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We consider a class of a convection-dispersion equation describing the transport model of oil spilled in sea water with concentration-dependent dispersion and spatial velocity. A potential variable is introduced to the governing equation such that the governing equation is written in conserved form (or auxiliary system). The conserved form admits a number of classical symmetries which induce potential symmetries for the governing equation. We construct group invariant solutions using admitted potential symmetries for the governing equation. The potential symmetry reduction is coupled with invariant surface condition to construct solutions via classical reduction method. The solutions are new and exact.

We conducted a small soil column penetration test and lysimeter experiment in four soils (Isohumosol, Ferrosol, Primosol, and Sandy soil), to generate a migration model for antimony (Sb) in soil based on the convection-diffusion equation. Variation of the Sb concentration was also simulated by the Artificial neural networks model (ANN) model. The convection-dispersion model will provide a reference for the transfer of Sb in the environment. The results agreed with the laboratory measurements with an R2 of 0.79–0.98 in the Isohumosol, Ferrosol, and Primosol. It was found that the convection-dispersion model could be used to explain the migration of Sb in soil. The areas of high potential ecological risk for the Ferrosol, Isohumosol, and Primosol soils were at the soil surface, and there was a low to moderate potential ecological risk for the deep soil profile. Artificial neural network model fitting results showed that Sb migration was rapid in Sandy soil because these soils have a weak Sb-adsorption capacity. The Sb concentration in the surface soil was much higher than at other soil depths. Little variation was observed in the Sandy soil and Isohumosol, indicating a weak Sb-adsorption capacity for these soils. The Sb migration model will help us to describe and predict Sb pollution in the soil environment, providing the basis for managing Sb-contaminated soil.

The convection–dispersion equation has always been a classic equation for studying pollutant migration models. There are certain deviations in scientific research because of the existence of the impurity of the medium and the nonsmooth boundary. In this paper, we introduced the one-dimensional convection–dispersion equation with fractal derivatives in fractal space, and established the fractal variational formula of the equation through the semi-inverse method. The fractal variational formula we have obtained can provide the conservation laws in an energy form in the fractal space and possible solution structures of the given equation. An analytical solution is obtained through the two-scale transform method and Laplace transform.

In this paper, we extend the novel integral transform of some functions by the duality relationship between it and Laplace transform. Additionally, applying the novel integral transform, we solve a 1-D convection-dispersion equation describing the dispersion process of chemical additives in porous rocks during the hydraulic fracturing. The results indicate that the novel integral transform can provide a new idea to obtain more exact solutions of different convection-dispersion problems.

Introducing boundary-layer (BL) theory to solve problems of solute transport provides a simple and accurate alternative method to estimate transport parameters. Most BL solutions to the convection–...

International audience; It is established that solute transport in homogenous porous media follows a classical ’S’ shape breakthrough curve that can easily be modelled by a convection dispersion equation. In this study, we designed a Model Heterogeneous Porous Medium (MHPM) with a high degree of heterogeneity, in which the breakthrough curve does not follow the classical ’S’ shape. The contrast in porosity is obtained by placing a cylindrical cavity (100% porosity) inside a 40% porosity medium composed with 1mm glass beads. Step tracing experiments are done by injecting salty water in the study column initially containing deionised water, until the outlet concentration stabilises to the input one. Several replicates of the experiment were conducted for n = 1 to 6 MHPM placed in series. The total of 116 experiments gives a high-quality database allowing the assessment of experimental uncertainty. The experimental results show that the breakthrough curve is very different from the ’S’ shape for small values of n, but the more n increases, the more the classical shape is recovered.

In this paper, we consider a form of convection-dispersion equation given in terms of the stream functions. The governing equation describing movements of contaminants under radial water flow background is given in the conserved form. As such, the conserved form of the governing equation may be written as a system of first order partial differential equations referred to as an auxiliary system, by the introduction of the nonlocal variable (or the potential variable). The resulting system of equations admits a number of (local) point symmetries which induce the nonlocal symmetries for the original governing equation. We construct classes of exact solutions using admitted genuine nonlocal symmetries, which include the invariant solutions obtained via corresponding point symmetries of the governing equation.

The dispersion coefficient (D) and retardation factor (R) are key parameters in convection–dispersion equation (CDE). The boundary-layer theory provides a simple and convenient method to solve the ...

Concern about the transport of chemicals in groundwater systems has stimulated the development of many models to describe transport in porous media, the most common of which under steady-state cond...

The convection-dispersion equation (CDE) is the classical approach for modeling solute transport in porous media. So, estimating parameters became a key problem in CDE. For statistical method, some problems such as parameter uniqueness are still unsolved because of more factors. Due to the advantage of clear physical concept and unique parameter values, the simple deterministic method became very useful alternatives. In this paper, a simple method was proposed to estimate both D and R, and the validity was verified by experiment, which can be applied in agriculture and environmental fields for predicting soil quality property.Key words: Convection-dispersion equation (CDE), parameters estimation, agricultural system.

The aim of this study is to present a two dimensional model for describing the movement of water and solute transport in two types of rockwool slabs – Floriculture, a high density type; and Expert, which has a low density. Water movement is described by the Richards equation, which is implicitly solved using the control volume finite element method. The governing convection dispersion equation describing the solute transport is explicitly solved. Numerical solutions are obtained for the distribution of water and also for the concentration of solute. The model is validated by comparing the results of water transport developed by this model with numerical results obtained by CFD that are validated by experimental results. A composite substrate is tested and validated, and is shown to give good conditions for development of plants.

Models of systemic drug absorption after oral administration are frequently based on a direct or a delayed first-order rate process. In practice, the use of the first-order approach to predict drug concentrations in blood plasma frequently yields a considerable mismatch between predicted and measured concentration profiles. This is particularly true for the upswing of the plasma concentration after oral administration. The current investigation explores an alternative model to describe the absorption rate based on the convection–dispersion equation describing the transport of chemicals through the GI tract. This equation is governed by two parameters, transport velocity and dispersion coefficient. One solution of this equation for a specific set of initial and boundary conditions was used to model absorption of paracetamol in a 22-year-old man after oral administration. The GI-tract passage rate in this subject was influenced by co-administration of drugs that stimulate or delay gastric emptying. The transport-limited absorption function is more accurate in describing the plasma concentration versus time curve after oral administration than the first-order model. Additionally, it provides a mechanistic explanation for the observed curve through the differences in GI-tract passage rate.

Based on the homotopy mapping, a globally convergent method of parameter inversion for non-equilibrium convection-dispersion equations (CDEs) is developed. Moreover, in order to further improve the computational efficiency of the algorithm, a properly smooth function, which is derived from the sigmoid function, is employed to update the homotopy parameter during iteration. Numerical results show the feature of global convergence and high performance of this method. In addition, even the measurement quantities are heavily contaminated by noises, and a good solution can be found.

The paper presents analytical solutions for 3D boundary value problems with time dependent parameters. Generally the problems considered describe diffusion and transport phenomena, but chemical reactions and other processes possibly accompanying the dispersion are also taken into account. The solutions are presented for 3D, 2D, and 1D flow domains with specific boundary conditions. In addition a relationship expressing the time evolution of the longitudinal dispersion coefficient and a method of analyzing its asymptotic behaviour at large time are deduced. The analytical results are used to interpret experimental breakthrough curves obtained from a natural gradient tracer experiment in an aquifer.

Soil structure and preferential flow are thought to be highly related, but quantification of this relationship has remained problematic. A combination of experiments on two soils at local and field scale was used to study the relationships between the degree of structural development, saturated conductivities Ks, moisture release, breakthrough parameters, and observations of preferential flow at the field scale. The convection dispersion equation and mobile-immobile model were fitted to breakthrough curves, and flow channels were stained with dye. Well-developed structure was associated with high dispersivities, a high coefficient of variation of Ks, and low mobile water contents, low exchange coefficients, and a low dyed area. Field-scale observations matched the differences in structure and preferential flow parameters observed at the local scales. Local-scale preferential flow appeared to highly influence field-scale solute transport, indicating that this is an important local-scale process which is not dampened at the field scale. A conceptual model of subsoil structural development was expanded with physical parameters matching the structural development stage to predict the likelihood of preferential flow on the basis of the saturation regime of the horizon or pedon.

A numerical model to predict the migration rate of submerged mound constructed by dredged materials due to waves is developed in this paper. The model is based on Bailard's sediment transport rate formula, Stokes' second-order wave theory, and the sediment balance equation. Particularly, the suspended load is included into the present numerical model, so that it may be possible to estimate the migration of mound composed of the mixture sediment accurately. The numerical model is satisfactorily verified by comparison of the numerical results with the field data measured both at Silver Strand mound and at Tauranga mound. The convection-dispersion processes, by which the crest of mound is flatted and moved shoreward at the same time, are generated by the present numerical model very well. In addition, it is found that the migration of mound depends directly on the wave height, the sediment size and the wave period through the analyses on the characteristics of mound migration. The dependence of mound migration on the wave height and the sediment size is relatively stronger than that on the wave period.

This paper uses a boundary-layer method to solve the convection-dispersion equation (CDE) to predict solute transport in soil. The boundary-layer solution describing chemical transport for a semi-infinite soil column or field soil profile has the advantages over the corresponding exact solution of simplicity in expression and flexibility in manipulation. Comparisons of the boundary-layer solution to the exact solution are conducted for a range of parameter values. Results show that the boundary layer solution is in good agreement with the exact solution. An important application of the boundary-layer solution is the estimation of transport parameters of solute movement through soils under both laboratory and field conditions. This leads to a new method for estimating parameters of solute transport in soils. The new method requires observation of the advance of the depth of boundary layer (solute front) with time. This can be done visually by using a tracer solution with dye in it. The new method is simple, saves time, and is applicable to both laboratory soil columns and field soils.

Most of the drilled wells are conducted on acidizing in order to eliminate the permeability reduction due to mud filtration. This type of formation damage is more sever in naturally fractured wells due to the high conductivity of fracture. If the radius of the damage due to lost circulation of drilling fluid is determined, the volume of acid needed to conduct any acid job could be calculated. Also, prediction of the near-wellbore conditions, such as mud filtrate radius, is important for accurate interpretation of the well-logs used for measuring and monitoring the properties of the near-wellbore formations. In this paper a mathematical model is developed to investigate the amount of mud filtration during drilling operation in a well with a finite conductivity vertical fracture in an infinite slab reservoir. Convection-dispersion filtrate transport model is developed in which the drilling mud can be mixed with the formation fluid. A systematic step-by-step procedure illustrating the methodology of the proposed model for the analysis of mud filtration in a finite conductivity fractured well is presented. This model is formulated for linear flow in both fracture and matrix. The effect of the filter cake is simulated by means of an empirically decaying filter rate equation. The developed model is solved by numerical analysis. The consistency of the numerical solution is checked and the best situation is considered. The sensitive analysis has been done on all the parameters of the model and the effect of each parameter such as wellbore pressure, fracture permeability, fracture diffusion coefficient, matrix permeability, matrix diffusion coefficient, matrix porosity, viscosity, constants related to the filter cake formation and fracture width on the amount of filtration are investigated. By means of the proposed model, the mud filtration can be plotted against position in both fracture and matrix for different wellbore and reservoir properties. The position in the fracture and matrix at which the curve of concentration reaches zero could be considered to represent skin radius. This radius could be used for the determination of the acid volume which is needed for the acidizing operation. Also, this model is a useful tool for accurate interpretation of the resistivity profiles which are essential for the development of efficient well-log interpretation.

The study selected DEHP from the Likeng landfill leachate in Guangzhou as the characteristic contaminant. The relevant parameters of the model were obtained by static adsorption, degradation and dispersion experiments, the convection dispersion mathematical mode of DEHP migration in vadose zone were established. And its risk to groundwater in vadose zone were predicted by finite difference numerical solution method. The results showed that: the most long distances where DEHP of landfill leachate polluted the groundwater in the vadose zone is 0.11m, which means that DEHP will have little affect on the groundwater.

Soil is a key element in human survival and can be described by various definitions according to its main utility. Nowadays, it is accepted that soil is a very complex heterogeneous medium which consists of solid phases containing minerals and organic matter and soil solution, defined as the fluid phase where soil reactions, transport and adsorption occurs (Alloway, 1995). Unfortunately, in the last decades, soils have been used as a deposit of residues. Therefore, the contamination and pollution of soils, sediments, surface and groundwater are of main concern. There are several ions and compounds of organic or inorganic nature that can contaminate soil and water. The inorganic contaminants include heavy metals like cadmium, chromium, cooper, lead and zinc, which are focused in this study. These metals are commonly present in industrial wastes (electroplating, textile and dyes, etc.), in fertilizers or in sewage sludge, but they can also contaminate soils through atmospheric deposition or runoff water (Serrano et al., 2005). Since adsorption is well recognized as the main process affecting the mobility of heavy metals through soils (Bradl, 2004), the aim of this work is the study the non competitive adsorption of these five metals, as well as the competitive adsorption in two different typical soils of the North of Portugal The soil samples were collected in Póvoa de Varzim, from O-horizon and A-horizon (0 cm -30 cm), one from a culture land and another from a beach land. To evaluate the adsorption equilibrium, batch tests were performed by adding to 2 g of air dried soil, 20 mL of 0.01 M CaCl2 solution, containing different initial concentrations of Pb [Pb(NO3)2], Cd [CdN2O6·4H2O], Cu [Cu(NO3)2·3H2O], Zn [Zn(NO3)2·6H2O] and Cr [K2Cr2O7], isolated (non-competitive system) and coupled (competitive system).To eliminate the mass effect, the multi-metal solutions were prepared in terms of molar concentrations (Echeverría et al., 1998). To avoid precipitation the pH of initial solution was adjusted to 2. Isotherm models like Freundlich and Langmuir equations were adjusted to experimental data. Batch tests are useful for the understanding of the adsorption process. However, advection and dispersion may occur in dynamic systems justifying the need of flow experiments (Miretzky et al., 2006). For continuous experiments, soil samples were packed in a column (25 cm × 3.2 cm) and a single – or multi – element solution of 50 mg/L was pass upwards during seven days. Then, using the CXTFIT code, the convection-dispersion equation was adjusted to the breakthrough curves in order to determine the retardation factor (R) and to compare the behavior of the different ions and soils.

In this paper, we consider the initial boundary value problem of Convection-Dispersion-Reaotion system which arise in solute transport or thermal transport in groundwater. Also, we prove the existence of the classical solutions of these systems.

The stochastic convection-dispersion equation (CDE) and its application to field scale subsurface solute transport are investigated with respect to their physical and mathematical foundations. It is shown that even the best-developed stochastic CDE models involve physical approximations and mathematical assumptions which stand in need of much more rigorous theoretical study and detailed field experimentation. The stochastic convection-dispersion model does not yet resolve clearly the mathematical conditions necessary in order to relate the mean solute concentration calculated with a field scale stochastic CDE to measured values of the solute concentration determined in a single field experiment. The “scale effect” in respect to solute dispersion coefficients has been studied only under conditions in which results of little predictive value can be obtained. Without the clarification of these and other fundamental issues raised in the present paper, it seems prudent to conclude that additional theoretical research on field scale solute dispersion is needed and that the stochastic convection-dispersion model does not yet warrant unqualified use as a quantitative predictive tool in the management of solute movement in aquifers and field soils.

The one-dimensional convection-dispersion equation subject to a nonlinear adsorption isotherm is formulated and solved by using a predictor-corrector finite difference scheme. The effects of convection, dispersion, and adsorption are illustrated for some typical values of variables encountered in ‘real systems.’ A comparison between the numerical results and some experimental data is also presented.

An upstream weight multiple-cell balance finite-element method is developed to simulate the three-dimensional contaminant transport in aquifers. The method combines the concepts of local mass balance and finite-element approximations. When convective transport dominates the dispersive transport, the proposed technique effectively eliminates the oscillation of numerical solutions with only a nominal increase of formulation effort over that of the general Galerkin finite-element method. Two numerical examples for which analytical solutions exist are used to verify the proposed numerical solutions. An additional example is presented to demonstrate the practical applicability of the proposed method.

Using the theory of volume averaging, we have shown1 that molecular diffusion, mass tortuosity, and mechanical mixing contribute to the mass-dispersion coefficient. A series of experiments were conducted on the system Triton X-100TM surfactant, n-decane oil, and water to determine the contribution of each mechanism to the total-dispersion matrix for flow through fired Berea sandstone. The dynamics of adsorption and the effect of dead-space volume are considered for the single-phase transport of surfactant through fired Berea. A new dynamic asdorption model is developed which considers both mass transfer to the fluid/solid surface and a kinetic surface-adsorption mechanism. Both kinetic adsorption and mass-transfer rate mechanisms are shown to be important over a wide range of injection rates.

The one-dimensional dispersion equation with nonlinear adsorption isotherm was formulated with the assumption that equilibrium exists between the adsorbed and the solution phase. The equation was solved numerically, using the extrapolated Crank-Nicolson finite-difference scheme. The numerical results are compared with the analytical solution, numerical results and some experimental data. The agreement is good.

The one-dimensional convection-dispersion model is investigated with respect to the proper boundary conditions to impose in applications to tracer movement in freely draining soil columns. It is shown that the model can be interpreted probabilistically and that this interpretation leads to physical restrictions on solute molecule transport through the inlet and exit boundaries of a soil column. These constraints, in turn, show that the correct exit boundary condition for a freely draining soil column is zero resident concentration or, equivalently, zero gradient in the flux concentration. A new solution is presented for the resident concentration under a closed inlet boundary.

Summary New analytical and numerical solutions are presented for the radial convection-dispersion differential equation with a velocity-dependent dispersion coefficient. The finite-difference numerical scheme is shown to have negligible numerical dispersion for this problem. Very close agreement is obtained between the analytical and numerical solutions. The effect of the type of boundary condition used at the wellbore is discussed.

Two models of hepatic elimination, the distributed sinusoidal perfusion model, and the convection-dispersion model, are extended and then compared for first order kinetics in the steady-state. The sinusoidal perfusion model is extended by the inclusion of intrahepatic sites of mixing between sinusoids. The degree of such mixing is estimated for taurocholate elimination by isolated perfused rat livers by a comparison of anatomical and kinetic estimates of uptake heterogeneity, using previously published data. The dispersion model is generalized by the inclusion of distributions of enzyme activity along the flow. Direct comparison of the two models in the limit in which the degree of dispersion is small, allows the flow-dependence of the dispersion coefficient to be determined, thereby greatly extending the explanatory power of the convection-dispersion model. Finally, the effect of intrahepatic mixing sites on uptake by Michaelis-Menten kinetics is quantified in terms of the distributed sinusoidal perfusion model, with results which may be applicable to capillary beds in general.