## 1. Introduction

[2] The dispersion of dissolved solutes through natural porous media is usually “anomalous” in that it does not follow Fick's 2nd law [e.g., *Berkowitz et al.*, 2006]. Typical anomalous features of large-scale, multi-dimensional plumes include 1) different growth rates along different directions (due to an inherently anisotropic depositional or structural geologic environment), 2) “heavy,” non-Gaussian leading edges and irregular channelling of the plume front, 3) solute sequestration in relatively immobile rock, and 4) local variation of either mean transport speed or dispersion rate. Simulating these large-scale plume characteristics using a traditional local (Fickian) advection-dispersion equation (ADE) can be computationally intensive, if not technically impossible, due to the influence of decimeter-scale rock properties that often have very long-range correlation [*Zheng and Gorelick*, 2003]. Nonlocal equations [e.g., *Haggerty et al.*, 2000; *Benson et al.*, 2001; *Berkowitz et al.*, 2006] have been developed to characterize the anomalous dispersion at a much coarser scale, but their successful application to field plumes is limited to 1-dimension (1-*d*). This study develops a solution of the nonlocal, multi-dimensional, spatiotemporal fractional ADE (fADE) with spatially dependent transport coefficients in a Lagrangian framework using a novel random walk method. This method allows us to simulate many of the characteristics of anomalous dispersion that are missed by a 1-*d* nonlocal model (such as direction-dependent spreading rates) without the burden of explicitly representing the fine-scale subsurface heterogeneity in a numerical model. We are not aware of any other nonlocal method with the same capability.

[3] We evaluate our model by simulating the tritium plumes measured at the highly heterogeneous Macrodispersion Experiment (MADE) test site [*Rehfeldt et al.*, 1992] where the above four main anomalous features are conspicuous and cannot be captured fully by previous transport models. It is noteworthy that *Lu et al.* [2002] extended *Benson et al.*'s [2001] 1-*d* space-only fADE to capture the 3-*d* MADE tritium plumes; however, their model restricted heavy-tailed motion to 1-*d*, did not account for transfer of mass to a relatively immobile phase, and did not account for the spatial variation of transport parameters. Their preliminary model did not adequately simulate the observed resident concentrations. The present study allows us to overcome all of the limitations of *Lu et al.*'s [2002] model.