Multirate Mass Transfer Model Research Articles

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.

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.

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.

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.

Mass-transfer impact on solute mobility in porous media: A new mobile-immobile model

The theory for modeling non-equilibrium solute transport in porous media is still based on approximations to a model proposed by Lapidus and Amundson in 1952 that has not been updated. This Mobile–Immobile Model (MIM) is based on the definition of a mass-transfer coefficient (α), which has been proven subject to some severe limitations. Measurements at both laboratory and field scales have demonstrated the scale-dependency of α values. This means that the MIM theory fails in real applications, since α is not constant, as defined in the kinetic model theory, but is a time-residence (or distance) dependent coefficient. Multi-rate mass-transfer models have been proposed in recent literature to capture real-world solute transport with a multiple mass transfer. In this study, we propose a novel model, which implements the analytical solution of Fick's second law of diffusion directly in the nonequilibrium advection/dispersion equation of solute transport in porous media. New model solutions properly fitted data collected during tracer tests carried out at the CNR-IRSA Laboratory (Bari, Italy) in a horizontal sandbox, 2 m of length, by using sodium chloride as the conservative tracer. Selected breakthrough curves at specific positions were used to validate the proposed model solution and estimate both conventional and proposed coefficients of mass transfer. Results have shown a decreasing trend of α from 0.09 to 0.04 h−1 after about 1.2 m of filtration for the investigated sand, whereas new solutions provide two scale-invariant tracer coefficients of rate of tracer mass-transfer (0.004 ± 0.005 h−1) and of tracer time delay (1.19 ± 0.01). The proposed model performs very well, since it provides a readily solved analytical solution with respect to the conventional MIM. Results of the proposed MIM are very similar to those provided by the conventional MIM. The new model solution can be implemented in particle tracking or random walk software in order to solve two-dimensional nonequilibrium solute transport in groundwater.

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.

Conservative transport upscaling based on information of connectivity

AbstractConnected structures in highly heterogeneous hydraulic conductivity fields lead to channels and preferential pathways for the main fluid flux and fastest solute particles. Their spatial complement is zones of slow advection, where solutes are delayed, causing tailing of solute breakthrough curves. These delays depend on the inclusion's size and the hydraulic conductivity contrast between inclusion and channel. The interplay between channels and small‐scale low conductivity inclusions leads to anomalous transport at larger scales. We test whether a simple separation of transport processes between channels and inclusions could be used to parameterize an effective transport model accounting for anomalous transport. Effective transport is represented by a multirate mass transfer model (MRMT): fast channel transport is controlled by parameters of the mobile zone, while slow advective delays are controlled by parameters of the mobile‐immobile exchange. We delineate the connected channels and analyze their connectivity followed by characterizing the low conductivity inclusions. We parameterize a MRMT model using connectivity and the statistics of the low permeable inclusions. Finally, we compare the parameterized MRMT with detailed numerical simulations in heterogeneous hydraulic conductivity fields with a clear separation between connected channel network and inclusions. In intermediately connected hydraulic conductivity fields only the cut‐off time of the tails is represented while early and intermediate time behavior is not reproduced. We suggest that an effective model for the latter case should account for additional processes like variability in advective velocity.

On the validity of effective formulations for transport through heterogeneous porous media

Abstract. 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.

Mathematical equivalence between time‐dependent single‐rate and multirate mass transfer models

AbstractThe often observed tailing of tracer breakthrough curves is caused by a multitude of mass transfer processes taking place over multiple scales. Yet, in some cases, it is convenient to fit a transport model with a single‐rate mass transfer coefficient that lumps all the non‐Fickian observed behavior. Since mass transfer processes take place at all characteristic times, the single‐rate mass transfer coefficient derived from measurements in the laboratory or in the field vary with time . The literature review and tracer experiments compiled by Haggerty et al. (2004) from a number of sites worldwide suggest that the characteristic mass transfer time, which is proportional to , scales as a power law of the advective and experiment duration. This paper studies the mathematical equivalence between the multirate mass transfer model (MRMT) and a time‐dependent single‐rate mass transfer model (t‐SRMT). In doing this, we provide new insights into the previously observed scale‐dependence of mass transfer coefficients. The memory function, g(t), which is the most salient feature of the MRMT model, determines the influence of the past values of concentrations on its present state. We found that the t‐SRMT model can also be expressed by means of a memory function . In this case, though the memory function is nonstationary, meaning that in general it cannot be written as . Nevertheless, the full behavior of the concentrations using a single time‐dependent rate is approximately analogous to that of the MRMT model provided that the equality holds and the field capacity is properly chosen. This relationship suggests that when the memory function is a power law, , the equivalent mass transfer coefficient scales as , nicely fitting without calibration the estimated mass transfer coefficients compiled by Haggerty et al. (2004).

Application of nonequilibrium fracture matrix model in simulating reactive contaminant transport through fractured porous media

Nonequilibrium and nonlinear sorption of the contaminants in the fractured porous media could significantly influence the shape of the breakthrough curve (BTC). For the fracture-matrix system, there are very few studies which consider these processes. In this study, the nonequilibrium fracture-matrix model with two different nonlinear sorption isotherms, namely nonlinear Freundlich and Langmuir sorption isotherms were developed. The effect of sorption nonlinearity and nonequilibrium conditions on the shape of the BTC was studied using the temporal moments. The developed models along with the linear equilibrium, linear nonequilibrium fracture matrix models, and the multirate mass transfer model were used to simulate the BTC, which were compared with the experimental data available in the literature. Both sorption nonequilibrium and nonlinearity were found to significantly influence the shape of the BTC. Presence of sorption nonlinearity reduces the solute spreading, whereas presence of nonequilibrium conditions increases the solute spreading. Considering the sorption nonequilibrium along with the sorption nonlinearity leads to an improved simulation of the BTC. The nonequilibrium nonlinear sorption models could simulate the extended BTC tailing resulting from sorption nonlinearity and rate-limited interaction in the fracture-matrix system.

Multi-Rate Mass Transfer (MRMT) models for general diffusive porosity structures

We determine the relevance of Multi-Rate Mass Tansfer (MRMT) models (Haggerty and Gorelick, 1995) to general diffusive porosity structures. To this end, we introduce Structured INteracting Continua (SINC) models as the combination of a finite number of diffusion-dominated interconnected immobile zones exchanging with an advection-dominated mobile domain. It directly extends Multiple INteracting Continua framework (Pruess and Narasimhan, 1985) by introducing a structure in the immobile domain, coming for example from the dead-ends of fracture clusters or poorly-connected dissolution patterns. We demonstrate that, whatever their structure, SINC models can be made equivalent in terms of concentration in the mobile zone to a unique MRMT model. We develop effective shape-free numerical methods to identify its few dominant rates, that comply with any distribution of rates and porosities. We show that differences in terms of macrodispersion are not larger than 50% for approximate MRMT models with only one rate (double porosity models), and drop down to less than 0.1% for five rates MRMT models. Low-dimensional MRMT models accurately approach transport in structured diffusive porosities at intermediate and long times and only miss early responses.

Linking aquifer spatial properties and non-Fickian transport in mobile–immobile like alluvial settings

Summary Time-nonlocal transport models can describe non-Fickian diffusion observed in geological media, but the physical meaning of parameters can be ambiguous, and most applications are limited to curve-fitting. This study explores methods for predicting the parameters of a temporally tempered Levy motion (TTLM) model for transient sub-diffusion in mobile–immobile like alluvial settings represented by high-resolution hydrofacies models. The TTLM model is a concise multi-rate mass transfer (MRMT) model that describes a linear mass transfer process where the transfer kinetics and late-time transport behavior are controlled by properties of the host medium, especially the immobile domain. The intrinsic connection between the MRMT and TTLM models helps to estimate the main time-nonlocal parameters in the TTLM model (which are the time scale index, the capacity coefficient, and the truncation parameter) either semi-analytically or empirically from the measurable aquifer properties. Further applications show that the TTLM model captures the observed solute snapshots, the breakthrough curves, and the spatial moments of plumes up to the fourth order. Most importantly, the a priori estimation of the time-nonlocal parameters outside of any breakthrough fitting procedure provides a reliable “blind” prediction of the late-time dynamics of subdiffusion observed in a spectrum of alluvial settings. Predictability of the time-nonlocal parameters may be due to the fact that the late-time subdiffusion is not affected by the exact location of each immobile zone, but rather is controlled by the time spent in immobile blocks surrounding the pathway of solute particles. Results also show that the effective dispersion coefficient has to be fitted due to the scale effect of transport, and the mean velocity can differ from local measurements or volume averages. The link between medium heterogeneity and time-nonlocal parameters will help to improve model predictability for non-Fickian transport in alluvial settings.