Modeling solute transport in one-dimensional homogeneous and heterogeneous soil columns with continuous time random walk

In this paper, we used the continuous time random walk (CTRW) framework to characterize the transport process in 1250-cm long one-dimensional homogenous and heterogeneous soil columns at the experiments conducted by Huang et al. [Huang, K., Toride, N., van Genuchten, M.Th., 1995. Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns. Trans. Porous Media. 18, 283-302]. The transport process was also simulated by using the advection-dispersion equation (ADE) and the spatial fractional advection-dispersion equation (FADE) for comparison. In the homogeneous soil column, the non-Fickian behavior is found at the distances less than 1000cm with beta values larger than 1.60, but less than 2, and Fickian form transport is obtained at distances larger than 1000cm with beta values larger than 2. In the heterogeneous soil column, we found the most anomalous behavior at distances from 200cm to 700cm with beta values ranging from 0.894 to 0.958, and non-Fickian transport process is observed at distances larger than 800cm with beta values in the range between 1 and 1.3. More significant non-Fickian behavior is found for transport in the heterogeneous soil column than that in the homogeneous soil column. The CTRW fits to the breakthrough curves (BTCs) have lower values of root mean square error (RMSE) and higher values of determination coefficient (r(2)), with respect to the fits of ADE and FADE. The CTRW model also is better captures the full evolution of BTCs, and especially their tails.     

Non-Fickian dispersion in a single fracture

Solute transport in fractured rocks is of major interest in many applications, from the petroleum industry to ground water management. This work focuses on the dispersion process in a transparent replica of a real single fracture. The fracture exhibits strong changes in heterogeneity, with the first half very heterogeneous and the second half fairly homogeneous. Three models have been used to interpret the tracer experiments: the classical advection-dispersion equation (ADE), the continuous time random walk (CTRW), and the stratified model. The main goals were to test these models and to study possible correlations between fitting parameters and heterogeneities. As expected, the solution derived from the ADE equation appears to be unable to model long-time tailing behavior. On the other hand, the results confirm the CTRW robustness and the coefficient beta seems well correlated to heterogeneities. Finally, the stratified model is also able to describe non-Fickian dispersion. The parameters defined by this model are correlated to the heterogeneities of the fracture.     

Evidence of one-dimensional scale-dependent fractional advection-dispersion

A semi-analytical inverse method and the corresponding program FADEMain for parameter estimation of the fractional advection-dispersion equation (FADE) were developed in this paper. We have analyzed Huang et al.'s [Huang, K., Toride, N., van Genuchten, M.Th., 1995. Experimental investigation of solute transport in large homogeneous and heterogeneous saturated soil columns. Trans. Porous Media 18, 283-302.] laboratory experimental data of conservative solute transport in 12.5-m long homogeneous and heterogeneous soil columns to test the non-Fickian dispersion theory of FADE. The dispersion coefficient was calculated by fitting the analytical solution of FADE to the measured data at different transport scales. We found that the dispersion coefficient increased exponentially with transport scale for the homogeneous column, whereas it increased with transport scale in a power law function for the heterogeneous column. The scale effect of the dispersion coefficient in the heterogeneous soil was much more significant comparing to that in the homogeneous soil. The increasing rate of dispersion coefficient versus transport distance was smaller for FADE than that for the advection-dispersion equation (ADE). Finite difference numerical approximations of the scale-dependent FADE were established to interpret the experimental results. The numerical solutions were found to be adequate for predicting scale-dependent transport in the homogeneous column, while the prediction for the heterogeneous column was less satisfactory.     

The effect of entrapped nonaqueous phase liquids on tracer transport in heterogeneous porous media: laboratory experiments at the intermediate scale

This work considers the applicability of conservative tracers for detecting high-saturation nonaqueous-phase liquid (NAPL) entrapment in heterogeneous systems. For this purpose, a series of experiments and simulations was performed using a two-dimensional heterogeneous system (10x1.2 m), which represents an intermediate scale between laboratory and field scales. Tracer tests performed prior to injecting the NAPL provide the baseline response of the heterogeneous porous medium. Two NAPL spill experiments were performed and the entrapped-NAPL saturation distribution measured in detail using a gamma-ray attenuation system. Tracer tests following each of the NAPL spills produced breakthrough curves (BTCs) reflecting the impact of entrapped NAPL on conservative transport. To evaluate significance, the impact of NAPL entrapment on the conservative-tracer breakthrough curves was compared to simulated breakthrough curve variability for different realizations of the heterogeneous distribution. Analysis of the results reveals that the NAPL entrapment has a significant impact on the temporal moments of conservative-tracer breakthrough curves.     

Application of continuous time random walk theory to nonequilibrium transport in soil

Continuous time random walk (CTRW) formulations have been demonstrated to provide a general and effective approach that quantifies the behavior of solute transport in heterogeneous media in field, laboratory, and numerical experiments. In this paper we first apply the CTRW approach to describe the sorbing solute transport in soils under chemical (or) and physical nonequilibrium conditions by curve-fitting. Results show that the theoretical solutions are in a good agreement with the experimental measurements. In case that CTRW parameters cannot be determined directly or easily, an alternative method is then proposed for estimating such parameters independently of the breakthrough curve data to be simulated. We conduct numerical experiments with artificial data sets generated by the HYDRUS-1D model for a wide range of pore water velocities (υ) and retardation factors (R) to investigate the relationship between CTRW parameters for a sorbing solute and these two quantities (υ, R) that can be directly measured in independent experiments. A series of best-fitting regression equations are then developed from the artificial data sets, which can be easily used as an estimation or prediction model to assess the transport of sorbing solutes under steady flow conditions through soil. Several literature data sets of pesticides are used to validate these relationships. The results show reasonable performance in most cases, thus indicating that our method could provide an alternative way to effectively predict sorbing solute transport in soils. While the regression relationships presented are obtained under certain flow and sorption conditions, the methodology of our study is general and may be extended to predict solute transport in soils under different flow and sorption conditions.     

Analysis of field observations of tracer transport in a fractured till

We analyze a set of observations from a recently published, field-scale tracer test in a fractured till. These observations demonstrate a dominant, underlying non-Fickian behavior, which cannot be quantified using traditional modeling approaches. We use a continuous time random walk (CTRW) approach which thoroughly accounts for the measurements, and which is based on a physical picture of contaminant motion that is consistent with the geometric and hydraulic characterization of the fractured formation. We also incorporate convolution techniques in the CTRW theory, to consider transport between different regions containing distinct heterogeneity patterns. These results enhance the possibility that limitations in predicting non-Fickian modes of contaminant migration can be overcome.     

Eulerian derivation of the fractional advection-dispersion equation

A fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. In contrast to the classical ADE, the fractional ADE has solutions that resemble the highly skewed and heavy-tailed breakthrough curves observed in field and laboratory studies. These solutions, known as alpha-stable distributions, are the result of a generalized central limit theorem which describes the behavior of sums of finite or infinite-variance random variables. We use this limit theorem in a model which sums the length of particle jumps during their random walk through a heterogeneous porous medium. If the length of solute particle jumps is not constrained to a representative elementary volume (REV), dispersive flux is proportional to a fractional derivative. The nature of fractional derivatives is readily visualized and their parameters are based on physical properties that are measurable. When a fractional Fick's law replaces the classical Fick's law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE. Fractional ADEs are ergodic equations since they occur when a generalized central limit theorem is employed.     

Continuous time random walk model better describes the tailing of atrazine transport in soil

Contaminant transport in soils is complicated and involves some physical and chemical nonequilibrium processes. In this research, the soil column displacement experiments of Cl⁻ and atrazine under different flow velocities were carried out. The data sets of Cl⁻ transport in sandy loam fitted to the convection dispersion equation (CDE) and the two-region model (TRM) indicated that the effects of physical nonequilibrium process produced by immobile water on the breakthrough curves (BTCs) of Cl⁻ and atrazine transport through the repacking soil columns were negligible. The two-site model (TSM) and the continuous time random walk (CTRW) were also used to fit atrazine transport behavior at the flow rate of 19.86 cm h⁻¹. The CTRW convincingly captured the full evolution of atrazine BTC in the soil column, especially for the part of long tailing. However, the TSM failed to characterize the tailing of atrazine BTC in the soil column. The calculated fraction of equilibrium sorption sites, F, ranging from 0.78 to 0.80 for all flow rates suggested the contribution of nonequilibrium sorption sites to the asymmetry of atrazine BTCs. Furthermore, the data sets for the flow rates of 6.68 cm h⁻¹ and 32.81 cm h⁻¹ were predicted by the TSM and the CTRW. As to the flow rate of 6.68 cm h⁻¹, the CTRW predicted the entire BTC of atrazine transport better than the TSM did. For the flow rate of 32.81 cm h⁻¹, the CTRW characterized the late part of the tail better, while the TSM failed to predict the tailings of atrazine BTC.     

Diffusive partitioning tracer test for the quantification of nonaqueous phase liquid (NAPL) in the vadose zone: Performance evaluation for heterogeneous NAPL distribution

A partitioning tracer test based on gas-phase diffusion in the vadose zone yields estimates of the residual nonaqueous phase liquid (NAPL) saturation. The present paper investigates this technique further by studying diffusive tracer breakthrough curves in the vadose zone for a heterogeneous NAPL distribution. Tracer experiments were performed in a lysimeter with a horizontal layer of artificial kerosene embedded in unsaturated sand. Tracer disappearance curves at the injection point and tracer breakthrough curves at some distance from the injection point were measured inside and outside of the NAPL layer. A numerical code was used to generate independent model predictions based on the physicochemical sand, NAPL, and tracer properties. The measured and modeled tracer breakthrough curves were in good agreement confirming the validity of important modeling assumptions such as negligible sorption of chlorofluorocarbon (CFC) tracers to the uncontaminated sand and their fast reversible partitioning between the soil air and the NAPL phase. Subsequently, the model was used to investigate different configurations of NAPL contamination. The experimental and model results show that the tracer disappearance curves of a single-well diffusive partitioning tracer test (DPTT) are dominated by the near-field presence of NAPL around the tip of the soil gas probe. In contrast, breakthrough curves of inter-well tracer tests reflect the NAPL saturation in between the probes, although there is no unique interpretation of the tracer signals if the NAPL distribution is heterogeneous. Numerical modeling is useful for the planning of a DPTT application. Simulations suggest that several cubic meters of soil can be investigated with a single inter-well partitioning tracer test of 24-hour duration by placing the injection point in the center of the investigated soil volume and probes at up to 1 m distance for the monitoring of gaseous tracers.     

Modelling of nonreactive tracer dipole tests in a shear zone at the Grimsel test site

The investigation of the migration of a high pH plume in a fractured shear zone is foreseen by a long-term experiment at the Grimsel rock laboratory. In order to characterise the initial conditions for the long-term experiment and to evaluate an optimal hydraulic in situ set-up, several dipole experiments with nonreacting tracers have been performed. The dipole experiments differ in geometry, pumping rates and orientation to the background water flow. Several single and double-porosity models have been applied to fit the results of these dipole tracer tests in order to extract values for some transport parameters and discriminate for certain transport processes. A two-dimensional porous medium approach was successfully used to fit tracer breakthrough curves measured for a dipole experiment. A model based on a one-dimensional dual porous medium approach was also successful, although the applied hydraulic dipole, with similar injection and extraction rates, suggests the existence of an extended two-dimensional flow field. For the two-dimensional porous medium approach, tracer breakthrough could only be fitted with a complex flow field geometry within the heterogeneous fractured shear zone. The heterogeneity was generated by heterogeneous porosity and hydraulic permeability distributions. Predictions for further dipole geometries and a sorbing tracer have been calculated by means of both models using the flow and transport parameters deduced from fits for a single dipole experiment. This allows for comparison with the measured breakthrough of sorbing tracers. The foreseen experiment with sorbing (radionuclide) tracers will help decide on the appropriate approach that should be used to describe such dipole experiments in this shear zone. Additionally, the migration and spreading of a solution with high pH has been calculated taking into account mineral dissolution and precipitation in a two-dimensional porous medium approach in order to estimate the amount and character of the mineral reactions induced by the interaction between the high pH solution and the rock.     

Characterization of pore scale NAPL morphology in homogeneous sands as a function of grain size and NAPL dissolution

In this study, we investigate pore scale morphology of nonaqueous phase liquids (NAPLs) trapped in different pore sizes using tracer techniques. Specific interfacial area and saturation of NAPL trapped in homogeneous sands were measured using the interfacial and partitioning tracer techniques. The observed NAPL-water interfacial areas increased in a log-linear fashion with decreasing sand grain size, but showed no clear trend with residual NAPL saturation formed in the various grain sizes. The measured values were used to calculate the NAPL morphology index, which characterizes the spatial NAPL distribution within the pore space. The NAPL morphology indices, increased exponentially with decreasing grain size, indicating that the NAPL becomes smaller, but more blobs. For a fixed grain size, the specific interfacial area and saturation of the NAPL were measured following changes caused by dissolution using alcohol. The observed interfacial areas showed a decrease linearly as a function of the NAPL saturation.     

Transport of Escherichia coli through variably saturated sand columns and modeling approaches

A sand column leaching system with well-controlled suction and flow rate was built to investigate the effects on bacterial transport of air-water interface effects (AWI) correlated to water content, particle size, and column length. Adsorption of Escherichia coli strain D to silica sands was measured in batch tests. The average % adsorption for coarse and fine sands was 45.9+/-7.8% and 96.9+/-3.2%, respectively. However, results from static batch adsorption experiments have limited applicability to dynamic bacterial transport in columns. The early breakthrough of E. coli relative to bromide was clear for all columns, namely c. 0.15 to 0.3 pore volume earlier. Column length had no significant effects on the E. coli peak concentration or on total recovery in leachate, indicating retention in the top layer of sands. Tailing of breakthrough curves was more prominent for all fine sand columns than their coarse sand counterparts. Bacterial recovery in leachate from coarse and saturated sand columns was significantly higher than from fine and unsaturated columns. Observed data were fitted by the convection-dispersion model, amended for one-site and two-site adsorption to particles, and for air-water interface (AWI) adsorption. Among all models, the two-site+AWI model achieved consistently high model efficiency for all experiments. Thus it is evident from experimental and modeling results that AWI adsorption plays an important role in E. coli transport in sand columns.     

Application of the method of temporal moments to interpret solute transport with sorption and degradation

In this note, we applied the temporal moment solutions of [Das and Kluitenberg, 1996. Soil Sci. Am. J. 60, 1724] for one-dimensional advective-dispersive solute transport with linear equilibrium sorption and first-order degradation for time pulse sources to analyse soil column experimental data. Unlike most other moment solutions, these solutions consider the interplay of degradation and sorption. This permits estimation of a first-order degradation rate constant using the zeroth moment of column breakthrough data, as well as estimation of the retardation factor or sorption distribution coefficient of a degrading solute using the first moment. The method of temporal moment (MOM) formulae was applied to analyse breakthrough data from a laboratory column study of atrazine, hexazinone and rhodamine WT transport in volcanic pumice sand, as well as experimental data from the literature. Transport and degradation parameters obtained using the MOM were compared to parameters obtained by fitting breakthrough data from an advective-dispersive transport model with equilibrium sorption and first-order degradation, using the nonlinear least-square curve-fitting program CXTFIT. The results derived from using the literature data were also compared with estimates reported in the literature using different equilibrium models. The good agreement suggests that the MOM could provide an additional useful means of parameter estimation for transport involving equilibrium sorption and first-order degradation. We found that the MOM fitted breakthrough curves with tailing better than curve fitting. However, the MOM analysis requires complete breakthrough curves and relatively frequent data collection to ensure the accuracy of the moments obtained from the breakthrough data.     

Migration of uranium(IV)/(VI) in the presence of humic acids in quartz sand: a laboratory column study

The migration behavior of U(IV) and U(VI) in the presence of humic acid was studied in a quartz sand system. Laboratory column experiments were performed using humic acid, U(VI) in humic acid absence, U(IV) and U(VI) in humic acid presence, and for comparison a conservative tracer. In experiments using humic acid, both redox species of U migrate nearly as fast as the conservative tracer. Humic acid accelerates the U(VI) breakthrough compared to the humic acid-free system. There are strong indications for a similar effect on the U(IV) transport. At the same time, a part of U(IV) and U(VI) associated with the humic acid is immobilized in the quartz sand due to humic colloid filtration thus producing a delaying effect. Tailing at a low concentration level was observed upon tracer elution. The experimental breakthrough curves were described by reactive transport modeling using equations for equilibrium and kinetic reactions. The present study demonstrates that humic acids can play an important role in the migration of actinides. As natural organic matter is ubiquitous in aquifer systems, the humic colloid-borne transport of actinides is of high relevance in performance assessment.     

Colloid transport in unsaturated porous media: the role of water content and ionic strength on particle straining

Packed column and mathematical modeling studies were conducted to explore the influence of water saturation, pore-water ionic strength, and grain size on the transport of latex microspheres (1.1 microm) in porous media. Experiments were carried out under chemically unfavorable conditions for colloid attachment to both solid-water interfaces (SWI) and air-water interfaces (AWI) using negatively charged and hydrophilic colloids and modifying the solution chemistry with a bicarbonate buffer to pH 10. Interaction energy calculations and complementary batch experiments were conducted and demonstrated that partitioning of colloids to the SWI and AWI was insignificant across the range of the ionic strengths considered. The breakthrough curve and final deposition profile were measured in each experiment indicating colloid retention was highly dependent on the suspension ionic strength, water content, and sand grain size. In contrast to conventional filtration theory, most colloids were found deposited close to the column inlet, and hyper-exponential deposition profiles were observed. A mathematical model, accounting for time- and depth-dependent straining, produced a reasonably good fit for both the breakthrough curves and final deposition profiles. Experimental and modeling results suggest that straining--the retention of colloids in low velocity regions of porous media such as grain junctions--was the primary mechanism of colloid retention under both saturated and unsaturated conditions. The extent of stagnant regions of flow within the pore structure is enhanced with decreasing water content, leading to a greater amount of retention. Ionic strength also contributes to straining, because the number of colloids that are held in the secondary energy minimum increases with ionic strength. These weakly associated colloids are prone to be translated to stagnation regions formed at grain-grain junctions, the solid-water-air triple point, and dead-end pores and then becoming trapped.     

Colloid transport in a geochemically heterogeneous porous medium: aquifer tank experiment and modeling

To examine colloid transport in geochemically heterogeneous porous media at a scale comparable to field experiments, we monitored the migration of silica-coated zirconia colloids in a two-dimensional layered porous media containing sand coated to three different extents by ferric oxyhydroxides. Transport of the colloids was measured over 1.65 m and 95 days. Colloid transport was modeled by an advection-dispersion-deposition equation incorporating geochemical heterogeneity and colloid deposition dynamics (blocking). Geochemical heterogeneity was represented as favorable (ferric oxyhydroxide-coated) and unfavorable (uncoated sand) deposition surface areas. Blocking was modeled as random sequential adsorption (RSA). Release of deposited colloids was negligible. The time to colloid breakthrough after the onset of blocking increased with increasing ferric oxyhydroxide-coated surface area. As the ferric oxyhydroxide surface area increased, the concentration of colloids in the breakthrough decreased. Model-fits to the experimental data were made by inverse solutions to determine the fraction of surface area favorable for deposition and the deposition rate coefficients for the favorable (ferric oxyhydroxide-coated) and unfavorable sites. The favorable deposition rate coefficient was also calculated by colloid filtration theory. The model described the time to colloid breakthrough and the blocking effect reasonably well and estimated the favorable surface area fraction very well for the two layers with more than 1% ferric oxyhydroxide coating. If mica edges in the uncoated sand were considered as favorable surface area in addition to the ferric oxyhydroxide coatings, the model predicted the favorable surface area fraction accurately for the layer with less than 1% ferric oxyhydroxide coating.     

Experimental investigation of pneumatic soil vapor extraction

Soil Vapor Extraction (SVE) is a common remediation technique for removing volatile organic compounds from unsaturated contaminated soils. Soil heterogeneities can however cause serious limitations to the applicability of SVE due to air bypassing low permeable areas of the soil, leading to diffusion limitation of the remediation. To enhance removal from areas subject to diffusion limitation a new remediation technique, pneumatic soil vapor extraction, is proposed. In contrast to traditional SVE, in which soil vapor is extracted continuously by a vacuum pump, pneumatic SVE is based on enforcing a sequence of large pressure drops on the system to enhance the recovery from the low-permeable areas. The pneumatic SVE technique was investigated in the laboratory using TCE as a model contaminant. 2D-laboratory tank experiments were performed on homogeneous and heterogeneous sand packs. The heterogeneous packs consisted of a fine sand lens surrounded by a coarser sand matrix. As expected when using traditional SVE, the removal of TCE from the low permeable lens was extremely slow and subject to diffusion limitation. In contrast when pneumatic venting was used removal rates increased by up to 77%. The enhanced removal was hypothesized to be attributed to mixing of the contaminated air inside the lens and generation of net advective transport out of the lens due to air expansion.     

An experimental study of flow and transport in fractured slate

A series of laboratory tracer migration experiments in a single rock fracture have been performed, and the breakthrough curves have been interpreted using mathematical modelling. Discrepancies were observed between the experimental data and the predictions made using a simple advection-dispersion model. The potential reasons for these discrepancies have been investigated by applying more complex models: one model incorporates channelling of flow within the fracture, the other couples dispersion and advection in the fracture with rock-matrix diffusion. It is concluded that chanelling of flow can adequately explain the observed spreading behaviour; rock-matrix diffusion is not a significant mechanism influencing transport in these experiments.     

Determination of bromacil transport as a function of water and carbon content in soils

This study was conducted to determine the significance of bromacil transport as a function of water and carbon content in soils and to explore the implications of neglecting sorption when making assessments of travel time of bromacil through the vadose zone. Equilibrium batch sorption tests were performed for loamy sand and sandy soil added with four different levels of powdered activated carbon (PAC) content (0, 0.01, 0.05, and 0.1%). Column experiments were also conducted at various water and carbon contents under steady-state flow conditions. The first set of column experiments was conducted in loamy sand containing 1.5% organic carbon under three different water contents (0.23, 0.32, and 0.41) to measure breakthrough curves (BTCs) of bromide and bromacil injected as a square pulse. In the second set of column experiments, BTCs of bromide and bromacil injected as a front were measured in saturated sandy columns at the four different PAC levels given above. Column breakthrough data were analyzed with both equilibrium and nonequilibrium (two-site) convection-dispersion equation (CDE) models to determine transport and sorption parameters under various water and carbon contents. Analysis with batch data indicated that neglect of the partition-related term in the calculation of solute velocity may lead to erroneous estimation of travel time of bromacil, i.e. an overestimation of the solute velocity by a factor of R. The column experiments showed that arrival time of the bromacil peak was larger than that of the bromide peak in soils, indicating that transport of bromacil was retarded relative to bromide in the observed conditions. Extent of bromacil retardation (R) increased with decreasing water content and increasing PAC content, supporting the importance of retardation in the estimation of travel time of bromacil even at small amounts of organic carbon for soils with lower water content.