## 1. Introduction

[2] Transverse mixing is a critical factor in the transport of continuously introduced contaminants that react with compounds from outside the plume [e.g., *Cirpka et al.*, 2008; *Cirpka and Valocchi*, 2007; *Willingham et al.*, 2008]. In heterogeneous formations, transverse mixing controls the reduction of peak concentrations [*Fiori*, 1996, 2001; *Tonina and Bellin*, 2008] and transfers longitudinal spreading to longitudinal mixing in transient transport [*Cirpka and Kitanidis*, 2000; *Cirpka*, 2002], thus reducing the concentration variance [*Pannone and Kitanidis*, 1999; *Vanderborght*, 2001; *Fiori*, 2001; *Kapoor and Gelhar*, 1994a, 1994b]. Its importance in risk assessment [e.g., *Rubin et al.*, 1994; *de Barros et al.*, 2009] and in the evolution of statistical concentration distributions with distance [*Schwede et al.*, 2008; *Bellin and Tonina*, 2007; *Caroni and Fiorotto*, 2005; *Fiorotto and Caroni*, 2002] has been demonstrated. In all these applications, the exchange of solute mass between neighboring stream tubes acts as a controlling mechanism, either for mixing-controlled reactions or for smoothing of properties that vary in the lateral direction and are transported in the longitudinal direction by advection.

[3] In the past 3 decades, the influence of spatial variability in hydraulic properties on solute transport has intensively been analyzed by stochastic theory [e.g., *Dagan*, 1989; *Gelhar*, 1993; *Rubin*, 2003]. The main focus was on longitudinal dispersion phenomena, describing longitudinal spreading of solute plumes and mixing in the flow direction. To a lesser extent, dispersion theories were also applied to transverse solute transfer. Traditional ensemble dispersion quantifies how quickly the second central spatial moments of the ensemble-averaged concentration increase [e.g., *Gelhar and Axness*, 1983; *Dagan*, 1984; *Neuman et al.*, 1987; *Dagan*, 1988]. The classical transverse ensemble dispersion coefficient starts at the value of the local transverse dispersion coefficient, increases with travel time or distance, exhibits a peak when a few integral scales have been sampled by the plume, decreases again, and approaches an asymptotic value, which is slightly higher than the local transverse dispersion coefficient [*Dagan*, 1994; *Attinger et al.*, 2004]. The intermediate maximum, however, expresses the uncertainty of determining the transverse first moment of individual plumes and thus does not quantify mass exchange between neighboring stream tubes. In effective dispersion, also denoted relative dispersion [*Kitanidis*, 1988; *Dagan*, 1990; *Andricevic and Cvetkovic*, 1998; *Attinger et al.*, 1999; *Dentz et al.*, 2000a, 2000b; *Cirpka and Attinger*, 2003; *Dentz and Carrera*, 2003], the order of taking second central moments and ensemble averages is exchanged, such that the uncertainty in estimating the first moment has been factored out. For longitudinal mixing, *Cirpka* [2002] claimed that effective dispersion for point-like injection is a good predictor of solute mixing. In the transverse direction, however, values of effective dispersion coefficients are very small and most likely prone to higher-order effects [*Dagan*, 1994; *Attinger et al.*, 2004]. Also, individual plumes can get wider and narrower by spatial variability of advection alone, so that traditional spatial moments analysis can be misleading with respect to actual mixing across streamlines [*Rahman et al.*, 2005; *Rolle et al.*, 2009].

[4] Contrasts in hydraulic conductivity lead to convergence and divergence of streamlines, which locally focuses and defocuses the flow field [*Cirpka and Kitanidis*, 2000; *Werth et al.*, 2006; *Chiogna et al.*, 2011]. This phenomenon changes locally the characteristic length and time scales for transverse mass transfer by diffusion and hydromechanical (local-scale) dispersion whenever the solute plume travels through focused and defocused regions of the flow field. Furthermore, most of the mass transfer takes place along the fringe of the plume, where high-concentration gradients can be found [see *Rubin*, 1991; *Cirpka et al.*, 2008; *Bauer et al.*, 2009; *Prommer et al.*, 2009]. Flow focusing (or flow channeling) enhances mass transfer by diffusion and by hydromechanical dispersion to a large degree. When flow is unfocused, mass transfer by these two processes is slowed down since streamlines are farther apart. The general idea by *Werth et al.* [2006] is that on the average, these two effects do not cancel out but lead to an overall enhancement of transverse mass transfer by hydromechanical dispersion and diffusion. Depending on the degree of contrast between different permeability zones, the mass transfer induced by flow focusing can yield a substantial increase in transverse dilution and mixing.

[5] Experimental evidence provided by *Rahman et al.* [2005], *Rolle et al.* [2009], and *Bauer et al.* [2009], as well as field-scale analysis, supports the concepts and initial models provided by *Cirpka and Kitanidis* [2000] and *Werth et al.* [2006]. The results from *Rolle et al.* [2009] clearly illustrate the significant overall enhancement in lateral mixing of a contaminant plume due to inclusion of high-permeability materials within the experimental setup. The increase in lateral spread due to flow focusing is also well documented in numerical studies by *Salandin and Fiorotto* [1998], *de Dreuzy et al.* [2007], and *Beaudoin et al.* [2010]. The impact of permeability inclusions located within contaminant source zones (and related flow focusing effects) on far-field prediction was shown by *de Barros and Nowak* [2010], analytically as well as numerically.

[6] Classical linear stochastic theory does not account for effects of flow focusing and defocusing on transverse mixing because (1) the stochastic-analytical equations have been developed for uniform local dispersion tensors at best and (2) integrations of velocity fluctuations are performed along mean trajectories. This implies that squeezing and stretching of solute plumes and their effect on transverse dispersive mass exchange are neglected in these theories. Higher-order extensions to dispersion theory are restricted to the behavior of the ensemble-averaged concentration in normal coordinates [e.g., *Hsu et al.*, 1996; *Neuman*, 1993] and thus appear insufficient to express the effects of flow focusing on mean mixing and its uncertainty.

[7] The key objective of the present contribution is to derive a stochastic framework of transverse mixing. The effects of plume meandering, squeezing, and stretching in two-dimensional heterogeneous domains will be accounted for by switching from Cartesian spatial coordinates to streamline coordinates before going into stochastic analysis. As a key target, we analyze the probability of a solute particle to jump into a neighboring stream tube. This is expressed in terms of the second central moment in the stream function value. Because mixing is taking place at the fringes of solute plumes, we analyze the squared width of the plume fringe in streamline coordinates, which is identical to the second central moment for a point-like injection. We propose to formulate dispersion in terms of streamline coordinates in individual realizations. We take expected values only in a succeeding step, thus providing an effective mixing-related dispersion coefficient for transport in streamline coordinates.

[8] Figure 1 schematically illustrates the concept as a series of steps: (1) A theoretically infinitely large ensemble of log conductivity fields with identical statistical properties is generated. (2) Flow and (3) steady state transport are simulated on each of these fields. (4) The concentration fields are spatially transformed into streamline coordinates, and (5) second central transverse moments are evaluated (in streamline coordinates) in each realization. Figure 1 shows the injection of the solute over half the volume flux at the inlet; the moments, however, are computed for an equivalent point-like injection at the transition point (for an explanation, see section 5.1). (6) After the steps illustrated in Figure 1, ensemble averaging is performed. (7) Finally, we define an equivalent transverse dispersion coefficient by matching the second central transverse moments in streamline coordinates obtained in heterogeneous domains with those obtained in a homogeneous domain with identical effective hydraulic conductivity.

[9] This paper is organized as follows: Section 2 reviews the governing equations in different systems of coordinates and derives the equivalent transverse dispersion coefficient and the uncertainty of cumulative mixing in principal terms. In sections 3 and 4, stochastic properties of the log conductivity field are propagated to flow and transport in order to obtain closed-form analytical expressions of mixing and its corresponding uncertainty. In section 5, the steps visualized in Figure 1 are explicitly followed with finite ensembles of conductivity fields in numerical simulations, and the results are compared to the analytical expressions derived in sections 3 and 4. Finally, we present our conclusions in section 6. Details of the analytical derivations are included in Appendix A.