Classical Advection-dispersion Equation Research Articles

Numerical method for fractional Advection–Dispersion equation using shifted Vieta–Lucas polynomials

In the pursuit of creating more precise and flexible mathematical models for complex physical phenomena, this study constructs a unique fractional model for the Advection–Dispersion equation. The Advection–Dispersion equation is a fundamental mathematical tool for analyzing fluid dynamics and mass transfer processes, but traditional integer-order models often fail to accurately capture anomalous transport behaviors. To overcome this limitation, we employ a non-singular non-integer operator based on the Heydari–Hosseininia notion. This operator allows us to transform the classical advection–dispersion equation into a more robust and flexible fractional model, which can better represent a variety of transport phenomena across different scales and mediums.The numerical approximation of the proposed fractional Advection–Dispersion equation model is achieved using a set of specific polynomials known as shifted Vieta–Lucas polynomials. Aided by their unique properties, the shifted Vieta–Lucas polynomials enable us to transform the original differential system into a more computationally tractable algebraic system via a non-integer derivative matrix. Furthermore, we undertake a detailed analysis of convergence and truncation error associated with the shifted Vieta–Lucas polynomials, underpinning the validity and stability of our numerical method. To illustrate the efficiency and robustness of the derived method, we provide several example problems which are solved using our method. The corresponding solutions are extensively presented with the aid of figures and tables, demonstrating the method’s remarkable performance in solving the fractional Advection–Dispersion equation model. The proposed method shows promising potential for further application and development in solving other fractional differential equations and related scientific problems.

Study on CO2 transport in fractal porous media for a Hausdorff fractal derivative advection-dispersion model

PurposeThe classical advection-dispersion equation (ADE) model cannot accurately depict the gas transport process in natural geological formations. This paper aims to study the behavior of CO2 transport in fractal porous media by using an effective Hausdorff fractal derivative advection-dispersion equation (HFDADE) model.Design/methodology/approachAnomalous dispersion behaviors of CO2 transport are effectively characterized by the investigation of time and space Hausdorff derivatives on non-Euclidean fractal metrics. The numerical simulation has been performed with different Hausdorff fractal dimensions to reveal characteristics of the developed fractal ADE in fractal porous media. Numerical experiments focus on the influence of the time and space fractal dimensions on flow velocity and dispersion coefficient.FindingsThe physical mechanisms of parameters in the Hausdorff fractal derivative model are analyzed clearly. Numerical results demonstrate that the proposed model can well fit the history of gas production data and it can be a powerful technique for depicting the early arrival and long-tailed phenomenon by incorporating a fractal dimension.Originality/valueTo the best of the authors’ knowledge, first time these results are presented.

Improving predictions of solute transport in a laboratory sandbox aquifer through high-resolution characterization with hydraulic tomography

Hydraulic tomography (HT) has been developed over the last two decades to image subsurface heterogeneity at high resolution that is critically important to robust solute transport predictions. In this study, HT based on two concepts (i.e., geological zonation and geostatistical inversion models) was utilized to characterize the K distribution of a laboratory sandbox aquifer at different parameter resolutions. We compared the estimated K fields for predicting independent pumping tests and solute transport using the classical advection–dispersion equation (ADE). The impact of using geological zonation models of varying accuracy for HT analyses was investigated. Results revealed that, when being calibrated to datasets from multiple pumping tests, geological zonation models that faithfully represented the true stratigraphy predicted groundwater flow satisfactorily. The deterministic K field of the calibrated reliable geological zonation models satisfactorily reproduced the preferential migration path of the observed tracer plume, and the breakthrough curves (BTCs) at sampling ports. With further resolving the inter- and intra-layer heterogeneity, the reconstructed K fields by HT better predicted both migrating plumes and BTCs for two conservative tracer tests. The estimated dispersivity values for ADE only resulted in slight improvements in the fitting between observed and simulated solute tracer data. It also demonstrated the need of integrating reliable geological information into the geostatistical inversion approach to reproduce geological structures in K fields. Our study suggested the proper consideration of geological zonation between boreholes during field implementation of HT to yield improved solute transport results.

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.

Applicability of the Time Fractional Derivative Model on Fickian and Non-Fickian Transport in the Single Fracture: An Experimental Investigation

The understanding of solute transport in rock fractures is of great importance in many engineering fields. In this study, two groups of experiments on artificial and natural single fractures with different fracture apertures and roughness were conducted to investigate the changes of solute transport regimes. The time fractional advection–dispersion equation (tFADE) as a promising model was applied to describe the anomalous transport. The performance of the classical advection-dispersion equation (ADE) and tFADE was compared according to the fitting precision of breakthrough curves (BTCs). The responses of the fitting parameters in the tFADE to the experimental conditions were also discussed. The results indicated that the non-Fickian transport more likely occurred in the short distance transport, and the larger Peclet number ( Pe ) led to the increase of the exponent of the power-law function in the phase of concentration decline. The tFADE was superior to ADE in capturing the non-Fickian transport especially the tailing behavior. The fractional order of time α in the tFADE was the key parameter to describe the anomalous transport process, and its responding mechanisms of were revealed: the best-fit α decreased with the increase of flow velocity and the decrease of the fracture aperture. The roughness of the single fracture which leads to a complex flow field had a significant effect on the best-fit α . The findings of this study can help for better understanding the effectiveness and physical significance of the tFADE.

Migration modelling of As(V) loaded by humic acid and nano iron oxide composite colloids affected by various environmental factors

• Adsorption of nano-iron oxide on arsenic diminishes with increasing humic acid, pH, and iron oxide. • Increasing ionic strength has an inhibitory effect on the arsenic migration. • The fractal model can accurately capture the arsenic migration in porous media. Nanoparticles have been applied to remediate heavy metal pollutants (e.g., arsenic) in groundwater and soil, where the migration mechanism of pollutants in this multi-component system has not been fully understood. To better understand how nano-iron oxide affects arsenic migration under multivariate influence, this study prepared arsenic loaded composite colloids using arsenic (As(V)), humic acid, and nano iron oxide (n-α-Fe 2 O 3 ). Then we explored the migration behavior of the composite colloids (HA-n-α-Fe 2 O 3 ) loaded As(V) in a quartz sand column affected by n-α-Fe 2 O 3 under various humic acid concentrations, pH values, ionic strengths, and ferric oxide contents. Two migration models, including a classical advection-dispersion equation (ADE) and a Hausdorff fractal advection-dispersion equation (HADE), were used to quantify the observed co-migration of arsenic conveyed by HA-n-α-Fe 2 O 3 colloids loaded As(V) in the column. Our analysis indicates that the adsorption capability of nano-iron oxide on arsenic decreases with pH, humic acid and iron oxide concentrations, thus accelerating arsenic movement. While, the increasement of ionic strength enhances the adsorption force of nano-iron oxide on arsenic and suppresses arsenic migration. Furthermore, simulation results suggest that the HADE model outperforms the traditional ADE model in characterizing arsenic migration, where the time derivative index is an indicator of anomalous diffusion.

Multi-gene genetic programming expressions for simulating solute transport in fractures

In lieu of process-based models, evolutionary artificial intelligence techniques can yield accurate expressions describing complex phenomena. In the current study, closed-form expressions are developed to predict solute transport in a fracture-matrix system as a function of the parameters that describe relevant physical and chemical processes. The study adopts a multi-gene genetic programming approach to approximate a solution of the classical advection–dispersion equation for reactive transport in single, parallel-plate fractures. The approach is employed to obtain an accurate relationship between the hydraulic, geological, and chemical parameters of the fracture-matrix system as inputs and an ensemble of breakthrough curves as outputs. Solutions generated by the developed model showed good agreement with those of corresponding analytical and numerical models. Computationally, the developed approach is highly efficient, particularly when compared with the analytical solution, which typically requires relatively fine discretization to calculate the long-tailed breakthrough curves. Therefore, future work could extend the developed model to simulate field-scale networks and include additional and more complex transport phenomena. This approach advances solute transport behavior predictions through being simpler and computationally more efficient than currently adopted techniques, which is important as the scale of simulation increases from that of a single fracture to a network.

Natural Transform along with HPM Technique for Solving Fractional ADE

The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ 0 , 1 . The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.

A field evidence model: how to predict transport in heterogeneous aquifers at low investigation level

Abstract. Aquifer heterogeneity in combination with data scarcity is a major challenge for reliable solute transport prediction. Velocity fluctuations cause non-regular plume shapes with potentially long-tailing and/or fast-travelling mass fractions. High monitoring cost and a shortage of simple concepts have limited the incorporation of heterogeneity into many field transport models up to now. We present an easily applicable hierarchical conceptualization strategy for hydraulic conductivity to integrate aquifer heterogeneity into quantitative flow and transport modelling. The modular approach combines large-scale deterministic structures with random substructures. Depending on the modelling aim, the required structural complexity can be adapted. The same holds for the amount of monitoring data. The conductivity model is constructed step-wise following field evidence from observations, seeking a balance between model complexity and available field data. The starting point is a structure of deterministic blocks, derived from head profiles and pumping tests. Then, subscale heterogeneity in the form of random binary inclusions is introduced to each block. Structural parameters can be determined, for example, from flowmeter measurements or hydraulic profiling. As proof of concept, we implemented a predictive transport model for the heterogeneous MADE site. The proposed hierarchical aquifer structure reproduces the plume development of the MADE-1 transport experiment without calibration. Thus, classical advection–dispersion equation (ADE) models are able to describe highly skewed tracer plumes by incorporating deterministic contrasts and effects of connectivity in a stochastic way without using uni-modal heterogeneity models with high variances. The reliance of the conceptual model on few observations makes it appealing for a goal-oriented site-specific transport analysis of less well investigated heterogeneous sites.

Fractional Derivative Modeling on Solute Non-Fickian Transport in a Single Vertical Fracture

Solute transport in a single vertical fracture (SVF) cannot be reliably described by the classical advection-dispersion equation (ADE) model, due to the heterogeneity nature of fracture. This study conducted a group of experiments to investigate chloride ion transport in the SVFs under different rough-walled conditions, and then applied a time fractional advection-dispersion equation (F-ADE) model to offer an accurate description. A comparison between F-ADE model and a classical ADE model in describing experimental data, was also carried out. Results show that the FADE model is better than the ADE model in describing the breakthrough curve and heavy-tail phenomenon of solute transport in the fracture. Especially in the experiments with lower flow rate and higher roughness fracture, the FADE model can offer an accurate description, indicating that it is a promising tool for characterizing solute transport heterogeneous vertical fracture.

Comparison of Surface Roughness and Transport Processes of Sawed, Split and Natural Sandstone Fractures

In single fractures, dispersion is often linked to the roughness of the fracture surfaces and the resulting local aperture distribution. To experimentally investigate the effects of diverse fracture types and surface morphologies in sandstones, three fractures were considered: those generated by sawing and splitting, and a natural sedimentary fracture. The fracture surface morphologies were digitally analyzed and the hydraulic and transport parameters of the fractures were determined from Darcy and the tracer tests using a fit of a continuous time random walk (CTRW) and a classical advection–dispersion equation (ADE). While the sawed specimen with the smoothest surface had the smallest dispersivity, the natural fracture has the largest dispersivity due to strong anisotropy and non-matching fracture surfaces, although its surface roughness is comparable to the split specimen. The parameterization of the CTRW and of the ADE agree well for β > 4 of the truncated power law. For smaller values of β, non-Fickian transport processes are dominant. Channeling effects are observable in the tracer breakthrough curves. The transport behavior in the fractures is controlled by multiple constraints such as several surface roughness parameters and the equivalent hydraulic aperture.

Nitrate-leaching and groundwater vulnerability mapping in North Bihar, India

Aquifer vulnerability assessment is crucial for studying the impact of increasing pollution load scenarios on the quantification of contaminant concentration with movement of plume for protecting groundwater resources. Only a few recent studies have focused on the performance evaluation of vulnerability assessment methods using process-based modeling approach. However, the moisture flow and pollutant transport through partially saturated zone plays a crucial role under varying hydrogeological conditions, which are generally ignored in index-based methods. Thus, the objective of this research is to evaluate the vulnerability of groundwater resources to nitrate in Samastipur, Darbhanga and Madhubani districts of Bihar State, India, using soil moisture flow and solute transport modeling. Richard’s equation integrated with the classical advection dispersion equation is simulated using HYDRUS 1D by incorporating a constant head and atmospheric boundary conditions. The time taken to reach the nitrate peak concentration at groundwater table is considered to estimate vulnerability index (VI). Results have shown that high risk in terms of nitrate-leaching vulnerability in southern part of study area is dominated by Gangetic kankar in subsurface. Further, high pollution risk was reported in eastern north part of study area having alluvial deposition in subsurface. The main causes of high risk were due to the short depth of water table, little discharge and more hydraulic conductivity presence in the subsurface media. Moreover, comparatively low vulnerability was observed in area having clay capping of 2–4 m from surface. This research may help in better implementation of agricultural, soil–water conservation practices and urban/industrial infrastructure development in and around the study area.

An investigation on the fractional derivative model in characterizing sodium chloride transport in a single fracture⋆

Contaminant transport in a single fracture cannot be reliably captured by the classical advection-dispersion equation model, due to the heterogeneity nature of the fracture. This study applied a time fractional advection-dispersion equation (ADE) model to quantify chloride ion transport in a single fracture observed in the lab. We further explored the solute transport characteristics by comparing them with the simulation results using a classical ADE model and a space-time fractional ADE model. Comparison results show that the time fractional derivative model is better than the classical ADE model in describing the blocking process of solute transport in the fracture (the tailing phenomenon of the breakthrough curve or BTC). The space-time fractional ADE model is not needed to describe the observed BTC, because only time evolution of solute concentration is considered in this case. Experimental analysis also indicates that the order of the time fractional derivative increases with the fracture aperture, and sub-diffusion is enhanced by a slower flow velocity.

Augmented upwind numerical schemes for a fractional advection-dispersion equation in fractured groundwater systems

The anomalous transport of particles within non-linear systems cannot be captured accurately with the classical advection-dispersion equation, due to its inability to incorporate non-linearity of geological formations in the mathematical formulation. Fortunately, fractional differential operators have been recognised as appropriate mathematical tools to describe such natural phenomena. The classical advection-dispersion equation is adapted to a fractional model by replacing the time differential operator by a time fractional derivative to include the power-law waiting time distribution. The advection component is adapted by replacing the local differential by a fractional space derivative to account for mean-square displacement from normal to super-advection. Due to the complexity of this new model, new numerical schemes are suggested, including an upwind Crank-Nicholson and weighted upwind-downwind scheme. Both numerical schemes are used to solve the modified fractional advection-dispersion model and the conditions of their stability established.

A Lattice Boltzmann model for 2D fractional advection-dispersion equation: Theory and application

Natural soils and aquifers are inherently heterogenous and chemical transport through them is anomalous characterized by an early arrival followed by a persistent tail. How to describe such anomalous phenomena has been an interest and, as a result, a number of approaches have been proposed over the past few decades. Among others, the fractional advection-dispersion equation (FADE) is a model able to describe anomalous transport when the dispersion is regional rather than local as presumed in the classical advection-dispersion equation. Practical application of FADE needs numerical solutions, which is challenging because its spatial discretization gives rise to a full coefficient matrix. In this paper, we propose a Lattice Boltzmann model to solve the two-dimensional FADE. Given that the anomalous dispersion in soils and aquifers is hydrodynamic and caused by spatial variation in water velocity across the pore space and that the chemical plume spreads preferentially along the mean water-flow direction, the dispersion coefficient and the order of the fractional derivative should be both anisotropic. The anisotropies are solved by the two-relaxation time Lattice Boltzmann model using different relaxation parameters in different directions. Compared with existing numerical methods for the FADE, the proposed model has advantages that it is explicit, second-order accurate in both time and space, mass-conservative, and free of numerical dispersion; its stability is independent of dispersion coefficient. We verify the model against the analytical solution of a benchmark problem and then apply it to simulate Cl− movement in a tracer experiment conducted on an up-catchment hillslope. The results show that the FADE with backward skewness can reproduce the breakthrough curves measured from the experiment.

Assessing Groundwater Resource Vulnerability by Coupling GIS-Based DRASTIC and Solute Transport Model in Ajmer District, Rajasthan

Abstract Groundwater aquifer vulnerability has been assessed by incorporating the major geological and hydrogeological factors that affect and control the groundwater contamination using GIS-based DRASTIC model along with solute transport modeling. This work demonstrates the potential of GIS to derive a vulnerability map by overlying various spatially referenced digital data layers (i.e., depth to water, net recharge, aquifer media, soil media, topography, the impact of vadose zone and hydraulic conductivity) that portrays cumulative aquifer sensitivity ratings in Kishangarh, Rajasthan. It provides a relative indication of groundwater aquifer vulnerability to contamination. The soil moisture flow and solute transport regimes of the vadose zone associated with specific hydrogeological conditions play a crucial role in pollution risk assessment of the underlying groundwater resources. An effort has been made to map the vulnerability of shallow groundwater to surface pollutants of thestudy area, using soil moisture flow and contaminant transport modeling. The classical advection-dispersion equation coupled with Richard’s equation is numerically simulated at different point locations for assessing the intrinsic vulnerability of the valley. The role of soil type, slope, and the land-use cover is considered for estimating the transient flux at the top boundary from daily precipitation and evapotranspiration data of the study area. The time required by the solute peak to travel from the surface to the groundwater table at the bottom of the soil profile is considered as an indicator of avulnerability index. Results show a high vulnerability in the southern region, whereas low vulnerability is observed in the northeast and northern parts. The results have recognized four aquifer vulnerability zones based on DRASTIC vulnerability index (DVI), which ranged from 45 to 178. It has been deduced that approximately 18, 25, 34, and 23% of the area lies in negligible, low, medium and high vulnerability zones, respectively. The study may assist in decision making related to theplanning of industrial locations and the sustainable water resources development of the selected semi-arid area.

Non-Fickian Solute Transport in a Single Fracture of Marble Parallel Plate

Accurate prediction of solute transport processes in a fracture aquifer is an important task not only for proper management of the groundwater but also for pollution control. A key issue of this task is how to accurately obtain the experimental data and to analyze the solute transport in fracture in subsurface hydrology, which would greatly help us to understand the releasing mechanism and transport of the solute in a fracture. In this study, a fracture experiment is conducted in a laboratory based on previous studies. The fracture used with a length of 60 cm and a width of 10 cm is sealed with glass glue to avoid leakage of tracer due to uneven fracture walls. The sodium chloride (NaCl) solute is injected from the left of the fracture. And an electrical conductivity monitoring system is installed on the right of the fracture. Then breakthrough curves (BTCs) of solute transport are fitted using the classical advection-dispersion equation (ADE) and the truncation power-law function (TPL) model in the package of the continuous time random walk (CTRW). The results show that the flow satisfies non-Darcian law in the experimental conditions, which can be better fitted using the Forchheimer equation and Izbash equation. The solute transport presents non-Fickian phenomena and shows a long tailing. The fitting results of the TPL model are far better than ADE in fitting the long tailing at three different flow velocities. Furthermore, electrical conductivity monitoring method not only is effective but also has an advantage of no disturbance to water and concentration fields in a fracture.

Effect of Roughness on Conservative Solute Transport through Synthetic Rough Single Fractures

Understanding solute transport in fractured rocks is of particular importance in many applications. Aperture values ranging from 4.7 to 8.7 mm and Reynolds number (Re) values at 9.38~1743.8 were set for investigating fluid flow through synthetic horizontal single smooth and rough fractures. The Brilliant Blue FCF dye was chosen as the tracer to visualize the transport process. This paper focuses on the dispersion process in rough single fractures under non-Darcian flow conditions. Non-Darcian flow existed in both smooth and rough single fractures and the average flow velocity–hydraulic gradient (V–J) relationships were best described by the Forchheimer equation. The main objectives were to check the existing flow and transport models and to study possible correlations between fitting parameters and heterogeneities. The classical advection dispersion equation (ADE) model failed to capture the long-tailing of breakthrough curves (BTCs). Instead, the continuous time random walk (CTRW) model was better at explaining BTCs in both smooth and rough fractures, especially in capturing the long-tailing feature. The non-Darcian coefficient βc in the Forchheimer equation and the coefficient β in the CTRW model appeared to be most relevant for characterizing the heterogeneity of the rough single fractures.

Solute transport along a single fracture in a porous rock: a simple analytical solution and its extension for modeling velocity dispersion

A simple and robust solution is developed for the problem of solute transport along a single fracture in a porous rock. The solution is referred to as the solution to the single-flow-path model and takes the form of a convolution of two functions. The first function is the probability density function of residence-time distribution of a conservative solute in the fracture-only system as if the rock matrix is impermeable. The second function is the response of the fracture-matrix system to the input source when Fickian-type dispersion is completely neglected; thus, the effects of Fickian-type dispersion and matrix diffusion have been decoupled. It is also found that the solution can be understood in a way in line with the concept of velocity dispersion in fractured rocks. The solution is therefore extended into more general cases to also account for velocity variation between the channels. This leads to a development of the multi-channel model followed by detailed statistical descriptions of channel properties and sensitivity analysis of the model upon changes in the model key parameters. The simulation results obtained by the multi-channel model in this study fairly well agree with what is often observed in field experiments—i.e. the unchanged Peclet number with distance, which cannot be predicted by the classical advection-dispersion equation. In light of the findings from the aforementioned analysis, it is suggested that forced-gradient experiments can result in considerably different estimates of dispersivity compared to what can be found in natural-gradient systems for typical channel widths.

Benchmarking numerical codes for tracer transport with the aid of laboratory-scale experiments in 2D heterogeneous porous media

We present a combined experimental and numerical modeling study that addresses two principal questions: (i) is any particular Eulerian-based method used to solve the classical advection-dispersion equation (ADE) clearly superior (relative to the others), in terms of yielding solutions that reproduce BTCs of the kind that are typically sampled at the outlet of a laboratory cell? and (ii) in the presence of matches of comparable quality against such BTCs, do any of these methods render different (or similar) numerical BTCs at locations within the domain? To address these questions, we obtained measurements from carefully controlled laboratory experiments, and employ them as a reference against which numerical results are benchmarked and compared. The experiments measure solute transport breakthrough curves (BTCs) through a square domain containing various configurations of coarse, medium, and fine quartz sand. The approaches to solve the ADE involve Eulerian-Lagrangian and Eulerian (finite volume, finite elements, mixed and discontinuous finite elements) numerical methods. Model calibration is not examined; permeability and porosity of each sand were determined previously through separate, standard laboratory tests, while dispersivities are assigned values proportional to mean grain size. We find that the spatial discretization of the flow field is of critical importance, due to the non-uniformity of the domain. Although simulated BTCs at the system outlet are observed to be very similar for these various numerical methods, computed local (point-wise, inside the domain) BTCs can be very different. We find that none of the numerical methods is able to fully reproduce the measured BTCs. The impact of model parameter uncertainty on the calculated BTCs is characterized through a set of numerical Monte Carlo simulations; in cases where the impact is significant, assessment of simulation matches to the experimental data can be ambiguous.