Classical Convection-dispersion Research Articles

Estimation of soil hydraulic and solute transport properties from field infiltration experiments

To model water flow and solute transport in soils, hydrodynamic and hydrodispersive parameters are required as input data in the mathematical models. This work aims to estimate the soil hydraulic and solute transport properties using a ponded axisymmetric infiltration experiment using a single-ring infiltrometer along with a conservative tracer (Cl-) in the field. Single ring infiltration experiments were accomplished on an Oxisol in Areia in the state of Paraíba, Brazil (6o 58' S, 35o 41' W, and 645 m), in a 50 x 50 m grid (16 points). The unsaturated hydraulic conductivity (K) and the sorptivity (S) were estimated for short or long time analysis of cumulative three-dimensional infiltration. The single tracer technique was used to calculate mobile water fraction (Ф) by measuring the solute concentration underneath the ring infiltrometer at the end of the infiltration. Two solute transfer numerical models based on the mobile-immobile water concept were used. The mobile water fraction (Ф), the dispersion coefficient (D), and the mass transfer coefficient (a) between mobile and immobile were estimated from both the measured infiltration depth and the Cl- concentration profile underneath the infiltrometer. The classical convection-dispersion (CD) and the mobile-immobile (MIM) models showed a good agreement between calculated and experimental values. However, the lowest standard errors to the fitted parameters were obtained by the CD model.

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.

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.

One-dimensional solution of the transport equation in porous media in transient state by a new numerical method for the management of particle track

For the last fifteen years or so, the random-walk methods have proved their worth in solving the transport equation in porous and fractured media. Their principal shortcomings remain their relatively slow calculation speed and their lack of precision at low concentrations. This paper proposes a new code which eliminates these disadvantages by managing the particles not individually but in the form of numerical values (representing the number of particles in each phase, mobile and immobile) assigned to each cell in a 1-D system. The calculation time then is short, and it is possible to introduce as many particles as desired into the model without increasing the calculation time. A large number of injection types can be simulated, and to the classical convection-dispersion phenomenon can be added a process of exchange between the mobile and immobile phase according to first-order kinetics. Because the particles are managed as numbers, the analytical solution obtained for the exchange during a time step reduces the calculation to a simple assignation of numerical values to two variables, one of which represents the mobile and the other the immobile phase; the calculation is then almost instantaneous. Because the program is developed in C, it leaves much room for graphic interaction which greatly facilitates the fitting of tracer experiments with a limited set of parameters.

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.

Parameter estimation for simulating binary homovalent cation transport in aggregated soils at variable ionic strength

Miscible displacement of calcium by magnesium in column experiments with aggregate packings was fairly well described by the classical convection-dispersion transport model combined with an equilibrium ion-exchange isotherm (one-region model) when the model was fitted to the experimental breakthrough curves. Goodness of fit was not significantly better when a two-region model with first-order mass-transfer kinetics between mobile and immobile regions was used instead of the one-region model. However, the best-fit model parameter values for the cation-exchange capacity and the selectivity coefficient did not agree satisfactory with values determined independently from batch experiments of the two soils studied. Model calculations using independently determined ion-exchange as well as transport parameter values resulted in good predictions of the early phases of experimental breakthrough curves but did not adequately account for the tailing behaviour during later phases. In another approach parameters were calibrated by fitting the models to the first of two subsequent break-through steps of the experimental effluent curves. Extrapolation of the calibrated models to the second breakthrough step then resulted in improved predictions of the second breaktrough step, as compared to independent parameter estimation. However, neither of the two models did fully account for the immediate reaction of the effluent concentrations upon the second input step. It is suggested that, additional to classical ion-exchange processes, other cation retention and release mechanisms may have been involved in the observed transport behaviour. These processes, which have not been identified yet, appear to be effective on larger time scales than those of batch experiments and upon changes in the total ionic strength.

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.

Experimental and theoretical analysis of solute transport from a diffuse source of pollution

Transport processes governing soil and groundwater contamination by a diffuse (non-point) source were studied by means of a field tracer experiment in the Hupselse Beek catchment area in The Netherlands. A bromide tracer was applied to the intake area (11m × 260m) of a drain, and subsequently monitored in the unsaturated zone, as well as in the drain effluent. Data from the unsaturated zone were analyzed using both the classical convection-dispersion solute transport equation (CDE) and a regional stochastic model (RSM) which assumes lognormal distributions of the pore water velocity and the dispersion coefficient across the field. Both models described the data equally well, with the CDE model giving a somewhat better mass balance of the recovered bromide. Calculated solute fluxes leaving the unsaturated were used as input into a mixing cell type model for transport in the saturated zone. Drain effluent concentrations predicted with the coupled unsaturated-saturated transport model were compared with the observed breakthrough curve; results indicate the presence of preferential flow in the unsaturated zone.