A Three-Dimensional Flow and Transport Modeling of an Aquifer Contaminated by Perchloroethylene Subject to Multi-PRB Remediation

Although subsurface contamination by organic chemicals is a pervasive environmental problem, a permeable reactive barrier (PRB) as a typical in-situ remediation technology is often successful at many sites. Laboratory tests have shown that perchloroethylene (PCE) can be dechlorinated by the combination of zero-valent iron and anaerobic microbial communities (FeMB), and the degradation pathway was: PCE → TCE → 1, 1-DCE → ethylene→  ethane (Ma amd Wu Environ Geol 55(1):47–54, 2008). Based on Ma’s experimental results, we have extended MT3DMS to simulate mother-daughter chain reactions using MODFLOW-2005 V1.7/MT3DMS V5.2. Second, using FeMB as multi-PRB’s reactive media, a 5-component transport model for a three-dimensional aquifer contaminated by PCE subject to multi-PRB remediation was built. Third, the adsorption and degradation parameters of reactive media were estimated by means of genetic algorithm. Finally, the three-dimensional, homogeneous aquifer contaminated by PCE subject to multi-PRB remediation was simulated. Overall, the purpose of this paper was to use FeMB as multi-PRB’s reactive media, and develop a modified MODFLOW/MT3DMS that can simulate a three-dimensional aquifer contaminated by PCE and its daughters. Results demonstrated that FeMB could be a potential reactive media for PCE-contaminated groundwater. Multi-PRB could be a preferred option for secondly pollution caused by application of one-stage PRB. The modified MODFLOW/MT3DMS can effectively simulate multispecies mother–daughter chain kinetic reactions. Sensitivity analysis showed that losses of reactivity and permeability may have a significant effect on remediation’s success.     

Parameter Determination for Reductive Dechlorination of Chlorinated Solvents

The contamination of subsurface due to the chlorinated solvents such as tetrachlorethylene (PCE) and trichlorethylene (TCE) is one of the most difficult environmental problems to treat. Bioremediation has been shown by many researchers to be a remedial alternative for this type of contamination. Chlorinated solvents are not directly mineralized but rather are transformed by microorganisms into one or more intermediate compounds before converting into a final compound. These sequential reactions, termed “reductive dehalogenation”, consist of replacing a chlorine atom by a hydrogen atom. The pathway of degradation of PCE can be expressed by the following scheme PCE → TCE → DCE → VC → ETH, where dichloroethene (DCE), vinyl chloride (VC) and finally ETH is ethylene. Since the biotransformation rate coefficients of each intermediate compound are different, they have to be determined very precisely to establish an effective treatment operation. The sequential decay can be described by Michaelis–Menten’s kinetics, which constitutes a highly nonlinear system of ordinary differential equations (ODEs). This is very sensitive to the changes of biotransformation rate coefficients. In this study we introduce a methodology how to numerically estimate the rate coefficients for Michaelis–Menten’s equations from the knowledge of the concentrations of PCE, TCE, DCE, VC and ETH. The efficiency of the proposed method is demonstrated on some examples. Estimated biotransformation coefficients are employed to predict the concentrations of chlorinated solvents. Computations and measurements show a very good agreement.     

Study of the influence of isomerism on the curing properties of the epoxy system DGEBA(n=0)/1,2-DCH by rheology

Rheology was used to characterise the curing process of the epoxy system consisting of a diglycidyl ether of bisphenol A [DGEBA(n=0)] and one, or a mixture of the two isomers of the curing agent 1,2-diaminocyclohexane (1,2-DCH). The trans isomer and a mixture cis–trans (30–70% respectively) of 1,2-DCH were used to investigate their different behaviour. In a previous article (L. Núñez et al., Therm Anal Calorim, 2005), it was reported that the values of gel time measured by dielectric analysis for the systems DGEBA(n=0)/trans-1,2-DCH and DGEBA(n=0)/cis–trans −1,2-DCH were very similar, while times to vitrification, as measured by modulated scanning calorimetry differed widely. In the present study, rheometry was used to corroborate the results obtained in the previous article for gel time and activation energy and also to analyse differences in the cure of the two epoxy systems studied by comparing the results reported here with those previously measured by DEA and DSC.     

Chemical Compositions in Salinity Waterflooding of Carbonate Reservoirs: Theory

Higher oil recovery after waterflood in carbonate reservoirs is attributed to increasing water wettability of the rock that in turn relies on complicated surface chemistry. In addition, calcite mineral reacts with aqueous solutions and can alter substantially the composition of injected water by mineral dissolution. Carefully designed chemical and/or brine flood compositions in the laboratory may not remain intact while the injected solutions pass through the reactive reservoir rock. This is especially true for a low-salinity waterflood process, where some finely tuned brine compositions can improve flood performances, whereas others cannot. We present a 1D reactive transport numerical model that captures the changes in injected compositions during water flow through porous carbonate rock. We include highly coupled bulk aqueous and surface carbonate-reaction chemistry, detailed reaction and mass transfer kinetics, 2:1 calcium ion exchange, and axial dispersion. At typical calcite reaction rates, local equilibrium is established immediately upon injection. In SI, we validate the reactive transport model against analytic solutions for rock dissolution, ion exchange, and longitudinal dispersion, each considered separately. Accordingly, using an open-source algorithm (Charlton and Parkhurst in Comput Geosci 37(10):1653–1663, 2011. https://doi.org/10.1016/j.cageo.2011.02.005), we outline a design tool to specify chemical/brine flooding formulations that correct for composition alteration by the carbonate rock. Subsequent works compare proposed theory against experiments on core plugs of Indiana limestone and give examples of how injected salinity compositions deviate from those designed in the laboratory for water-wettability improvement.

     

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

Vasculature based model for characterizing the oxygen transport in skin tissues – analogy to the Weinbaum-Jiji bioheat equation

Based on the conceptual three-layer microvascular structure of skin tissues proposed by Weinbaum et al. [20–25] and in analogy to the well known Weinbaum-Jiji (W-J) bioheat equation, a new oxygen transport model was established in this paper, which collectively included the contributions of the vascular geometry and the blood flow condition. The new one-dimensional three-layer oxygen transport model was then applied to predict the average oxygen concentration distribution in skin tissues and numerical solutions for the boundary value problem coupling the three layers were obtained. A simple expression for the tensor diffusivity (D_eff) of oxygen transport over the deep tissue layer was presented, which was orders of magnitude higher than the intrinsic diffusivity (D_t) in tissue without blood flow. Effects of blood flow velocity and vascular geometry to the oxygen transport were investigated. Calculations indicated that the vascular geometry had significant effects on oxygen transport. The oxygen exchange between the arteries and veins was relatively small for the deep tissue layer. Further, the average oxygen concentration gradient appears low in intermediate layer due to large capillary perfusion. The theoretical results were implemented to interpret some previous experimental results and a better understanding on the oxygen transport across the vascularized living tissues was obtained. The strategy proposed in this paper may provide a feasible way to comprehensively characterize the oxygen transport behaviors in living tissues with real and complex vasculature.     

Analytical Solutions for Reactive Transport of Multiple Volatile Contaminants in the Vadose Zone

Groundwater contamination usually originates from surface contamination. Contaminants then move downward through the vadose zone and finally reach the groundwater table. To date, however, analytical solutions of multi-species reactive transport are limited to transport only in the saturated zone. The motivation of this work is to utilize analytical solutions, which were previously derived for single-phase transport, to describe the reactive transport of multiple volatile contaminants in the unsaturated zone. A mathematical model is derived for describing transport with phase partitioning of sequentially reactive species in the vadose zone with constant flow velocity. Linear reaction kinetics and linear equilibrium partitioning between vapor, liquid, and solid phases are assumed in this model.     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

Parameter Determination of Sequential Reductive Dehalogenation Reactions of Chlorinated Hydrocarbons

Chlorinated hydrocarbons such as tetrachloroethylene (PCE) and trichloroethylene (TCE) are not directly mineralized, but rather are transformed into one or more intermediate compounds before converting into a final compound. Since the biotransformation rate coefficients of each intermediate compound are different, the coefficients in each step of reductive dehalogenation must be determined to establish an effective treatment operation. These parameters can be estimated by fitting the experimental data to Michaelis–Menten equation. In this study, we introduce a methodology, using both the curve-fitting and graphical methods, to estimate the rate of maximum biotransformation and half-saturation coefficients of parent and intermediate compounds. First-order rate coefficients are also estimated by simplifying the Michaelis–Menten equation for both curve-fitting and graphical methods. The results show that both methods produce similar parameter values for each rate equation. Estimated first-order kinetic parameters are employed to predict the compound concentrations from the analytical solutions of governing equations for sequential dehalogenation. Comparisons of predicted and experimental values show favorable agreement.     

A new r‐ratio formulation for TVD schemes for vertex‐centered FVM on an unstructured mesh

The r‐ratio is a parameter that measures the local monotonicity, by which a number of high‐resolution and TVD schemes can be formed. A number of r‐ratio formulations for TVD schemes have been presented over the last few decades to solve the transport equation in shallow waters based on the finite volume method (FVM). However, unlike structured meshes, the coordinate directions are not clearly defined on an unstructured mesh; therefore, some r‐ratio formulations have been established by approximating the solute concentration at virtual nodes, which may be estimated from different assumptions. However, some formulations may introduce either oscillation or diffusion behavior within the vertex‐centered (VC) framework. In this paper, a new r‐ratio formulation, applied to an unstructured grid in the VC framework, is proposed and compared with the traditional r‐ratio formulations. Through seven commonly used benchmark tests, it is shown that the newly proposed r‐ratio formulation obtains better results than the traditional ones with less numerical diffusion and spurious oscillation. Moreover, three commonly used TVD schemes—SUPERBEE, MINMOD, and MUSCL—and two high‐order schemes—SOU and QUICK—are implemented and compared using the new r‐ratio formulation. The new r‐ratio formulation is shown to be sufficiently comprehensive to permit the general implementation of a high‐resolution scheme within the VC framework. Finally, the sensitivity test for different grid types demonstrates the good adaptability of this new r‐ratio formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

A nonlinear ALE‐FCT scheme for non‐equilibrium reactive solute transport in moving domains

In this paper, we present a conservative, positivity‐preserving, high‐resolution nonlinear ALE‐flux‐corrected transport (FCT) scheme for reactive transport models in moving domains. The mathematical model is a convection–diffusion equation with a nonlinear flux equation on the moving channel wall. The reactive transport is assumed to have dominant Péclet and Damköhler numbers, a phenomenon that often results in non‐physical negative solutions. The scheme presented here is proven to be mass conservative in time and positive at all times for a small enough Δt. Reactive transport examples are simulated using this scheme for its validation, to show its convergence, and to compare it against the linear ALE‐FCT scheme. The nonlinear ALE‐FCT is shown to perform better than the linear ALE‐FCT schemes for large time steps. Copyright © 2014 John Wiley & Sons, Ltd.     

Marangoni instabilities in small circular containers under microgravity

Circular containers of various aspect ratios a with flat free upper liquid surfaces were heated from below under microgravity to generate the Marangoni instability (MI). We realized “liquid lateral sidewalls” for the containers to come near to the “slippery sidewalls” introduced by Rosenblat et al. (J Fluid Mech 120:91–122, 1982a) and Echebarría et al. (Physica D 99:487–502, 1997), henceforth referred to as RHD and EKP, respectively. The flow structure was visualized by aluminium flakes and recorded on videotape. The MI was clearly observed in all containers above a critical Marangoni number Ma ^c which depends on a. In the first microgravity experiment in a container with a=7.5, we found significant convective heat transport and reported a Nusselt number Nu=1.8 for Ma=4×Ma ^c . In a second microgravity experiment with containers with a=0.5, 0.75, 1.0, 1.5, 2.0, 4.0 and 5.0, various flow structures (azimuthal and radial wave numbers) were observed, depending on a and Ma. The observed scenario compares qualitatively well with the stability curves calculated by RHD and EKP. Frequent switching between modes (2,1) and (1,1) was observed in the container with a=2 at supercritical Ma that is exactly the case for which this behaviour was predicted by EKP for reduced gravity.     

Transfert du trichloréthylène en milieu poreux à partir d'un panache de vapeursMass transfer of trichloroethylene in a porous medium from a vapor plume

A large scale experiment has been carried out on an experimental facility to study the mass transfer of trichloroethylene (TCE) in a partially saturated porous medium. 5 liters of TCE have been infiltrated in the vadose zone of the site. The mass transfer of TCE from the vapor plume in the unsaturated zone towards the top of groundwater was quantified based on an analytical and a numerical approach. The mass of the pollutant measured at the exit of the model is well represented by the two mathematical approaches. It is found that the transfer of TCE towards the groundwater from the vapor plume is weak, which corresponds to 285 g of TCE, approximately 4% of the initial mass. To cite this article: H. Benremita, G. Schäfer, C. R. Mecanique 331 (2003).     

Anomalous Reactive Transport in the Framework of the Theory of Chromatography

The anomalous reactive transport considered here is the migration of contaminants through strongly sorbing permeable media without significant retardation. It has been observed in the case of heavy metals, organic compounds, and radionuclides, and it has critical implications on the spreading of contaminant plumes and on the design of remediation strategies. Even in the absence of the well-known fast migration pathways, associated with fractures and colloids, anomalous reactive transport arises in numerical simulations of reactive flow. It is due to the presence of highly pH-dependent adsorption and the broadening of the concentration front by hydrodynamic dispersion. This leads to the emergence of an isolated pulse or wave of a contaminant traveling at the average flow velocity ahead of the retarded main contamination front. This wave is considered anomalous because it is not predicted by the classical theory of chromatography, unlike the retardation of the main contamination front. In this study, we use the theory of chromatography to study a simple pH-dependent surface complexation model to derive the mathematical framework for the anomalous transport. We analyze the particular case of strontium (Sr²⁺) transport and define the conditions under which the anomalous transport arises. We model incompressible one-dimensional (1D) flow through a reactive porous medium for a fluid containing four aqueous species: H⁺, Sr²⁺, Na⁺, and Cl⁻. The mathematical problem reduces to a strictly hyperbolic 2 × 2 system of conservation laws for effective anions and Sr²⁺, coupled through a competitive Langmuir isotherm. One characteristic field is linearly degenerate while the other is not genuinely nonlinear due to an inflection point in the pH-dependent isotherm. We present the complete set of analytical solutions to the Riemann problem, consisting of only three combinations of a slow wave comprising either a rarefaction, a shock, or a shock–rarefaction with fast wave comprising only a contact discontinuity. Highly resolved numerical solutions at large Péclet numbers show excellent agreement with the analytic solutions in the hyperbolic limit. In the Riemann problem, the anomalous wave forms only if: hydrodynamic dispersion is present, the slow wave crosses the inflection locus, and the effective anion concentration increases along the fast path.     

An algebraic factorisation scheme for spectral element solution of incompressible flow and scalar transport

A spectral element algorithm for solution of the unsteady incompressible Navier–Stokes and scalar (species/heat) transport equations is developed using the algebraic factorisation scheme. The new algorithm utilises Nth order Gauss–Lobatto–Legendre points for velocity and the scalar, while (N-2)th order Gauss–Legendre points are used for pressure. As a result, the algorithm does not require inter-element continuity for pressure and pressure boundary conditions on solid surfaces. Implementations of the algorithm are performed for conforming and non-conforming grids. The latter is accomplished using both the point-wise matching and integral projection methods, and applied for grids with both polynomial and geometric non-conformities. Code validation cases include the unsteady scalar convection equation, and Kovasznay flow in two- and three-dimensional domains. Using cases with analytical solutions, the algorithm is shown to achieve spectral accuracy in space and second-order accuracy in time. The results for the Boussinesq approximation for buoyancy-driven flows, and the species mixing in a continuous flow micro-mixer are also included as examples of applications that require long-time integration of the scalar transport equations.     

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c ^–1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.Some recent exceptions to the neglect of segmental stretch can be found in Marrucci and Grizzuti (1988), Pearson et al. (1991), Mead et al. (1992).     

k‐Version of finite element method in 2‐D polymer flows: Oldroyd‐B constitutive model

In this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process of two‐dimensional polymer flows utilizing Oldroyd‐B constitutive model. Alternate forms of the choices of dependent variables in the governing differential equations (GDEs) are considered and is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious over others. It is shown that: (a) since, the differential operator in the GDEs is non‐linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non‐symmetric and hence may have partial or completely complex basis and thus the resulting computational processes may be spurious. (b) Since the VC of the VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches always have possibility of spurious solutions. (c) Least squares process utilizing GDEs in u, v, p, τ (strong form of the GDEs) variables (as well as others) is variationally consistent. The coefficient matrices are always symmetric and positive definite and hence always have a real basis and thus naturally yield computational processes that are free of spurious solutions. (d) The theoretical solution of the GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solution are preserved, undoubtedly requires local approximations in higher order scalar product spaces . (e) LSP with local approximations in spaces provide an incomparable mathematical and computational framework in which it is possible to preserve desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of the Oldroyd‐B model or lack of it is examined for both model problems for increasing De. A mathematical idealization of the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when h_d→0. For both model problems shear rate is examined in the flow domain to establish validity of the Oldroyd‐B constitutive model. Copyright © 2006 John Wiley & Sons, Ltd.     

Determination of Chlorinated Solvent Sorption by Porous Material—Application to Trichloroethene Vapor on Cement Mortar

Experiments have been performed to investigate the sorption of trichloroethene (TCE) vapor by concrete material or, more specifically, the cement mortar component. Gas-flow experiments were conducted using columns packed with small pieces of cement mortar obtained from the grinding of typical concrete material. Transport and retardation of TCE at high vapor concentrations (500 mg L \(^{-1})\) was compared to that of a non-reactive gas tracer (Sulfur Hexafluoride, SF \(_{6})\) . The results show a large magnitude of retardation (retardation factor \(=\) 23) and sorption (sorption coefficient \(=\) 10.6 cm \(^{3}\)  g \(^{-1})\) for TCE, compared to negligible sorption for SF \(_{6}\) . This magnitude of sorption obtained with pollutant vapor is much bigger than the one obtained for aqueous-flow experiments conducted for water-saturated systems. The considerable sorption exhibited for TCE under vapor-flow conditions is attributed to some combination of accumulation at the air-water interface and vapor-phase adsorption, both of which are anticipated to be significant for this system given the large surface area associated with the cement mortar. Transport of both SF \(_{6}\) and TCE was simulated successfully with a two-region physical non-equilibrium model, consistent with the dual-medium structure of the crushed cement mortar. This work emphasizes the importance of taking into account sorption phenomena when modeling transport of volatile organic compounds through concrete material, especially in regard to assessing vapor intrusion.     

Lyapunov Functional Method for 1D Radiative and Reactive Viscous Gas Dynamics

We consider the compressible Navier–Stokes system for 1D-flows of a viscous heat-conducting gas, with the pressure law and a one-order kinetics to include radiative effects and reactive processes. The mass force and the ignition phenomenon are also taken into account. For large data and under general assumptions on the heat conductivity, we establish global-in-time bounds and exponential stabilization for solutions in L^q and H¹ norms. To this end, we construct new global Lyapunov functionals and show that they describe the dynamics of solutions for any t≧0. A short proof of the corresponding global existence is also included for completeness.