Analytical models of steady-state plumes undergoing sequential first-order degradation

An exact, closed‐form analytical solution is derived for one‐dimensional (1D), coupled, steady‐state advection‐dispersion equations with sequential first‐order degradation of three dissolved species in groundwater. Dimensionless and mathematical analyses are used to examine the sensitivity of longitudinal dispersivity in the parent and daughter analytical solutions. The results indicate that the relative error decreases to less than 15% for the 1D advection‐dominated and advection‐dispersion analytical solutions of the parent and daughter when the Damköhler number of the parent decreases to less than 1 (slow degradation rate) and the Peclet number increases to greater than 6 (advection‐dominated). To estimate first‐order daughter product rate constants in advection‐dominated zones, 1D, two‐dimensional (2D), and three‐dimensional (3D) steady‐state analytical solutions with zero longitudinal dispersivity are also derived for three first‐order sequentially degrading compounds. The closed form of these exact analytical solutions has the advantage of having (1) no numerical integration or evaluation of complex‐valued error function arguments, (2) computational efficiency compared to problems with long times to reach steady state, and (3) minimal effort for incorporation into spreadsheets. These multispecies analytical solutions indicate that BIOCHLOR produces accurate results for 1D steady‐state, applications with longitudinal dispersion. Although BIOCHLOR is inaccurate in multidimensional applications with longitudinal dispersion, these multidimensional multispecies analytical solutions indicate that BIOCHLOR produces accurate steady‐state results when the longitudinal dispersion is zero. As an application, the 1D advection‐dominated analytical solution is applied to estimate field‐scale rate constants of 0.81, 0.74, and 0.69/year for trichloroethene, cis‐1,2‐dichloroethene, and vinyl chloride, respectively, at the Harris Palm Bay, FL, CERCLA site.     

Composite analytical solutions for a soil vapour extraction system

Soil vapour extraction (SVE) is a common remediation technique for cleaning up unsaturated soils contaminated by volatile organic compounds (VOCs). Analytical solutions, which result from simple mathematical models, can allow the fast approximation of the time‐dependent effluent concentration and the gaining of insight into the processes that take place during soil remediation. Deriving the analytical solutions to advection–dispersion equations that simultaneously take into account the mechanical dispersion and molecular diffusion is very difficult because of the variable dependence of governing equations' coefficients. In this study, we first present two simplified analytical solutions that only consider mechanical dispersion or molecular diffusion. The two developed analytical solutions are compared with the numerical solution that simultaneously considers both mechanical dispersion and molecular diffusion to examine the applicability of the two simplified analytical solutions and distinguishes the individual contribution of the mechanical dispersion and molecular diffusion to total VOCs transport in an SVE system. Results show that dispersion plays an important role during SVE decontamination and neither the diffusion‐dominated solution nor the dispersion‐dominated solution can agree well with the numerical solution when both mechanical dispersion and molecular diffusion have significant contributions to the total VOCs transport flux. A composite analytical solution that linearly couples the diffusion‐ and dispersion‐dominated analytical solutions, which is proposed herein to eliminate the discrepancy between the analytical solutions and the numerical solution. Results indicate that the proposed composite analytical solution agrees well with the numerical solution and is an effective tool for quickly and accurately evaluating the time‐dependent effluent concentration for parameters of the different ranges of interest in an SVE remedial system. Copyright © 2006 John Wiley & Sons, Ltd.     

Issues and Options in the Delineation of Well Capture Zones under Uncertainty

The delineation of wellhead protection areas (WHPAs) under uncertainty is still a challenge for heterogeneous porous media. For granular media, one option is to combine particle tracking (PT) with the Monte Carlo approach (PT‐MC) to account for geologic uncertainties. Fractured porous media, however, require certain restrictive assumptions under this approach. An alternative for all types of media is the capture probability (CP) approach, which is based on the solution of the standard advection‐dispersion equation in a backward mode, making use of the analogy between forward and backward transport processes. Within this context, we review the current controversy about the correct form of the conceptual model for transport, finding that the advection‐diffusion model, which represents the diffusive interchange between streamtubes with differing velocities, is more physically realistic than the conventional advection‐dispersion model. For mildly to moderately heterogeneous materials, stochastic theories and simulation experiments show that this process converges at the field scale to an effective advection‐dispersion process that can be simulated with conventional transport models using appropriate macrodispersivity values. For highly heterogeneous materials, stochastic theories do not yet exist but there is no reason why the process should not converge naturally as well. Macrodispersivities appear to be formation‐specific. The advection‐dispersion model can be used for capture zone delineation in heterogeneous granular media. For fractured porous systems, hybrid equivalent porous medium and discrete fracture network or CP‐based approaches may have potential. In general, capture zones delineated by PT without MC will always be too small and should not be used as a basis for land‐use decisions.     

Vectorized simulation of groundwater flow and contaminant transport using analytic element method and random walk particle tracking

Groundwater contaminant transport processes are usually simulated by the finite difference (FDM) or finite element methods (FEM). However, they are susceptible to numerical dispersion for advection‐dominated transport. In this study, a numerical dispersion‐free coupled flow and transport model is developed by combining the analytic element method (AEM) with random walk particle tracking (RWPT). As AEM produces continuous velocity distribution over the entire aquifer domain, it is more suitable for RWPT than FDM/finite element methods. Using the AEM solutions, RWPT tracks all the particles in a vectorized manner, thereby improving the computational efficiency. The present model performs a convolution integral of the response of an impulse contaminant injection to generate concentration distributions due to a permanent contaminant source. The RWPT model is validated with an available analytical solution and compared to an FDM solution, the RWPT model more accurately replicates the analytical solution. Further, the coupled AEM‐RWPT model has been applied to simulate the flow and transport in hypothetical and field aquifer problems. The results are compared with the FDM solutions and found to be satisfactory. The results demonstrate the efficacy of the proposed method.     

Error analysis of finite difference methods for two-dimensional advection–dispersion–reaction equation

In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution.     

Closed‐Form Approximations for Two‐Dimensional Groundwater Age Patterns in a Fresh Water Lens

Simple closed‐form approximations are presented for calculating the steady‐state groundwater age distribution in two‐dimensional vertical cross sections of idealized fresh water lenses overlying salt water, for aquifers that are vertically semi‐infinite and of finite thickness. The approximations are developed on the basis of existing one‐dimensional analytical solutions for travel‐time calculation in fresh water lenses and approximate streamline formulations. The two‐dimensional age distributions based on the closed‐form solutions match convincingly with numerical simulations. As expected, notable deviations from the numerical solution are encountered at the groundwater flow divide and when submarine groundwater discharge occurs. Ratios of recharge over hydraulic conductivities are varied to explore how the magnitude of the deviations changes, and it is found that the approximate closed‐form solutions perform well over a range of conditions found in natural systems.     

Plume Detachment and Recession Times in Fractured Rock

The influence of source zone concentration reduction on solute plume detachment and recession times in fractured rock was investigated using new semianalytical solutions to transient solute transport in the presence of advection, dispersion, sorption, matrix diffusion, and first-order decay. Novel aspects of these solutions are: (1) the source zone concentration behavior is simulated using a constant concentration with the option for either an instantaneous reduction to zero concentration or an exponentially decaying source zone concentration initiated at some time (t^*) after the source is introduced, and (2) different biodegradation rates in the fracture and rock matrix. These solutions were applied for sandstone bedrock and revealed that biodegradation in the matrix, not the fracture, may be the most significant attenuation mechanism and therefore may dictate remediation time scales. Also, instantaneous and complete source concentration reduction in aged plumes may not be beneficial with respect to plume response because back-diffusion can sustain plume migration for long periods of time. Moderate source zone concentration reduction has a similar impact on the rate of advance of the leading edge of the plume as aggressive concentration reduction. If the source zone concentration reduction half-life is less than the plume decay half-life, then volatile organic compound (VOC) mass sequestered in the rock matrix will ultimately dictate plume persistence and not the presence of the source zone.     

A Practical Modeling Framework for Non‐Fickian Transport and Multi‐Species Sequential First‐Order Reaction

Many studies indicate that small‐scale heterogeneity and/or mobile–immobile mass exchange produce transient non‐Fickian plume behavior that is not well captured by the use of the standard, deterministic advection‐dispersion equation (ADE). An extended ADE modeling framework is presented here that is based on continuous time random walk theory. It can be used to characterize non‐Fickian transport coupled with simultaneous sequential first‐order reactions (e.g., biodegradation or radioactive decay) for multiple degrading contaminants such as chlorinated solvents, royal demolition explosive, pesticides, and radionuclides. To demonstrate this modeling framework, new transient analytical solutions are derived and are inverted in Laplace space. Closed‐form, steady‐state, multi‐species analytical solutions are also derived for non‐Fickian transport in highly heterogeneous aquifers with linear sorption–desorption and matrix diffusion for use in spreadsheets. The solutions are general enough to allow different degradation rates for the mobile and immobile zones. The transient solutions for multi‐species transport are applied to examine the effects of source remediation on the natural attenuation of downgradient plumes of both parent and degradation products in highly heterogeneous aquifers. Results for representative settings show that the use of the standard, deterministic ADE can over‐estimate cleanup rates and under‐predict the cleanup timeframe in comparison to the extended ADE analytical model. The modeling framework and calculations introduced here are also applied for a 30 year groundwater cleanup program at a site in Palm Bay, Florida. The simulated plume concentrations using the extended ADE exhibited agreement with observed long concentration tails of trichloroethene, cis 1,2 DCE, and VC that remained above cleanup goals.     

Semi-analytical solutions for solute transport in fractured porous media using a strip source of finite width

Transient and steady-state analytical solutions are derived to investigate solute transport in a fractured porous medium consisting of evenly spaced, parallel discrete fractures. The solutions incorporate a finite width strip source, longitudinal and transverse dispersion in the fractures, source decay, aqueous phase decay, one-dimensional diffusion into the matrix, sorption to fracture walls, and sorption within the matrix. The solutions are derived using Laplace and Fourier transforms, and inverted by interchanging the order of integration and utilizing a numerical Laplace inversion algorithm. The solutions are verified for simplified cases by comparison to solutions derived by Batu [Batu V. A generalized two-dimensional analytical solution for hydrodynamic dispersion in bounded media with the first-type condition at the source. Wat Resour Res 1989;25(6):1125] and Sudicky and Frind [Sudicky EA, Frind EO. Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Wat Resour Res 1982;18(6):1634]. The application of the solutions to a fractured sandstone demonstrates that narrower source widths and larger values of transverse dispersivity both lead to lower downstream concentrations in the fractures and shorter steady-state plumes. The incorporation of aqueous phase decay and source concentration decay both lead to lower concentrations and shorter plumes, with even moderate amounts of decay significantly shortening the persistence of contamination.     

Can a Time Fractional‐Derivative Model Capture Scale‐Dependent Dispersion in Saturated Soils?

Time nonlocal transport models such as the time fractional advection‐dispersion equation (t‐fADE) were proposed to capture well‐documented non‐Fickian dynamics for conservative solutes transport in heterogeneous media, with the underlying assumption that the time nonlocality (which means that the current concentration change is affected by previous concentration load) embedded in the physical models can release the effective dispersion coefficient from scale dependency. This assumption, however, has never been systematically examined using real data. This study fills this historical knowledge gap by capturing non‐Fickian transport (likely due to solute retention) documented in the literature (Huang et al. 1995) and observed in our laboratory from small to intermediate spatial scale using the promising, tempered t‐fADE model. Fitting exercises show that the effective dispersion coefficient in the t‐fADE, although differing subtly from the dispersion coefficient in the standard advection‐dispersion equation, increases nonlinearly with the travel distance (varying from 0.5 to 12 m) for both heterogeneous and macroscopically homogeneous sand columns. Further analysis reveals that, while solute retention in relatively immobile zones can be efficiently captured by the time nonlocal parameters in the t‐fADE, the motion‐independent solute movement in the mobile zone is affected by the spatial evolution of local velocities in the host medium, resulting in a scale‐dependent dispersion coefficient. The same result may be found for the other standard time nonlocal transport models that separate solute retention and jumps (i.e., displacement). Therefore, the t‐fADE with a constant dispersion coefficient cannot capture scale‐dependent dispersion in saturated porous media, challenging the application for stochastic hydrogeology methods in quantifying real‐world, preasymptotic transport. Hence improvements on time nonlocal models using, for example, the novel subordination approach are necessary to incorporate the spatial evolution of local velocities without adding cumbersome parameters.     

Comparison of alternative upscaled model formulations for simulating DNAPL source dissolution and biodecay

This paper compares the performance of analytical and numerical approaches for modeling DNAPL dissolution with biodecay. A solution derived from a 1-D advective transport formulation (“Parker” model) is shown to agree very closely with high resolution numerical solutions. A simple lumped source mass balance solution in which with decay is assumed proportional to DNAPL mass (“Falta1” model) over- or underpredicts aqueous phase biodecay depending on the magnitude of the exponential factor governing the relationship between dissolution rate and DNAPL mass. A modification of the Falta model that assumes decay proportional to the source exit concentration is capable of accurately simulating source behavior with strong aqueous phase biodecay if model parameters are appropriately selected or calibrated (“Falta2” model). However, parameters in the lumped models exhibit complex interdependencies that cannot be quantified without consideration of transport processes within the source zone. Combining the Falta2 solution with relationships derived from the Parker model was found to resolve these limitations and track the numerical model results. A method is presented to generalize the analytical solutions to enable simulation of partial mass removal with changes in source parameters over time due to various remedial actions. The algorithm is verified by comparison with numerical simulation results. An example application is presented that demonstrates the interactions of partial mass removal, enhanced biodecay, enhanced mass transfer and source zone flow reduction applied at various time periods on contaminant flux reduction. Increasing errors that arise in numerical solutions with coarse discretization and high decay rates are shown to be controlled by using an adjusted decay coefficient derived from the Parker analytical solution.     

Error estimation of closed‐form solution for annual rate of structural collapse

With the increasing emphasis of performance‐based earthquake engineering in the engineering community, several investigations have been presented outlining simplified approaches suitable for performance‐based seismic design (PBSD). Central to most of these PBSD approaches is the use of closed‐form analytical solutions to the probabilistic integral equations representing the rate of exceedance of key performance measures. Situations where such closed‐form solutions are not appropriate primarily relate to the problem of extrapolation outside of the region in which parameters of the closed‐form solution are fit. This study presents a critical review of the closed‐form solution for the annual rate of structural collapse. The closed‐form solution requires the assumptions of lognormality of the collapse fragility and power model form of the ground motion hazard, of which the latter is more significant regarding the error of the closed‐form solution. Via a parametric study, the key variables contributing to the error between the closed‐form solution and solution via numerical integration are illustrated. As these key variables cannot be easily measured, it casts doubt on the use of such closed‐form solutions in future PBSD, especially considering the simple and efficient nature of using direct numerical integration to obtain the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Mixing cell method for solving the solute transport equation with spatially variable coefficients

The advection–dispersion equation with spatially variable coefficients does not have an exact analytical solution and is therefore solved numerically. However, solutions obtained with several of the traditional finite difference or finite element techniques typically exhibit spurious oscillation or numerical dispersion when advection is dominant. The mixing cell and semi-analytical solution methods proposed in this study avoid such oscillation or numerical dispersion when advection dominates. Both the mixing cell and semi-analytical solution methods calculate the spatial step size by equating numerical dispersion to physical dispersion. Because of the spatial variability of the coefficients the spatial step size varies in space. When the time step size Δt→0, the mixing cell method reduces to the semi-analytical solution method. The results of application to two cases show that the mixing cell and semi-analytical solution methods are better than a finite difference method used in the study. © 1998 John Wiley & Sons, Ltd.     

Stochastic analysis of biodegradation fronts in one-dimensional heterogeneous porous media

We consider a one-dimensional model biodegradation system consisting of two reaction–advection equations for nutrient and pollutant concentrations and a rate equation for biomass. The hydrodynamic dispersion is ignored. Under an explicit condition on the decay and growth rates of biomass, the system can be approximated by two component models by setting biomass kinetics to equilibrium. We derive closed form solutions for constant speed traveling fronts for the reduced two component models and compare their profiles in homogeneous media. For a spatially random velocity field, we introduce travel time and study statistics of degradation fronts via representations in terms of the travel time probability density function (pdf) and the traveling front profiles. The travel time pdf does not vary with the nutrient and pollutant concentrations and only depends on the random water velocity. The traveling front profiles are expressed analytically or semi-analytically as functions of the travel time. The problem of nonlinear transport by a random velocity reduces to two subproblems: one being nonlinear transport by a known (unit) velocity, and the other being linear (advective) transport by a random velocity. The approach is illustrated through some examples where the randomness in velocity stems from the spatial variability of porosity.     

A linear systems approach to watershed transport simulation

Models that simulate loadings of pollutants from agricultural landscapes to surface waters often operate at time scales that are relatively coarse (e.g. daily) compared with how fast water moves in streams, suggesting a commensurate physical scale that is substantially larger than typical agricultural fields. In general, as pollutants enter water and move downstream, longitudinal dispersive effects and travel time de‐synchronization tend to cause flattening and broadening of concentration peaks—an effect with implications for potential impacts on ecological and human health, and for which adequate representation is thus important for risk assessment. In‐stream transport is often approximated in practice using numerical implementation of the one‐dimensional advection–dispersion equation (ADE), with streams discretized into linked homogeneous segments. However, when a daily time step is employed, limitations inherent in the finite difference methodology may constrain simulated dispersion in lotic waters to unrepresentative or unrealistic magnitudes. In this paper, a convolution‐based approach to surface water transport is suggested as an alternative to the ADE, for use in combination with daily input loading models. This approach offers the advantage of greater flexibility in representing longitudinal mixing by using impulse response functions (IRF) to represent inter‐segment transport. Networks of stream segments are represented using nested convolutions, implemented using forward and inverse discrete Fourier transform to simplify calculations. Enhanced representational flexibility arises from the freedom afforded the modeller in selecting each segment's IRF, which may be chosen to represent dispersive regimes ranging from pure advection (plug flow) to compete mixing, and beyond to the sort of long‐tailed mixing characterized by fractal inverse frequency power‐law scaling. The approach is explored in proof‐of‐concept exercises that make use of atrazine monitoring data sets collected over common time periods from upstream and downstream locations within the same watersheds. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Quantitative analysis of numerically induced mixing in a coastal model application

In recent years, various attempts have been made to estimate the amount of numerical mixing in numerical ocean models due to discretisation errors of advection schemes. In this study, a high-resolution coastal model using the ocean circulationmodel GETM is applied to the Western Baltic Sea, which is characterised by energetic and episodic inflows of dense bottom waters originating from the Kattegat. The model is equipped with an easy-to-implement diagnostic method for obtaining the numerical mixing which has recently been suggested. In this diagnostic method, the physical mixing is defined as the mean tracer variance decay rate due to turbulent mixing. The numerical mixing due to discretisation errors of tracer advection schemes is defined as the decay rate between the advected square of the tracer variance and the square of the advected tracer, which can be directly compared to the physical variance decay. The source and location of numerical mixing is further investigated by comparing different advection schemes and analysing the amount of numerical mixing in each spatial dimension during the advection time step. The results show that, for the setup used, the numerically and physically induced mixing have the same orders of magnitude but with different vertical and horizontal distributions. As the main mechanism for high numerical mixing, vertical advection of tracers with strong vertical gradients has been identified. The main reason for high numerical mixing is due to bottom-following coordinates when density gradients, especially for regions of steep slopes, are advected normal to isobaths. With the bottom-following coordinates used here, the horizontal gradients are reproduced by a spurious sawtooth-type profile where strong advection through, but not along, the vertical coordinate levels occurs. Additionally, the well known relation between strong tracer gradients and high velocities on the one and high numerical mixing on the other hand is approved quantitatively within this work.     

Importance of advective zone in longitudinal mixing experiments

One-dimensional Fickian dispersion models such as the advection diffusion equation (ADE) are commonly used to analyse and predict concentration distributions downstream of contamination events in watercourses. Such models are only valid once the tracer had entered the equilibrium zone. This paper compares previous theoretical, experimental and numerical estimates of the distance to reach the equilibrium zone with new experimental values, obtained by examining the change of skewness in a tracer profile, downstream of a cross-sectionally well mixed source. Closer agreement was found with Fischers’ theoretical estimate than prior experimental and numerical studies.     

Inversion of potential field data using the structural index as weighting function rate decay

Nonparametric inverse methods provide a general framework for solving potential‐field problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth‐weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998) , differing significantly in the definition of the depth‐weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth‐weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth‐weighting function and to the consequent solutions. The allowed values for the exponent of the depth‐weighting function depend on the range of N for sources: 0 ≤N≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth‐weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi‐source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonov‐form including a depth‐weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.     

Modeling Variably Saturated Subsurface Solute Transport with MODFLOW‐UZF and MT3DMS

The MT3DMS groundwater solute transport model was modified to simulate solute transport in the unsaturated zone by incorporating the unsaturated‐zone flow (UZF1) package developed for MODFLOW. The modified MT3DMS code uses a volume‐averaged approach in which Lagrangian‐based UZF1 fluid fluxes and storage changes are mapped onto a fixed grid. Referred to as UZF‐MT3DMS, the linked model was tested against published benchmarks solved analytically as well as against other published codes, most frequently the U.S. Geological Survey's Variably‐Saturated Two‐Dimensional Flow and Transport Model. Results from a suite of test cases demonstrate that the modified code accurately simulates solute advection, dispersion, and reaction in the unsaturated zone. Two‐ and three‐dimensional simulations also were investigated to ensure unsaturated‐saturated zone interaction was simulated correctly. Because the UZF1 solution is analytical, large‐scale flow and transport investigations can be performed free from the computational and data burdens required by numerical solutions to Richards' equation. Results demonstrate that significant simulation runtime savings can be achieved with UZF‐MT3DMS, an important development when hundreds or thousands of model runs are required during parameter estimation and uncertainty analysis. Three‐dimensional variably saturated flow and transport simulations revealed UZF‐MT3DMS to have runtimes that are less than one tenth of the time required by models that rely on Richards' equation. Given its accuracy and efficiency, and the wide‐spread use of both MODFLOW and MT3DMS, the added capability of unsaturated‐zone transport in this familiar modeling framework stands to benefit a broad user‐ship.     

Developing a Lagrangian sediment transport model for open channel flows

A three-dimensional stochastic Lagrangian particle tracking sediment transport model is developed to solve the discrete advection-dispersion equation using a combination of empirical dispersion equations.The performance of three widely-used longitudinal dispersion coefficient equations was examined to select one of them as the primary dispersion equation term in the developed model. Also, a conditional empirical equation was used to consider the effect of vertical dispersion term in top layers n...