Tailing Of Breakthrough Curves Research Articles

The bimolecular reactive transport in heterogeneous porous media: Sub-diffusion in interpretation of laboratory experiment

This present work consists of investigating the effects of particle size heterogeneity and flow rates on transport-reaction kinetics of CuSO4 and Na2EDTA2− in porous media, via the combination of a bimolecular reaction experiment and model simulations. In the early stages of transport, a peak is observed in the concentration breakthrough curve of the reactant CuSO4, related to the delayed mixing and reaction of the reactants. The numerical results show that an increase in flow rate promotes the mixing processes between the reactants, resulting in a larger peak concentration and a slighter tail of breakthrough curves, while an increase in medium heterogeneity leads to a more significant heavy tail. The apparent anomalous diffusion and heavy-tailing behavior can be effectively quantified by a novel truncated fractional derivative bimolecular reaction model. The truncated fractional-order model, taking into account the incomplete mixing, offers a satisfactory reproduction of the experimental data.

Dispersion of artificial tracers in ventilated caves

Artificial CO2 was used as a tracer along ventilated karst conduits to infer airflow and investigate tracer dispersion. In the karst vadose zone, cave ventilation is an efficient mode of transport for heat, gases and aerosols and thus drives the spatial distribution of airborne particles. Modelling this airborne transport requires geometrical and physical parameters of the conduit system, including the cross-sectional areas, the airflow and average air speed, as well as the longitudinal dispersion coefficient which describes the spreading of a solute. Four gauging tests were carried out in one mine (artificial conduit) and two ventilated caves (natural conduits). In this paper, we demonstrate that it is possible to gain reliable airflow rates and geometric information of ventilated karst conduits using CO2 as a tracer. Airflow was gauged along two caves and one mine and compared with punctual measurements made with a hot-wire anemometer. Cross-sectional areas estimated with CO2 tests were compared with those measured in situ. Moreover, breakthrough curve (BTC) analysis displayed an accentuated tailing along the investigated natural conduits due to the presence of dispersive singularities which possibly enable aerosol deposition. The long tailing observed in Milandre and Longeaigue Caves is probably due to cross-section variations. A 1-D advection-dispersion model tested for these sites was unable to fit BTC tailing in natural conduits. In Baulmes artificial conduit, where long tailing is not observed, the dispersion coefficient has been estimated using Chatwin’s method, and compared with the prediction of Taylor’s theory. Despite the regular geometry of Baulmes Mine, Taylor’s correlation significantly underestimates the dispersion coefficient deduced from field data, showing the need for more theoretical work on turbulent dispersion in mines. This paper gives a first insight into air motion and matter dispersion along ventilated karst conduits, preparing for proper aerosol dispersion modelling.

Recharge dynamic and flow-path geometry controls of solute transport in karst aquifer

Solute transport is complex in karst aquifer due to its heterogeneous structure and polytropic hydrodynamic condition. Hydrological monitoring and artificial tracer test were conducted in two flow paths under different recharge conditions, and the Advection–Dispersion Equation and Dual Region Advection–Dispersion models were successfully calibrated to simulate the breakthrough curves. Intermittent concentrated recharge at sinkholes and quick hydrological response at spring outlet often occur after rainstorms. Single symmetrical shape and bimodal shape with long tail were characterized in the breakthrough curves. Stronger recharge hydrodynamic condition drives a larger transport velocity with higher solute transport efficiency. The tracer injection time during a concentrated recharge event is found to significantly affect the solute transport process. Constantly pushed by the follow-up recharge water, the piston transport process of solute in conduit enables higher recovery rate for the tracer injected at the initial recharge process. Solute transient storage in fissures occurs while tracer is injected at the peak flow. A negative correlation between dispersivity and flow velocity is found in karst conduits. The reduction in cross-section area of karst conduits causes high dispersivity and obvious tailing in breakthrough curves due to the blocking function of solute by the uneven bottom surface in karst conduits. The solute transport process is significantly controlled by the changes in hydrodynamic and flow cross-section due to concentrated recharge events, which deepen our understanding of solute transport in karst groundwater.

Effects of Topological Properties with Local Variable Apertures on Solute Transport through Three-Dimensional Discrete Fracture Networks

In this study, the effects of topological properties with local variable apertures on fluid flow and solute transport through three-dimensional (3D) discrete fracture networks (DFNs) were investigated. A series of 3D DFNs with different fracture density, length, and aperture distribution were generated. The fluid flow and solute transport through the models were simulated by combining the MATLAB code and COMSOL Multiphysics. The effects of network topology and aperture heterogeneity on fluid flow and transport process were analyzed. The results show that the fluid flow and solute transport exhibit a strong channeling effect even in the DFNs with identical aperture, in which the areas of fast and slow migration fit well with the high- and low-flow regions, respectively. More obvious preferential paths of flow and migration are observed in individual fractures for the DFN with heterogeneous aperture than the model with identical aperture. Increasing the fracture length exponent reduces the available flow and transport paths for sparse fracture networks but does not significantly change the flow and transport channels for dense fracture networks. The breakthrough curves (BTCs) shift towards the right and slightly lag behind as the fracture density decreases or the aperture heterogeneity increases. The advection–diffusion equation (ADE) model cannot properly capture the evolution of BTCs for 3D DFNs, especially the long tails of BTCs. Compared to the ADE model, the mobile-immobile model (MIM) model separating the liquid phase into flowing and stagnate regions is proven to better fit the BTCs of 3D DFNs with heterogeneous aperture.

An efficient LSTM network for predicting the tailing and multi-peaked breakthrough curves

As typical characteristics of solute transport in the complex flow field, the tailing and the multi-peaked phenomena of the breakthrough curve (BTC) have been widely studied. Due to a large number of model parameters and the complex structure, traditional numerical models are difficult to carry out in practical applications. In this study, we proposed a prediction method of BTC time series based on long short-term memory (LSTM) network. The method was validated by the field tracer experiment data conducted in a subterranean river. The model performance using multivariable and univariate time series as input was compared, and the prediction of the tailing and multi-peaked phenomena of BTC was realized. The results showed that the proposed method can effectively and accurately predict and reproduce different tailing and multi-peaked BTC. In addition, the prediction in different seasons also showed that LSTM could well describe the tracer data in the natural flow with various discharges.

The coupling effects of the matrix thickness and Peclet number on the late time transport tailing in the fracture-matrix systems

Solute transport process in fractured rocks is central to many engineering applications. However, how matrix diffusion affects non-Fickian transport behavior, which is typically characterized by the late time tailing in breakthrough curves and residence time distributions, remains underexplored, especially considering the coupling effects of the finite matrix thickness (h) and transport regimes quantified by Peclet number (Pe). In this work, the fluid flow and solute transport were simulated by directly solving the Navier-Stokes equations and advection-diffusion equation in a two-dimensional fracture-matrix system considering matrix diffusion. In total, eighty cases considering variations of h and Pe were conducted to investigate the impacts of h and Pe on the late time solute tailing characterized by the exponent (n) in the power law function. Numerical results demonstrate that n decreases with an increase of h and Pe, where n eventually reduces to 1.5 that is a theoretical value for an infinite matrix domain. Moreover, the relationship between n and (h, Pe) is established via a polynomial function, which is further validated by the three-dimensional heterogeneous fracture-matrix system. The newly-established function can be used to accurately assess the power law tailing process given the knowledge of measurable h and Pe.

The Effects of Pore Geometry on Late Time Solute Transport with the Presence of Recirculation Zone

The solute transport process in porous media is central to understanding many geophysical processes and determines the success of engineered applications. However, fundamental understanding of solute transport in heterogeneous porous media remains challenging especially when inertial effects are significant. To address this challenge, we employed direct numerical simulations in a variety of intrapore geometries at a high Reynolds number (Re = 10) flow regime, where recirculation zones (RZs) are present with significant inertial effects. We find that the volume of RZs depends on pore geometries. Moreover, RZs serve as an immobile domain that can trap and release solutes that lead to non-Fickian transport, characterized by the early arrival and heavy tailing of breakthrough curves and bimodal residence time distributions (RTDs). Lastly, the late time portion of RTDs is fitted to the power law function with determined exponent n, where n depends on the pore geometries and consequently the volume of RZs. Our study sheds light on the mechanisms of an immobile zone on the solute transport, especially improving our understanding of late time transport tailing in pressurized heterogeneous porous media.

Stepping beyond perfectly mixed conditions in soil hydrological modelling using a Lagrangian approach

Abstract. A recent experiment of Bowers et al. (2020) revealed that diffusive mixing of water isotopes (δ2H and δ18O) over a fully saturated soil sample of a few centimetres in length required several days to equilibrate completely. In this study, we present an approach to simulate such time-delayed diffusive mixing processes, on the pore scale, beyond instantaneously and perfectly mixed conditions. The diffusive pore mixing (DIPMI) approach is based on a Lagrangian perspective on water particles moving by diffusion over the pore space of a soil volume and carrying concentrations of solutes or isotopes. The idea of DIPMI is to account for the self-diffusion of water particles across a characteristic length scale of the pore space using pore-size-dependent diffusion coefficients. The model parameters can be derived from the soil-specific water retention curve, and no further calibration is needed. We test our DIPMI approach by simulating diffusive mixing of water isotopes over the pore space of a saturated soil volume using the experimental data of Bowers et al. (2020). Simulation results show the feasibility of the DIPMI approach for reproducing the measured mixing times and concentrations of isotopes at different tensions over the pore space. This result corroborates the finding that diffusive mixing in soils depends on the pore size distribution and the specific soil water retention properties. Additionally, we perform a virtual experiment with the DIPMI approach by simulating mixing and leaching processes of a solute in a vertical, saturated soil column and compare the results against simulations with the common perfect mixing assumption. The results of this virtual experiment reveal that the frequently observed steep rise and long tailing of breakthrough curves, which are typically associated with non-uniform transport in heterogeneous soils, may also occur in homogeneous media as a result of imperfect subscale mixing in a macroscopically homogeneous soil matrix.

Advanced workflow for multi-well tracer test analysis in a geothermal reservoir

Interpretation of tracer tests in geothermal reservoirs is carried out by fitting the measured data either with simplified two-dimensional (2-D) analytical solutions or with complex numerical models. Available analytical solutions commonly only describe isotropic conditions in 1-D or 2-D, which is generally unsatisfactory to construct realistic reservoir models. Moreover, due to the large spatial and temporal scale of a tracer test in deep reservoirs, the concentration levels measured in the field are relatively low due to dispersion that complicates the assessment of breakthrough curve tailing and residence time. Fitting tracer data with fully resolving 3-D numerical models thus may be more appropriate, but even with a rich set of field data, reliable calibration is often compromised by the high computational effort and data hunger of such models. In this study, an advanced workflow is presented for evaluating tracer test data. Firstly, a 3-D analytical model in which solute transport is considered in anisotropic porous media is developed. Green's function method is applied to obtain a moving line source solution of the convection-dispersion-diffusion equation for solute transport in a 3-D porous medium. In addition, Green's function equation is analytically convoluted with a rectangular pulse function, which represents tracer injection. Secondly, the analytical model results are fitted to the tracer test data by Monte-Carlo simulations to obtain feasible ranges of flow velocities, as well as longitudinal and transversal dispersivities. Finally, the derived parameter values are implemented in a 3-D numerical model to evaluate the solute residence time in a large-scale reservoir. The results applied to field data from the Kızıldere field in Turkey demonstrate that the workflow provides robust estimates of effective parameters from well-to-well data in complex reservoir systems with anisotropic flow paths. Thus, despite the higher effort of applying convolution and stochastic parameter estimation, the preceding analytical step of the workflow substantially eases numerical model set-up.

Experimental Research on the Thresholds of Scale-Dependent Dispersivity of Solute Transport

Abstract The conventional advection-dispersion equation (ADE) has been widely used to describe the solute transport in porous media. However, it cannot interpret the phenomena of the early arrival and long tailing in breakthrough curves (BTCs). In this study, we aim to experimentally investigate the behaviors of the solute transport in both homogeneous and heterogeneous porous media. The linear-asymptotic model (LAF solution) with scale-dependent dispersivity was used to fit the BTCs, which was compared with the results of the ADE model and the conventional truncated power-law (TPL) model. Results indicate that (1) the LAF model with linear scale-dependent dispersivity could better capture the evolution of BTCs than the ADE model; (2) dispersivity initially increases linearly with the travel distance and is stable at some limited value over a large distance, and a threshold value of the travel distance is provided to reflect the constant dispersivity; and (3) compared with the TPL model, both the LAF and ADE models can capture the behavior of solute transport as a whole. For fitting the early arrival, the LAF model is less than the TPL; however, the LAF model is more concise in mathematics and its application will be studied in the future.

Investigating the relationships between parameters in the transient storage model and the pool volume in karst conduits through tracer experiments

The tailing of breakthrough curves (BTCs) commonly observed in the field is often associated with the existence of pools in karst conduits. The transient storage model was always used to simulate the BTC tailing caused by the storage zone, i.e. pool, however, the relationship between parameters in this model and pool volume is not clear, which is very useful for the understanding of the internal geometry of the conduit using this model or prediction of the contaminant transport. In this paper, we performed a series of indoor tracer experiments in a pool-pipe system involving two different pool structures, symmetrical pool (SP) and asymmetrical pool (ASP) with different positions, sizes, and numbers, to explore the relationship between model parameters and the pool volume. Several experiments were conducted in the ASP of 20 cm with different discharges so as to investigate the effect of turbulent flow condition on model parameters. The experimental results indicate that the duration time of the BTC, and the mean residence time of the solute increase linearly with the pool volume (pool size or pool number). For the model parameters, the cross-sectional area of the storage zone (As) or the calculated volume (Vs) of the storage zone using As shows a good linear relationship with the pool volume (Vp) (As = 0.0078Vp + 3E−06; Vs = 0.8195Vp + 0.0003, R2 = 0.95) indicating a potential to use this parameter, which is almost constant with the discharge, to roughly estimate the pool volume. The cross-sectional area (A) of the main channel does not change with the pool volume and changes little with the discharge; it has a close value to the actual cross-sectional area. The remaining parameters, dispersion coefficient (D) and exchange coefficient (α), have strong connections with the pool type, pool distribution or flow condition, which can hardly be used to deduce the internal conduit structure directly. The relationships between model parameters and the pool volume are helpful to use the model parameters to identify the conduit geometry roughly.

Study of Dynamic Concentration Gradient on Mass Transfer Coefficient: New Approach to Mobile–Immobile Modeling

The theory of mobile–immobile partitioning to capture a medium’s heterogeneity for simulating the interaction of contaminant mass between these two regions is still limited to the lump value of mass transfer coefficient (MTC) that fails to capture the long tails of breakthrough curves (BTCs). For a time-dependent solute source, BTCs consists of two parts, for example, rising and falling limbs. During the rising part, the concentration in the mobile region is higher and mass transfer occurs from the mobile to immobile region. However, during falling limb concentration in the immobile region have higher values, resulting in the reverse diffusive mass transfer process. This study focuses on overcoming the reported limitations of the mobile–immobile model (MIM) in the prediction of long tails of BTC during the falling limb. To achieve this objective, we propose an approach that is based on the dynamics of time resident concentration and its gradient between hydraulically coupled mobile and immobile regions. In this modified MIM, we estimated two distinct diffusive MTCs for rising and falling limbs (RFMT) of BTCs using a nonlinear least square optimization algorithm. Two experimental data sets available in the literature were simulated using a numerical solution of the proposed model and asymptotic time-dependent dispersion function. The estimated parameters supported the hypothesis that for pulse type input, liquid phase transport during the rising limb of BTCs is governed by advection and dispersion, whereas during the falling limb it is majorly diffusive dominated that can be represented by the new MTCs. Simulated results of RFMT are then compared with continuous-time random walk (CTRW) and constant mass transfer (CMT) approaches to compare the quality of the simulation. A better overall simulation of experimental BTCs was obtained using RFMT in comparison with other models. Sensitivity analysis is also carried out to evaluate the capabilities of RFMT over the rising and falling portions of BTCs. This theory finds its application in quantifying persistent chemical residuals in the immobile region that acts as a source when purging and subsequently helps when designing appropriate cleansing operations.

Evaluating the Impact of Turbulence Closure Models on Solute Transport Simulations in Meandering Open Channels

River meanders form complex 3D flow patterns, including secondary flows and flow separation. In particular, the flow separation traps solutes and delays their transport via storage effects associated with recirculating flows. The simulation of the separated flows highly relies in the performance of turbulence models. Thus, these closure schemes can control dispersion behaviors simulated in rivers. This study performs 3D simulations to quantify the impact of the turbulence models on solute transport simulations in channels under different sinuosity conditions. The 3D Reynolds-averaged Navier-Stokes equations coupled with the k − ε , k − ω and SST k − ω models are adopted for flow simulations. The 3D Lagrangian particle-tracking model simulates solute transport. An increase in sinuosity causes strong transverse gradients of mean velocity, thereby driving the onset of the separated flow recirculation along the outer bank. Here, the onset and extent of the flow separation are strongly influenced by the turbulence models. The k − ε model fails to reproduce the flow separation or underestimates its size. As a result, the k − ε model yields residence times shorter than those of other models. In contrast, the SST k − ω model exhibits a strong tailing of breakthrough curves by generating more pronounced flow separation.

Improving Predictions of Fine Particle Immobilization in Streams

AbstractFine particles are critical to stream ecosystem functioning, influencing in‐stream processes from pathogen transmission to carbon cycling, all of which depend on particle immobilization. However, our ability to predict particle immobilization is limited by (1) availability of combined solute and particle tracer data and (2) identifying parameters that appropriately represent fine particle immobilization, due to the myriad of objective functions and model formulations. We found that improved predictions of the full distribution of possible fine particle residence times requires using an objective function that assesses both the peak and tailing of breakthrough curves together with solute tracers to constrain in‐stream transport processes. The representation of immobilization processes was significantly improved when solute tracer data were combined with a particle model, starkly contrasting the common assumption that fine particles transport as washload. We develop a clear strategy for improving fine particle transport predictions, reshaping the potential role of fine particles in water quality management.

How Dynamic Boundary Conditions Induce Solute Trapping and Quasi‐stagnant Zones in Laboratory Experiments Comprising Unsaturated Heterogeneous Porous Media

AbstractThe vadose zone is subject to dynamic boundary conditions in the form of infiltration and evaporation. A better understanding of implications for flow and solute transport, arising from these dynamic boundary conditions in combination with heterogeneous structure, will help to improve the prediction of the fate of solutes. We present laboratory experiments and numerical simulations of heterogeneous porous media under unsaturated conditions where controlled, temporally varying precipitation and evaporation are applied to study the effect of dynamic boundary conditions on solute transport in the presence of material interfaces. Dye tracers Eosine Y and Brilliant Blue FCF are utilized to visualize solute transport and analyze redistribution processes in a flow cell. Water and solute fluxes in and out of the flow cell are quantified. While in dynamic experiments application of small infiltration rates (significantly below the saturated hydraulic conductivities of the materials) led to a reversal of transport paths between infiltration and succeeding evaporation, larger infiltration rates altered downward transport such that flow and transport paths differed from those observed during evaporation. Differences in transport paths ultimately led to a redistribution and trapping of solute in one material which manifested as pronounced tailing in breakthrough curves. Trapping was induced not by the formation of a stagnant zone as result of large parameter contrast but by an interplay of dynamic boundary conditions and material heterogeneity. This study thereby highlights the importance to consider dynamic boundary conditions in predictions of solute leaching.

Anomalous transport through free-flow-porous media interface: Pore-scale simulation and predictive modeling

Pore-scale velocity and turbulence structures near streambeds may control solute transport and dispersion in streams. In this study, pore-scale flow and transport simulations are performed to investigate the effects of pore-scale processes on anomalous transport, which is often manifested by remarkably long residence times of solute particles, in coupled free-flow and porous media systems. By solving the 2D Reynolds-averaged Navier–Stokes equations integrated with a renormalization group k − ε turbulence model, we resolve complex pore-scale flows featured by vortices and preferential flows. Then, we incorporate the simulated velocity and turbulence fields into a Lagrangian particle tracking model that considers advection, turbulent diffusion, and molecular diffusion. Simulation results reveal that the interplay between pore-scale vortices and turbulence structures near the free-flow-permeable bed interface controls anomalous transport. High porosity induces strong turbulence penetration and preferential flow paths within permeable beds. The enhanced subsurface turbulence facilitates the escape of solute particles from recirculation zones via turbulent diffusion, causing steep power-law slopes in breakthrough curves (BTCs). In contrast, low porosity introduces heavy tailing in BTCs from particles that are trapped in the near-interface recirculation zones characterized by low velocities and limited turbulence. We upscale and predict particle transport via a Spatial Markov model (SMM) honoring the interplay between Lagrangian velocity distribution and velocity correlation. The SMM reproduces anomalous transport behaviors obtained from the numerical simulations. These results demonstrate that Lagrangian velocity statistics effectively encode anomalous transport mechanisms in the coupled systems.

Mechanisms, Upscaling, and Prediction of Anomalous Dispersion in Heterogeneous Porous Media

AbstractWe study the upscaling and prediction of large‐scale solute dispersion in heterogeneous porous media with focus on preasymptotic or anomalous features such as tailing in breakthrough curves and spatial concentration profiles as well as nonlinear evolution of the spatial variance of the concentration distribution. Spatial heterogeneity in the hydraulic medium properties is represented in a stochastic modeling approach. Direct numerical Monte Carlo simulations of flow and advective particle motion combined with a Markov model for streamwise particle velocities give insight in the mechanisms of preasymptotic and asymptotic solute transport in terms of the statistical signatures of the medium and flow heterogeneity. Based on the representation of equidistantly sampled particle velocities as a Markov process, we derive an upscaled continuous time random walk approach that can be conditioned on the flow velocities and thus hydraulic conductivity in the injection region. In this modeling framework, we identify the Eulerian velocity distribution, advective tortuosity, and the correlation length of particle velocities as the key quantities for large‐scale transport prediction. Thus, the upscaled model predicts the spatial concentration profiles, their first and second centered moments, and the breakthrough curves obtained from direct numerical Monte Carlo simulations in spatially heterogeneous conductivity fields. The presented approach allows to relate the medium and flow properties to large‐scale preasymptotic and asymptotic solute dispersion.

Effects of flow rate variation on solute transport in a karst conduit with a pool

To investigate the effects of flow rate variation on solute transport in a karst conduit, three pipe structures of a constant diameter pipe, the pipe connected to a symmetrical pool and an asymmetrical pool respectively were chosen, and several tracer experiments were conducted separately in each of the three pipe structures at nine flow rates. Experimental results show that the peak of the breakthrough curve (BTC) increased and the tailing decreased with increasing discharge. Three models, the advection–dispersion equation (ADE), the two-region nonequilibrium model (TRNM) and the transient storage model (TSM), were used to simulate BTCs and explore the change of transport parameters with increasing flow rate. Simulations show that ADE was capable of replicating the almost symmetrical BTCs of the single pipe but incapable of fitting the appreciable BTC tails for the pools. Nevertheless, TRNM and TSM can reproduce all BTCs of single pipe and pipe with a pool very well. The research demonstrates the significant effect of the pool on solute transport. The parameters in the two models (TRNM and TSM) exhibited similar trends with increasing discharge in either pool. In the TRNM, a clear positive correlation with discharge emerged for the partition coefficient and mass transfer coefficient. Meanwhile, the main channel cross-sectional area and exchange coefficient in TSM increased gradually with discharge. The storage zone area decreased generally with increasing flow rate. The relationship between solute transport and the flow rate is more complex in the asymmetrical pool than in the symmetrical pool.

Hydrodynamic dispersion of sodium by equilibrium and non-equilibrium methods through undisturbed sandy soil columns from the Adamantina Formation

Column tests using undisturbed samples of residual sandy soil from the Adamantina Formation (K) were performed to determine the sodium hydrodynamic coefficient by equilibrium and non-equilibrium methods. Analyzing the porosity and soil characteristics, as well as the breakthrough curves, it was possible to note a typical non-equilibrium transport behavior, related to the soil dual-porosity, particularly at early and late time behavior in the breakthrough curve tails. The experimental data were best fitted to the non-equilibrium method than to the equilibrium one, and the sum of the square errors was up to five-fold lower. The conceptual model analysis of the methods led to observations on the limitations of each method and a careful analysis of the obtained values, which are also related to the soil characteristics.

Influence of eddies on conservative solute transport through a 2D single self-affine fracture

The solute transport regime is highly dependent upon the heterogeneity of the flow field. In this study, the influence of eddies on conservative solute transport through a two-dimensional (2D) single self-affine fracture was investigated by the Navier-Stokes flow and solute transport simulations. The self-affine rough fracture was generated by the successive random additions (SRA) technique. The simulations showed that eddies had significant influence on the spatial evolution of the solute plume through the fracture. Analysis of breakthrough curves (BTCs) and residence time distributions (RTDs) presented that the solute transport through the fracture was non-Fickian and the developing eddies enhanced the typical non-Fickian characteristics (i.e., “early arrival” and “heavy tail”) in BTCs for a step tracer injection. Increasing the Reynolds number caused the increasing exponent of the power-law tail with the developing eddies. Fitting the non-Fickian BTCs with the classical inverse model (advection–dispersion-equation, ADE) led to a non-negligibleerror due to the presence of eddies. The continuous time random walk (CTRW) inverse model with truncated power law transition rate probability was alternatively employed and the fitting results showed the robust capability of CTRW in capturing the eddy-induced non-Fickian transport. It was found that the value of parameter β of CTRW decreased as the total eddy volume increased, indicating that the developing eddies might increase the magnitude of non-Fickian transport. Furthermore, the dilution index presenting the exponential of the Shannonentropyof a concentration probability distribution was used to quantify the uniformity of concentration distribution within the fracture. We could conclude that eddies provided strong resistance for solute transport and significantly delayed the mass exchange between the main flow channel and the eddy-controlled zones. Consequently, the delayed mass exchange process directly determined the magnitude of tails in BTCs. The results may enhance our understanding of the role of eddies in the solute transport through fractures.