Classical Convection-dispersion Equation Research Articles

Fractional convection-dispersion equation with conformable derivative approach

Abstract In this present work, a well-structured and limit-based derivative definition of fractional derivative term, known as conformable derivative, is employed to develop a local generalized form of time variable fractional convection-dispersion equation (FCDE). The fractional models of convection-dispersion equation have been widely established as more authentic mode to characterize the pollutant transport in geological structures. In this work, models are formulated corresponding to spatial dependency of velocity and dispersion coefficient and also for the temporally varying decay rate, that dealt with more realistic phenomenon of the pollutant transport in groundwater reservoir. Due to the non-linearity of the problem, homotopy analysis method (HAM) is adopted to investigate the complex solutions of FCDE. As the FCDE comprises the classical convection-dispersion equation (CDE) as a special case corresponding to the fractional order α = 1 , so the obtained solutions are validated by the corresponding numerical solution and exact analytical solution for α = 1 and solutions are also verified for different fractional values of α . The effect of conformable derivative order α is properly visible over concentration strength. Hence, this investigation helps to interpret the accurate description of time dependent behaviour of contaminant transport in porous structure.

Effects of nanocarbon on the hydraulic parameters and the solute transport process for disturbed loessial soil

Due to its strong absorbability and significant specific surface area, the existence of nanocarbon in loessial soil could significantly affect the physical and chemical properties of the soil. The effects of nanocarbon on the hydraulic properties and solute transport parameters of the soil were investigated in this study. The soil water characteristic curves (SWCCs) were obtained using the centrifuge method in the loessial soil with various nanocarbon contents by weight (0, 0.1, 0.5, 0.7, and 1 %). Based on the soil columns experiment (5 cm in inner diameter and 20 cm in length), the saturated hydraulic conductivity (Ks) and the solute transport process for soil with different nanocarbon contents were also measured. The primary results were shown as follows: the water retention capacity increased with an increase in nanocarbon content, whereas the Ks value and the unsaturated hydraulic conductivity (K) decreased. The relationship between the Ks value and the nanocarbon content could be described by an exponential function. The Brooks–Corey and the van Genuchten models provided a good description of the SWCC of the disturbed loessial soil with various nanocarbon contents. Generally, the residual water content (θr) and the empirical parameter (α) increased with an increasing nanocarbon content, whereas the pore size distribution parameter (n) decreased. Application of nanocarbon in the soil could enhance available water capacity, and the obtained nanocarbon use amount was 0.746 % from the study. The classical convection–dispersion equation (CDE) and the mobile and immobile model (MIM) fitted the chloride transport process in the soil well. Furthermore, the average pore water velocity (v), the Peclet number (Pe), the retardation factor (R), the mobile water fraction (β), and the dimensionless mass transfer coefficient (ω) decreased, whereas the dispersion (λ) increased with the nanocarbon contents. In addition, ω could be expressed by a multiple regression equation related to the nanocarbon content and the v and β values.

Migration of fallout radiocaesium in a grassland soil from 1986 to 2001: Part II: Evaluation of the activity–depth profiles by transport models

The vertical migration of 134Cs, deposited by the Chernobyl fallout (1986), and 137Cs, deposited by the Chernobyl and the global fallout, in the soil of an undisturbed Bavarian grassland in Germany was investigated from 1986 to 2001. The activity–depth profiles of both isotopes at ten sampling dates were evaluated by the classical convection–dispersion equation and a random walk particle model. In both models, the apparent migration velocity v and the apparent dispersion coefficient D were assumed to be independent of time. However, optimized values of v and D were significantly different for the different locations sampled at different times. If nevertheless constant values of v and D were used, the simulated activity densities per soil layer were out of the range of the spatial variability of the observed activity densities determined in 2001. It is concluded that without further simultaneous investigations e.g. on bioturbation at the study site, migration parameters of radiocaesium determined by classical transport models based on convection and dispersion during the first years after the deposition of the activity cannot be used for predictive purposes.

Vertical random variability of the distribution coefficient in the soil and its effect on the migration of fallout radionuclides

In the field, the distribution coefficient, Kd, for the sorption of a radionuclide by the soil cannot be expected to be constant. Even in a well defined soil horizon, Kd will vary stochastically in horizontal as well as in vertical direction around a mean value. The horizontal random variability of Kd produce a pronounced tailing effect in the concentration depth profile of a fallout radionuclide, much less is known on the corresponding effect of the vertical random variability. To analyze this effect theoretically, the classical convection-dispersion model in combination with the random-walk particle method was applied. The concentration depth profile of a radionuclide was calculated one year after deposition assuming (1) constant values of the pore water velocity, the diffusion/dispersion coefficient, and the distribution coefficient (Kd = 100 cm3.g-1), and (2) exhibiting a vertical variability for Kd according to a log-normal distribution with a geometric mean of 100 cm3.g-1 and a coefficient of variation of CV = 0.53. The results show that these two concentration depth profiles are only slightly different, the location of the peak is shifted somewhat upwards, and the dispersion of the concentration depth profile is slightly larger. A substantial tailing effect of the concentration depth profile is not perceivable. Especially with respect to the location of the peak, a very good approximation of the concentration depth profile is obtained if the arithmetic mean of the Kd-values (Kd = 113 cm3.g-1) and a slightly increased dispersion coefficient are used in the analytical solution of the classical convection-dispersion equation with constant Kd. The evaluation of the observed concentration depth profile with the analytical solution of the classical convection-dispersion equation with constant parameters will, within the usual experimental limits, hardly reveal the presence of a log-normal random distribution of Kd in the vertical direction in contrast to the horizontal direction.

Exact solutions for one-dimensional transport with asymptotic scale-dependent dispersion

A general analytical solution is developed for one-dimensional solute transport in heterogeneous porous media with scale-dependent dispersion. The solution assumes that the dispersivity, α, increases linearly with distance, x, that is, α( x) = ax, until some distance x 0, after which a reaches an asymptotic value, α L = ax 0. The parameters a and x 0 characterize the nature of the scale-dependent dispersion process. The general solution contains as special cases the solutions of the classical convection-dispersion equation (CDE) assuming a constant dispersivity, and a recent solution by Yates assuming a linearly increasing dispersivity with distance. A simplified solution is also derived for cases where diffusion can be neglected. In addition, a solution for steady-state transport is presented. Results obtained with the proposed solutions demonstrate several features of scale-dependent dispersion in nonhomogeneous media which differ from those predicted with the CDE model and the model of Yates. 1

Transect study on solute transport in a macroporous soil

Solute transport experiments using a non-reactive tracer were conducted on short, undisturbed, saturated columns of a sandy loam soil. All columns, 20 cm in diameter and 20 cm long, were collected along a transect of 35 m. Most of the soil columns had pre-existing macropores. The columns were leached at a steady flow-rate under ponding conditions. The resulting breakthrough curves (BTCs) showed a large heterogeneity. Several of the BTCs displayed early breakthrough and long tailing. All the data were interpreted in terms of dimensional time moments, the classical convection-dispersion equation (CDE) and the mobile-immobile transport model (MIM). Experimental time moments were found to vary significantly among the different BTCs. Analysis of the time moments also revealed that the variance of the field-scale BTC was several times larger than the average of the local-scale variance. The pore water velocity v and dispersion coefficient D were obtained by fitting the CDE to the local-scale BTCs, resulting in an average dispersivity of 7.4 cm. Frequency distributions for the CDE parameters v and D were equally well described by a normal or log-normal probability density function (pdf). When a log-normal pdf for D is considered, the variance of the loge transformed D values (σ 1nD 2 ) was found to be 2.1. For the MIM model, two additional parameters were fitted : the fraction of mobile water, θ m /θ, and the first-order mass transfer coefficient, a. The MIM was more successful in describing the data than the CDE transport model. For the MIM model, the average dispersivity was about 2 cm. The MIM parameters v, D and θ m /θ were best described by a log-normal pdf rather than a normal pdf. Only the parameter α was better described by a normal pdf. Mobile water fractions, θ m /θ ranged from 0.01 to 0.98, with a mean of 0.43 (based on a log-normal pdf). When the CDE and MIM were applied to the data, the fitted pore water velocities, v, compared favourably with the effective pore water velocities, v eff , obtained from moment analysis.

Estimating solute transport in undisturbed soil columns using time-domain reflectometry

Time-domain reflectometry (TDR) was used to monitor solute breakthrough curves (BTC's) in 30 saturated undisturbed soil columns collected along a 35-m-long transect in the field. The BTC's were obtained by relating the bulk soil electrical conductivity, EC a, to the relative concentration of a KCl solute pulse applied to the soil surface. Values of EC a were estimated by measuring the soil's impedance to an electromagnetic wave generated by a cable tester. Parallel two-rod TDR probes inserted horizontally at a depth of 10 cm were used to monitor the soil's impedance during transport of the KCl solute pulse. Calculated experimental time moments indicated that the BTC data were very variable in time and space. This variability was attributed in part to the relatively small volume of soil sampled with the TDR probes, and in part to the natural heterogeneity of the sandy loam soil. The observed BTC's were classified into three groups. One group showed bell-shaped curves consistent with the classical convection-dispersion equation (CDE). A second group was characterized by early breakthrough and long tailing. The BTC's in this group could be described by a mobile-immobile transport model (MIM). A third group of BTC's showed irregular shapes with several peaks. Time moments were used to compare the estimated (from the moments), fitted (CDE and MIM) and independently measured pore-water velocities. The disparities between the observed and fitted velocities suggest that for structured soil several TDR probes may be necessary in order to obtain reasonable estimates of column-scale solute transport behaviour.

Solute movement through homogeneous and heterogeneous soil columns

The classical convection-dispersion equation (CDE) and the stochastical convection-dispersion equation (SCDE) were utilized to study solute transport in homogeneous and heterogeneous porous media. Extensive experimental data collected in 1250-cm soil columns were used for testing validity of the CDE and SCDE. Analyses of the experimental data showed that the observed dispersion coefficient increased with the travel distance for both homogeneous and heterogeneous soils. The range of the dispersion coefficients was from 0·05 to 20 cm 2 min −1 for the homogeneous soil, and from 3 to 200 cm 2 min −1 for the heterogeneous soil. A linear relationship between the dispersion coefficient and travel distance, assumed in the SCDE, described the scale-dependent dispersion coefficient reasonably well. Therefore, the SCDE predicted solute transport in the porous media more accurately than the CDE for these examples. In addition, the SCDE provided more consistent estimation of breakthrough curves than the CDE, using information obtained at a location either close to or far from the predicting place.

LYSIMETER STUDY OF ANION TRANSPORT DURING STEADY FLOW THROUGH LAYERED COARSE-TEXTURED SOIL PROFILES

Transport of Cl and NO3 was studied during steady-state unsaturated water flow in several lysimeters filled with a coarse sand or sandy loam. Breakthrough curves were obtained with suction cups at six different depths. The transport parameters were determined by fitting the classical convection-dispersion equation to the data assuming a homogeneous or a two-layer soil profile. As expected, the two-layer solution described the data better than the solution assuming a homogeneous soil, especially for the sand where the fitted pore-water velocities were about 60% higher in the subsoil compared with the topsoil. The average dispersivity for the sandy loam was 1.9 cm for the topsoil and 1.3 cm for the subsoil, whereas a value of 0.58 cm could be used for the entire profile for the coarse sand. Solute outflow from the lysimeters was reasonably well predicted using the transport parameters estimated from breakthrough curves obtained with the suction cups. The dispersion coefficients estimated from the effluent concentrations were generally somewhat larger than those derived from suction cup data. The total concentration had little effect on the dispersion coefficient and the pore-water velocity. No significant differences were found between NO3 and Cl, although dispersivities for Cl were always slightly higher than those for NO3.

The form of the dispersion equation under recharge and variable velocity, and its analytical solution

In this article an attempt is made to describe field scale solute transport parameters in terms of regional hydrologic and aquifer hydraulic properties, such as recharge rate, transmissivity, hydraulic gradient, aquifer thickness and soil porosity. Aquifers subject to natural recharge from rainfall exhibit groundwater velocities which vary with distance and with the recharge intensity. This in turn generates an evolving transport dispersion coefficient which increases with distance even in a homogeneous aquifer with constant dispersivity. The dispersion equation in an aquifer subject to recharge and variable groundwater velocity is one with coefficients given as variable functions of distance. A new stable analytical solution of this equation is presented along with numerical comparisons with the classical convection dispersion equation and sensitivity tests on the effect of hydrologic‐hydraulic variables on the contaminant evolution. It was found that the recharge rate substantially affects the contaminant distribution and may partially explain the scale dependence of dispersion parameters. Transmissivity and hydraulic gradient values also determine the velocity distribution and therefore the rate of migration. It would appear that constant, laboratory scale, dispersivities may be sufficient for the modeling of field scale concentrations if an equation which accounts for the effects of hydrologie and hydraulic variables is used.

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.

Calcium and magnesium transport in aggregated soils at variable ionic strength

Miscible displacement experiments were performed under conditions of variable total concentration and saturated steady-state water flow in order to study the transport of CaCl 2 and MgCl 2 with simultaneous cation exchange through aggregates (diameter: 0.2–0.4 cm) of two soils (Abist loam and Laegern clay loam). Using independently measured parameter values, good predictions of the experimental CaMg breakthrough curves (BTC) were obtained with a transport model based on the classical convection-dispersion equation and on the assumption of local ion exchange equilibrium. The agreement between experimental and model curves was best in the early parts of the BTC. Some tailing was observed in the later parts which was not successfully accounted for by the model. Calculations with an extended model, based on the mobile-immobile water (or two-region) concept, showed that the tailing effect may be partially due to physical non-equilibrium conditions governing the solute transfer between inter- and intra-aggregate solution. In a previous sensitivity analysis, predictions with the two-region model showed great sensitivity to the choice of parameter values determining mass transfer and partitioning of the total cation exchange capacity between mobile and immobile region. Unfortunately, however, no satisfactory methods were found to be available to determine these parameters independently. Estimation of the mass transfer coefficient from BTC of non-reactive tracers ( 3H 2O and 3Cl) failed, because the latter were in perfect agreement with the local equilibrium assumption. In addition to physical non-equilibrium effect, the results suggest that cation exchange capacities during transport were significantly underestimated with the standard NH 4OAc method.

An experimental study of solute transport in a stony field soil

The transport of two conservative tracers, bromide and chloride, was studied in a stony field soil under natural soil, vegetative, and climatic conditions. Small tracer pulses were applied evenly over the 94‐m2 plot area in April 1982 (bromide) and October 1982 (chloride). A 15‐m‐long, 3‐m‐deep transect through the plot area was extensively sampled in May 1983, approximately 400 days after the bromide and 200 days after the chloride was applied. Tracer concentrations were obtained from 842 samples taken at 61 locations along the transect. Two‐dimensional contour plots of the data showed a relatively uniform displacement in the vertical direction, as well as a significant horizontal redistribution during the study period. The data were analyzed with the classical convection‐dispersion equation (CDE) and with a regional stochastic model that assumes logarithmic distributions of the pore water velocity and the dispersion coefficient across the field. Both models successfully described momentary field‐averaged concentration distributions but failed to predict observed concentration data in October 1984 when another, less intensive sampling was carried out. Pore water velocities could be estimated reasonably well with as few as five vertical sampling lines, whereas dispersion coefficients and solute loads required numerous additional samplings. A field scale dispersivity of about 8 cm was obtained for the CDE model. Between 75 and 130% of the applied tracers were recovered in 1982. The effects of sampling size (0.3 versus 3 kg dry soil weight) were found to be relatively small.