Coupled Consolidation and Contaminant Transport in Compressible Porous Media

This paper presents an experimental and numerical investigation of coupled consolidation and contaminant transport in compressible porous media. Numerical simulations were performed using the CST2 computational model, in which a dual-Lagrangian framework is used to separately follow the motions of fluid and solid phases during consolidation. Diffusion and large strain consolidation-induced transport tests were conducted on composite specimens of kaolinite slurry consisting of an upper uncontaminated layer and a lower layer contaminated with potassium bromide. Assessment of the importance of the consolidation process on solute transport is based on measured and simulated solute breakthrough curves and final contaminant concentration profiles. CST2 simulations closely match the experimental data for three different loading conditions. Diffusion and consolidation-induced advection made important contributions to solute transport and mass outflow in this study. Additional simulations indicate that consolidation-induced contaminant transport may also be affected by specimen boundary drainage and initial concentration conditions.     

Coupled Large Strain Consolidation and Solute Transport. II: Model Verification and Simulation Results

The results of numerical simulations for coupled large strain consolidation and solute transport, obtained using the CST1 model, are presented. CST1 accounts for advection, longitudinal and transverse dispersion, first-order decay reactions, and linear equilibrium sorption. Verification checks of CST1 show excellent agreement with analytical solutions for one-dimensional (1D) transport in rigid porous media, including various Peclet numbers and concentration boundary conditions. Similarly excellent agreement is observed for two-dimensional advection-dispersion transport in rigid media and 1D advection-dispersion transport in compressible media undergoing large strain consolidation. CST1 is then used to investigate consolidation-induced solute transport for a single composite liner system and a confined disposal facility for dredged contaminated sediments. In both cases, solute transport was found to be strongly affected by consolidation-induced advection both during and after the consolidation period. Consolidation has a lasting effect on solute migration because transient advective flows change the distribution of solute mass, which then becomes the initial condition for subsequent transport processes.     

Model for Consolidation-Induced Solute Transport with Nonlinear and Nonequilibrium Sorption

A numerical model, called CST2, is presented for coupled large strain consolidation and solute transport in saturated porous media. The consolidation and solute transport algorithms include the capabilities of a previous model, CST1, with the addition of a variable effective diffusion coefficient during consolidation and nonlinear nonequilibrium sorption. The model is based on a dual-Lagrangian framework that tracks separately the motions of fluid and solid phases. Verification checks of CST2 show excellent agreement with analytical and numerical solutions for solute transport in rigid porous media. A parametric study illustrates that, for the test cases considered, variation of effective diffusion coefficient during consolidation has an important effect on solute transport, whereas nonlinearity of the sorption isotherm has a less important effect. Additional simulations show that nonequilibrium sorption can have a strong effect on consolidation-induced solute transport and that this effect becomes more important as the rate of consolidation increases. The simulations also corroborate previous findings that consolidation can have a lasting effect on solute migration because transient advective flows change the distribution of solute mass which then becomes the initial condition for subsequent transport processes.     

Coupled Large Strain Consolidation and Solute Transport. I: Model Development

The development of a numerical model, CST1, for coupled large strain consolidation and solute transport in saturated porous media is presented. The consolidation algorithm is one-dimensional and includes the capabilities of a previous code, CS2, with the addition of time-dependent loading, unload/reload effects, and an externally-applied hydraulic gradient. The solute transport algorithm is two-dimensional and accounts for advection, longitudinal and transverse dispersion, first-order decay reactions, and linear equilibrium sorption. Solute transport is consistent with temporal and spatial variations of porosity and seepage velocity in the consolidating layer. The key to the transport model is the definition of two Lagrangian fields of elements that follow the motions of fluid and solid phases separately. This reduces numerical dispersion and simplifies transport calculations to that of dispersion mass flow between contiguous fluid elements. The effect of relative numerical resolution of fluid and solid elements on the accuracy of sorption/desorption is also discussed. This paper presents the theoretical and numerical development of the CST1 model. A companion paper presents verification checks of CST1 and the results of simulations that illustrate the significance of consolidation-induced solute transport for some interesting numeric examples.     

Analytical Solutions for Contaminant Diffusion in Double-Layered Porous Media

Analytical solutions for conservative solute diffusion in one-dimensional double-layered porous media are presented in this paper. These solutions are applicable to various combinations of fixed solute concentration and zero-flux boundary conditions (BC) applied at each end of a finite one-dimensional domain and can consider arbitrary initial solute concentration distributions throughout the media. Several analytical solutions based on several initial and BCs are presented based on typical contaminant transport problems found in geoenvironmental engineering including (1) leachate diffusion in a compacted clay liner (CCL) and an underlying stratum; (2) contaminant removal from soil layers; and (3) contaminant diffusion in a capping layer and underlying contaminated sediments. The analytical solutions are verified against numerical solutions from a finite-element method based model. Problems related to leachate transport in a CCL and an underlying stratum of a landfill and contaminant transport through a capping layer over contaminated sediments are then investigated, and the suitable definition of the average degree of diffusion is considered.     

Investigation of Consolidation-Induced Solute Transport. I: Effect of Consolidation on Transport Parameters

This paper presents an experimental investigation of the effect of clay consolidation on parameters that govern the advective-dispersive transport of an inorganic solute. Batch, diffusion, dispersion, and solute transport tests were conducted using kaolinite clay and dilute solutions of potassium bromide (KBr). Batch tests produced the highest levels of K+ sorption and indicated that equilibrium sorption was achieved in approximately 10–30 min. The increase in sorption observed in the batch tests, as compared to the dispersion or solute transport tests, reflects the significantly lower solids-to-solution ratio and more efficient mixing process. By comparison, kaolinite consolidation had little effect on sorption due to the relatively small change in porosity. Values of hydrodynamic dispersion coefficient (Dh), effective diffusion coefficient (D?), and apparent tortuosity factor decreased with decreasing porosity. Values of D? obtained for Br? were generally larger than for K+, whereas Dh values for Br? were significantly smaller than for K+. Values of longitudinal dispersivity (α) were larger for K+ than Br? and showed no clear trend with decreasing void ratio. In general, the experimental results suggest that changes in D? and Dh should be taken into account during clay consolidation whereas the sorption isotherm and α may be considered as unchanged during the consolidation process.     

Retardation of Nonlinearly Sorbed Solutes in Porous Media

The impact of the assumption of linear sorption on retardation of nonlinearly sorbed solutes in porous media is numerically explored in this paper. Breakthrough data of nonlinearly sorbed solutes are generated using the BIO1D simulation code along with the Freundlich-type nonlinear sorption model. Retardation coefficients (R) from generated breakthrough curves are estimated using first-moment analysis. Variations of R with experimental conditions revealed that R of a nonlinearly sorbed solute is a function of the input concentration, the injection period and the pore-water velocity but is independent of the length scale. This study also showed that it is appropriate to estimate R of a nonlinearly sorbed solute using a linearized isotherm if all soil particles experience sorption with liquid concentration equal to the induced concentration. Otherwise, the estimated linearized R will be either under- or overestimated depending on the applied experimental conditions and Freundlich parameters. The study further revealed that inability to account for sorption nonlinearity may in some cases erroneously be interpreted as evidence of the presence of transport nonequilibrium. A method is suggested to determine nonlinear sorption parameters from miscible displacement experiments.     

Coupled Surface–Subsurface Solute Transport Model for Irrigation Borders and Basins. II. Model Evaluation

The development of a coupled surface–subsurface solute transport model for surface fertigation management is presented in a companion paper (Part I). This paper discusses an evaluation of the coupled model. The numerical solution for pure advection of solute in the surface stream was evaluated using test problems with steep concentration gradients. The result shows that the model can simulate advection without numerical diffusion and oscillations, an important problem in the solution of the advection–dispersion equation in advection dominated solute transport. In addition, a close match was obtained between the numerical solution of the one-dimensional advection–dispersion equation and a simplified analytical solution. A comparison of field data and model output show that the overall mean relative residual between field observed and model predicted solute breakthrough curves in the surface stream is 16.0%. Excluding only two outlier (in the graded basin data) reduces the over all mean relative residual between field observed and model predicted breakthrough curves to 5.2%. Finally, potential applications of the model in surface fertigation and salinity management are highlighted.     

Radial Pore Diffusion with Nonuniform Intraparticle Porosities

Effects of radially dependent intraparticle pore sizes on solute fate and transport are examined for batch systems with spherical particles using a recently developed numerical model. The model can accommodate multiple particles distributed in size, mass transfer resistance at particle boundaries, intraparticle reversible sorption kinetics, and first-order decays. Two applications are examined. In the first application, random or deterministic intraparticle porosities across a spherical particle are considered. In the second application, multiple particles distributed in sizes with particle size-dependent intraparticle porosities are studied. Results from these applications indicate that concentration profiles are largely determined by interplays between B, η, and ε that incorporate the effects of intraparticle pore structures. Steady-state concentration values in both applications are determined by the volume-averaged intraparticle porosities. These results could be useful for understanding solute tailing behavior in natural porous media and the design of synthetic sorbents for treatment of contaminated waters.     

Retardation of Nonlinearly Sorbed Solutes in Porous Media

The impact of the assumption of linear sorption on retardation of nonlinearly sorbed solutes in porous media is numerically explored in this paper. Breakthrough data of nonlinearly sorbed solutes are generated using the BIO1D simulation code along with the Freundlich-type nonlinear sorption model. Retardation coefficients (R) from generated breakthrough curves are estimated using first-moment analysis. Variations of R with experimental conditions revealed that R of a nonlinearly sorbed solute is a function of the input concentration, the injection period, and the pore–water velocity but is independent of the length scale. This study also showed that it is appropriate to estimate R of a nonlinearly sorbed solute using a linearized isotherm if all soil particles experience sorption with liquid concentration equal to the induced concentration. Otherwise, the estimated linearized R will be either under- or overestimated depending on the applied experimental conditions and Freundlich parameters. The study further revealed that inability to account for sorption nonlinearity may in some cases erroneously be interpreted as evidence of the presence of transport nonequilibrium. A method is suggested to determine nonlinear sorption parameters from miscible displacement experiments.     

Multiple-Porosity Contaminant Transport by Finite-Element Method

An exponential finite-element model for multiple-porosity contaminant transport in soils is proposed. The model combines three compartments for dissolved contaminants: a primary compartment of diffusion–advection transport with nonequilibrium sorption, a secondary compartment with diffusion in rectangular or spherical soil blocks, and a tertiary compartment for immobile solutions within the primary compartment. Hence the finite-element model can be used to solve four types of mass-transfer problems which include: (1) intact soils, (2) intact soils with multiple sources of nonequilibrium partitioning, (3) soils with a network of regularly spaced fissures, and (4) structured soils. Hitherto, mobile/immobile compartments, fissured soils, and nonequilibrium sorption have been treated separately or in pairs. A Laplace transform is applied to the governing equations to remove the time derivative. A Galerkin residual statement is written and a finite-element method is developed. Both polynomial and exponential finite elements are implemented. The solution is inverted to the time domain numerically. The method is validated by comparison to analytical and boundary element predictions. Exponential elements perform particularly well, speeding up convergence significantly. The scope of the method is illustrated by analyzing contamination from a set of four waste repositories buried in fissured clay.     

One-Dimensional Modeling of Dam-Break Flow over Movable Beds

A one-dimensional model has been established to simulate the fluvial processes under dam-break flow over movable beds. The hydrodynamic model adopts the generalized shallow water equations, which consider the effects of sediment transport and bed change on the flow. The sediment model computes the nonequilibrium transport of bed load and suspended load. The effects of sediment concentration on sediment settling and entrainment are considered in determining the sediment settling velocity and transport capacity. In particular, a correction factor is proposed to modify the Van Rijn formulas of equilibrium bed-load transport rate and near-bed suspended-load concentration for the simulation of sediment transport under high-shear flow conditions. The governing equations are solved by an explicit finite-volume method with the first-order upwind scheme for intercell fluxes. The model has been tested in two experimental cases, with fairly good agreement between simulations and measurements. The sensitivities of the model results to parameters such as the sediment nonequilibrium adaptation length, Manning’s roughness coefficient and the proposed correction factor have been verified. The proposed model has also been compared to an existing model and the results indicate the new model is more reliable.     

Coupled Surface–Subsurface Solute Transport Model for Irrigation Borders and Basins. I. Model Development

Surface fertigation is widely practiced in irrigated crop production systems. Lack of design and management tools limits the effectiveness of surface fertigation practices. The availability of a process-based coupled surface–subsurface hydraulic and solute transport model can lead to improved surface fertigation management. This paper presents the development of a coupled surface–subsurface solute transport model. A hydraulic model described in a previous paper by the writers provided the hydrodynamic basis for the solute transport model presented here. A numerical solution of the area averaged advection–dispersion equation, based on the split-operator approach, forms the surface solute transport component of the coupled model. The subsurface transport process is simulated using HYDRUS-1D, which also solves the one-dimensional advection–dispersion equation. A driver program is used for the internal coupling of the surface and subsurface transport models. Solute fluxes calculated using the surface transport model are used as upper boundary conditions for the subsurface model. Evaluation of the model is presented in a companion paper.     

Chlorinated Ethene Reduction by Cast Iron: Sorption and Mass Transfer

Tetrachloroethylene (PCE) and trichloroethylene (TCE) exhibited significant nonlinear sorption to nonreactive sites when exposed to four cast irons. Cast iron is a reactive material that promotes reductive dechlorination and has recently been used for in-situ remediation of chlorinated solvent contaminated ground water. Comparisons between PCE sorption to cast iron, graphite, and iron-containing minerals indicate that nonreactive sorption is due to exposed graphite inclusions in the cast iron. Sorption of the homologous series of chloroethenes to a cast iron adheres to Traube's rule; thus, the extent of sorption is related primarily to compound hydrophobicity. An analytical model incorporating rate-limited sorption∕desorption to nonreactive sites was used to assess sorption nonequilibrium. Effective sorption and desorption rate coefficients determined how significant mass transfer limitations to nonreactive sorption sites exist for PCE and not for TCE. The nonreactive sorption observed indicates that flow-through cast iron treatment systems will exhibit significant delayed attainment of steady-state conditions for chlorinated ethenes, particularly PCE and TCE.     

Simulation of Bed-Load Dispersion Process

Two general dispersion models suitable for nonequilibrium bed-load transport were constructed. The first one, called the P model, is based on the probability of migration for specific groups of sediment particles. The second one, called the D model, is derived from the advection equation discretized in finite-difference form, which is equivalent to the general dispersion equation. By comparing these models, it is found that the D model can be treated like the P model in some respects. The Courant number, Cr, in the D model has the same physical meaning as the probability of migration, P, in the P model. Although the D model and P model were based on different concepts, the simulated bed-load transport rates, which result from their application, are the same. Therefore, the dispersion equation was replaced by the numerical algorithm of the advection equation (D model) to examine several dispersion phenomena of bed-load transport. To explore further the nonequilibrium dispersion process, a series of flume experiments was conducted by using color-painted fine gravels. Having compared model simulation results and experimental data, it is shown that the models derived in this study have a reasonably good agreement with the experimental results. In summary, this study has indirectly proven that the D model, which is equivalent to the dispersion equation, is capable of simulating the nonequilibrium bed-load transport.     

Predicting the Performance of Activated Carbon-, Coke-, and Soil-Amended Thin Layer Sediment Caps

In situ capping manages contaminated sediment on-site without creating additional exposure pathways associated with dredging, e.g., sediment resuspension, and potential human exposure during transport, treatment, or disposal of dredged material. Contaminant mass is not immediately removed in sediment capping, which creates concerns over its long-term effectiveness. Groundwater seepage can also decrease the effectiveness of in situ capping. This study compares the effectiveness of commercially available sorbents that can be used to amend sand caps to improve their ability to prevent contaminant migration from the sediments into the bioactive zone. Amendments evaluated include coke, activated carbon, and organic-rich soil. The properties relevant to advective-dispersive transport through porous media (sorption, porosity, dispersivity, and bulk density) are measured for each material, and then used as inputs to a numerical model to predict the flux of 2,4,5-polychlorinated biphenyl (PCB) through a sand cap amended with a thin (1.25-cm) sorbent layer. Systems with and without groundwater seepage are considered. Isolation times provided by the sorbent layers increased with increasing sorption strength and capacity (activated carbon?coke ≈ soil?sand). The effective porosity, dispersivity, and bulk density of the sorbent layer had little effect on cap performance compared to sorption strength (Kf). In the absence of seepage, all sorbents could isolate PCBs in the underlying sediment for times greater than 100?years and would be effective for most cap applications. With groundwater seepage (Darcy velocity = 1?cm/day), activated carbon was the only sorbent that provided contaminant isolation times greater than 60?years. Long isolation times afforded by sorbent-amended caps allow time for inherently slow natural attenuation processes to further mitigate PCB flux.     

Influence of Sorption Intensity on Solute Mobility in a Fractured Formation

Diffusive mass transfer between fracture and matrix accompanied with sorption significantly influences the efficiency of natural attenuation in hard rocks. While these processes have extensively been studied in a fractured formation, limited information exists on the sorption nonlinearity. For this purpose, a numerical model is developed that couples matrix diffusion and nonlinear sorption at the scale of a single fracture using the dual-porosity concept. The study is limited to a constant continuous solute source boundary condition. The influence of both favorable and unfavorable sorption intensities on solute mobility is investigated using the method of spatial moments. The differing capacities of available sorption sites between fracture surfaces at the fracture-matrix interface and the solid grain surfaces within the rock matrix result in a slower migration of solutes along the fracture, and a larger amount of diffusive mass transfer away from the high permeability fracture.     

Accurate Two-Dimensional Simulation of Advective-Diffusive-Reactive Transport

The present paper presents an accurate numerical algorithm for the simulation of 2D solute∕heat transport by unsteady advection-diffusion-reaction. The model was specifically developed for the study of convective exchange processes in a cross section of lakes and ponds, when the currents are predominantly driven by density (temperature) gradients. The numerical scheme is based on the split-operator approach, in which advection and diffusion with chemical∕biological kinetic processes are calculated separately at each time step. Special attention is given to the advection operator in order to avoid excessive numerical damping or oscillations, as well as to the source∕sink term, which may cause numerical instability and inaccuracy if improperly treated. The model has been verified on standard test problems for a wide range of Courant, Fourier, Péclet, and Thiele numbers, and found to produce stable results of high accuracy.     

Dynamic Mathematical Modeling of an Isothermal Three-Phase Reactor: Model Development and Validation

A catalytic reactor model (CatReac) that describes the transport and series reactions of compounds in a three-phase fixed-bed catalytic reactor is developed for the purpose of describing the volatile assembly reactor system within the potable water processor on-board the International Space Station. CatReac includes these mechanisms: advective flow, axial dispersion, gas-to-liquid and liquid-to-solid mass transport, intraparticle mass transport with pore and surface diffusion, and series reactions of multiple compounds. A dimensional analysis of CatReac revealed the following seven dimensionless groups may be used to determine the controlling transport and/or reaction mechanisms: (1) the Peclet number is the ratio of the advective to the dispersive transport; (2) the Stanton number is the ratio of the external mass transfer rate to the advective rate; (3) the Damk?hler number compares the reaction rate to the advective transport rate; (4) the surface diffusion ratio equals the rate of transport by surface diffusion divided by the rate of transport by advection; (5) the pore diffusion modulus is the ratio of the rate of transport by pore diffusion to the rate of transport by advection; (6) the ratio of the gas to liquid advective rates; and, (7) the Biot numbers for surface and pore diffusion compare the external mass transfer rate to the intraparticle mass transfer rate. These dimensionless numbers are used to evaluate the impacts of the different mechanisms on the overall performance of the reactor. The numerical solution using orthogonal collocation was validated for a wide range of controlling mechanisms by comparing model simulations with several analytical solutions: (1) Gas-to-Liquid mass transfer controlling the overall mass transfer-reaction mechanisms, for a wide range of Pe number values; (2) Liquid-phase dispersion controlling the overall process; (3) Liquid-to-solid mass transfer resistance controlling the overall mass transfer-reaction process; (4) Reactions in series with two possibilities (4a): No intraparticle mass transfer resistance, and (4b): Significant intraparticle mass transfer resistance; (5) Langmuir isotherm (5a): single compound, no mass transfer resistance, and (5b): multicomponent competitive adsorption without mass transfer resistance; (6) Unsteady state operation: Plug flow with mass transfer and no reaction. These validations systematically examine all the mechanisms that are included in the general model and examine the model limitations based on the controlling mechanisms.     

Reducing Numerical Diffusion Effects with Pycnocline Filter

Numerical or artificial diffusion is the unintentional smoothing of gradients associated with the discretization of the transport equations. In lakes and reservoirs where through-flow is small, the effects of numerical diffusion of mass are cumulative, leading to a progressive weakening of vertical density stratification. This density field misrepresentation precludes accurate, long-term, three-dimensional (3D), hydrodynamic simulations on fixed grids in closed basins with an active thermocline. An ad hoc technique to limit the destratifying effects of numerical diffusion of mass is presented and tested for a 3D, hydrostatic, Z-coordinate numerical model. The technique quantifies the domain-integrated numerical diffusion by assessing the change in the background potential energy Eb. At each time step, the change in Eb associated with numerical diffusion is calculated, then removed using a sharpening filter applied to each water column. In idealized test cases, the filtering technique is effective in maintaining density stratification over one year while undergoing periodic, large-amplitude forcing by internal waves. Forty-day simulations of Lake Kinneret compared to field measurements demonstrate improved representation of density stratification using the filtering technique.