path for anomalous superdiffusion across an interface, where the empirical reflection probability can only be obtained by fitting the solution of the governing fADE model due to the lack of an analytical solution. As demonstrated by Sokolov and Metzler , the spatial fADE with constant parameters may not uniquely connect to a trajectory (see also the discussion by Zhang et al. [2006a]). To the best of our knowledge, no other alternative particle tracking scheme is presently available to build a trajectory that can be compared to the results derived in this section. In the following, we explore the physical explanation of the reflection approach, and compare the sample path of superdiffusion to that of normal dispersion.
 For the case of pure Fickian diffusion in a macroscopically homogeneous medium, the solute particle moves back and forth with the same transition probability and the same density of jump size (thin tailed). Correspondingly, the reflection approach tracks equally the particle passing through, or returning from, the current location, and the simulated stochastic process in continuous time exactly describes the Brownian motion process. For a macroscopically discontinuous media represented by two dispersion coefficients D1 and D2 (assuming that the particle is originated from the D1 zone and D1 > D2), the interface retards the dispersion of particles, which is analogous to the case of a continuously decreasing D. If the reflective probability is still the same as that for the homogeneous case (i.e., 0.5), particles can accumulate around the interface (at the side of lower dispersion coefficient) (Figure 4b). More particles, therefore, need to be reflected back, to overcome the local mass balance problem and reserve a continuous concentration profile, resulting in a sample path (Figures 4b and 4d).
 For the case of symmetric superdiffusion (which can be described by the Riesz-Feller operator ∂α/∂∣x∣α, a symmetric space fractional derivative), the generalized reflection approach has the same physical reliability as the classical reflection approach for normal dispersion. Particularly, particles are reflected back and forth with the same probability (0.5) and density of (heavy tailed) strength, and the stochastic process recovers the symmetric Lévy motion. When the CTRW jump distribution is positively skewed (for example, due to the heavy tailed distribution of velocities at the microscopic scale), however, the forward movement is apparently heavier than the backward dispersion (Figure 4c) and the transition probability is no longer symmetric. Corrections in reflection probability and jump strength must be taken to account for the nonsymmetric jump distribution and overcome the local mass balance error. One simulated sample path for a single particle is shown in Figure 4c.
2.5. Discontinuity of Effective Porosity
 The Monte Carlo approximation developed above is not limited to discontinuous dispersion coefficient. Composite media can certainly have other discontinuous properties that affect contaminant transport, including effective porosity [Molz et al., 2004; Marica et al., 2006; Tilke et al., 2006]. We select the superdiffusion model with constant D and variable n as an example
The above Monte Carlo simulation composed of the two-step jump (described by (16a) and (16b)) and the empirical reflection probability (17) provides a reasonable approximation of (18) with discontinuous n, as shown by Figure 5 for different index α. The best fit reflection probability generally matches the empirical value estimated by (17) (see Figure 6).
3. Subdiffusion Across an Interface
 The anomalous subdiffusive process can be simulated by the time nonlocal transport model, including the SA-ADE, CTRW, MRMT, and the time fADE models. This study focuses on the following time fADE proposed by Schumer et al.  to characterize the subdiffusion resulting from multiple-rate mass transfer between mobile and (parallel) immobile phases
where γ [dimensionless] (0 < γ < 1) is the scale index, β [Tγ−1] is the capacity coefficient, and Ctot is the total (mobile plus immobile) concentration. Here the solute is assumed to have been placed initially in the mobile zone, to be consistent to the typical tracer test. When γ = 1, (19) reduces to the advection-dispersion equation with a retardation factor 1 + β; when β = 0, (19) reduces to the classical second-order ADE (where all particles are in the mobile state at any time).
 The dispersive flux in (19) is a Fickian type, which is different from the Fokker-Planck flux used in the CTRW model developed by Berkowitz et al. [2002, 2006] and Cortis et al. [2004b]. Whether the Fickian flux or the Fokker-Planck flux should be used to describe the Brownian motion is beyond the focus of this study, and remains an open question [Berkowitz et al., 2009]. Particularly, the proper definition for the transition probability density describing the microscopic physics is an open question. For example, LaBolle et al.  derived the second-order ADE using principles of probabilities from a microscopic point of view, to successfully generalize a theory for transport in composite porous media. Therein, the definition for velocity replaces the usual drift coefficient. If one assumes that diffusions (dispersions) are Fickian, then the physical parameters velocity and dispersion tensor are defined as written by LaBolle et al. . Although Fick's first law is generally accepted as descriptive of microscopic physical behavior, the microscopic behavior may in fact follow a different form. Recent bench-scale experiments by Berkowitz et al.  suggest that longitudinal dispersion at an interface may be non-Fickian. In this case, the definitions for coefficients change. More study is clearly needed to better understand this subtle issue. Note that the macroscopic form that we have chosen in our equations is Fickian.
3.1. Lagrangian Mobile/Immobile Decomposition and Markovian Analysis
where the concentration Ctot follows the conditioning argument
Here m and t denote the motion time and the total time, respectively. Equation (22) describes a motion process with a number density p, while equation (23) governs a hitting time process with a density h.
 Measurement of concentrations in the field may preferentially sample a relatively mobile phase (such as the MADE site; see the analysis by Harvey and Gorelick ) or both the mobile and immobile phases [Zhang et al., 2006b]. For the former case, a model that can distinguish the solute status is critical. The mobile/immobile decomposition of the time fractional derivative model can therefore be applicable, such as the simulation of the MADE site plumes discussed in section 5.
 The fully Lagrangian approach has been developed by various researchers to approximate the solutions of the multirate mass transfer model. For example, Salamon et al.  used the time-dependent particle mass [see, e.g., Salamon et al., 2006, equations (13) and (14)], which is the phase transition probability [Michalak and Kitanidis, 2000], to determine the status of particles at each time step. The time fADE (equation (19)), as explained above, contains an infinite number of rate coefficients, and therefore requires a different scheme than the one proposed by Salamon et al. . As will be shown below, the Lagrangian scheme developed in this section can be computationally efficient, since the particle stays immobile for a random period of time and the number of jumps can be significantly reduced. A two-step random walk scheme (jump followed by waiting) was proposed by Dentz and Berkowitz  to approximate the MRMT model; see also Dentz et al. [2004, equations (13) and (14)] and the brief description of methodology by Berkowitz and Scher . This two-step, fully Lagrangian scheme was applied by Le Borgne and Gouze [2008, equation (8)] (with continuous dispersion coefficient and the solution for the total phase) and was extended by Le Borgne et al. [2008, equation (1)] to describe non-Markovian transport. This section supports Dentz and Berkowitz's  two-step scheme by reviewing the mobile/immobile decomposition and the Markovian process underlying the waiting time process.
 The backward version of (23) is needed to develop its Lagrangian description. The particle tracking method we use relies on identifying the associated Markov process, whose generator is the adjoint of the fractional derivative operator in time [Zhang et al., 2006a]. What we actually simulate here is the temporal Langevin equation, which is equation (29) [see also Zhang et al., 2008]. The Markov process defined by a Langevin equation corresponds to a generator, which is the backward version of the target governing equation. Here the time fADE (19) is a forward equation, and its backward equation (mapping probabilities backward) is needed to build the Markov process. The backward counterpart of (23) can be evaluated through a fractional adjoint transform proposed first by Zhang et al. [2006a]:
where T1 and T2 denote the starting and ending point of t, respectively, and Γ is the Gamma function. Changing the order of integral and substituting τ = t − z (denoting the backward travel time) in the integral on the right-hand side (RHS) of (25), we have
which can be further simplified by using partial differential
The above integrals have not previously been presented in the literature. The hitting time process (23), therefore, has the following backward equation
where dLγ(dm) = (dm)1/γdSγ(β = +1, σ = 1, μ = 0), dSγ [dimensionless] is a standard γ stable Lévy random variable (see also dSα in equation (6)), and sign(u) denotes the sign function (which is +1 if u > 0; otherwise −1). Note here the sign function is −1 since β cos(πγ/2) is always positive for 0 < γ < 1, and thus the second term on the RHS of (29) is always positive (note dLγ > 0 since 0 < γ < 1). So here we always have dt > dm. The backward equation (28) and the temporal Langevin equation (29) are simplified versions of Zhang et al. [2008, equations (5) and (6)].
 The time for a particle to finish one jump, denoted as dt in (29), therefore contains two parts. The first part dm is exactly the same as the period of time used by the motion process (22) during the jump. During the remaining part dt − dm, no movement occurs, and this period must be corresponding to an immobile state. After defining the state of particle at a given time, we obtain the concentration for solute in each phase. This Lagrangian framework decomposes efficiently the mobile and immobile states embedded in the time fractional derivative model (19).
 It is noteworthy that the time fADE model (19) assumes Fickian diffusion for particle jumps in operational time (as indicated by equation (22)), which differs significantly from the superdiffusion (super-Fickian) characterized by the space fADE model (1). Cautions are needed, however, when evaluating the variance of solute particle displacement. As reviewed by Metzler and Klafter [2000, equations (39) and (92)], the variance of displacement for the simplified time fADE model (21) can increase faster than linear in time if 0.5 < γ < 1 and v ≠ 0. The same behavior can be found for (19) when time is relatively large. We prefer to call the retention process defined by the time fADE models as “subdiffusion” in this study because (1) the actual transport remains as a retention process (due to the always positive trapping time in the relatively immobile zones) and (2) the mean displacement of solute particles always increases slower than linear in time for both models (21) and (19). Such treatment also helps us to distinguish the dynamics underlying the time fADE and the space fADE.
3.2. Numerical Examples
 We first compare the above Lagrangian decomposition to the Eulerian decomposition. For a simplified operator Ax = −v∂/∂x + D∂2/∂x2, the Fourier (x → k) and Laplace (t → s) solution of (19) is
which can be rewritten as
Here M and IM denote the concentration in the mobile and immobile phase, respectively. It is easy to show that the Fourier-Laplace inverse transform of M satisfies the mobile-phase governing equation proposed by Schumer et al. [2003, equation (15)]:
where gγ is the stable subordinator. Inverse Laplace transform can be used to approximate gγ first, and then CM(x, t) can be calculated by approximating the integral (33) numerically. A similar procedure is used to solve CIM(x, t) [see also Schumer et al., 2003, equation (22)]:
 Numerous tests have been performed to numerically compare the above Lagrangian and Eulerian decompositions. Two examples are shown in Figure 7, where the two decompositions have similar solutions. Note the motion process (22) and the hitting time process (23) describe the dynamics of particles in the total phase. They are not the governing equations for particles in the mobile and immobile phases, and the Lagrangian decomposition discussed in 3.1 therefore differs from the Eulerian decomposition discussed in this section.
 Three more examples are shown in Figure 8. Figure 8a plots results from the Monte Carlo approximations for the time fADE (19) with constant dispersion coefficient. The simulated snapshots of concentrations for contaminant in different phases match well the Eulerian decomposition solutions. Then we test the full capability of the MC approximation using different index γ and capacity coefficient β. When β → 0, the MC solutions of fADE (19) are close to the analytical solutions of the classical second-order ADE with discontinuous D [Carslaw and Jaeger, 1959], just as expected (Figure 8b).
 From this we can draw the intermediate conclusion that the discontinuity of dispersion coefficient modifies the distribution of particle jump size, since it changes the trajectory of particles crossing the discontinuity. The density of particle waiting time (i.e., the trapping process), however, is space-independent and should not be affected (note again D is time-independent in this study).
 Note that the normal dispersion on the RHS of the time fADE (19) can be generalized by the fractional-order dispersion, resulting in a mixture of subdiffusion and superdiffusion. The corresponding Monte Carlo approximation is the combination of the schemes discussed in sections 2.4 and 3. More specifically, the motion of a particle follows (16a) and (16b), and the subsequent trapping process can be simulated by (29). In section 5 we will demonstrate application of this Monte Carlo tool to the MADE site plume [Boggs et al., 1992; Adams and Gelhar, 1992; Rehfeldt et al., 1992; Boggs and Adam, 1992].
4. Monte Carlo Approximation of the Mixed Superdiffusion and Subdiffusion With Continuous Parameters
 If the transport parameters, including velocity v, dispersion coefficient D, and porosity n, are smooth functions in space, the mixed superdiffusion and subdiffusion can be captured by the following spatiotemporal fADE
When β = 0, (35) reduces to the space fADE. When α = 2, (35) reduces to the time fADE. When γ = 1 and α = 2, (35) reduces to the second-order ADE with a retardation factor 1 + β.
 The probability density of particles, denoted as Q, can be transferred to the solute concentration via the relationship [LaBolle et al., 1996]
where n0 denotes the porosity at the source location, and m0 is the total initial mass. Using (36) and the product rule for the space dispersion term to rewrite (35) in the form
 We then need to expand the second term on the RHS of (37). According to the fractional Leibniz rule [Osler, 1971], the following fractional derivatives can be expanded into an infinite series of fractional derivatives and integrals
 The fractional-order adjoint method developed by Zhang et al. [2006a], by expanding Feller's [1971, page 338] integer-order adjoint method, reveals the following transform
where the circle with a cross denotes adjoint operation. Note here the range of x is the whole real axis and the range of α is between 1 and 2, which are different from the ranges of t and γ in (25). Based on (45), we build the backward equation of (44):
and the corresponding Markov process X(m) can be specified in terms of the following Langevin equation
where dLα(m) = (dm)1/αdSα(+1, 1, 0). Similarly, dLα−1(m) = (dm)1/(α−1)dSα−1(+1, 1, 0) and dLα−1(m) = (dm)1/(α−1)dSα−1(+1, 1, 0) are independent random noises underlying an (α − 1)-order Lévy motion. The three parameters B1, B2, and B3 rescale the random noises dLα, dLα−1 and dLα−1
where D(x) > 0 and n(x) > 0. Therefore, the last three terms on the RHS of (47) denote the random motion of each particle driven by the dispersion coefficient D, the gradient of D, and the gradient of ln(n), respectively, at the staring point x. Note the influence of porosity on solute transport is therefore secondary, compared to D.
 The random walk scheme for the hitting time process is the same as the one discussed in section 3. Numerous tests using the above Monte Carlo method to approximate the generalized fADE model (35) have been done, and the results were validated by the Eulerian solutions developed in Appendix A. A few examples are shown in Figures 9 and 10, and it can be seen from these plots that the Monte Carlo approximations generally match the Eulerian solutions. In section 5 we test the applicability of the MC scheme.
 It is noteworthy that the MC scheme discussed in this section can be analogous to the interpolation method discussed in sections 2.3 and 2.4, if the interpolation zone is the whole model domain (so the parameter becomes continuous).
5. Field Application
 The technique that tracks a particle across a discontinuous interface for space and time nonlocal transport can be practically flexible in modeling realistic plumes. We simulate the MADE-1 bromide plume as an example. The observed bromide plumes at the well-studied MADE test site (which is an alluvial depositional system) show both superdiffusive (i.e., heavy leading edges) and subdiffusive (i.e., apparent mass decline) behaviors [Adams and Gelhar, 1992; Schumer et al., 2003] that can be captured with a Fickian diffusion process, but only if the fine-scale heterogeneity (probably at decimeter scale; see the discussion by Zheng and Gorelick ) characteristic of the site is explicitly represented in the model. The resolution necessary to simulate realistic plume behavior is impractical for most modeling efforts. A nonlocal transport model (such as the fADE) may be able to simulate plume behavior with a fraction of the computational effort. However, Lu et al.  further demonstrated that the space fADE (2) with constant parameters cannot capture the apparent space-dependent spreading of the MADE-2 tritium plumes or any observed mass decay, as further confirmed by Zhang et al. [2007a].
 In the following we first apply the space-time fADE model to fit the MADE bromide snapshot. Then we explore the applicability of various fractional derivative models for MADE plumes. The influence of porosity heterogeneity and the prediction of model parameters are also discussed.
5.1. Application of the Mixed Superdiffusive and Subdiffusive Model and the Solver
 Given the reflection technique discussed above, we now can build a zonal model with mixed superdiffusion and subdiffusion, containing as few parameters as possible, to capture the complex anomalous behaviors at the MADE site. Results (Figure 11) show that a simplified, two-zone model (i.e., equation (35) with discontinuous parameters) can capture the observed anomalous behaviors simultaneously, including the near source peak, the downstream front, the space-dependent spreading rate along the longitudinal direction, and the mobile mass decay. The simulated interface for D at location x = 55 m corresponds to an apparent change in depositional properties near the center section of the test site [Rehfeldt et al., 1992], although the measured bromide concentrations do not reflect directly the presence of this interface, probably due to the measurement noise. The space-time fADE with constant D (note that the average velocity is roughly constant, conserving the water flux) cannot capture simultaneously the peak and front, verifying the conclusion of Lu et al.  and favoring the application of a space-time fADE with space-dependent parameters.
 We then apply the Monte Carlo approximation for the spatiotemporal fADE (35) with continuous D to simulate again the MADE-1 bromide snapshot. As discussed in section 4, this is analogous to the interpolation method over the whole model domain. The result (grey line in Figure 11) shows that the Monte Carlo scheme developed in section 4 also characterizes the main anomalous behaviors of the plume.
 The second tracer, tritium, tested at the MADE site is used to double check the applicability of the space-time fADE model (35). Tritium exhibits anomalous plume behavior similar to that of bromide, though the measurements of tritium contain relatively less noise than those of bromide at the leading edge of the plumes (Figure 12). These results show that the tritium snapshot can also be simulated reasonably well using the space-time fADE (35) with space-dependent dispersion coefficient D (note the radioactive decay of tritium is also considered in the modeling).
5.2. Comparison of Different Transport Models
 There are different types of fractional derivative models having different physical meaning and hydrologic interpretation (see the review by Zhang et al. ). For comparison purposes, here we investigate three more time nonlocal transport models for describing the MADE plumes: (1) the time fADE model (19), (2) the simplified time fADE model (21), and (3) the single-rate mass transfer model (which is also the classical dual-porosity model) , where Cm and Cim denote the solute concentration in the mobile and immobile phases, respectively.
 The best fit mass evolution of bromide using all these models is shown in Figure 13. Note the velocity and dispersion coefficient do not affect the evolution of mobile mass (they affect only the spatial moments higher than the zeroth order). Results show that the simplified time fADE model (21) cannot describe the mass decline (since it cannot distinguish the status of particles, and thus the solute mass remains constant). The single-rate mass transfer model captures the decline of measured mass, but it significantly underestimates the declining rate of bromide at late times. It is well known that the mobile mass estimated by the single-rate mass transfer model eventually becomes constant [see, e.g., Quinodoz and Valocchi, 1993; Michalak and Kitanidis, 2000], since the transfer probability from mobile to immobile phases approaches the constant β/(1 + β). Only the time fADE model (19) and the space-time fADE model (21) successfully recover the temporal evolution of the measured tracer mass.
 We then test the applicability of the time fADE model (19) in fitting the MADE data. Numerical experiments show that model (19) underestimates the heavy (power law) leading edge of the bromide snapshot if the dispersion coefficient D(x) increases linearly with transport distance (Figure 14). Only when D(x) increases faster than linear in space (for example, D(x) ∼ x2.1), will the simulated leading edge be heavy enough to approximate the measured one. This implies that the large displacement of solute particles should be a Lévy motion (where the jump size PDF is a power law function), supporting the application of the space-time fADE model (35).
 If the tracer breakthrough curve exponentially declines at later time for a passive tracer transport in heterogeneous porous media, then the two-domain model (or “dual-porosity” model) with a single mass transfer rate between the mobile and the relatively immobile phases may provide a reasonable fit. To model the slower than exponential decline of the late time solute concentrations, the MRMT or the time fADE model is a better option than the single-rate mass transfer model. In natural media, it is expected that the mass exchange rate can vary significantly in space, as measured for example by Dean and Reimus . This favors the application of the MRMT and the time fADE models. In addition, the space fADE can describe the space nonlocal transport, where the movement of solute particles has a broad distribution (as suggested by the MADE plumes and analysis by Zhang et al. [2007a]). This is beyond the capability of the two-domain model.
 This study focuses on both space and time nonlocal transport processes on the point view of fractional dynamics. As reviewed recently by Neuman and Tartakovsky , the CTRW framework developed by Berkowitz et al.  is a time nonlocal transport model. The CTRW stochastic process can be made space nonlocal (as demonstrated by Meerschaert et al. ), which is, however, beyond the topic of this study. The (simplified) time fractional derivative model derived by Berkowitz et al.  is different from the time fADE (19) considered in this study. In addition, this study considers a Fickian-type of flux for the time fADE model (19), while Berkowitz et al.'s  CTRW framework [Berkowitz et al., 2002; Cortis et al., 2004b] used the Fokker-Planck flux. These two diffusive fluxes define quite different sample paths for random walking solute particles (for instance, the definition of particle velocity is quite different). Although the fADE models are scaling limits of heavy tailed CTRWs and the random walk provides a useful approximation for the fADE [Zhang et al., 2009], the fADE and the numerical approach developed in this study are not the same as Berkowitz et al.'s  CTRW framework. In other words, the numerical methods developed by Cortis et al. [2004b] and Hornung et al.  cannot be used for the models considered in this study.
5.3. Influence of Porosity Heterogeneity
 The impact of porosity variability on MADE site plumes remains unknown. The average porosity and standard deviation from 84 core samples collected from the MADE site were 0.31 and 0.08, respectively [Boggs et al., 1992]. A mean porosity, 0.35, was suggested by Adams and Gelhar  to account for potential consolidation during core extraction. This mean porosity was then used by various researchers in transport simulation of MADE site tracers [Feehley et al., 2000; Harvey and Gorelick, 2000; Julian et al., 2001; Salamon et al., 2007] (where n = 0.32), while heterogeneity of porosity was typically not considered since its variability is believed to be relatively small compared to that of hydraulic conductivity (K). This assumption has been supported by various studies [see, e.g., Eggleston and Rojstaczer, 1998; Zheng and Gorelick, 2003; Molz et al., 2006; Salamon et al., 2007], where numerical experiments suggested that the local-scale, heterogeneous K distribution in the alluvial depositional system is the main reason for anomalous dispersion at the MADE site.
 Porosity in natural porous systems was also found to be a random variable with a fractal distribution (see the review by Molz et al. ). Monte Carlo simulation and stochastic analysis conducted by Hassan et al.  and Hassan  showed that heterogeneity of porosity can affect plume behaviors in porous media, especially when it is correlated to conductivity. The recent work of Riva et al. , where extensive stochastic modeling was performed to distinguish the relative importance of aquifer heterogeneity and conceptual transport models in describing the non-Fickian plumes observed at the Lauswiesen forced gradient tracer test, revealed that the spatial variation of porosity plays an important role in the proper interpretation of non-Fickian plumes. In particular, they found that the spatial distribution of porosity, in addition to hydraulic conductivity, dominates the early arrival of tracer breakthrough curves.
 The influence of porosity variation on MADE bromide plumes is beyond the main focus of this study, but we explore it briefly here. Accurate simulation of porosity distribution in a regional-scale aquifer is almost impossible, owing to limited field measurements. As a preliminary numerical experiment focusing specifically on porosity, we assume that (1) porosity varies linearly along the longitudinal direction and (2) all other transport parameters are constant. We also normalize the tracer concentration, so that the space-only fADE model can be applied (as demonstrated by Zhang et al. [2007a, 2009]). The following space fADE model therefore can be used
In a distance of 300 m, porosity varies uniformly between 10% and 60%, with values of mean and standard deviation of 0.35 and 0.14, respectively (similar to the measured ones). Hydrogeologic investigations at the MADE site reveal that hydraulic conductivity generally increases in the direction of flow [Boggs et al., 1992]. Therefore, porosity may decline in the direction of flow if it possesses a negative correlation to conductivity. Results show that the simulated concentration at the plume front increases and matches the measurements better than the model with a constant porosity (Figure 15). However, if porosity in fact increases in the direction of flow, the simulated peak concentration increases and the concentration at the leading edge apparently underestimates the observations (Figure 15).
 It is noteworthy that the variation of concentration does not necessary reflect the dynamics of solute particles. For example, when porosity declines in the direction of flow, the gradient of porosity causes additional upward particle jumps (because ∂n/∂x < 0 in equation (48c)), resulting in relatively more mass moving upward. A decline in porosity increases the local concentration, which counterbalances (actually is much more apparent than) the upward movement of solute particles. Therefore, additional consideration is needed when interpreting tracer concentrations from field sites having apparent porosity variation. This conclusion is consistent to the finding of Riva et al. , who showed that the observed non-Fickian transport behaviors were primarily controlled by heterogeneities that cannot be explicitly captured by a conceptual transport model. This study shows that porosity heterogeneity affects both the peak and leading edge concentrations, owing to the specific function of porosity in space (i.e., large difference of porosities between the source location and the plume leading edge). A detailed investigation of porosity variation in natural porous media and its influence on tracer test will be conducted in a future study.
5.4. Prediction of Model Parameters
 Preliminary research has been conducted to explore the predictability of the fADE parameters from field measurements. For example, Benson et al.  successfully estimated the space index α based on hydraulic conductivity statistics, where the estimation formula was proposed by Meerschaert and Scheffler . Zhang et al. [2007b] found that the time index γ and the capacity coefficient β might be related to the spatial distribution of immobile zones in heterogeneous media. A qualitative relationship between the fADE parameters and the degree of medium heterogeneity (represented by the variance of hydraulic conductivity) was also recently given by Zhang et al. . Further studies are needed to connect the fADE parameters to filed measurements, such as hydraulic conductivity and/or borehole logs.
 1. The 1-D fractional derivative models with space-dependent parameters can describe anomalous dispersion in the macroscopically heterogeneous medium, which may contain a transition from one environment to another. Each environment has its own distinct geologic character. One example is the MADE site (on the scale of ∼ 300 m) in an alluvial depositional system. The area near the tracer source contains mainly low-permeability materials, and the middle to the edge of the testing area contains relatively high permeability deposits (corresponding to a former river meander in this area).
 2. The Monte Carlo simulation of uncoupled continuous time random walks yields a stochastic solution for the fractional advection-dispersion equation. For superdiffusion across an interface where the dispersion coefficient D is a step function, the modified reflection method is the only tool that can track the particle sample path correctly through the discontinuity. Treatments different from the classical reflection method are required, due to the discrepancy between the maximum positively skewed Lévy motion and the symmetric Brownian motion.
 3. For the subdiffusive process crossing a discontinuity, the governing equation can be decomposed into a motion process and a hitting time process, allowing an efficient Monte Carlo simulation containing a modified particle jump size distribution (corresponding to the motion process) and an unaffected waiting time distribution (corresponding to the hitting time process). If the motion process is Fickian, then the traditional approaches, including the SDE approach and the classical reflection approach, can be used. If the motion is non-Fickian, then the modified reflection method can be used to track the movement of contaminant particles.
 4. For anomalous superdiffusion and subdiffusion in heterogeneous media with spatially continuous D and n, the corresponding particle dynamics contain three independent stable Lévy noises scaled by D, the gradient of D, and the gradient of ln(n).
 5. The classical interpolation method is not applicable for simulating superdiffusion across a discontinuous interface owing to the contradiction between the large jump of particles and the limited size of the artificial interpolation zone. However, if the step function of the transport parameter is interpolated (even as simply as linear) across the entire model domain, the interpolation method (which is analogous to the MC scheme for the fADE with continuous parameters) can be a useful alternative to simulate the anomalous bromide plume observed at the MADE-1 test site. We suggest using this computationally efficient interpolation method to provide preliminary estimation of the influence of discontinuities on anomalous diffusion.
 6. The implicit Euler finite difference method is developed to solve the superdiffusion equation (i.e., the space fractional fADE) with space-dependent (continuous or discontinuous) velocity, dispersion coefficient, and effective porosity. Combined with the time subordination technique, the Eulerian solution can be converted to the numerical solution for the space and time fADE (i.e., the governing equation for a combined superdiffusion and subdiffusion). However, the Eulerian solver should not replace the grid-free Lagrangian solver to simulate large-scale transport with local-scale physical/chemical heterogeneity and sharp concentration fronts. Extensive numerical examples show that the Monte Carlo approximations generally match the Eulerian solutions of the fADE. The Monte Carlo approach can distinguish the particle state conveniently based on the Langevin analysis, where the contaminant concentrations in different phases need not to be calculated separately, showing the computational efficiency of the MC approach.
 7. The discontinuity of effective porosity affects both the solute transport dynamics and the concentration calculation, but its application is overshadowed by the dispersion coefficient. In the case of continuous parameters, the influence of porosity on solute transport is secondary compared to the influence of dispersion coefficient. The spatial variation and influence of porosity on realistic, anomalous plumes remains to be shown, and the Monte Carlo scheme developed in this study can serve as the fundamental numerical method for such an investigation. A preliminary application at the MADE site reveals that the increase of local concentration may not be due to the increase of solute mass, but rather due to the decrease of effective porosity. In other words, the spatial variation of porosity can cause different effects in particle displacement and concentration variation. Therefore cautions are needed when interpreting tracer concentrations from field sites having apparent porosity variation.
 8. The Monte Carlo method developed in this study is limited to 1-D model. Actual solute transport in real heterogeneous media is typically high dimensional. Preliminary tests (not discussed here) show that the MC simulation for discontinuous parameters might be extended to high dimensions, while the extension of the MC simulation for continuous parameters to high dimensions is relatively straightforward. Further extension of the MC method in a future study is needed.
Appendix A:: Implicit Eulerian Finite Difference Solution of the FADE
 The spatiotemporal fractional advection-dispersion equation with space-dependent parameters is
which takes the same form as equation (35). Here we assume that (1) the velocity v(x) and D(x) are functions in space but not in time (corresponding to the steady state flow) and (2) v(x) and D(x) are positive and they are monotonously increasing with x. To the best of our knowledge, the fADE (A1) has not been approximated by any Eulerian solver yet. The lack of numerical solutions motivates this appendix. The Eulerian solution of (A1) is also used extensively in the text.
The effective porosity is deleted from both sides of equation (A3), since the Riemann-Liouville fractional derivative is a linear operator [Miller and Ross, 1993]. In the following we first develop the implicit Euler finite difference solution for the motion process (A2).
 Using the product rule for the advection term, (A2) can be expanded and rearranged as
 Using the implicit Euler method, the fADE (A4) can be solved approximately as
where fk is the weight for a zero-shift Grünwald approximation for (α − 1)-order fractional derivative [Meerschaert and Tadjeran, 2004]. h = Δx denotes the space discretization, and Δm is the time interval.
 Because fi+1 < 0, ni > ni−1, and vi > vi−1, we have
 Similarly, one can derive (we leave it to the readers)
This implies that the eigenvalues of the matrix A are all larger than or equal to 1 in magnitude. Hence the spectral radius of the inverse matrix A−1 is less than or equal to 1. Thus any error in pL is not magnified, and therefore the method is unconditionally stable.
where gγ is the stable subordinator. The limit for m is from 0 to t, since the density gγ located out of this range is strictly zero for 0 < γ < 1. Equation (A14) shows that the concentration Ctot in total time t is the weighted sum of density p at each motion time m. In this study, the integral (A14) is calculated approximately using the trapezoid rule.
 This work was supported by the National Science Foundation (NSF) under EAR-0748953 and EAR-0748984. Y. Z. was also partially supported by the Desert Research Institute (DRI) (IR&D funds). Any opinions, findings, conclusions, or recommendations do not necessary reflect the views of the NSF or DRI. The authors thank the Associate Editor Alberto Guadagnini, the Editor Scott W. Tyler, and three anonymous reviewers for their insightful suggestions, which improved this work significantly.