Multirate Mass Transfer Research Articles

Biofilm Growth in Porous Media Well Approximated by Fractal Multirate Mass Transfer With Advective‐Diffusive Solute Exchange

AbstractBiofilm growth in porous media changes not only the hydrodynamic properties of the medium (reduction in porosity and permeability, and increase in dispersivity), but also the transport itself (breakthrough curves display increasingly fast first arrivals and long tails). These features are well reproduced by multicontinuum models (Multi‐Rate Mass Transfer, MRMT) which can be used to describe reactive transport in heterogeneous porous media and facilitate the simulation of reactions that are localized within biofilms. Here, we present a conceptual and numerical model of biochemical reactive transport with dynamic biofilm growth based on MRMT formulations. Mass exchange between mobile water and immobile biofilm aggregates is represented by a memory function, which simplifies definition of MRMT parameters. We successfully tested this model on two sets of laboratory data and found that (a) a basic model based on the growth of uniformly sized biofilm aggregates fails to reproduce laboratory tracer tests and rate of biofilm growth, while a fractal growth model, which we obtain by integrating the memory functions of biofilm aggregates with a power law distribution, does; (b) the biofilm memory function evolves as the biofilm grows; and (c) the early time portion of eluted volume tracer breakthrough curves are independent of flow rate, whereas the tail becomes heavier when the flow rate is decreased, which implies that both advection controlled and diffusion controlled mass exchange coexist in biofilms. These findings imply that porous media biofilms are essentially different from those developing in human tissues or open spaces.

Multirate mass transfer simulation of denitrification in a woodchip bioreactor

Denitrifying woodchip bioreactors (DWBs) have proven to be an efficient nature-based solution for nitrate removal. Modeling DWBs is required for improving their design and operation, but is hindered by the complexity of the modeled system where numerous chemical species and model parameters are needed. Reactions inside the woodchips are different from those at the edges, causing chemical localization (i.e., apparent simultaneous occurrence of incompatible reactions). We used the Multi Rate Mass Transfer (MRMT) approach to overcome these problems when simulating reactive transport processes in a DWB located at Kiruna, Sweden. Besides denitrification, other nitrogen-cycling processes (e.g., nitrification, dissimilatory nitrate reduction to ammonium, anammox) and alternative electron donors (e.g. oxygen, sulfate) were also considered. Biomass concentration is incorporated into the biochemical reaction rates, including growth and decay, to characterize microbial catalysis. We found that the MRMT model: 1) can account for the heterogeneity of the porous woodchips; 2) was capable of reproducing the nitrogen species evolution in the DWB with kinetic parameters from the literature; and 3) allows reproducing localized biochemical reactions (e.g., aerobic reactions on the woodchip edges, near the DWB entrance and anaerobic reactions inside); and 4) reproduces the full denitrification reactions sequence, but with the different reactions occurring in different portions of the woodchip (e.g., nitrate to nitrite near the edges and nitrite to nitrous oxide further inside). The latter observation suggests that increasing woodchip size may reduce the outflow of these undesired species.

On the localization of chemical reactions in multicontinuum media

Reactive transport (RT) through heterogeneous media, may cause chemical heterogeneity if water flux is slow through portions of the medium. In such cases, chemical localization (i.e., the occurrence of reactions that would not occur in well mixed media) may develop, which is especially relevant for biochemical reactions occurring in biofilms. The objective of this work is to study the conditions for chemical localization. We represent the impact of heterogeneity by means of the non-local multirate mass transfer (MRMT) model, which views the porous media as consisting of one mobile and many immobile zones. A dimensional analysis of the governing equations shows that the problem is characterized by reaction times and the distribution of residence times in immobile zones, relative to transport time. To analyze the interplay between them, we simulated simple RT problems in multicontinuum media. Results indicate that immobile zones with residence times much smaller than transport can be lumped together with the mobile zone by modifying the reaction rates, which reduces computations. More importantly, reactions driven by species that are not present in the inflowing water but are the result of previous reactions will take place preferentially in immobile zones, whose residence time is comparable to or larger than reaction times. In fact, daughter species may take a long time and distance to build up. That is, daughter species will not be largest near the inflow, where parent species display largest concentrations, but further downstream at isolated (long residence times) immobile zones.

Advective Trapping in the Flow Through Composite Heterogeneous Porous Media

We study the mechanisms of advective trapping in composite porous media that consist of circular inclusions of distributed hydraulic conductivity embedded in a high conductivity matrix. Advective trapping occurs when solute enters low velocity regions in the media. Transport is analyzed in terms of breakthrough curves measured at the outlet of the system. The curve’s peak behavior depends on the volume fraction occupied by the inclusions, while the tail behavior depends on the distribution of hydraulic conductivity values. In order to quantify the observed behaviors, we derive two equivalent upscaled transport models. First, we derive a Lagrangian trapping model using the continuous-time random walk framework that is parameterized in terms of volume fraction and the distribution of conductivities in the inclusions. Second, we establish a non-local partial differential equation for the mobile solute concentration by volume averaging of the microscale transport equation. We show the equivalence between the two models as well as (first-order) multirate mass transfer models. The upscaled approach parameterized by medium and flow properties captures all features of the observed solute breakthrough curves and sheds new light on the modeling of advective trapping in heterogeneous media.

Time‐Fractional Flow Equations (t‐FFEs) to Upscale Transient Groundwater Flow Characterized by Temporally Non‐Darcian Flow Due to Medium Heterogeneity

AbstractUpscaling groundwater flow is a fundamental challenge in hydrogeology. This study proposed time‐fractional flow equations (t‐FFEs) for upscaling long‐term, transient groundwater flow and propagation of pressure heads in heterogeneous media. Monte Carlo simulations showed that, with increasing variance and correlation of the hydraulic conductivity (K), flow dynamics gradually deviated from Darcian flow and exhibit sub‐diffusive, time‐dependent evolution which can be separated into three major stages. At the early stage, the interconnected high‐K zones dominated flow, while at intermediate times, the transverse flow due to mixed high‐ and low‐K zones caused delayed rise of the piezometric head. At late times when flow in the relatively high‐K domains reached stability, cells with very low‐K continued to block the entry of water and generate “islands” with low piezometric head, significantly extending the temporal evolution of the piezometric head. The elongated water breakthrough curve cannot be quantified by the flow equation with an effective K, the space‐fractional flow equation, or the multi‐rate mass transfer (MRMT) flow model with a few rates, motivating the development of t‐FFEs assuming temporally non‐Darcian flow. Model applications showed that both the early and intermediate stages of flow dynamics can be captured by a single‐index t‐FFE (whose index is the exponent of the power‐law probability density function of the random operational time for water parcels), but the overall evolution of flow dynamics, especially the enhanced retention of flow at later times, required a distributed‐order t‐FFE with variable indexes for different flow phases that can dominate flow dynamics at different stages. Therefore, transient groundwater flow in aquifers with spatially stationary heterogeneity can be temporally non‐Darcian and non‐stationary, due to the time‐sensitive, combined effects of interconnected high‐K channels and isolated low‐K deposits on flow dynamics (which is the hydrogeological mechanism for the temporally non‐Darcian flow and sub‐diffusive pressure propagation), whose long‐term behavior can be quantified by multi‐index stochastic models.

A general and efficient numerical solution of reactive transport with multirate mass transfer

The presence of low permeability regions within porous media impacts solute transport and the distribution of species concentrations. Therefore, (bio)chemical reactions are equally affected. Multirate Mass Transfer (MRMT) models can be used to represent this anomalous transport process. MRMT conceptualizes the medium as a set of multiple continua: one mobile zone and multiple immobile zones. It simulates species transport in mobile and immobile zones simultaneously, which are related by first-order mass exchange. Numerical modeling of reactive transport in this kind of multicontinua media is complex and demanding because of the high dimensionality of the problem. In this paper, we establish the governing equations of reactive transport in multicontinuum media incorporating chemical kinetics into the governing equations. We propose a general numerical solution of reactive transport with MRMT by applying direct substitution approach (DSA) based on Newton-Raphson method. The efficiency of the proposed algorithm benefits of the block structure of the system, which allows us to eliminate immobile zones equations and leads to significant savings in CPU time. We test the validity of the developed solution by comparison with other numerical and analytical solutions.

Multirate Mass Transfer Approach for Double‐Porosity Poroelasticity in Fractured Media

AbstractNatural and anthropogenic fractured aquifers and reservoirs are dual porosity matrix‐fracture systems, where the fracture network provides highly–conductive flow pathways and the low–permeability matrix stores most of the fluid. The coupling between flow and mechanical deformation in fractured media is often modeled using the classical theory of dual‐porosity poroelasticity (DPP), based on Barenblatt's hypothesis of pressure equilibrium inside the rock matrix blocks. Equilibrium can be expected if the matrix blocks are small and the matrix diffusion time is comparable to the flow time scales along the fractures. In practice, matrix blocks may be large enough so that diffusion time scales are long, and the equilibrium hypothesis breaks down. Here, we study nonequilibrium effects in coupled flow and deformation in fractured media. We compare analytical predictions and modeling results of coupled flow and deformation in heterogeneous fractured porous media. The theoretical analysis is a nonequilibrium, dual‐porosity model. We use this theory to (a) Reveal the limitations of classical DPP formulations. (b) Obtain the scalings for drainage and displacement to be expected for coupled flow and deformation in highly heterogeneous, fractured media. (c) Identify what behavior to expect in fractured aquifers and reservoirs regarding flow and deformation. We observe strong tailing in fluid fluxes and land subsidence that cannot be captured by the classical DPP approach or a single porosity effective medium approach. We show that theoretical predictions from the multirate DPP model and high‐fidelity models agree, even for highly heterogeneous matrix‐fracture systems, and reproduce the observed nonequilibrium effects.

Heterogeneous Multi-Rate mass transfer models in OpenFOAM®

We implement the Multi-Rate Mass Transfer (MRMT) model for mobile–immobile transport in porous media (Haggerty and Gorelick, 1995; Municchi and Icardi, 2019 [1]) within the open-source finite volume library OpenFOAM® (Foundation, 2014). Unlike other codes available in the literature (Geiger et al., 2011 [2]; Silva et al., 2009), we propose an implementation that can be applied to complex three-dimensional geometries and highly heterogeneous fields, where the parameters of the MRMT can arbitrarily vary in space. Furthermore, being built over the widely diffused OpenFOAM® library, it can be easily extended and included in other models, and run in parallel. We briefly describe the structure of the multiContinuumModels library that includes the formulation of the MRMT based on the works of Haggerty and Gorelick (1995) and Municchi and Icardi (2020a). The implementation is verified against benchmark solutions and tested on two- and three-dimensional random permeability fields. The role of various physical and numerical parameters, including the transfer rates, the heterogeneities, and the number of terms in the MRMT expansions, is investigated. Finally, we illustrate the significant role played by heterogeneity in the mass transfer when permeability and porosity are represented using Gaussian random fields. Program summaryProgram Title: mrmtFoamCPC Library link to program files:https://dx.doi.org/10.17632/fkywm44j3y.1Developer’s repository link:https://github.com/multiform-UoN/mrmtFOAMLicensing provisions: GPL 3.0Programming language: C++Nature of problem: Large scale dynamics of heat and mass transfer in heterogeneous media where one mobile region coexists with multiple immobile regions.Solution method: The multi-rate mass transfer model is employed to described pre-asymptotic (i.e., non equilibrium ) transfer between regions. This method is implemented using the opensource finite volume library OpenFOAM®

A multirate mass transfer model to represent the interaction of multicomponent biogeochemical processes between surface water and hyporheic zones (SWAT-MRMT-R 1.0)

Abstract. Surface water quality along river corridors can be modulated by hyporheic zones (HZs) that are ubiquitous and biogeochemically active. Watershed management practices often ignore the potentially important role of HZs as a natural reactor. To investigate the effect of hydrological exchange and biogeochemical processes on the fate of nutrients in surface water and HZs, a novel model, SWAT-MRMT-R, was developed coupling the Soil and Water Assessment Tool (SWAT) watershed model and the reaction module from a flow and reactive transport code (PFLOTRAN). SWAT-MRMT-R simulates concurrent nonlinear multicomponent biogeochemical reactions in both the channel water and its surrounding HZs, connecting the channel water and HZs through hyporheic exchanges using multirate mass transfer (MRMT) representation. Within the model, HZs are conceptualized as transient storage zones with distinguished exchange rates and residence times. The biogeochemical processes within HZs are different from those in the channel water. Hyporheic exchanges are modeled as multiple first-order mass transfers between the channel water and HZs. As a numerical example, SWAT-MRMT-R is applied to the Hanford Reach of the Columbia River, a large river in the United States, focusing on nitrate dynamics in the channel water. Major nitrate contaminants entering the Hanford Reach include those from the legacy waste, irrigation return flows (irrigation water that is not consumed by crops and runs off as point sources to the stream), and groundwater seepage resulting from irrigated agriculture. A two-step reaction sequence for denitrification and an aerobic respiration reaction is assumed to represent the biogeochemical transformations taking place within the HZs. The spatially variable hyporheic exchange rates and residence times in this example are estimated with the basin-scale Networks with EXchange and Subsurface Storage (NEXSS) model. Our simulation results show that (1), given a residence time distribution, how the exchange fluxes to HZs are approximated when using MRMT can significantly change the amount of nitrate consumption in HZs through denitrification and (2) source locations of nitrate have a different impact on surface water quality due to the spatially variable hyporheic exchanges.

Adaptive Multirate Mass Transfer (aMMT) Model: A New Approach to Upscale Regional‐Scale Transport Under Transient Flow Conditions

AbstractThe long‐term evaluation of regional‐scale groundwater quality needs efficient upscaling methods for transient flow. Upscaling techniques, such as the Multirate Mass Transfer (MRMT) method with constant upscaling parameters, have been used for transport with steady‐state flow, yet the upscaling parameters (i.e., rate coefficients) may be time dependent. This study proposed and validated an adaptive MRMT (aMMT) method by allowing the mass transfer coefficients in MRMT to change with the flow field. Advective‐dispersive contaminant transport simulated in a 3‐D heterogeneous medium was used as a reference solution. Equivalent transport under homogeneous flow conditions was evaluated by applying the MRMT and aMMT models for upscaling. The relationship between mass transfer coefficients and flow rates was fitted under steady‐state flow driven by various hydraulic gradients. A power law relationship was obtained, which was then used to update the mass transfer coefficients in each stress period under transient flow conditions in the aMMT method. Results indicated that for advection‐dominated transport, both the MRMT and aMMT methods can upscale the anomalous transport dynamics affected by subgrid heterogeneity under transient flow conditions. Whereas for diffusion‐dominated systems, the MRMT model failed to capture the tails of tracer breakthrough curves after the boundary condition changed, but the results from the aMMT model were significantly improved. However, if the overall flow direction changed, both MRMT and aMMT failed to represent the breakthrough curve tail generated by the heterogeneous system. The results point toward a promising path for upscaling transport in complex aquifers with transient flow.

Generalized multirate models for conjugate transfer in heterogeneous materials

We propose a novel macroscopic model for conjugate heat and mass transfer between a \emph{mobile region}, where advective transport is significant, and a set of \emph{immobile regions} where diffusive transport is dominant. Applying a spatial averaging operator to the microscopic equations, we obtain a \emph{multi-continuum} model, where an equation for the average concentration in the mobile region is coupled with a set of equations for the average concentrations in the immobile regions. Subsequently, by mean of a spectral decomposition, we derive a set of equations that can be viewed as a generalisation of the multi-rate mass transfer (MRMT) model, originally introduced by Haggerty & Gorelick. This new formulation does not require any assumption on local equilibrium or geometry. We then show that the MRMT can be obtained as the leading order approximation, when the mobile concentration is in local equilibrium. The new Generalised Multi-Rate Mode (GMRM) has the advantage of providing a direct method for calculating the model coefficients for immobile regions of arbitrary shapes, through the solution of appropriate micro-scale cell problems. An important finding is that a simple re-scaling or re-parametrisation of the transfer rate coefficient (and thus, the memory function) is not sufficient to account for the flow field in the mobile region and the resulting non-uniformity of the concentration at the interfaces between mobile and immobile regions.

Comparison of negative skewed space fractional models with time nonlocal approaches for stream solute transport modeling

Continuous time random walks (CTRW), multi-rate mass transfer (MRMT), and fractional advection-dispersion equations (FADEs) are three promising models of anomalous transport as commonly found in natural streams. Although these paradigms are mathematically related, understanding their advantages and limitations poses a challenge for model selection. In this paper, we quantitatively evaluate the advection-dispersion equation (ADE), fractional-mobile-immobile (FMIM), fractional-in-space ADE (sFADE), fractional in space transient storage (FSTS), truncated time-fractional model (TTFM), and CTRW models with truncated power-law waiting time distribution (CTRW-TPL) by fitting them first to synthetic data. We then applied these models to observations from tracer experiments conducted in several rivers. Based on the extensive analysis, we conclude that the performance of the FSTS model (in particular, the model with negative skewness β=-1) is comparable or superior to the other nonlocal models evaluated in the paper; therefore, the model represents an alternative to existing models for simulating stream solute transport for spatially-homogeneous flows.

Regressed Models for Multirate Mass Transfer in Heterogeneous Media

AbstractSeveral applications to flow and transport in sorptive porous media are fundamentally dependent on the heterogeneous medium properties. This poses a challenge for direct numerical simulations, as fine grids are needed. One alternative approach is the application of the multirate mass transfer (MRMT) method. In this approach, rather than explicitly modeling immobile regions, such as clay layers, the solute concentration in these regions is treated by explicit state variables, and the transport between mobile and immobile regions is modeled by first‐order exchange terms. In the implementation of this approach, fine‐scale simulations on subdomains are often necessary to develop appropriate parameters, since regressed models for the estimation of these parameters, based on a priori data, are unavailable. To facilitate applications of MRMT to three‐dimensional heterogeneous media at the field scale, in this work, upscaled parameters needed for a two‐rate MRMT model are fit to several hundred three‐dimensional fine‐scale simulations by employing a two‐step fitting procedure. The fitted parameters are then regressed with respect to geostatistical, sorptive, and flow parameters. Upscaled simulations using the fitted and the regressed parameters produce breakthrough curves that are very similar to fine‐scale curves. The data used to regress the parameters span a range of heterogeneous properties and flow conditions, and the model is demonstrated to be robust to untested parameter values and initial and boundary conditions. These developed regressions can be employed to efficiently account for back‐diffusion and desorption in future modeling of transport in heterogeneous media without the need to perform fine‐scale simulations.

Upscaling of Regional Scale Transport Under Transient Conditions: Evaluation of the Multirate Mass Transfer Model

AbstractRegional scale transport models are needed to support the long‐term evaluation of groundwater quality and to develop management strategies aiming to prevent serious groundwater degradation. The purpose of this study is to evaluate the capacity of a previously developed upscaling approach to adequately describe the main solute transport processes, including the capture of late‐time tails under changing boundary conditions. Potential factors that impact the performance of upscaling methods, including temporal variations in mass transfer rates and mass distributions, were investigated. Advective‐dispersive contaminant transport in a 3‐D heterogeneous domain was simulated and used as a reference solution. The equivalent transport under homogeneous flow conditions was then evaluated by applying the multirate mass transfer (MRMT) model. The random walk particle tracking method was used to solve the solute transport for heterogeneous and homogeneous MRMT scenarios under steady state and transient conditions. The results indicate that the MRMT model can capture the tails satisfactorily for plumes transported with ambient steady state flow fields at all studied scales using the same parameters. However, when the boundary conditions change in either local, plume, or regional scale, the mass transfer model calibrated for transport under steady state conditions cannot accurately reproduce the tailings observed for the heterogeneous scenario. The deteriorating impacts of transient boundary conditions on the upscaled model are more significant for regions where the flow fields are dramatically affected, which highlights the poor applicability of the MRMT approach for complex field settings. This finding also has implications for the suitability of other potential upscaling approaches.

Measuring, imaging and modelling solute transport in a microporous limestone

The analysis of dispersive flows in heterogeneous porous media is complicated by the appearance of anomalous transport. Novel laboratory protocols are needed to probe the mixing process by measuring the spatial structure of the concentration field in the medium. Here, we report on a systematic investigation of miscible displacements in a microporous limestone over the range of Péclet numbers, 20<Pe<600. Our approach combines pulse-tracer tests with the simultaneous imaging of the flow by Positron Emission Tomography (PET). Validation of the experimental protocol is achieved by means of control experiments on random beadpacks, as well as by comparing observations with both brine- and radio-tracers (labelled with 11C or 18F). The application of residence time distribution functions reveals mass transport limitations in the porous rock in the form of a characteristic flow-rate effect. Two transport models, namely the Advection Dispersion Equation (ADE) and the Multi-Rate Mass Transfer (MRMT) model, are thoroughly evaluated with both the experimental breakthrough curves and the internal concentration profiles. We observe that the dispersion coefficient scales linearly with the Péclet number for both porous systems. The tracer profiles acquired on the rock sample are successfully described upon application of the MRMT model that uses two representative grain sizes and a fraction of intra-granular pore space that is independent of the fluid velocity. The analysis of the PET images evidences the presence of macrodispersive spreading caused by subcore-scale heterogeneities, which contribute significantly to the value of the estimated core-scale dispersivity. This effect can be significantly reduced upon application of the ‘dispersion-echo’ technique, which enables decoupling the effects of spreading and mixing in heterogeneous porous media. These observations are likely to apply to any laboratory-scale rock sample and the approach presented here provides a practical means to study anomalous transport in these systems.

Quantifying Transport of Arsenic in Both Natural Soils and Relatively Homogeneous Porous Media using Stochastic Models

Core Ideas A tempered time fractional‐derivative equation to model Arsenic transport in soil is proposed. Arsenic kinetics of leaching and flushing was affected by multi‐rate sorption–desorption. Model parameters changed with soil properties, such as clay content and pH condition. The widespread distribution of arsenic in soils is a pollution source that jeopardizes human health, and the transport of arsenic (As) under various conditions is not fully understood and quantified. This study proposes a tempered time fractional advection‐dispersion equation (fADE), to model the rate‐limited diffusion and sorption–desorption of As in soil. Applications show that the time fADE can effectively capture skewed breakthrough curves (BTCs) of As leaching from natural soils, which contain multiple stages of desorption that cannot be fully described by the single‐rate mass transfer (SRMT) model. The time fADE model parameters change with soil properties, such as clay content and pH condition. For comparison purposes, a series of As injection column experiments packed with glass beads were conducted to generate BTCs under various hydrologic conditions. Applications show that the time fADE and the multi‐rate mass transfer (MRMT) model perform better than the SRMT model in characterizing non‐Fickian dynamics, especially for prolonged retention, in most runs of leaching and flushing experiments. Parameters in the MRMT model and the time fADE can be linked to simplify the fitting procedure. Such simplification and representation of experimental conditions demonstrate the promise and applicability of the time fADE model for capturing As transport in soil.

Comparison of Time Nonlocal Transport Models for Characterizing Non-Fickian Transport: From Mathematical Interpretation to Laboratory Application

Non-Fickian diffusion has been increasingly documented in hydrology and modeled by promising time nonlocal transport models. While previous studies showed that most of the time nonlocal models are identical with correlated parameters, fundamental challenges remain in real-world applications regarding model selection and parameter definition. This study compared three popular time nonlocal transport models, including the multi-rate mass transfer (MRMT) model, the continuous time random walk (CTRW) framework, and the tempered time fractional advection–dispersion equation (tt-fADE), by focusing on their physical interpretation and feasibility in capturing non-Fickian transport. Mathematical comparison showed that these models have both related parameters defining the memory function and other basic-transport parameters (i.e., velocity v and dispersion coefficient D) with different hydrogeologic interpretations. Laboratory column transport experiments and field tracer tests were then conducted, providing data for model applicability evaluation. Laboratory and field experiments exhibited breakthrough curves with non-Fickian characteristics, which were better represented by the tt-fADE and CTRW models than the traditional advection–dispersion equation. The best-fit velocity and dispersion coefficient, however, differ significantly between the tt-fADE and CTRW. Fitting exercises further revealed that the observed late-time breakthrough curves were heavier than the MRMT solutions with no more than two mass-exchange rates and lighter than the MRMT solutions with power-law distributed mass-exchange rates. Therefore, the time nonlocal models, where some parameters are correlated and exchangeable and the others have different values, differ mainly in their quantification of pre-asymptotic transport dynamics. In all models tested above, the tt-fADE model is attractive, considering its small fitting error and the reasonable velocity close to the measured flow rate.

Equivalence of Finite Dimensional Input–Output Models of Solute Transport and Diffusion in Geosciences

We show that for a large class of finite dimensional input-output positive systems that represent networks of transport and diffusion of solute in geological media, there exist equivalent multirate mass transfer and multiple interacting continua representations, which are quite popular in geosciences. Moreover, we provide explicit methods to construct these equivalent representations. The proofs show that controllability property is playing a crucial role. These results contribute to our fundamental understanding on the effect of fine-scale geological structures on the transfer and dispersion of solute.

Inherent relevance of MRMT models to concentration variance and mixing-induced reactivity

Several anomalous transport approaches have been developed to model the interaction between fast advectively-dominated transport in well-connected porosity and fracture structures and slow diffusively-dominated transport in poorly-connected or low-permeability ones. Among them, the Multi-Rate Mass Transfer approach (MRMT) represents the anomalous dispersion along the main flow paths (mobile zone) induced by a large distribution of first-order exchanges with immobile zones. Even though MRMTs have been developed for conservative transport processes in the mobile zone, we demonstrate that they also conserve the variance of the concentration distribution in the immobile zones, and, hence, pertain to mixing induced reactivity. This property is established whatever the organization of the immobile zones and whatever the injection and sampling conditions in the mobile zone. It inherently derives from the symmetry properties of the diffusion operator in the immobile zones, but cannot be directly extended to heterogeneous dispersive processes in the mobile zone.

On the Properties of the Parameter Space of the Generalized Continuum Transport Model for Description of Fluid Flow in Porous Networks

Generalized transport models, such as Dual and Multiple Continua Models, Global Random Walk, Multirate Mass Transfer and Continuous Time Random Walk are widely used for description of anomalous transport in fractured and porous media. For these models the form of the parameter space is crucial for the most accurate description of anomalous effects as well as the mean transport phenomenon. Constraining of the parameter space is required for the proper interpretation of the physical properties taking place. In this study the Generalized Continuum Transport model is considered as a versatile tool for the parameter space selection as well as better quantification of anomalous (non-Fickian) transport. Different variants of the parameter space are applied to the GCT model and the breakthrough curves obtained from the pore-network models with strong anomalities are fitted. Flexibility of the model is demonstrated through its static and dynamic adaptivity to network structure and transport complexity. The beneficial results of the curve fitting are also compared with the classical models. It is thus demonstrated that the complexity of the model as well as the model parameters can be directly determined based on fine-scale simulations.