## 1. Introduction

[2] Solute transport has been traditionally described in terms of advection and dispersion. Advection represents the mean displacement of the solute and is expressed in terms of the mean fluid velocity. Dispersion quantifies the rate of spreading of the solute plume and is expressed in terms of the dispersion coefficient. This description is sufficient for many purposes (e.g., risk analysis), but insufficient when mixing-driven chemical reactions come into play. The rate of reaction is often controlled by the rate at which reactants mix. Hence, it is not surprising that mixing has been thoroughly studied in porous media [*Cirpka and Kitanidis*, 2000; *de Simoni et al.*, 2005; *Kitanidis*, 1994] as well as in complex flows [*Pope*, 2000; *Tennekes and Lumley*, 1972].

[3] Assuming that transport is locally Fickian, *de Simoni et al.* [2005] found that the rate of fast reactions is given by *r* = *f*_{Q} · *f*_{m}, where *f*_{Q} is a chemical factor, dependent on reaction parameters, and *f*_{m} is a mixing factor given by with *d*_{0} the diffusion coefficient and the local concentration. This expression points out that mixing results from the effect of diffusion. As such, it tends to homogenize concentration distributions. However diffusion alone yields limited mixing. It can however be strongly enhanced by flow heterogeneity that generates complex concentration patterns and increases [*Chiogna et al.*, 2011; *Le Borgne et al.*, 2010; *Rolle et al.*, 2009; *Tartakovsky et al.*, 2008].

[4] The key question is how to characterize and evaluate mixing. Using the mixing factor *f*_{m}, a natural choice for the overall mixing rate is:

where Ω is the overall domain. This magnitude is called scalar dissipation rate because it quantifies the rate of decay of the second moment of concentration *M*(*t*) [*Pope*, 2000]. Indeed assuming that concentration obeys the advection diffusion equation and that solute flux through the boundaries is zero, then

where

[5] Therefore, *M*(*t*) characterizes a mixing state in the sense that the better mixed, the smaller *M*(*t*). It is clear that a proper description of transport should honor not only advection and spreading (dispersion), but also mixing. This work is motivated by the need to find a quantitative evaluation of mixing (i.e., *M*(*t*)) to test candidate transport equations.

[6] The problem lies in the intricate link between spreading and mixing. In fact, late time mixing rate can be explained in terms of spreading rate [*Le Borgne et al.*, 2010]. This together with the view of local mixing as controlled by diffusion shows that the mixing stat *M*(*t*) results from both the local diffusion processes and the more global dispersion processes. Therefore, it is natural to distinguish these two origins of mixing by expressing concentration as the sum of the average concentration and the deviation about it *c*′:

[7] Schematically and *c*′ are expected to rate the contributions to mixing of spreading (i.e., dispersion) and diffusion, respectively. More precisely, using equations (3) and (4) and the fact that , the mixing state *M*(*t*) can then be decomposed in

with

[8] That is, the overall actual mixing state *M*(*t*) is the sum of what we call the dispersive mixing state *M*_{1}(*t*) and the local mixing state *M*_{2}(*t*). For stratified flows, *Bolster et al.* [2011] show that the actual mixing *M*(*t*) departs from dispersive mixing, which represents the effect of spreading, at times lower than the diffusion time. At this stage, *M*_{2}(*t*) is significantly larger than 0 showing that concentration perturbations triggered by advective heterogeneities are not yet wiped out by diffusion. Asymptotically however, mixing becomes dominated by spreading and *M*(*t*) becomes equal to *M*_{1}(*t*).

[9] The decomposition of mixing given by equation (5) is not only natural, it is also appealing because the local mixing state *M*_{2}(*t*) turns out to be precisely the spatially integrated concentration variance , a quantity that has been extensively studied for uncertainty quantification, solute plume characterization and risk analysis to cite only few applications [*Andricevic*, 1998; *Caroni and Fiorotto*, 2005; *Fiori and Dagan*, 2000; *Fiori*, 2001; *Fiorotto and Caroni*, 2002; *Kapoor and Gelhar*, 1994a, 1994b; *Kapoor and Kitanidis*, 1998; *Pannone and Kitanidis*, 1999; *Tennekes and Lumley*, 1972; *Tonina and Bellin*, 2008; *Vanderborght*, 2001]. Unfortunately, although appealing, this decomposition cannot generally be used for relating mixing to spreading because most results on concentration variance are based on the approximation that the mean concentration is Gaussian [*Fiori and Dagan*, 2000; *Kapoor and Gelhar*, 1994a; *Kapoor and Kitanidis*, 1998]. Under this closure assumption, can effectively be derived from spreading. The assumption of Gaussianity is however valid only at late times (i.e., much larger than the advection time) or for low levels of heterogeneity, i.e., in the conditions that have been classically investigated [*Kapoor and Kitanidis*, 1998; *Tonina and Bellin*, 2008; *Vanderborght*, 2001]. Actually, the mean concentration becomes significantly non Gaussian at early times and high heterogeneity (Figure 1). Since most mixing occurs then, it is clear that the deviation of actual from dispersive mixing may play an important role and needs to be studied. The objective of our work is precisely to analyze such deviation so as to propose a quantitative evaluation of the evolution of mixing.