Transverse Dispersion Coefficients Research Articles

Evolution of pore-scale concentration PDFs and estimation of transverse dispersion from numerical porous media column experiments

Knowing local concentration distributions is important for transport and mixing, particularly in porous media, yet a comprehensive understanding of them remains a challenge. Computing advancements have enabled high-resolution pore-scale simulations, offering an unprecedented opportunity for in-depth investigation of mixing. In this study we use simulation data to examine concentration distributions at the pore scale in the context of longitudinal (pseudo-one-dimensional) solute transport through a porous column. These distributions arise in a single column from heterogeneous flow at the pore-scale, which gets averaged out when upscaled and are not with reference to statistics across multiple random realizations. To measure these distributions, we first devise a semi-analytical approach to estimate the mean effective transport velocity profile for a non-uniform Darcy-scale fluid velocity, which unavoidably occurs due to the presence of lateral boundaries. This development allows sampling micro-scale concentrations over a moving surface that possesses a well defined Darcy-scale mean concentration, enabling empirical computation of the local concentration distribution. As an added benefit we find that our approach allows for the estimation of transverse dispersion coefficients, which is not typical in traditional column experiments. The implemented approach can estimate it via inverse modeling, and it agrees closely with previously published experimental data across the range of Peclet numbers we studied. We found that the measured pore-scale concentration probability density functions are best represented by a beta distribution, thus validating this longstanding hypothesis with direct evidence. Furthermore, we propose a model to describe the temporal and spatial evolution of the local concentration pdf, as well as its Péclet number dependence.

Application of Oversampling Techniques for Enhanced Transverse Dispersion Coefficient Estimation Performance Using Machine Learning Regression

The advection–dispersion equation has been widely used to analyze the intermediate field mixing of pollutants in natural streams. The dispersion coefficient, manipulating the dispersion term of the advection–dispersion equation, is a crucial parameter in predicting the transport distance and contaminated area in the water body. In this study, the transverse dispersion coefficient was estimated using machine learning regression methods applied to oversampled datasets. Previous research datasets used for this estimation were biased toward width-to-depth ratio (W/H) values ≤ 50, potentially leading to inaccuracies in estimating the transverse dispersion coefficient for datasets with W/H > 50. To address this issue, four oversampling techniques were employed to augment the dataset with W/H > 50, thereby mitigating the dataset’s imbalance. The estimation results obtained from data resampling with nonlinear regression method demonstrated improved prediction accuracy compared to the pre-oversampling results. Notably, the combination of adaptive synthetic sampling (ADASYN) and eXtreme Gradient Boosting regression (XGBoost) exhibited improved accuracy compared to other combinations of oversampling techniques and nonlinear regression methods. Through the combined ADASYN–XGBoost approach, it is possible to enhance the transverse dispersion coefficient estimation performance using only two variables, W/H and bed friction effects (U/U*), without adding channel sinuosity; this represents the effects of secondary currents.

Improvement of the Two-Dimensional Routing Procedure for Observing Dispersion Coefficients in Open-Channel Flow

The dispersion coefficients are crucial in understanding the spreading of pollutant clouds in river flows, particularly in the context of the depth-averaged two-dimensional (2D) advection–dispersion equation (ADE). Traditionally, the 2D stream-tube routing procedure (2D STRP) has been the predominant method for determining both the longitudinal and transverse dispersion coefficients of the 2D ADE under transient concentration conditions. This study aims to quantitatively analyze and address the limitations of the 2D STRP using hypothetically generated data. The findings of these evaluations revealed that the existing 2D STRP failed to accurately reproduce reliable results when the tracer clouds reached wall boundaries. This limitation prompted the development of the 2D STRP-i, which effectively resolves this drawback. The newly developed routing-based observation method, 2D STRP-i, enables the reliable estimation of dispersion coefficients, considering the effect of the wall boundary. The results indicated that the existing 2D STRP yielded 2D dispersion coefficients with relative errors ranging from 40% to 200%, while 2D STRP-i consistently yielded relative errors of 3% to 5% on average. When applied to tracer test data obtained through remote sensing, the 2D STRP-i demonstrated its ability to accurately observe temporal concentration distributions, even when wall boundaries have a significant impact on contaminant transport.

Pore-network modelling of transverse dispersion in porous media under non-Darcy flow conditions

This work is concerned, for the first time, with estimating the transverse dispersion coefficient under non-Darcy laminar flow conditions in porous media using pore-network modeling. The pore-network modelling approach and the mixed cell method are adopted to simulate both the steady laminar flow and the transient transport of solute for Berea and Bentheimer sandstone samples. For non-Darcy flow, the inertial effect is attributed to the quadratic increase of the pore throat mean velocity, which is embedded in the pressure head losses at either expansion or contraction located at the two ends of any pore throat. A time dependent function, based on both Péclet number and average residence time within the pore throat, is used to incorporate the effect of nonuniform pore throat velocity in the dispersion coefficient. A restrictive upper limit of the time step interval is introduced to prevent numerical overshoots of the concentrations calculated over time. For the same pressure gradient through the sample, the results show a decrease in both Péclet number and the coefficient of transverse dispersion in case of adopting the non-Darcy flow condition instead of the Darcy flow one. The percentage of decrease is up to 20 % for the maximum applied pressure gradient through the sample.

Solution of Two-Dimensional Solute Transport Model for Heterogeneous Porous Medium Using Fractional Reduced Differential Transform Method

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy.

Development of Simple Formula for Transverse Dispersion Coefficient in Meandering Rivers

This study aims to develop a straightforward and practical formula to estimate transverse dispersion coefficients in meandering natural rivers, a critical factor for predicting solute transport. We present a novel expression for the transverse dispersion coefficient based on dispersion and hydraulic data sets obtained from tracer experiments conducted in natural rivers. A distinctive feature of the formula is its reliance on one dimensionless hydraulic parameter, u¯u*hRc. To assess the effectiveness and accuracy of our proposed formula, we compare it with previously established equations commonly employed in the field. Furthermore, we apply the formula to natural river bends situated in the Nakdong River of Korea. This equation serves to estimate the initial value of the dispersion coefficient in two-dimensional solute transport modeling. As a result, the calibrated value of the dimensionless transverse dispersion coefficient is 0.97, which is only a 16% difference between the initial value of 0.81 as obtained from the formula. The formula presented in this study, simplifying and utilizing the dimensionless hydraulic parameter, offers a promising approach to estimating transverse dispersion in natural meandering rivers in cases where tracer and secondary flow data are unavailable. Additionally, the formula can be refined with more recent dispersion data, leading to a clearer, more straightforward, and validated formulation that captures the intricate interplay between topography and transverse dispersion.

Resolution limits of size exclusion chromatography columns identified from flow reversal and overcome by recycling liquid chromatography to improve the characterization of manufactured monoclonal antibodies

The flow reversal (FR) technique consists of reversing the flow direction along a chromatographic column. It is used to reveal the origin (such as poor column packing, active sites, or slow absorption/escape kinetics) for the resolution limit of 4.6 mm × 150 mm long columns packed with 1.7 μm 200 Å Bridge-Ethylene-Hybrid (BEHTM) Particles. These columns are used to separate manufactured monoclonal antibodies (mAb, ∼ 150 kDa) from their close impurities (or IdeS fragments, ∼ 100 kDa) by size exclusion chromatography (SEC). FR unambiguously demonstrates that the resolution limit of these SEC columns is primarily due to long-range flow velocity biases covering distances of at least 500 μm across the column diameter. This confirms the existence of center-to-wall flow heterogeneities which cause undesirable tailing for the mAb peak. Because the transverse dispersion coefficient (Dt=1.1 × 10−6 cm2/s) of mAbs across the column diameter is intrinsically low, the bandspreading of the mAb in a single flow direction is in part reversible upon reversing the flow direction. For the very same residence time in the column, the column efficiency is found to increase by +85% relative to that observed under conventional elution mode. The observed peak tailing of the mAb and its sub-units is not caused by active surface sites or by slow absorption/escape from the BEH Particles. Therefore, the most critical mAb impurities (hydrolytic degradation Fab/c and IdeS F(ab′)2 fragments) can only be successfully separated and quantified with acceptable accuracy by adopting alternate pumping recycling liquid chromatography (APRLC). APRLC enables the full baseline separation of the mAb and 100 kDa mAb fragments and partial separation of Fab/c and F(ab′)2 fragments after increasing the number of cycles to ten. It was made possible to accurately measure the relative abundances of the mAb (99.0 ± 0.1%), F(ab′)2 fragment (0.88 ± 0.03%), and Fab/c immunogenic fragment (0.13 ± 0.02%) in less than 45 min for a total mAb sample load of only 5 μg. Still, further improvements are needed to increase the sensitivity of the APRLC method and to reduce the solvent consumption by adopting narrow-bore 2.1 mm i.d. SEC columns.

Importance of turbulent diffusion in transverse mixing in rivers

ABSTRACT An expression for the transverse mixing coefficient in a rectangular channel is derived to evaluate the importance of transverse turbulent diffusion. Although some formulas for the transverse mixing coefficient account for both turbulent diffusion and shear dispersion, most calculations of the transverse dispersion coefficient do not include the effect of transverse turbulent diffusion on shear dispersion. Both vertical and transverse turbulent diffusion contribute appreciably to transverse shear dispersion, and the direct contribution of transverse turbulent diffusion to transverse mixing is important for a range of conditions commonly observed in natural channels. A comparison of predicted and measured transverse mixing coefficients supports including turbulent diffusion in analyses of transverse mixing – both in its effect on shear dispersion and as a direct mechanism of transport.

Influences of Momentum Ratio on Transverse Dispersion for Intermediate-Field Mixing Downstream of Channel Confluence

This study aims to analyze the influences of momentum ratio (Mr) and confluence angle (α) on the transverse dispersion in an urban scale confluence channel from the numerical simulation results using the Environmental Fluid Dynamics Code model. By changing the momentum flux and confluence angle from the simulation results, the analysis focused on the relations between the vertical variations of transverse velocity and transverse dispersion. The high momentum tributary aligned the mixing interface toward the outer bank and created a strong helical motion, which transported the contaminated water along the channel bed and inflows into the recirculation zone. The high momentum ratio induced the large vertical shear in transverse velocity with a strong helical motion and increased the transverse dispersion. However, the helical motion persistence rapidly decreased as the flow reached downstream and led to a decrease in the transverse dispersion for the large confluence angle. Thus, the transverse dispersion coefficient increased with a high momentum ratio and low confluence angle, and the dimensionless transverse dispersion coefficient was in the range of 0.39-0.67, which is observed in meandering channels, for Mr > 1 and α = 45°.

Contaminant transport in the surf zone

Dispersion of dissolved contaminants introduced at various locations within and just outside the surf zone are investigated. It is shown that the Longuet-Higgins model of surf-zone hydrodynamics adequately describes the distribution of longshore currents measured at the laboratory scale. Relations are derived between the longitudinal and transverse dispersion coefficients and the influencing parameters. The maximum longitudinal dispersion coefficients are associated with tracer releases near the breaker line, and longitudinal dispersion coefficients generally increase with travel time for distances up to at least 10 surf-zone widths. In contrast, transverse dispersion coefficients remain relatively constant for increasing travel time. The longitudinal and transverse dispersion coefficients can be significantly influenced by assumed values of local turbulent diffusion and cross-shore shear dispersion.

Evaluation of longitudinal / transverse dispersion coefficients and prediction of concentration at river confluence in two-dimensional solute transport analysis

Evaluation of longitudinal / transverse dispersion coefficients and prediction of concentration at river confluence in two-dimensional solute transport analysis

Modelling transverse solute mixing across a vegetation generated shear layer

ABSTRACT Transverse solute mixing across a vegetation generated horizontal shear layer was quantified using laser induced fluorometry techniques for artificial and real vegetation. A two-dimensional finite difference model (FDM) was developed to describe transverse concentration profiles for flows containing transverse variations in velocity and transverse dispersion, from a steady solute input. The FDM was employed inversely, to optimize the parameters describing the transverse distribution of the transverse dispersion coefficient for vegetation generated shear layers. When laboratory data are available, continuous function descriptions produce slightly improved FDM modelled solute concentration profiles compared with simplified step discontinuity velocity and dispersion inputs. When laboratory data are not available, estimates of step or continuous transverse distributions from other work enable concentration profiles to be predicted with a similar goodness of fit. This paper presents a validated, simple, robust finite difference model to describe the mixing of solutes in a channel containing marginal vegetation.

Dispersion Characteristics in a Counter-Current Microstructured Slurry Bubble Column and Its Analysis Based on the Turbulence and Circulation

The slurry bubble column has gained the attention of researchers over the years because of its wide application in various processes in the chemical and mineral industries. This work aims to quantify the degree of axial and transverse slurry dispersion in a counter-current microstructured slurry bubble column within the ranges of particle concentration of 0.5–3.0 wt %, particle size of 63–408 μm, superficial gas velocity of 0.011–0.074 m/s, and slurry velocity of 0.018–0.058 m/s. The residence time distribution (RTD) technique by online conductivity measurement was used to study the dispersion phenomena at different axial and transverse positions in the column. The axial and transverse slurry dispersion coefficients were found within the ranges of 9.12–137.23 and 12.63–141.83 cm2/s, respectively. Both the coefficients increase with the superficial gas and slurry velocities and the particle diameter but decreases with the particle concentration. The transverse slurry dispersion coefficient was 3.35–38.79% higher than the axial slurry dispersion coefficient. A model was developed to interpret the intensity of the dispersion based on liquid turbulence and circulation. The dispersion due to circulation strongly depends on the slurry velocity relative to the superficial gas velocity. The degree of dispersion was also analyzed by the velocity distribution model. The present study gives insight into process intensification and modeling of the counter-current slurry bubble column.

Velocity distributions, dispersion and stretching in three-dimensional porous media

Using index matching and particle tracking, we measure the three-dimensional velocity field in an isotropic porous medium composed of randomly packed solid spheres. This high-resolution experimental dataset provides new insights into the dynamics of dispersion and stretching in porous media. Dynamic-range velocity measurements indicate that the distribution of the velocity magnitude, , is flat at low velocity (probability density function ). While such a distribution should lead to a persistent anomalous dispersion process for advected non-diffusive point particles, we show that the dispersion of non-diffusive tracers nonetheless becomes Fickian beyond a time set by the smallest effective velocity of the tracers. We derive expressions for the onset time of the Fickian regime and the longitudinal and transverse dispersion coefficients as a function of the velocity field properties. The experimental velocity field is also used to study, by numerical advection, the stretching histories of fluid material lines. The mean and the variance of the line elongations are found to grow exponentially in time and the distribution of elongation is log-normal. These results confirm the chaotic nature of advection within three-dimensional porous media. By providing the laws of dispersion and stretching, the present study opens the way to a complete description of mixing in porous media.

Alternative method to study the radial dispersion in liquid chromatography columns. Part I: Theory

We report on an alternative experimental method to determine the transversal or radial dispersion coefficient (Drad) in packed bed columns for liquid chromatography. The method uses a recently developed type of column end-fitting with an impermeable segmentation ring that splits the incoming flow in a central and peripheral part. Using this device and continuously sending a tracer-laden flow through the central inlet and a tracer-less flow through the peripheral inlet, a steady-state radial dispersion pattern is established which can be used to determine the degree of radial dispersion.The present part of the study lays the theoretical foundation for the method and shows the parameter sensitivity of the different possible data analysis variants. In addition, computational fluid dynamics simulations have been used to validate the established procedure (agreement between true and determined Drad-value better than 0.4%) as well as to study the effect of the most important error sources: the presence of the annular flow segmentation ring and the almost inevitable occurrence of mismatches between the central and the peripheral in- and outlet flows. In the latter case, a combined read-out method can be proposed that nearly perfectly compensates for the flow mismatch error (remaining error on the Drad-value smaller than 3%).

Relation between transverse dispersion and diffusion at meandering channel in two-dimensional mixing based on tracer tests

It is fundamental task to reveal the mechanism of the dispersion and the diffusion on solute transport in open channel flow. In this study, the dispersion and the diffusion coefficients were observed and compared with each other based on the tracer tests at a prototype channel in the Andong experimental site, South Korea. The channel has sinuosity of 1.2 and can control the flow rate. For each experimental condition, measurements of average velocity by wading and the turbulence intensity were performed with a fixed device. The dispersion and diffusion coefficients were calculated using a dissolved tracer (Rhodamine WT) and a surface particle tracer (GPS floater), respectively. For each channel section, based on the tracer experiments, the transverse dispersion coefficient and transverse diffusion coefficient were calculated. The observed diffusion and dispersion coefficients were laid on the reliable ranges. The relation between dispersion and diffusion was analyzed by comparing the transverse diffusion coefficient, calculated by the diffusion of surface particles, and the transverse dispersion coefficient, determined by the surface flow velocity, which was calculated using the vertical distribution equation. Based on the analysis, the results that the turbulent diffusion coefficient is proportional to the turbulence intensity, and the transverse turbulence intensity is proportional to the transverse flow velocity. Also, the transverse flow velocity is relative to the transverse dispersion coefficient. Thus, the correlation between dispersion and diffusion is confirmed by the experimental results, and a formula that could be converted based on the relation is proposed.

Longitudinal and transverse dispersion coefficients of 2D contaminant transport model for mixing analysis in open channels

In this study, longitudinal and transverse dispersion coefficients of the two-dimensional advection-dispersion model were calculated using both the velocity and concentration data obtained through tracer tests in large-scale meandering channels. The velocity-based dispersion coefficients calculated by applying the theoretical equations to the velocity data were compared with concentration-based dispersion coefficients that were calculated by applying the routing procedure to the measured concentration curves. The results showed that the cross-sectional averaged values of the dimensionless longitudinal dispersion coefficient, DL/HU∗, calculated from the velocity profile data ranged 4.1–6.5, which corresponded to the theoretical value, whereas DL/HU∗ by concentration data ranged 14.7–35.5, which was 4–6 times larger than the velocity-based coefficient. The cross-sectional maximum values of the dimensionless transverse dispersion coefficient, DT/HU∗, by the velocity-based method ranged 0.1–3.3, whereas DT/HU∗ by concentration-based method was 0.9–2.9. These results revealed that the velocity-based longitudinal dispersion coefficient is a fractional value of the concentration-based result, which contained mixing caused by channel irregularities throughout the reach, in addition to shear dispersion by vertical velocity variations, while transverse dispersion coefficients by both methods represented the same dispersion mechanism which was dominated by the secondary currents developed in highly sinuous channels.

Determination of binary diffusion coefficients between hot liquid solvents and bitumen with X-ray CT

Abstract The Steam Assisted Gravity Drainage (SAGD) process has been used for in-situ thermal bitumen recovery in Canada. Nevertheless, there are several drawbacks that could impair the SAGD performance. One of the main drawbacks of SAGD is nontrivial consumption of energy to generate steam. To curtail the consumption of steam, many SAGD variants have been proposed. For example, Vapor Extraction (VAPEX) was proposed as a solvent-based cold production method and Solvent-Assisted SAGD (SA-SAGD) was proposed as a solvent-assisted hybrid method. In those solvent-based/assisted methods, the dissolution of solvents in bitumen enhances the in-situ oil mobility. The speed of the mixing between solvents and bitumen in solvent-based/assisted recovery processes is dictated by dispersive mass flux which is proportional to the transverse dispersion coefficient. Diffusion coefficient, tortuosity and transverse dispersivity should be quantified to characterize the transverse dispersion coefficient. In this research, the diffusion coefficients of solvents in Athabasca bitumen were determined with X-ray computed tomography (CT). A medical X-ray CT scanner was used to capture the diffusion phenomenon occurring between fresh liquid solvents and bitumen. The experiments were designed to reproduce the in-situ high pressure, high temperature condition (3.5 MPa, 135C) where solvent and bitumen actually interact at the vapor chamber edge in the SA-SAGD process. Using our new °experimental system, we successfully measured molecular diffusion coefficients at the condition of the vapor chamber edge for the first time. In conclusion, the binary diffusion coefficients between solvents and bitumen were successfully measured at the condition where the mixing of solvents and bitumen actually occurs in the SA-SAGD process. The characterized diffusion coefficients will be input parameters for the reservoir simulation and semi-analytical models.

A Critical Analysis of Transverse Dispersivity Field Data.

Transverse dispersion, or tracer spreading orthogonal to the mean flow direction, which is relevant e.g, for quantifying bio-degradation of contaminant plumes or mixing of reactive solutes, has been studied in the literature less than the longitudinal one. Inferring transverse dispersion coefficients from field experiments is a difficult and error-prone task, requiring a spatial resolution of solute plumes which is not easily achievable in applications. In absence of field data, it is a questionable common practice to set transverse dispersivities as a fraction of the longitudinal one, with the ratio 1/10 being the most prevalent. We collected estimates of field-scale transverse dispersivities from existing publications and explored possible scale relationships as guidance criteria for applications. Our investigation showed that a large number of estimates available in the literature are of low reliability and should be discarded from further analysis. The remaining reliable estimates are formation-specific, span three orders of magnitude and do not show any clear scale-dependence on the plume traveled distance. The ratios with the longitudinal dispersivity are also site specific and vary widely. The reliability of transverse dispersivities depends significantly on the type of field experiment and method of data analysis. In applications where transverse dispersion plays a significant role, inference of transverse dispersivities should be part of site characterization with the transverse dispersivity estimated as an independent parameter rather than related heuristically to longitudinal dispersivity.

Transport in Porous Media with Nonlinear Flow Condition

We investigate local aspects and heterogeneities of porous medium morphology and relate them to the relevant mechanisms of momentum transfer. In the inertial flow range, there are very few experimental data that allow to recognize the effects of porous structure on the flow and transport through porous media. An experimental analysis was performed in order to understand above processes at different Reynolds numbers in randomly structured porous media. The objective of the analysis is to explore the effects of porous media particle size on inertial and viscous forces and determine range of the Reynolds numbers in which the inertial flow predominantly contributes in dispersive processes. Transport characteristics of the randomly structured porous media and the influence of inertial force on longitudinal and transverse dispersion coefficients were studied.