Analytical solution of two-dimensional solute transport in an aquifer-aquitard system

A novel analytical model of solute transport in a layered aquifer system with mixing processes in the reservoirs

Analytical models of solute transport have been widely used to aid the understanding of the physical and chemical processes undergone by substances introduced in a layered aquifer system. However, in previous studies, the advection component of transport was assumed to be one dimensional, while also ignoring the mixing processes that occur in the inlet and the outlet reservoirs. In this study, new sets of models describing those mixing processes are presented. Beyond that, these models were integrated into already existing models and the result is a novel analytical model of solute transport in aquifer-aquitard systems. The novel analytical solution was derived by the Laplace transform method and the finite-cosine Fourier transform method under the mobile-immobile (MIM) framework. The calculations take into account: the longitudinal and vertical dispersion, the molecular diffusion and the horizonal and vertical advection components of solute transport, as well as first-order chemical reaction, in both the aquifer and the aquitard. A finite-difference solution of the model is tested against experimental data in order to critique its reliability. Results indicate that the numerical and analytical solutions of the new model match well with experimental data. This new model outperforms the previous models in terms of interpreting experimental data. The mixing old and new water in the reservoirs during solute transport in aquifer-aquitard systems is important. Global sensitivity analysis demonstrates that the output concentration of solute in the aquifer-aquitard system is most sensitive to the volume of water in the inlet reservoir. The contribution of the molecular diffusion effect to the total mass flux of the tracer cross the aquifer-aquitard interface is much smaller than the contribution of the dispersive and advective effects.

Laboratory observations for two-dimensional solute transport in an aquifer-aquitard system

Low-permeability media such as clay appear in nearly all hydrogeological systems. To date, although significant efforts have been put forward by hydrologists, transport mechanism is still not well understood in such media, especially in an aquifer-aquitard system. In this study, two-dimensional experiments of groundwater flow and solute transport were conducted in a clay-sand two-layer system to investigate the characteristics of flow and transport in such a system. Sodium chloride (NaCl) (a conservative tracer) from a tank was injected after passing by the pre-inlet reservoir where the mixing effect and flow transiency were analyzed. A new numerical model considering the mixing effect and flow transiency was developed to interpret the experimental data based on the finite-element COMSOL Multiphysics platform. Transport parameters were assessed by best fitting the observed breakthrough curves (BTCs). Several important results were obtained. Firstly, aquitard advection was found to be non-negligible and should be considered in a proper mathematical model for describing the transport process. Secondly, advective velocities were temporally variable and showed decreasing trends in the sand and clay layers, mainly due to the impacts of physical and biological clogging. Thirdly, the mixing effect in the pre-inlet reservoir led to a lower tracer concentration in the sand layer at early times. Finally, the observed BTCs exhibited early arrivals in the clay layer, possibly resulting from preferential flow pathways. These findings can provide hints for contamination remediation works in aquifer-aquitard systems.

Impact of diffuse layer processes on contaminant forward and back diffusion in heterogeneous sandy-clayey domains

Low-permeability aquitards can significantly affect the transport, distribution, and persistence of contaminant plumes in subsurface systems. Although such low-permeability materials are often charged, the key role of charge-induced electrostatic processes during contaminant transport has not been extensively studied. This work presents a detailed investigation exploring the coupled effects of heterogeneous distribution of physical, chemical and electrostatic properties on reactive contaminant transport in field-scale groundwater systems including spatially distributed clay zones. We performed an extensive series of numerical experiments in three distinct heterogeneous sandy-clayey domains with different levels of complexity. The flow and reactive transport simulations were performed by explicitly resolving the complex velocity fields, the small-scale electrostatic processes, the compound-specific diffusive/dispersive fluxes and the chemical processes utilizing a multi-continua based reactive transport code (MMIT-Clay). In each particular domain, numerical experiments were performed focusing on both the forward and back diffusion through the sandy-clayey interfaces. The results illuminate the control of microscopic electrostatic mechanisms on macroscopic mass transfer. Coulombic interactions in the clay's diffuse layer can significantly accelerate or retard a particular species depending on its charge. Furthermore, the chemical heterogeneity plays a major role in mass storage and release during reactive transport. Neglecting such processes can lead to substantial over- or underestimation of the overall transport behavior, which underlines the need for integrated physical, chemical and electrostatic approaches to accurately describe mass transfer processes in systems including low-permeability inclusions.

Contaminant occurrence and migration between high- and low-permeability zones in groundwater systems: A review

In recent decades, water quality problems that impact human health, especially groundwater pollution, have been intensely studied, and this has contributed to new ideas and policies around the world such as Low Impact Development (LID) and Superfund legislation. The fundamental to many of these problems is pollutant occurrence and migration in saturated porous media, especially in groundwater. Such environments often contain contrasting zones of high and low permeability with significant differences in hydraulic conductivity (~10^-4 and 10^-8 m/s, respectively). High-permeability zones (HPZs) represent the primary pathways for pollutant transport in groundwater, while low-permeability zones (LPZs) are often diffusion dominated and serve as both sinks and sources (i.e., via back-diffusion) of pollutants over many decades. In this review, concepts and mechanisms of solute source depletion, contaminant accumulation, and back-diffusion in high- and low-permeability systems are presented, and new insights gained from both experimental and numerical studies are analyzed and summarized. We find that effluent monitoring and novel image analysis techniques have been adroitly used to investigate temporal and spatial evolutions of contaminant concentration; simultaneously, mathematical models are constantly upscaled to verify, optimize and extend the experimental data. However, the spatial concentration data during back-diffusion lacks diversity due to the limitations of pollutant species in studies, the microscopic mechanisms controlling pollutant transformation are poorly understood, and the impacts of these reactions on contaminant back-diffusion are rarely considered. Hence, most simulation models have not been adequately validated and are not capable of accurately predicting pollutant fate and cleanup in realistic heterogeneous aquifers. Based on these, some hypotheses and perspectives are mentioned to promote the investigation of contaminant migration in high- and low-permeability systems in groundwater.

A new solution to transient single-well push-pull test with low-permeability non-Darcian leakage effects

An accurate solute transport model is critical to the interpretation of single-well push-pull (SWPP) test. Previous studies of SWPP test generally consider solitary aquifer that is confined by impermeable layers. Also, existing solutions for solute transport in aquifer-aquitard systems only consider the injection phase and over-simplify the flow field by assuming uniformly distributed velocity in the aquitard. In this study, we developed a numerical model with Dirichlet boundary condition for SWPP test affected by leakage described by a low-permeability non-Darcian expression involving a threshold pressure gradient (I₀). Our SWPP test model considered transient flow in multi-phases, which include injection, chase, rest and extraction phases. Finite-difference scheme was adopted to solve the models of flow and solute transport. The results indicate that an increasing hydraulic diffusivity leads to a greater peak value of breakthrough curve (BTC) while a medium with larger grain size results in less estimation error when using steady-state flow model to interpret the transient SWPP test. Additionally, A greater I₀ makes the solute stored in aquitard more difficult to be extracted out due to dispersion dominance, which results in higher BTC values. For the purpose of application, a lumped dimensionless index called the non-Darcian index (NDI) was proposed to quantify the overestimation degree by neglecting leakage, and the underestimation degree by accounting for Darcian leakage, when interpreting the SWPP test with low-permeability non-Darcian leakage. The long-term slope of breakthrough curve coupled with the NDI can be employed to determine the cases in which the low-permeability non-Darcian leakage should be considered.

Contaminant transport in a largely-deformed aquitard affected by delayed drainage

This paper employed a one-dimensional large-deformation model in consideration of the coupling of mechanical consolidation and solute transport, to study the transport of contaminants in a largely-deformed aquitard. An analytical solution has been derived to describe the drawdown variation in a largely-deformed aquitard which is subjected to abrupt hydraulic head decline in adjacent confined aquifers. The pore water flux and void ratio variation were obtained on the basis of the analytical solution. The equation for transient contaminant flux was solved by the finite difference method. A hypothetical case study was done to explore the effect of consolidation on the contaminant transport in a largely-deformed aquitard. The transit time of contaminant transport in the aquitard is mainly determined by the hydraulic conductivity, thickness, partitioning coefficient, void ratio and effective diffusion coefficients of aquitard, as well as the drawdown in the adjacent confined aquifer. The impact of delayed drainage on the contaminant transport in the largely-deformed aquitard is mainly controlled by two factors: the transient water flow and the decrease of aquitard thickness, in the process of aquitard consolidation. The former increases the breakthrough time of contaminant transport in the aquitard, and the latter gives rise to an opposite case with its effect decreasing with increasing contaminant partitioning coefficient for soil particles sorption. A larger deformation, which may be induced by a larger thickness, higher specific storativity of aquitard or a larger drawdown of the adjacent confined aquifer, and a lower hydraulic conductivity of aquitard cause a more significant impact of delayed drainage on the contaminant transport in an aquitard.

Distribution of randomly diffusing particles in inhomogeneous media

Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct particle hopping rates. Starting from the master equations (MEs) governing diffusion in inhomogeneous media we derive here, for arbitrary spatial dimensions, the deterministic lattice equations (DLEs) specifying the average particle number at each lattice site for randomly diffusing particles in inhomogeneous media. We consider the case of free (Fickian) diffusion with no steric constraints on the maximum particle number per lattice site as well as the case of diffusion under steric constraints imposing a maximum particle concentration. We find, for both transient and asymptotic regimes, excellent agreement between the DLEs and kinetic Monte Carlo simulations of the MEs. The DLEs provide a computationally efficient method for predicting the (average) distribution of randomly diffusing particles in inhomogeneous media, with the number of DLEs associated with a given system being independent of the number of particles in the system. From the DLEs we obtain general analytic expressions for the steady-state particle distributions for free diffusion and, in special cases, diffusion under steric constraints in inhomogeneous media. We find that, in the steady state of the system, the average fraction of particles in a given domain is independent of most system properties, such as the arrangement and shape of domains, and only depends on the number of lattice sites in each domain, the particle hopping rates, the number of distinct particle species in the system, and the total number of particles of each particle species in the system. Our results provide general insights into the role of spatially inhomogeneous particle hopping rates in setting the particle distributions in inhomogeneous media.

Subsurface solute transport with one-, two-, and three-dimensional arbitrary shape sources

Solutions with one-, two-, and three-dimensional arbitrary shape source geometries will be very helpful tools for investigating a variety of contaminant transport problems in the geological media. This study proposed a general method to develop new solutions for solute transport in a saturated, homogeneous aquifer (confined or unconfined) with a constant, unilateral groundwater flow velocity. Several typical source geometries, such as arbitrary line sources, vertical and horizontal patch sources, circular and volumetric sources, were considered. The sources can sit on the upper or lower aquifer boundary to simulate light non-aqueous-phase-liquids (LNAPLs) or dense non-aqueous-phase-liquids (DNAPLs), respectively, or can be located anywhere inside the aquifer. The developed new solutions were tested against previous benchmark solutions under special circumstances and were shown to be robust and accurate. Such solutions can also be used as a starting point for the inverse problem of source zone and source geometry identification in the future. The following findings can be obtained from analyzing the solutions. The source geometry, including shape and orientation, generally played an important role for the concentration profile through the entire transport process. When comparing the inclined line sources with the horizontal line sources, the concentration contours expanded considerably along the vertical direction, and shrank considerably along the groundwater flow direction. A planar source sitting on the upper aquifer boundary (such as a LNAPL pool) would lead to significantly different concentration profiles compared to a planar source positioned in a vertical plane perpendicular to the flow direction. For a volumetric source, its dimension along the groundwater flow direction became less important compared to its other two dimensions.

Chloride Dispersion across Silt Deposits in a Glaciated Bedrock River Valley

Soil and groundwater from the Neponset River floodplain deposit that receive high concentrations of deicing agents from nearby highways were investigated. The silty sand floodplain is separated by a silty aquitard from the underlying aquifer that serves as a public water supply. We made a transport-based assessment of the capacity of the aquitard to protect the underlying aquifer. One hundred seventeen soil samples and 469 groundwater samples collected during a period of 4 yr from boreholes and 10 wells grouped in two well clusters were analyzed for dissolved Cl concentration. The soil characterization and groundwater monitoring results agreed, showing a very slow change in subsurface Cl contamination with time. These data also calibrated a vertical one-dimensional advective-dispersive transport model across the deposits. Advective transport dominated only in the top 3.37 m of the floodplain deposit, with dispersion being the main transport mechanism below this depth. Due to the silty nature of the aquitard, dispersion rather than diffusion was the main transport mechanism into the floodplain-aquitard system. Soil and groundwater quality data confirmed a Cl concentration at the floodplain surface near the highway runoff drainage outlets of 2450 mg L. The model estimated a vertical dispersivity at the site of 8 mm and a vertical hydrodynamic dispersion coefficient of 3.71 × 10 m s. These data confirmed the aquitard's capacity to contain deicing agents, protecting the underlying aquifer from contamination.

Impact of thin aquitards on two-dimensional solute transport in an aquifer

The influence of aquitards on solute transport in an aquifer is an important and often overlooked process for subsurface contaminant transport. In particular, slow advection (leakage) into an aquitard is often neglected in previous analytical treatment of solute transport, making such analytical solutions unsuitable for benchmarking numerical simulations of transport when aquitard leakage exists. In this study, a semi-analytical solution to the two-dimensional conservative solute transport in an aquifer bounded by thin aquitards is derived in the Laplace domain. The governing equation in the aquifer (not aquitard) incorporates terms accounting for advection, longitudinal dispersion, and transverse vertical dispersion. Both one-dimensional vertical advection and molecular diffusion are considered for aquitard transport. The solutions are derived under conditions of steady-state flow and the first- and third-type transport boundary conditions in the aquifer along with assuming the continuity of concentration and vertical mass flux at aquifer and aquitard interfaces. The solutions in the real time domain are obtained by numerically inverting the solutions in the Laplace domain using the Stehfest (1970) algorithm. The semi-analytical solutions are compared with those from Zhan et al. (2009b), which considered aquitard leakage in infinitively thick aquitards. The concentration profiles, breakthrough curves and distribution profiles in the aquifer are different from those of Zhan et al. (2009b) at small ratios of the aquitard/aquifer thickness; whereas, the results of both are consistent for thick bounding aquitards. This study reveals that the residence time distribution (RTD) in the main aquifer is related to the aquitard/aquifer thickness ratios, Peclet numbers and porosities of adjacent aquitards. The results also suggest that MT3DMS (a commonly applied transport code) cannot successfully simulate solute transport at the aquifer-aquitard interfaces. The presented solutions improve available solutions for transport processes in an aquifer bounded by thin aquitards with leakage. The developed solutions can be directly extended to cases when the vertical hydrodynamic dispersion of the aquitards is considered by simply replacing the effective molecular diffusion coefficients of the aquitard by the vertical hydrodynamic dispersion coefficients of the aquitards.

A mathematical solution and analysis of contaminant transport in a radial two-zone confined aquifer

An aquifer with a wellbore surrounded by a finite-thickness skin, such as a gravel pack, can be regarded as a radial two-zone system. In this study, a mathematical model is developed to describe contaminant transport in a radial two-zone confined aquifer system. The solution of the model equations can be used to delineate contaminant transport in such a two-zone aquifer system and to investigate the skin effect on the temporal and spatial concentration distribution. The present solution is shown to reduce to an existing solution for transport in a homogeneous aquifer in the case when there is no wellbore skin. The results predicted from the solution indicate that the skin effect has a significant impact on the concentration distribution at early time. But an abrupt change in the spatial concentration distribution may occur at the interface of the skin zone and aquifer formation zone. The results from a sensitivity analysis reveal that dispersivity in the formation zone has a more significant effect on the concentration distribution than do the effects of skin thickness and dispersivity in the skin zone.

Analytical solutions of one-dimensional multispecies reactive transport in a permeable reactive barrier-aquifer system

The permeable reactive barrier (PRB) remediation technology has proven to be more cost-effective than conventional pump-and-treat systems, and has demonstrated the ability to rapidly reduce the concentrations of specific chemicals of concern (COCs) by up to several orders of magnitude in some scenarios. This study derives new steady-state analytical solutions to multispecies reactive transport in a PRB-aquifer (dual domain) system. The advantage of the dual domain model is that it can account for the potential existence of natural degradation in the aquifer, when designing the required PRB thickness. The study focuses primarily on the steady-state analytical solutions of the tetrachloroethene (PCE) serial degradation pathway and secondly on the analytical solutions of the parallel degradation pathway. The solutions in this study can also be applied to other types of dual domain systems with distinct flow and transport properties. The steady-state analytical solutions are shown to be accurate and the numerical program RT3D is selected for comparison. The results of this study are novel in that the solutions provide improved modeling flexibility including: 1) every species can have unique first-order reaction rates and unique retardation factors, and 2) daughter species can be modeled with their individual input concentrations or solely as byproducts of the parent species. The steady-state analytical solutions exhibit a limitation that occurs when interspecies reaction rate factors equal each other, which result in undefined solutions. Excel spreadsheet programs were created to facilitate prompt application of the steady-state analytical solutions, for both the serial and parallel degradation pathways.

An analytical solution to contaminant diffusion in semi-infinite clayey soils with piecewise linear adsorption

The adsorption of contaminants onto soil particles typically is nonlinear if the contaminant concentration is sufficiently high. A simplified piecewise linear adsorption isotherm consistent with experimental results is proposed as an approximation for nonlinear adsorption behavior. This approximation allows for the use of analytical solution to model solute diffusion of contaminants that exhibit nonlinear adsorption. A moving boundary is introduced to represent significant changes in the retardation factor of clay with an increase in solute concentration. The proposed analytical solutions were validated using experimental data presented in the literature. There is negligible difference between the results obtained by the proposed analytical solution and those obtained by the linear model when C(m)/C(0) reached 0.5. The results also show that the model based on linear adsorption using the initial secant of the Freundlich isotherm leads to significantly lower estimated breakthrough time for the contaminant of interest than that obtained using the proposed model. The earlier breakthrough is due to an under-estimation of the amount of adsorption. The proposed method is relatively simple to apply and can be used for evaluating experimental results and verifying more complex numerical models.

Stepwise superposition approach for the analytical solutions of multi-dimensional contaminant transport in finite- and semi-infinite aquifers

Analytical solutions of contaminant transport in multi-dimensional media are significant for theoretical and practical purposes. However, due to the problems for which the solutions are sought which are complex in most of the cases, most available analytical solutions in multi-dimensional media are not given in their closed forms. Integrals are often included in the solution expressions, which may limit the practitioners to use the solutions. In addition, available multi-dimensional solutions for the third-type sources in bounded media are fairly limited. In this paper, a stepwise superposition approach for obtaining approximate multi-dimensional transport solutions is developed. The approach is based on the condition that the one-dimensional solution along the flow direction is known. The solutions are expressed in their closed forms without integrals. The transport media to the solutions are flexible and can be finite, semi-infinite, or infinite in the transverse directions. The solutions subject to the first- and third-type boundary conditions at the inlet with a distributed source over the domain are obtained. The integrals in some known solutions can also be evaluated by the approach if they can be derived to include known longitudinal integrals with respect to time. The accuracy and efficiency of the solutions proposed in this paper are verified through test problems and calculation examples.

Stepwise superposition approximation approach for analytical solutions with non-zero initial concentration using existing solutions of zero initial concentration in contaminate transport

Analytical solutions for contaminant transport are widely used for both theoretical and practical purposes. However, many existing solutions are obtained subject to an initial condition of zero concentration, which is often unrealistic in many practical cases. This article proposed a stepwise superposition approximation approach to solve the non-zero initial concentration problem for first-type and third-type boundary conditions by using the existing zero initial concentration solution. Theoretical examples showed that the approach was highly efficient if a proper superposition scheme with relative concentration increments was constructed. The key parameter that controlled the convergence speed was the time increment (lambdat) multiplied by the rate constant (lambda). The approach served also as an alternative way to make a convenient concentration calculation even if the non-zero initial concentration solution of a problem was known.