Geological heterogeneity enhances spreading of solutes and causes transport to be anomalous (i.e., non-Fickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through heterogeneous porous media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the multi-rate mass transfer (MRMT) model to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, in general non-dispersive mixing cannot.

Transport is anomalous in heterogeneous porous media. Anomalous transport observations include tailing in concentration breakthrough curves and plumes, or the strong increase in the rate of spreading of plumes. Several frameworks have been developed to generalize the advection–dispersion equation (ADE) and overcome its limitations (Frippiat and Holeyman, 2008). All these alternative frameworks share the goal to model complex permeability, velocity and concentration patterns in unified parsimonious effective equations. The limited number of parameters makes them efficient for the limited quantity of data usually available. In fact, they can be parameterized from breakthrough curves. They comply with the broad residence time distributions and non-local transport processes observed in reality (Gjetvaj et al., 2015; Le Borgne and Gouze, 2008; Willmann et al., 2008). They represent the consequences of complex concentration patterns, of simultaneous concentration trapping and fast progress on residence times while averaging out all the fine concentration structures in the upscaling process. These anomalous transport frameworks have proven to be highly effective for residence times, transport time distribution and effective spreading, both phenomenologically and practically (Berkowitz et al., 2006; Neuman and Tartakovsky, 2009). However, their ability to reproduce mixing, which is required for properly reproducing chemical reactions, has not been tested.

We argue that an effective transport formulation should honor not only the mean advection, and spreading observed in heterogeneous porous media (HPM), but also the evolution of mixing. This should not be understood as limiting anomalous transport frameworks but as extending them to handle broader ranges of physical and chemical processes, and at further promoting the approach of effective equations that upscale out the fine-scale structures to retain only their main consequences in terms of transport, reactivity and reactive transport couplings.

Here, we investigate the relevance of multi-rate mass transfer (MRMT) framework to model not only spreading but also mixing. MRMT is taken as a typical anomalous transport framework. Its advantage lies in providing local concentrations, which can be straightforwardly used to evaluate concentration variance, mixing and mixing-induced reactivity (Babey et al., 2014; Carrera et al., 1998; de Dreuzy et al., 2013; Haggerty and Gorelick, 1995), as well as the apparent reduction in the rate of kinetic reactions (Dentz et al., 2011). The question is whether its validity as a representation of transport through HPM can be extended to reproduce the effects of the evolution of mixing rates resulting from the stretching and folding associated with complex velocity structures (de Anna et al., 2014b; Jimenez-Martinez et al., 2015; Le Borgne et al., 2015).

This comparison is especially appropriate as anomalous transport processes are currently extended to simulate reactive transport processes (Cirpka and Valocchi, 2007; Clement, 2001; de Barros et al., 2012; Donado et al., 2009; Hochstetler et al., 2013; Luo et al., 2008; Luo and Cirpka, 2011; Orgogozo et al., 2013; Schneider et al., 2013). They deal with chemical reactivity either in a stochastic manner, representing reactivity with molecular analogies, or in classical approaches by means of concentrations (Bolster et al., 2010; Cirpka et al., 2012; Ding et al., 2013; Hayek et al., 2012; Knutson et al., 2007; Zhang et al., 2013). Extensions are both required for application purposes and attractive for capturing the consequences of anomalous transport to potential “anomalous” and enhanced reactivity (Battiato et al., 2009; Sadhukhan et al., 2014; Scheibe et al., 2015; Tartakovsky et al., 2009).

Some assessment of MRMT to model reactivity in HPM has been made in former works (Willmann et al., 2010). Equivalent reactivity has been evaluated at some well-defined travel distances on MRMT calibrated on residence time distributions. Here, we follow a different approach by analyzing the temporal development of spreading and mixing. We extend the integrated assessment of mixing-induced reactivity at given travel distances to its temporal development.

Our contribution concerns the comparison of different models much more than the HPM and MRMT models themselves. For the sake of completeness, we recall model equations and simulation methods in Sect. 2 (models and methods) and measures of spreading and mixing in Sect. 3. We use these measures to propose the conditions that should be met by effective (upscaled) transport formulations to be considered valid representations of transport through HPM (Sect. 4). We then test whether MRMT formulations meet the proposed conditions (Sect. 5). While this last section depends on the specific choice of the MRMT framework as an equivalent transport model, the comparison methodology is independent of it and can be used to assess transport equations respecting both spreading and mixing.

We present the MRMT and HPM models sequentially. As they are both well known, we present only the main equations and highlight the critical assumptions of importance in this study.

MRMT models express anomalous transport by the interaction between transport
in a mobile zone and a series of immobile zones (Carrera et al., 1998;
Haggerty and Gorelick, 1995). Transport in the mobile zone is advective and
dispersive with a mean solute velocity

Initial and boundary conditions will be described later for both MRMT and HPM
models. MRMT models differ by the distributions of characteristic rates

We simulate MRMT models with a standard time- and space-adaptative method
that preserves mass (de Dreuzy et al., 2013) and always complies with the CFL
conditions (Daus et al., 1985). The advective and the diffusive processes in
the mobile zone, as well as the exchange with the immobile zones, are treated
with a sequential non-iterative coupling method. These methods lead to
efficient simulations of large spatial domains and extended times with
initial refined resolutions. We have successfully compared them with a more
classical fixed-time Galerkin finite element method, integrated with the
fourth order Runge–Kutta method (ode45 function of Matlab) and found
relative differences less than 10

For reference purposes, we restrict the analysis to heterogeneity of
hydraulic conductivity (

Flow is solved with a finite volume scheme with permeameter-like boundary
conditions under a unit head gradient. Transport is simulated using the ADE,
with heterogeneous advection and homogeneous diffusion. Therefore, it is
characterized by the Peclet number

The same type of injection and boundary conditions are used for both models. Flow has a major flow direction imposed in HPM by a head gradient in the longitudinal direction and periodic boundary direction in the transverse direction. For transport, reflecting and absorbing boundary conditions are used upstream and downstream, respectively (Beaudoin and de Dreuzy, 2013). Injection is performed downstream to the inlet boundary to minimize boundary effects.

Concentration fields normalized by their maximal value

Extended injection conditions are used for the HPM and MRMT models.
Concentrations are homogeneous orthogonally to the main flow direction
within a square wave of longitudinal and transverse widths

For an extended plume, spreading is generally measured by the square root of
the second centered moment of the spatial distribution of concentration

In MRMT, spreading comes from the exchanges to the mobile zone. That is,
spreading results from trapping. Solutes are slowed down and dispersed by
the exchanges with the immobile zones. The resulting dispersivity is a
monotonously increasing function of the residence times in immobile zones
(both their mean

In HPM, spreading comes both from diffusive exchanges with low velocity
zones and from spatial fluctuations of the velocity field (de Dreuzy et al.,
2007; Salandin and Fiorotto, 1998). The asymptotic dispersivity increases
both with the correlation length

Any concentration plume can be approximated by a Gaussian concentration
profile

In summary, in HPM, dispersivity comes primarily form the velocity structure, which drives the generation of gradients in concentration, and thus, mixing. Instead, in MRMT, effective dispersivity is controlled by mobile–immobile exchanges and delays the actual mixing between the immobile and mobile solute concentrations.

Time evolution of the deviation from dispersive mixing

The Gaussian profile only gives a crude approximation of the concentration
field with a strong deviation on the distribution of concentration values,
especially at early times when diffusion has not homogenized the
concentration field in the transverse direction (Fig. 1). Actual
concentrations remain much higher and closer to the initial concentration
value than in the Gaussian profile prediction. That is, the initial
concentrations are much less diluted (i.e., mixed) than in the maximum
entropy Gaussian distribution. The Gaussian profile

We propose four conditions for any effective transport formulation to be
considered as a valid representation of transport through heterogeneous
media. In essence, an effective transport equation should yield the same
mean advection, spreading and mixing as the HPM and be sufficiently flexible
to represent real problems. Evaluation of these conditions can be done as
follows:

Mean advection simply requires mean water velocity (i.e., mean
plume velocity for non-reactive solutes) to equal

Spreading is characterized by dispersivity, which measures the
rate of growth of plume size (Eq. 10). In cases where asymptotic dispersion
is reached, this condition implies that dispersivity of the effective
equation should tend to the asymptotic dispersivity of the HPM. Otherwise,
dispersion (or directly, spread, as measured by

In addition, the time required to reach the above dispersion value should
also be honored by the effective formulation to ensure that the rate of
growth of the plume is reproduced. In our case, where asymptotic dispersion
is reached, we propose to define this criterion in terms of

where

Mixing is required for properly reproducing fast reactions
(slow reactions should be properly reproduced if the resident time
distribution is honored, which is assured if mean advection and dispersion
are reproduced). As discussed above, mixing is essentially dispersive and
well characterized by

To compare the timings of spreading and mixing, we define the additional
criterion

Most of the work on effective transport is of a
theoretical nature, but the ultimate goal should be application to real
problems. This implies that a valid transport formulation should be able to
accommodate different types of boundary conditions and flow regimes (i.e.,
transient flow) and dimensions. Most importantly, it should accommodate
characterization. Dispersion usually includes the effects of heterogeneity
and uncertainty. Whereas the latter is reduced by aquifer characterization,
the former is not. Specifically, hydrologists use geology, hydraulics,
geophysics, hydrochemistry and isotopes to figure out, among other things,
the patterns of spatial variability of hydraulic conductivity. The resulting
models display variability not only in the mean log

We consider it well established that MRMT, and other non-local in-time
formulations, can reproduce mean advection and spreading, as discussed in the
introduction. Mean advection in the MRMT approach is equivalent to that of
the HPM provided that flux and total porosity are equivalent. Additionally, the
distribution of residence times in immobile zones can be adapted so that the
asymptotic dispersivity of the MRMT model be equal to that of the HPM model
in Eq. (11). It is always possible, as dispersivity is an increasing
function of the residence times. This imposes a condition on the temporal
range of

In HPM, the temporal extension of the deviation
from the dispersive mixing regime

Comparison of

All MRMT models capture the sharp rise of

At larger times, progressive release of solute mass from the immobile zone
and equilibration with the concentration values in the mobile zone allow

While they are similar to HPM for the extension of the non-dispersive mixing regime,
MRMT models with slopes

Values of the maximum deviation to the dispersive mixing regime

MRMT models cannot match both the amplitude and the timing of

Dependence of the deviation from dispersive function

Concerning the non-dispersive mixing shapes of the scaled function

To qualify the memory effect in MRMT and HPM, we analyze their sensitivity to
the initial injection width

The concentration second moment

Results of the

We propose conditions to test anomalous transport
frameworks not only on spreading but also on mixing. We define a minimum set
of six essential constraints that should be respected in order to retain the
main transport, reactivity and reactive transport couplings. These
constraints involve the conservation of (1) the mean advection,
(2) dispersivity amplitude and (3) timing generally imposed. Beyond these
flow and spreading metrics, (4) amplitude and (5) timing of the deviation
towards the dispersive mixing regime should be respected. The last condition
concerns (6) the respective timings for mixing and spreading. Under ergodic
injection conditions, spreading is characterized by the standard dispersivity
describing the evolution of the plume size along the main flow direction.
Mixing is characterized by the deviation from the dispersive mixing regime

We use these criteria to evaluate MRMT models by comparison to
advective–diffusive transport simulations through HPM, represented by the
classical isotropic 2-D Gaussian correlated multi-Gaussian log-permeability
fields, characterized by variances between 1 and 9. A broad range of MRMT
models are considered. We conclude that MRMT models cannot match both the
amplitude and the timing of

Our study does not preclude, however, the existence of effective transport equations consistent with spreading and mixing of HPM. Nonetheless, we argue that the proposed criteria and existing results of HPM should be used as guidelines to set up effective transport equations that respect spreading, mixing and eventually reactive transport.

The European Union is acknowledged for its funding through the Marie-Curie
Fellowship PIEF-GA-2009-251710 and through the project
FP7-ENERGY-2012-1-2STAGE TRUST (High-resolution monitoring, real time
visualization and reliable modeling of highly controlled, intermediate and
upscalable size pilot injection tests of underground storage of CO