Constant Dispersion Coefficient Research Articles

Non‐invasive mass transfer measurements in complex biofilm‐coated structures

We demonstrate a novel application of (13)C pulsed field gradient (PFG) NMR to monitor mass transfer, due to both flow and diffusion, in a 3D complex porous support structure modified by biofilm growth. This enables timescales an order of magnitude larger than previously possible to be accessed with respect to displacement probability distribution (propagator) measurements. The evolution in the propagator shape with observation time to the Gaussian asymptote (constant dispersion coefficient) is consequently well resolved. We also simulated the measured displacement propagators with good agreement between experiment and prediction. The methodology has significant potential for the selective characterization of the transport of nutrients, metabolic products, pollutants and biocides in such complex biofilm-containing structures.

Migration Behaviour of Ferrous Ion in Compacted Bentonite Under Reducing Conditions Controlled With Potentiostat

AbstractCarbon steel overpack is corroded by consuming oxygen introduced by repository construction after closure of the repository and then maintains the reducing environment in the vicinity of the repository. The migration of iron corrosion products through the buffer material will affect the migration of redox-sensitive radionuclides. Therefore, it is important to study the migration of iron corrosion products through the buffer material because it may affect the corrosion rate of overpack, and migration of redox-sensitive radionuclides. Electromigration experiments have been conducted with the source of iron ions supplied by anode corrosion of the iron coupon in compacted bentonite. The carbon steel coupon was connected as the working electrode to the potentiostat and was held at a constant supplied potential between - 650 to +300 mV vs. Ag/AgCl electrode for up to 168 hours. The amount of iron penetrated into a bentonite specimen was in good agreement with the calculated value from the corrosion current under the assumption that iron is dissolved as ferrous ions. A model using dispersion and electromigration could explain the measured iron profiles in the bentonite specimens. The fitted value of electromigration velocity depended on the potential supplied. On the other hand the fitted value of the dispersion coefficient did not depend on the potential supplied but a constant. This constant dispersion coefficient could be due to the much larger diffusion coefficient of ferrous ion in bentonite compared with the effect of mechanical dispersion. The experimental configurations used in this study are applicable to the examination of the migration behaviour of cations with the source of iron ions under a reducing condition controlled with a potentiostat.

One‐Dimensional Modeling of Transport in Soils with Depth‐Dependent Dispersion, Sorption and Decay

Macroscopic spatial variations in advection velocity lead to an increase in dispersion with increasing travel distance or depth. In soils, this increase goes along with a decrease in decay and sorption of organic substances. We used three different one‐dimensional models that make different assumptions about the dispersion process to compare predicted leaching in a 1‐m‐deep soil profile with layers with different sorption and decay parameters. The first two convective–dispersive models assume that dispersion results from microscopic variations in solute particle velocities that are not correlated across soil layer boundaries. The third model, a stream tube model (STM), assumes that the particle velocity remains constant along its trajectory and is perfectly correlated in different layers. The three models were parameterized to predict the same inert tracer breakthrough curve (BTC) at 1‐m depth. The first convective–dispersive model assumes a constant dispersion coefficient (“homogeneous” convection–dispersion equation [CDE]). The second model uses different dispersion coefficients in the different layers (“layered” CDE) to predict the same inert tracer BTCs as the STM at the layer boundaries. Despite similar predictions of inert tracer BTCs, the models predicted different BTCs of reactive substances at 1‐m depth. The different predictions by the STM and layered CDE illustrate the importance of the correlation of solute particle velocities in different soil layers. They also point to a fundamental problem related to the use of a CDE with a depth‐dependent dispersion to mimic a dispersion process caused by macroscopic variations in particle velocities.

Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters

We present exact asymptotic similariton solutions of the generalized nonlinear Schrödinger equation (NLSE) with gain or loss terms for a normal-dispersion fiber amplifier with dispersion, nonlinearity, and gain profiles that depend on the propagation distance. Our treatment is based on the mapping of the NLSE with varying parameters to the NLSE with constant dispersion and nonlinearity coefficients and an arbitrary varying gain function. We formulate an effective procedure that leads directly, under appropriate conditions, to a wide range of exact asymptotic similariton solutions of NLSE demonstrating self-similar propagating regimes with linear chirp.

Specification of the dispersion coefficient in the modeling of gravity‐driven flow in porous media

Dispersion is an important component of transport in gravity‐driven flow in porous media. However, its detailed modeling requires large computer powers and long calculation times. In this paper, we theoretically and experimentally investigate the conditions for which dispersion may be treated (1) as a simple constant or (2) as a more complex velocity‐dependent term for flow in a homogeneous porous medium. The fundamental equations for fluid flow and solute transport in a porous medium are solved using an augmented Lagrangian algorithm. A general numerical code is developed in which the dispersion coefficient can be set as either a constant or as velocity‐dependent. The theoretical results predicted by these two models are compared with those of our new laboratory experiments. We also theoretically analyze case scenarios not easily realizable in the laboratory. On the basis of these results we propose a method for estimating representative constant dispersion coefficients and discuss the conditions for which such simplification is valid.

Hydraulic modeling and axial dispersion analysis of UASB reactor

In this work, the axial distribution of a conservative tracer in the upflow anaerobic sludge bed (UASB) reactor was modelled by a two-compartment model describing sludge bed and liquid zones. The orthogonal collocation algorithm was used for numerical solution of the model. The model parameters were estimated using on-line fluorescence-based measurements of an inert tracer rhodamine. Rhodamine concentrations were simultaneously measured at the bottom, middle and top of the sludge bed, as well as in the reactor effluent. The model used either a constant or variable axial dispersion coefficient in the sludge bed zone. A significantly better agreement between model outputs and observed tracer distribution was obtained when the dispersion coefficient was proportional to a constant raised to the power of the reactor height. In addition, a near linear dependence of the dispersion coefficient on the liquid upflow velocity was found. These findings, along with high efficiency of the orthogonal collocation algorithm used for model solution, enable future development of a comprehensive model, which accounts for both hydraulics and biokinetics of the anaerobic digestion process.

Factors affecting longitudinal dispersion in estuaries of different scale

Traditionally, the overall diluting capacity of an estuary is characterized using a coefficient of longitudinal dispersion, K xe , which is given by the ratio of the dispersive flux of a dissolved substance to its tidally averaged longitudinal gradient. A steady-state model, which assumes a balance between the dispersive and non-dispersive fluxes and an exponential increase in estuary cross-sectional area towards the sea, has been used to derive expressions for the axial salinity distribution and the dispersive flux of salt. The model was set up assuming either a constant dispersion coefficient along the estuary or one that increased with distance towards the sea. By comparing salinity predictions with data from five UK estuaries, estimates were made of the maximum dispersive salt flux and the corresponding maximum salinity gradient for each system. The results indicated that there was an approximately linear relation between the fluxes and gradients, and the slope of a line plotted through the origin provided an estimate of a common K xe for all five estuaries. The magnitude of K xe was found to be about 90 m2 s−1 with a standard deviation of approximately ±32 m2 s−1. It is concluded that a representative value of 100 m2 s−1 for K xe is a reasonable first choice when setting up a cross-sectionally averaged estuary model. The results also showed that larger systems, such as the Thames, had lower salinity gradients and lower dispersive salt fluxes, whilst smaller estuaries displayed the opposite characteristics. The model was used to predict the variation in the non-advective flux of salt along an estuary. The distribution was found to be similar to the corresponding flux distribution estimated from observations at the seaward end of the Tees estuary, despite appreciable spatial variations in the individual flux components. Allowing for a small decrease in freshwater flow, the model indicated that there was a decrease in the maximum dispersive flux between neap and spring tides. It is argued that such a reduction in flux can result in a seaward shift in the salinity distribution to a region of greater cross-section, where the freshwater transport per unit area again balances the reduced upstream dispersive flux, as found in the neap to spring response in the Tees estuary.

Salinity modelling for the Sabine?Neches Estuary, Southeast Texas

A two-dimensional, depth averaged, computer code was used to model salinities for the Sabine–Neches Estuary in Southeast Texas, USA. The model bathymetry was set to the mean sea level of 1996, and channel friction was adjusted to give an optimised Manning's n value for the annual, 6-minute water level data at eight stations.The simulated velocities were checked using flow discharges measured at six estuary sections for a three-day survey period. Three dispersion methods were used to calibrate the model for salinity. The data used were a 6 month record for 1996 at seven estuary stations, and a 7-year partial record from 1990 to 1996 for two stations at the Upper and Lower ends of Sabine Lake. The first method used a constant salinity dispersion coefficient and the second used a velocity related coefficient; they were accurate to within an annual mean value of about 5 ppt. For the third method, a two dimensional dispersion function was derived, which modelled to within an annual mean value of about 1 ppt. The ultimate purpose for the study was to estimate salinity increase for Sabine Lake with designed, fresh water flow extractions from the catchment. A maximum salinity increase of about 2 ppt for a 50 year return period was estimated.

Salinity modelling for the Sabine—Neches Estuary, Southeast Texas

A two-dimensional, depth averaged, computer code was used to model salinities for the Sabine–Neches Estuary in Southeast Texas, USA. The model bathymetry was set to the mean sea level of 1996, and channel friction was adjusted to give an optimised Manning's n value for the annual, 6-minute water level data at eight stations.The simulated velocities were checked using flow discharges measured at six estuary sections for a three-day survey period. Three dispersion methods were used to calibrate the model for salinity. The data used were a 6 month record for 1996 at seven estuary stations, and a 7-year partial record from 1990 to 1996 for two stations at the Upper and Lower ends of Sabine Lake. The first method used a constant salinity dispersion coefficient and the second used a velocity related coefficient; they were accurate to within an annual mean value of about 5 ppt. For the third method, a two dimensional dispersion function was derived, which modelled to within an annual mean value of about 1 ppt. The ultimate purpose for the study was to estimate salinity increase for Sabine Lake with designed, fresh water flow extractions from the catchment. A maximum salinity increase of about 2 ppt for a 50 year return period was estimated.

Characterization of macro-scale heterogeneity and homogeneity of porous media employing fractal geometry

Previous study in fractal geometry has demonstrated that porous media can be characterized in terms of a fractal exponent, β. It is found that for β⩽−100, the porous media approaches homogeneity. Increasing values of β up to −1 infer increasingly heterogeneous porous media. β can be obtained by a type curve matching procedure. This is accomplished by plotting the dimensionless concentration against dimensionless time and then compared with the curves from the approximate analytical solution to obtain the best fit curve. From this work, it has been verified that the assumption of constant dispersion coefficient is only valid for porous media that exhibits very small values of β (approaching homogeneity). In this work, the concentration profile of invading fluids is consistent with previous investigation by Ogata and Banks when β⩽−100 or K( t D)⩽0.010. It also supports the work of Ogata and Banks when t D is relatively long for smaller range of fractal exponent.

Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. I. Theory

Eulerian models developed to simulate dispersion in fluid mechanics often consider the flux of the contaminant species to be proportional to the concentration gradient via a constant or time-dependent dispersion coefficient. These models are crude approximations for systems with velocity fluctuations evolving over a hierarchy of scales on the scale of observation. We say a system behaves in a Fickian fashion if the dispersion coefficient is constant, it is quasi-Fickian if the dispersion coefficient is time dependent, and it is convolution-Fickian if the flux is a convolution. The fractional flux in the sense of fractional derivatives is a special case of a convolution-Fickian flux. More general forms of the flux are possible, and in any case we call all fluxes anomalous if there is not a constant coefficient of proportionality between the flux and the gradient of concentration. In paper I of this two-part sequence we present a theory with statistical mechanical origins for simulating anomalous dispersion. Under appropriate limiting conditions the theory gives rise to Fickian, quasi-Fickian, convolution-Fickian, and fractional-Fickian fluxes. The primary result is a dispersive flux of integral type which in its most general form is not a convolution on time (it is non-Markovian however), but it is always a convolution in space. The concentration is represented by the inverse Fourier transform of the self-part of the intermediate scattering function. In paper II we present an experimental procedure that uses this theory to examine if and when the Fickian limit is reached in porous media homogeneous on the Darcy-scale but heterogeneous on the pore-scale.

An analytical solution for two-dimensional contaminant transport during groundwater extraction

We obtain an analytical solution for two-dimensional steady-state transport of conservative contaminant between injecting and pumping wells. Flow and transport are considered in the vertical cross-section. The Dupuit approximation and conformal mapping onto the complex potential domain are employed to determine the velocity and concentration distributions, respectively. We use this solution to derive a priori conditions under which widely used 1-D analytical solutions with constant velocity and dispersion coefficients provide accurate approximations. These conditions are formulated in terms of aquifer parameters, such as hydraulic conductivity, porosity and dispersivities, and remediation strategy, e.g., well spacing and pumping regimes.

Mixing-cell boundary conditions and apparent mass balance errors for advective–dispersive solute transport

In many field or laboratory situations, well-mixed reservoirs like, for instance, injection or detection wells and gas distribution or sampling chambers define boundaries of transport domains. Exchange of solutes or gases across such boundaries can occur through advective or diffusive processes. First we analyzed situations, where the inlet region consists of a well-mixed reservoir, in a systematic way by interpreting them in terms of injection type. Second, we discussed the mass balance errors that seem to appear in case of resident injections. Mixing cells (MC) can be coupled mathematically in different ways to a domain where advective–dispersive transport occurs: by assuming a continuous solute flux at the interface (flux injection, MC-FI), or by assuming a continuous resident concentration (resident injection). In the latter case, the flux leaving the mixing cell can be defined in two ways: either as the value when the interface is approached from the mixing-cell side (MC-RI−), or as the value when it is approached from the column side (MC-RI+). Solutions of these injection types with constant or—in one case—distance-dependent transport parameters were compared to each other as well as to a solution of a two-layer system, where the first layer was characterized by a large dispersion coefficient. These solutions differ mainly at small Péclet numbers. For most real situations, the model for resident injection MC-RI+ is considered to be relevant. This type of injection was modeled with a constant or with an exponentially varying dispersion coefficient within the porous medium. A constant dispersion coefficient will be appropriate for gases because of the Eulerian nature of the usually dominating gaseous diffusion coefficient, whereas the asymptotically growing dispersion coefficient will be more appropriate for solutes due to the Lagrangian nature of mechanical dispersion, which evolves only with the fluid flow. Assuming a continuous resident concentration at the interface between a mixing cell and a column, as in case of the MC-RI+ model, entails a flux discontinuity. This flux discontinuity arises inherently from the definition of a mixing cell: the mixing process is included in the balance equation, but does not appear in the description of the flux through the mixing cell. There, only convection appears because of the homogeneous concentration within the mixing cell. Thus, the solute flux through a mixing cell in close contact with a transport domain is generally underestimated. This leads to (apparent) mass balance errors, which are often reported for similar situations and erroneously used to judge the validity of such models. Finally, the mixing cell model MC-RI+ defines a universal basis regarding the type of solute injection at a boundary. Depending on the mixing cell parameters, it represents, in its limits, flux as well as resident injections.

Effects of Infiltration on Agricultural Runoff Contamination

A procedure is presented for estimating the total pollutant mass loss from the interrill flow region of an agricultural field during a runoff event. It involves applications of an overland flow model and a mass transport model. A two-dimensional interrill flow model by Limback and other researchers provides lateral depth and velocity distributions that are used to determine the dispersion and entrainment coefficients required by the transport model. The transport model of Wallach and van Genuchten with variable rather than constant dispersion and entrainment coefficients served to model the transport process that carries solute from the soil interstices to the overland flow. Wallach's mass loss rate equation was integrated over space (laterally) and time to determine total mass loss. The effects of rainfall rate, infiltration rate, surface roughness, surface slopes, and soil properties are included in the analysis. An example application and an explicit sensitivity analysis of the effect of infiltration r...

Experimental and numerical validation of the dead-zone model for longitudinal dispersion in rivers

Experimental results of longitudinal dispersion in rivers are discussed in the light of new findings with respect to the “dead-zone” theory. Both the experiments made by authors as well as some literature results are taken into consideration. The dead-zone model parameters, namely the relative dead-zone volume, penetration time of the tracer into dead-zones, constant mean flow velocity and dispersion coefficient, are obtained with the use of the nonlinear least square technique applied to the image functions of concentration time-distributions. The statistical moments of the concentration time distributions as functions of distance are also analyzed. Expressions for statistical moments, recently obtained by authors, are found to agree well with both experimental and computational results.

A mathematical model for flow and solute transport in non-homogeneous rock fractures

An analytical solution presented in a previous paper is incorporated into a numerical algorithm for determining water flow and solute transport in a single rectangular rock fracture with variable aperture and surface roughness. The flow is laminar and steady. Head and velocity are predicted analytically at each point in the fracture using the integral transform method. It is assumed that the solute is non-reactive and its transport is entirely advective. The method of characteristics is used to determine the trajectories of fluid particles in the flow field. It is shown that heterogeneities in the void space of a fracture and the associated channel flow create “fingering” of the solute and an apparent dispersion even if dispersion is not explicitly accounted for in the transport equation. Numerical examples are presented for flow and transport in fractures with rough surfaces having fractal dimensions ranging between 2.0 and 2.3. It is found that the flow rate passing through those fractures is about 65–95% of the flow rate that would pass if the fractures were modeled as two parallel plates with a uniform aperture equal to their average aperture and the cubic law was valid. Also, estimating the time of breakthrough for solute transport in rough fractures using their average apertures can be misleading. The actual time of breakthrough and the breakthrough curves can be determined using the technique presented herein which requires a detailed description of the fracture void space. Finally, it is shown that the conventional one-dimensional advection-dispersion model using a constant dispersion coefficient does not simulate accurately the apparent dispersion created by channel flow in fractures if their hydraulic or average apertures are used to compute the seepage velocity.

The concept of the Dilution Index

In many applications, it is important to make the distinction between spreading and dilution of a plume in groundwater. Spreading is associated with the stretching and deformation of a contaminant plume, whereas dilution is associated with the increase in volume of the fluid occupied by the solute. The dilution and spreading of a Gaussian plume in a homogeneous porous medium with constant velocity are related in a simple fashion and are both characterized by the same parameters, the dispersion coefficients. However, the geological formations of interest in field applications are heterogeneous, and the plumes are irregular in shape. The dispersion coefficients that are deduced from tracer tests usually measure an overall rate at which a tracer plume spreads about its centroid and depend critically on the heterogeneity of the formation. These macroscopic dispersion coefficients are not reliable measures of the rate at which the maximum concentration is reduced because in heterogeneous formations the rates of dilution and spreading can be quite different. The main objective of this work is to introduce a new macroscopic measure of dilution, the dilution index E. Examples serve to demonstrate the usefulness of the measure. A general expression for the rate of dilution of a tracer plume is derived. The exact rate of increase of the dilution index under the idealized conditions of constant dispersion coefficients and a Gaussian plume is computed, and a lower bound is found to the same quantity for non‐Gaussian plumes. For the general heterogeneous case the analysis demonstrates that the instantaneous rate of increase of ln E is proportional to the small‐scale dispersion coefficients, everything else being the same. The rate of increase of ln E depends also on the degree of irregularity in the shape of the plume. Thus, in the long term, geologic heterogeneity should increase the rate of dilution because spatial variability in the flow velocity tends to deform plumes and make them less regular.

A SIMPLE, INVERSE MODEL FOR ESTIMATING NITROGEN REACTION RATES FROM SOIL COLUMN LEACHING EXPERIMENTS AT STEADY WATER FLOW

Many soil nitrogen studies have included estimations of nitrification or denitrification rates in soil column systems based on simple mass balance considerations. This approach does not take into account the effect of the solute dispersion. The present study presents a simple, inverse finite difference model (IFDM) for estimating temporal and spatial variations in nitrification or denitrification rates from measured nitrogen concentration profiles at steady water flow. The model takes into account both convective and dispersive transport of nitrogen by using a Crank-Nicolson finite-difference discretization of a paraphrased convection-dispersion equation for solute transport including a reaction term. A test procedure is suggested and used to find criteria for avoiding numerical errors in the calculations. Significant numerical errors are inherent in the IFDM but can be avoided if the depth increment and the ratio of time increment to depth increment are chosen carefully. The possible applications of the simple IFDM are illustrated by using the model on soil column data for rapid, continuous leaching of nitrate through water-saturated porous media at steady water flow. The IFDM-based analysis showed a large influence of soil composition and temperature on the temporal and spatial variations in nitrate removal rates. The IFDM assumes constant pore water velocity and solute dispersion coefficient. In order to use the IFDM not only for qualitative but also for quantitative predictions, these two parameters need to be carefully measured.

On the constancy of axial dispersion coefficients in liquid fluidized beds

The assumption of constant axial dispersion coefficient in residence time distribution (RTD) experiments is examined for liquid fluidized beds. Experimental results show this assumption to be violated in short beds. A quantitative criterion is proposed for the constraint required to achieve the true value of the axial dispersion coefficient in RTD experiments.

An analytical solution for one‐dimensional transport in heterogeneous porous media

An analytical solution for describing the transport of dissolved substances in heterogeneous porous media with a distance‐dependent dispersion relationship has been developed. The analytical solution can be used to characterize differences in the transport process relative to the classical convection‐dispersion equation which assumes that the hydrodynamic dispersion in the porous medium remains constant. The form of the hydrodynamic dispersion function used in the analytical solution is , where α(x) is the distance‐dependent dispersivity and is the average pore water velocity. It is shown that for models which differ only in how the dispersion function is expressed, erroneous model parameters may result from parameter estimation techniques which assume a constant hydrodynamic dispersion coefficient if the porous medium is more accurately characterized by a distance‐dependent dispersion relationship. For such situations, the proposed model could be used to provide an alternate means for obtaining these parameters. The overall features of the solution are illustrated by several examples for constant concentration and flux boundary conditions.