Nonlinear Reactive Transport Research Articles

On the Efficiency of the Direct Substitution Approach for Reactive Transport Problems in Porous Media

Nonlinear reactive transport problems can be solved using the Operator Splitting (OS) approach, where transport and reaction processes are separated or the Direct Substitution Approach (DSA) where chemical and transport equations are solved simultaneously. The OS techniques can be very attractive, but are known to introduce splitting errors with SNIA (Non Iterative OS) and have low convergence rate with SIA (Iterative OS). These problems are avoided with DSA which is more robust than OS schemes. On the other hand, DSA is more complicated and very demanding in terms of computing time and memory requirements. This can make DSA less efficient than OS schemes especially for fine discretizations and chemically simple problems. In this work, DSA, SIA and SNIA are combined with a new sparse direct (unifrontal/multifrontal) solver. The efficiency of this solver is not dependent on the matrix conditioning. The performance of the three approaches is studied for two transport problems with simple and difficult chemical reactions and for different number of unknowns. Results show that when combined with an efficient sparse direct solver, DSA is more efficient than SIA and SNIA even for chemically simple problems and large number of unknowns.

The influence of precipitate formation on the chemical oxidation of TCE DNAPL with potassium permanganate

A three-dimensional two-phase flow model is coupled to a non-linear reactive transport model to study the efficacy of potassium permanganate treatment on dense, non-aqueous phase liquid (DNAPL) source removal in porous media. A linear relationship between the soil permeability ( k) and concentration of manganese dioxide precipitate ([MnO 2(s)]), k = k o + S rind [MnO 2(s)], is utilized to simulate nodal permeability reductions due to precipitate formation. Using published experimental column studies, an S rind = −5.5 × 10 −16 m 2 L/mg was determined for trichloroethylene (TCE) DNAPL. This S rind was then applied to treatment simulations on three-dimensional TCE DNAPL source zones comprising either DNAPL at residual saturations, or DNAPL at pooled saturations. DNAPL dissolution without oxidation treatment, simulated using equilibrium and the Nambi and Powers [Nambi I, Powers S. Mass transfer correlations for non-aqueous phase liquid dissolution from regions with high initial saturations. Water Resour Res 2003;39(2):1–11, SBH 4] mass transfer expression, required 31 and 36 years, respectively, to eliminate the residual source zone. For equilibrium dissolution with continuous treatment and no precipitate influence ( S rind = 0 m 2 L/mg), the residual source zone was removed after 11 years. However, when considering the precipitate influence (i.e., S rind = −5.5 × 10 −16 m 2 L/mg), 21 years of treatment were necessary to remove the DNAPL. When considering pulse treatments of 1 and 2 years duration followed by only dissolution, approximately 36 and 38 years, respectively, were required before the source zone was depleted, suggesting that there is no benefit to pulse treatment. Similar trends were observed when allowing 10 years of dissolution prior to treatment initiation. The treatment behaviour of the pooled TCE source, while slightly more efficient than the residual saturation source, was similar. Based on simulation findings, the precipitate (rind) formation significantly influences DNAPL treatment with permanganate; the most significant reductions in efficacy were observed for single pulse treatments (of 1 and 2 years), which exhibited times to source depletion similar to the case of dissolution without treatment.

Stochastic finite element method for analysis of transport of nonlinearly sorbing solutes in three‐dimensional heterogeneous porous media

The probabilistic analysis by Monte Carlo simulation method (MCSM) for the transport of nonlinear reactive solutes in a three‐dimensional heterogeneous porous medium is a computationally prohibitive task. For linear transport problems, the perturbation‐based stochastic finite element method (SFEM) has been found to be computationally efficient with acceptable accuracy. This provides a motivation to develop the SFEM for the nonlinear reactive solute transport. In the present study, SFEM is developed for the transport of equilibrium nonlinear sorbing solutes, which follow the Langmuir‐Freundlich isotherm. This method produces a second‐order accurate mean and a first‐order accurate standard deviation of concentration. In this study, the governing medium propertis viz. hydraulic conductivity, dispersivity, molecular diffusion, porosity, sorption, and decay coefficients are considered to vary randomly in space. The performance of SFEM is compared to MCSM for both one‐ and three‐dimensional transport problems. The mean and the standard deviation of concentration for various test cases obtained with the SFEM compares well for the mild heterogeneity cases (standard deviation of log hydraulic conductivity less than 0.85) tested. SFEM produces a sharp front for the mean and the standard deviation of concentration while fronts obtained by MCSM are found to be dispersive. The error associated with the results obtained by SFEM is sensitive to the boundary conditions, the size of the domain, and the plume size. For a higher nonlinearity of sorption isotherm, the prediction uncertainty is higher. The pattern of the statistical moments of concentration is similar for cases with different correlation lengths of the parameters.

An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

Highly nonlinear advection–dispersion-reaction equations govern numerous transport phenomena. Robust, accurate, and efficient algorithms to solve these equations hold the key to the success of applying numerical models to field problems. This paper presents the development and verification of a computational algorithm to approximate the highly nonlinear transport equations of reactive chemical transport and multiphase flow. The algorithm was developed based on the Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture (LEZOOMPC) scheme. It consists of both backward and forward node tracking, rough element determination, peak/valley capturing, and adaptive local grid refinement. A second-order tracking was implemented to accurately and efficiently track all fictitious particles. Shanks’ method was introduced to deal with slowly converging case. The accuracy and efficiency of the algorithm were verified with the Burger equation for a variety of cases. The robustness of the algorithm to achieve convergent solutions was demonstrated by highly nonlinear reactive contaminant transport and multiphase flow problems.

Analysis of split operator methods applied to reactive transport with Monod kinetics

Split operator approaches are commonly used to solve coupled nonlinear reactive transport problems. The benefits of split operator algorithms come with the cost of an O ( Δ t ) truncation error. This error has two components: (i) an error strictly associated with the boundary, and (ii) an error occurring within the domain. Previous analyses of split operator methods for Monod reactions are incomplete and conflicting. First, previous analyses have focused only on the boundary error while ignoring the internal error. This is problematic as the boundary error is irrelevant in many practical applications. Second, conflicting results regarding the accuracy of alternating split operator methods have been reported. This new work exactly characterizes the net splitting error for an arbitrary system of nonequilibrium reactive transport equations showing that both the boundary and internal error can always be removed with alternating methods for sufficiently small Δ t. Numerical results are presented for reactive transport governed by single and multiple species with Monod kinetics.

A linearization technique for multi-species transport problems

We study a one-dimensional multi-species system of dispersive-advective contaminant transport equations coupled by nonlinear biological (kinetic reactions) and physical (adsorption) processes. To deal with the nonlinearities and the coupling, and to avoid additional computational costs, we propose a linearization technique based on first-order Taylor’s series expansions. A stabilized finite element in space, combined with an Euler implicit finite difference discretization in time, is used to approximate the dispersive-advective transport problem. Three computational tests are performed with different boundary conditions, retardation factors and kinetic parameters for a nonlinear reactive multi-species transport model. The proposed methodology is shown to be accurate and decrease computational costs in the numerical implementation of nonlinear reactive transport problems.

Monte Carlo simulation of nonlinear reactive contaminant transport in unsaturated porous media

In the current proposed solutions of radioactive waste repositories, the protective function against the radionuclide water-driven transport back to the biosphere is to be provided by an integrated system of engineered and natural geologic barriers. The occurrence of several nonlinear interactions during the radionuclide migration process may render burdensome the classical analytical–numerical approaches. Moreover, the heterogeneity of the barriers’ media forces approximations to the classical analytical–numerical models, thus reducing their fidelity to reality. In an attempt to overcome these difficulties, in the present paper we adopt a Monte Carlo simulation approach, previously developed on the basis of the Kolmogorov–Dmitriev theory of branching stochastic processes. The approach is here extended for describing transport through unsaturated porous media under transient flow conditions and in presence of nonlinear interchange phenomena between the liquid and solid phases. This generalization entails the determination of the functional dependence of the parameters of the proposed transport model from the water content and from the contaminant concentration, which change in space and time during the water infiltration process. The corresponding Monte Carlo simulation approach is verified with respect to a case of nonreactive transport under transient unsaturated flow and to a case of nonlinear reactive transport under stationary saturated flow. Numerical applications regarding linear and nonlinear reactive transport under transient unsaturated flow are reported.

Analysis of biodegradation in a 3-D heterogeneous porous medium using nonlinear stochastic finite element method

Probabilistic analysis by Monte Carlo Simulation method (MCSM) is a computationally prohibitive task for a reactive solute transport involving coupled PDEs with nonlinear source/sink terms in 3-D heterogeneous porous media. The perturbation based stochastic finite element method (SFEM) is an attractive alternative method to MCSM as it is computationally efficient and accurate. In the present study SFEM is developed for solving nonlinear reactive solute transport problem in a 3-D heterogeneous medium. Here the solution of the biodegradation problem involving a single solute by a single class of microorganisms coupled with dynamic microbial growth is attempted using this method. The SFEM here produces a second-order accurate solution for the mean and a first-order accurate solution for the standard deviation of concentrations. In this study both the physical parameters (hydraulic conductivity, porosity, dispersivity and diffusion coefficient) and the biological parameters (maximum substrate utilization rate and the coefficient of cell decay) are considered as spatially varying random fields. A comparison between the MCSM and SFEM for the mean and standard deviation of concentration is made for 1-D and 3-D problem. The effects of heterogeneity on the degradation of substrate and growth of biomass concentrations for a range of variances of input parameters are discussed for both 1-D and 3-D problems.

Advantages of using adaptive remeshing and parallel processing for modelling biodegradation in groundwater

Biodegrading plumes in groundwater are often typified by relatively reactive zones around the fringes and less reactive zones in the core. A high degree of refinement is required at the fringes if a model is to be of use in improving the conceptual understanding of plumes. Two strategies for dealing with the potentially high computational demands are (i) parallel processing, where the workload is shared between multiple processors, and (ii) locally adaptive remeshing, where a refined area of the grid tracks the moving plume fringes through the domain. The partial differential equation toolbox, UG (Unstructured Grids) offers advanced numerical tools including adaptive remeshing, sparse matrix storage schemes, and multigrid solvers. It embraces many of these features within a parallel processing environment. This paper reports on a recent development of UG to simulate field scale reactive biogeochemistry including Monod kinetics, NAPL dissolution, mineral precipitation/dissolution and ion exchange. The non-linear multicomponent reactive transport system is solved with the fully coupled method. Test cases have been used for verification of the new capability. The paper illustrates an application to a 3D field site. It is demonstrated that both adaptive remeshing and parallel processing can improve efficiency and in turn facilitate the incorporation of a more complex set of species and reactions such that understanding of plume processes is enhanced.

Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems

Numerical solutions to nonlinear reactive solute transport problems (NRTPs) are often computed using split-operator (SO) approaches, which separate the transport and reaction processes. This uncoupling introduces an additional source of numerical error, known as the splitting error. The iterative split-operator (ISO) algorithm removes the splitting error through iteration. Although the ISO algorithm is often used, there has been very little analysis of its convergence behavior. This work uses theoretical analysis and numerical experiments to investigate the convergence rate of the ISO approach for solving NRTPs. We show that under certain assumptions regarding smoothness, the convergence rate of the ISO algorithm applied NRTPs is O(Δ t 2). We demonstrate that the theoretical convergence rate can be achieved in practice if the numerical solution of the transport and reaction steps are carried out with sufficient accuracy. We also show that accurate estimation of the lagged operator in each step is crucial to obtaining the theoretical convergence rate.

Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems

Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems

Optimised Muskingum-Cunge solution method for solute transport with equilibrium Freundlich reactions

Because of its simplicity and efficiency, the Muskingum-Cunge (M-C) method is a widely used modelling tool applied to flood routing problems. However, in the M-C method, there is an infinite number of combinations of spatial and temporal step sizes for which the solution is feasible and not all of these lead to accurate results. This has been a major criticism of the method. The same approach has been adopted here to solve the problem of nonlinear reactive solute transport. An optimised M-C scheme is proposed and tested which is very simple and efficient as compared with existing standard numerical solutions such as the Crank-Nicolson scheme. The optimisation principles are: (1) that the column holdup (area above the breakthrough curve normalised by the column length) is maintained; and (2) that the truncation error produced by the numerical scheme is minimised. The conditions derived are shown to be satisfactory for the mildly nonlinear reactions of the Freundlich-type typical of reactive chemical transport in soil.

Modeling of multicomponent transport with microbial transformation in groundwater: The Fuhrberg Case

This paper serves the twofold purpose of demonstrating a numerical technique that appears to have potential in studies of nonlinear reactive transport problems and of presenting a case study where the implications of reactive transport processes are profound. The case study relates to the Fuhrberger Feld aquifer in northern Germany, which provides the bulk of the water supply for the half‐million city of Hannover. Part of the aquifer receives a strong influx of nitrate from agricultural activities. The nitrate in the aquifer is microbially transformed into sulfate, the transformation depending on the availability of reduced sulfur compounds present in the sediment, and the sulfate is also transformed further. The individual transformations occur within distinct reaction zones in the aquifer. The relevant dynamic processes can be formulated as a nonlinear transport problem involving multiple interacting species in dissolved or solid form. The system is simulated using a finite element‐based technique that generates a symmetric coefficient matrix for the transport equation, while providing second‐order accuracy in time. The form of the matrix allows the use of a highly efficient and robust symmetric conjugate gradient solver. The technique provides an ample spatial resolution capacity at reasonable cost and handles grids with irregular geometry in two or three dimensions. The convenience and versatility of conventional finite elements is retained. A vertical‐section simulation identifies different reaction zones in the aquifer and gives insight into the effect of the controlling parameters. The simulation study is a first step in the development of a prognosis for the fate of the groundwater resource.