Dependence Of Dispersion Coefficient Research Articles

Generalized Analytical Solutions of The Advection-Dispersion Equation with Variable Flow and Transport Coefficients

Demand has increased for analytical solutions to determine the velocities and dispersion coefficients that describe solute transport with spatial, temporal, or spatiotemporal variations encountered in the field. However, few analytical solutions have considered spatially, temporally, or spatiotemporally dependent dispersion coefficients and velocities. The proposed solutions consider eight cases of dispersion coefficients and velocities: both spatially dependent, both spatiotemporally dependent, both temporally dependent, spatiotemporally dependent dispersion coefficient with spatially dependent velocity, temporally dependent dispersion coefficient with constant velocity, both constant, spatially dependent dispersion coefficient with spatiotemporally dependent velocity, and constant dispersion coefficient with temporally dependent velocity. The spatial dependence is linear, while the temporal dependence may be exponential, asymptotical, or sinusoidal. An advection–dispersion equation with these variable coefficients was reduced to a non-homogeneous diffusion equation using the pertinent coordinate transform method. Then, solutions were obtained in an infinite medium using Green’s function. The proposed analytical solutions were validated against existing analytical solutions or against numerical solutions when analytical solutions were unavailable. In this study, we showed that the proposed analytical solutions could be applied for various spatiotemporal patterns of both velocity and the dispersion coefficient, shedding light on feasibility of the proposed solution under highly transient flow in heterogeneous porous medium.

An Analytical Solution of Spatially Dependent One-Dimensional Advection-Dispersion Equation for a Varying Pulse Type Input Source

In present study, solution of advection-dispersion equation is obtained to determine concentration distribution of solute introduced from a varying pulse type point source in one-dimensional heterogeneous semi-infinite porous medium. Heterogeneity of the medium gives rise to space dependent groundwater velocity, dispersion coefficient and retardation factor. Groundwater velocity is some exponent ξ to a linear function of space. The dispersion and retardation factor are also exponents of same linear function with exponents (ξ+1) and (ξ-1), respectively, where ξ takes the value 0 or 1. At one end of the domain, a time dependent varying nature source, which involves step-size increasing function of time, acts along the flow up to a certain time the n eliminated while concentration gradient is considered zero at the other end of the domain. Initially, medium is uniformly polluted. Firstly, the advection-dispersion equation is reduced into constant coefficients by using certain transformations and then Laplace Integral Transformation Technique is utilized to get the solution. The obtained result is illustrated with numerical examples to study the effect of various parameters.

New insights into tracer propagation in partially saturated porous media

This work deals with the influence of partial saturation on the transport process of a passive tracer. Transport experiments were done in a water-wet glass micromodel combined with specific optical techniques. Full water saturation was achieved by injecting initially the background solution and then the tracer, whereas for the partial saturation conditions, the micromodel was initially saturated with oil, and then sequential the background solution and the tracer were injected at the same flow rate. We have shown that in the investigated range of water saturations it exists a transition in the oil ganglia structure and size. For high water saturations oil ganglia have one or two pores in size, however for lower water saturations they comprise an important number of pores. Transport strongly depends on the size distribution of the oil ganglia as they create large percolating paths and stagnant zones. We also showed the existence of two different types of stagnant zones: zones accessible by diffusion into pores and zones only accessible by spatially limited diffusion in films. The major advantage of using glass micromodels lies in the fact that dispersion coefficients can be computed from concentrations averaged over the pore space or from concentrations at the outlet and simultaneously from spatial concentration profiles. Curves were fitted using the Advection–Dispersion Equation (ADE) with adequate boundary conditions. The fitting quality of the temporal evolution of the average and outlet concentration was very good. However, fitting of the concentration profiles could only be done for the higher water saturations. This is due to the fact that the Representative Elementary Volume (REV) of lower water saturations is larger than the micromodel. The results show that fitting the breakthrough curve in order to determine the dispersion coefficient in a partially saturated porous medium might be misleading. Indeed, when fitting the breakthrough curves we were able to compute a dispersion coefficient even in the case where the REV of the water saturation is larger than the micromodel. Consequently, the knowledge of the local concentration profiles as a function of time is necessary as it provides an additional information on the spatio-temporal behavior of the transport process and therefor a supplementary constraint of the fitting procedure. Finally, we observed a time dependent dispersion coefficient in the regime where oil ganglia comprise several pores. This fact might be attributed to the non-Gaussian nature of the transport.

ANALYTICAL SOLUTION OF A NON - HOMOGENEOUS ONE - DIMENSIONAL ADVECTION DIFFUSION EQUATION WITH TEMPORALLY VARYING COEFFICIENTS .

Advection Diffusion Equation is a partial differential equation that describes the transport of pollutants in rivers. Its coefficients (dispersion and velocity) can be constant, dependent on space or time or both space and time. This study presents an analytical solution of a one dimensional non - homogeneous advection diffusion equation with temporally dependent coefficients, describing one dimensional pollutant transport in a section of a river. Temporal dependence is accounted for by considering a temporally dependent dispersion coefficient along an unsteady flow assuming that dispersion is proportional to the velocity. Transformations are used to convert the time dependent coefficients to constant coefficients and to eliminate the advection term. Analytical solution is obtained using Fourier transform method considering an instantaneous point source. Numerical results are presented. The findings show that concentration monotonically decreases with increasing distance and increasing time.

Solute Transport due to Spatio-Temporally Dependent Dispersion Coefficient and Velocity: Analytical Solutions

AbstractThis paper presents an analytical solution of the advection-dispersion equation (ADE) with the dispersion coefficient and velocity being directly proportional to the spatial linear nonhomog...

Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method

In the present study analytical solutions of a two-dimensional advection–dispersion equation (ADE) with spatially and temporally dependent longitudinal and lateral components of the dispersion coefficient and velocity are obtained using Green’s Function Method (GFM). These solutions describe solute transport in infinite horizontal groundwater flow, assimilating the spatio-temporal dependence of transport properties, dependence of dispersion coefficient on velocity, and the particulate heterogeneity of the aquifer. The solution is obtained in the general form of temporal dependence and the source term, from which solutions for instantaneous and continuous point sources are derived. The spatial dependence of groundwater velocity is considered non-homogeneous linear, whereas the dispersion coefficient is considered proportional to the square of spatial dependence of velocity. An asymptotically increasing temporal function is considered to illustrate the proposed solutions. The solutions are validated with the existing solutions derived from the proposed solutions in three special cases. The effect of spatially/temporally dependent heterogeneity on the solute transport is also demonstrated. To use the GFM, the ADE with spatio-temporally dependent coefficients is reduced to a dispersion equation with constant coefficients in terms of new position variables introduced through properly developed coordinate transformation equations. Also, a new time variable is introduced through a known transformation.

Divergent dispersion behavior of ssDNA fragments during microchip electrophoresis in pDMA and LPA entangled polymer networks.

Resolution of DNA fragments separated by electrophoresis in polymer solutions ("matrices") is determined by both the spacing between peaks and the width of the peaks. Prior research on the development of high-performance separation matrices has been focused primarily on optimizing DNA mobility and matrix selectivity, and gave less attention to peak broadening. Quantitative data are rare for peak broadening in systems in which high electric field strengths are used (>150 V/cm), which is surprising since capillary and microchip-based systems commonly run at these field strengths. Here, we report results for a study of band broadening behavior for ssDNA fragments on a glass microfluidic chip, for electric field strengths up to 320 V/cm. We compare dispersion coefficients obtained in a poly(N,N-dimethylacrylamide) (pDMA) separation matrix that was developed for chip-based DNA sequencing with a commercially available linear polyacrylamide (LPA) matrix commonly used in capillaries. Much larger DNA dispersion coefficients were measured in the LPA matrix as compared to the pDMA matrix, and the dependence of dispersion coefficient on DNA size and electric field strength were found to differ quite starkly in the two matrices. These observations lead us to propose that DNA migration mechanisms differ substantially in our custom pDMA matrix compared to the commercially available LPA matrix. We discuss the implications of these results in terms of developing optimal matrices for specific separation (microchip or capillary) platforms.

Dispersion of Butane in Vapex: The Effect of Drainage Height~!2009-10-13~!2010-03-25~!2010-06-08~!

In this work, the dispersion of butane in heavy oil and bitumen is experimentally determined during the Vapex process with varying drainage heights. The experiments were performed at room temperature (22 o C) using butane as solvent at the dew point pressure. We used cylindrical models with different heights varying from 15 to 45 cm, packed with uniform mixture of heavy oil and glass beads. From the experiments, production rate dependency towards the drainage height is evaluated along with determination of butane gas solubility, live oil density, and viscosity. Conjugate gradient method is used to optimally determine concentration dependent dispersion coefficient as well as the solvent concentration at the solvent-oil interface by matching up experimental live oil production rates with ones predicted by model. A computational algorithm is developed to simultaneously solve process models of different heights, and optimally determine the corresponding dispersion coefficients.

One-dimensional modelling of the secondary clarifier-factors affecting simulation in the clarification zone and the assessment of the thickening flow dependence

This paper presents a one-dimensional (1-D) model of the secondary settling tank (SST), and the drive for model development is discussed using steady-state simulation results generated with a 2-D computational fluid dynamics (CFD) model. Concentration profiles measured in a flat-bottom circular (fbSST) that is equipped with suction-lift sludge removal, served for CFD model validation. We investigate the factors that may deteriorate the 1-D simulation performance of clarifier models, which incorporate dispersion both in terms of the effluent sludge concentration and in terms of the sludge blanket height. Furthermore, dispersion-models can account for the effect of clarifier flow conditions on thickening performance. However, correlations, found in the literature, are shown to have limited efficiency under the wide flow boundary conditions examined in this study. Optimisation of the 1-D clarifier model structure is proposed by using an overflow dependent dispersion coefficient and including a feed flow dependent reduction factor in the downward convective velocity term. Results obtained show improved simulation performance in the clarification zone and allow for an efficient flow-dependency formulation of the thickening performance.

One-Dimensional Clarifier Model with Sludge Blanket Heights

A one-dimensional clarifier model employing a concentration and feed velocity dependent dispersion coefficient was coupled with an algorithm for predicting sludge blanket height. The model was successfully calibrated using blanket heights; effluent suspended solids and return activated sludge concentrations; and the Vesilind parameters V0 and b . These data were collected from one of the secondary clarifiers at a full-scale activated sludge plant. Model validity was confirmed by comparing predictions of the calibrated model against separate sets of data collected from the same clarifier. The model outperformed other well-known models in predicting clarifier overloading. Better performance of the present model can be attributed to its ability to portray, through a dependence of dispersion coefficient on feed velocity, the detrimental effect of increasing hydraulic loading (for any given applied solids flux) on clarifier performance.

Stochastic‐Convective Transport with Nonlinear Reaction: Biodegradation With Microbial Growth

The representation of subsurface flow and reactive transport as an ensemble of one‐dimensional stream tubes is extended to account for nonlinear biodegradation with coupled microbial growth. The Stochastic‐convective reaction (SCR) model is derived for bioreaction of a single solute by a single class of microorganisms coupled with dynamic microbial growth. A new global variable, the integral of the solute degraded per unit length of system traversed, accounts for degradation. Dimensionless scaling and the method of characteristics are used to reduce the model, written for a single convecting reactor (stream tube), to a pair of coupled nonlinear functional equations for solute concentration and microbial biomass. Existence of a solution to the stream tube system is shown, both numerical and approximate analytical approaches to the solution are given, and example computations using both methods are presented. Conditions under which the stream tube solution is “canonical,” or scalable to fit any permissible stream tube travel time function, arise from requirements for invariance (over the stream tube ensemble) of effective one‐dimensional stream tubes used to represent transport along real stream tubes in three‐dimensional space. Averaging of the stream tube solution over travel time and reaction properties representative of physical and chemical heterogeneities is described as a way to separate and upscale the processes of macrodispersion and microbiological reaction. The approach is exercised to simulate Monte Carlo average behavior of bioreactive transport in physically heterogeneous two‐dimensional media. Results show that the method captures the ensemble average large‐scale effects of the nonlinear reactions more accurately than done in the classical reactive convection‐dispersion equation (CDR), even when the appropriate scale dependent dispersion coefficient is afforded to the CDR.

Modelling transport processes in the Ribble estuary

Field measurements of velocity and salinity at three stations along a reach of the Ribble estuary were used to evaluate the temporal variations of the vertical shear induced component of the longitudinal dispersion coefficient. The results yield clear temporal trends which lead to the conclusion that vertical shear-induced dispersion, resulting from differential advection and density gradients, is the most important component of the longitudinal mixing on the ebb tide. Turbulent diffusion, due to the high velocities, dominates the mixing mechanisms on the flood tide. The sensitivity to the dispersion coefficient in a one-dimensional model based on a schematised form of the channel geometry of the Ribble estuary revealed the need for a time dependent dispersion coefficient. Measurements of suspended solids revealed low concentrations compared to previous records. It is proposed that this is due to a pumping effect in the estuary, induced by a combination of the tidal asymmetry and the ebb tide vertical density gradients leading to upstream storage during low river flow spring tide periods. Modelling of the turbidity maximum is shown to be possible by adjustment of critical erosion and deposition shear stress terms.

Development of 2-D Highly Accurate Diffusion Simulation Model in Bay and Coastal Waters

Three different methods for estimating dispersion coefficient in 2-D diffusion simulation model are discussed. The basic ideas for constructing the dispersion coefficient are as follows:(1) applying k-e turbulence model to 2-D plane diffusion simulation.(2) taking the dependence of dispersion coefficient on instantaneous tidal current into account.(3) considering the difference between longitudinal dispersion and vertical eddy diffusion which is remarkable in 2-D treatment.Since the all estimations include the effects of stratification and drift current, the final formulae of dispersion coefficient are expected to be very practical and used widely in 2-D numerical calculation of pollutant diffusion.

The numerical analysis of water and solute movement in scale heterogeneous profiles

A computer based numerical method is presented for the analysis of water and solute movement in unsaturated heterogeneous porous materials. Such a method is necessary since, for those field studies where solute movement is of concern, the soil profiles under consideration are invariably heterogeneous. The numerical analysis is based on a general one-dimensional finite difference soil water flow model which includes a numerical technique combining the concepts of scale heterogeneity with an interpolative soil water hysteresis model. An explicit finite difference solute movement subroutine is incorporated into the unsaturated flow model to describe the transport of nonreactive solutes. A velocity dependent longitudinal dispersion coefficient is used in the solution of the hydrodynamic dispersion equation. The resulting hysteretic scale heterogeneous solute movement model permits the study of solute dynamics during infiltrating and redistribution in realistically complex spatially varying soil profiles. Results are presented for the leaching of both coarse grading to fine and fine grading to coarse sand profiles. Both vertical and horizontal profiles are studied using either a constant flux or a constant concentration input boundary condition. The four cases studied demonstrate the versatility of the numerical method and emphasise the substantial differences in transport behavior that can arise between heterogeneous and homogeneous profiles.

Long-term dispersion in unsteady skewed free surface flow

An analysis of two-dimensional horizontal plane shear dispersion in steady, periodic, almost-periodic and randomly forced skewed free surface flow is presented. A two-time scale perturbation analysis of the advection-diffusion equation is used to derive the two-dimensional advection-dispersion equation and the horizontal dispersion coefficient tensor. For combinations of steady, periodic and almost-periodic flow, the time dependent dispersion coefficient tensor contains steady terms and periodic terms at frequencies associated with the forcing frequencies and their sums and differences. For combinations of steady, periodic, almost-periodic and stationary random forcings, the expected value dispersion coefficient tensor contains terms associated with the steady forcings and terms associated with the unsteady forcings represented by the spectral density functions of the unsteady forcings. Estimates of the magnitude of the expected value dispersion coefficient tensor are presented for representative estuarine and continental shelf conditions.

Applications of Wiener’s Path Integral for the Diffusion of Brownian Particles in Shear Flows

Wiener’s path integral theory is revisited, stressing that it holds only when the condition of local equilibrium is fulfilled. The diffusion of a particle in a two and three-dimensional shear flow is studied, showing that for small time intervals the particle diffuses freely, without being influenced by the velocity gradient of the incident flow, while for long time intervals it exhibits a time dependent dispersion coefficient.

An Examination of Scale‐Dependent Dispersion Coefficients

ABSTRACTMany hydrologists have observed that dispersion coefficients, when measured in the field, turn out to be scale‐dependent. That is, the greater the travel distance in a tracer test used to measure dispersivity, the larger the dispersivity value that is calculated. Recently, Güven et al.(1983) presented a study which contains a basis for understanding the phenomenon of scale‐dependent dispersion within a deterministic framework. The results of that study are used as a basis for defining a scale‐dependent macro‐ dispersion coefficient for unidirectional flow in a stratified aquifer. Theoretical expressions are then obtained for the macrodispersivity, its various components, and its small time and large time limits. Using the data of Pickens and Grisak (1981a, b), numerical values are calculated for the macrodispersivity, and estimates are made of the travel distance required in order to reach Fickian conditions. Asymptotic large time macrodispersivity values of 49.5 m to 990 m and travel distances of 5.4 km to 109 km result for the particular aquifer studied. Based on these results it appears that Fickian conditions will seldom apply in practice to porous media flow. This study and previous studies show that the primary physical mechanism that causes spreading of a solute near the source is different advection rates at different elevations in the aquifer. Present results and comparisons with field data indicate that this phenomenon is not represented well by a scale‐ dependent dispersion coefficient. In modeling dispersion phenomena, it appears that more emphasis should be placed on field study and the accurate determination of hydraulic conductivity variations and other nonhomogeneities, and less on incorporating somewhat arbitrary dispersion coefficients into complex mathematical models.

Hydrodynamic dispersion during absorption in a fine sand: 2. The constant flux case

The conclusions of an earlier paper involving hydrodynamic dispersion during absorption in a fine sand under constant concentration conditions are discussed in relation to the support they provide for the concept of a velocity dependent dispersion coefficient. The present study analyzes the more exacting constant flux case using influent boundary fluxes of water ranging from 3.28 × 10−7 ms−1 to 1.05 × 10−5 ms−1. These values allow the ratio of the mechanical dispersion component of the dispersion coefficient to the molecular diffusion component to vary from 0.43 to 5.81. For all cases, computer‐based numerical solutions are compared with experimental results using reduced space and time variables in the presentation of the data. It is observed that the bandwidth of the scatter of the experimental data is generally greater than the spread of the computed concentration profiles. In keeping with the constant concentration results, this suggests that the form of the experimental data is not inconsistent with the concept of a hydrodynamic dispersion coefficient which is velocity dependent. In all cases a dispersion length of 0.0001 m was used.

Modified Fickian Model for Predicting Dispersion

The ordinary Fickian model for longitudinal dispersion is modified by using a time or distance dependent dispersion coefficient with a time-scale much greater than the Fischer time scale. From the three bulk flow properties U*, A, and d¯, the asymptotic dispersion coefficient D\dL and the peak concentration C\Dmax\N in natural streams can be predicted within a factor of 2.5 and 1.6, respectively. The model is useful for the prediction of longitudinal spread of pollutants in rivers over both short and long distances.

Analytical solution of a velocity dependent dispersion problem

A complete description of the convective‐dispersive transport of a contaminant from an injection well requires consideration of a velocity dependent dispersion coefficient. An analytical solution of this equation for steady fluid flow conditions can be obtained. The solution is significantly different from existing approximate analytical solutions. A numerical solution to this problem converges to the new analytical solution upon refinement of the finite difference net.