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Four sets of nonreactive solute transport experiments were conducted with micromodels. Each set consisted of three experiments with one variable, i.e., flow velocity, grain diameter, pore-aspect ratio, and flow-focusing heterogeneity. The data sets were offered to pore-scale modeling groups to test their numerical simulators. Each set consisted of two learning experiments, for which all results were made available, and one challenge experiment, for which only the experimental description and base input parameters were provided. The experimental results showed a nonlinear dependence of the transverse dispersion coefficient on the Peclet number, a negligible effect of the pore-aspect ratio on transverse mixing, and considerably enhanced mixing due to flow focusing. Five pore-scale models and one continuum-scale model were used to simulate the experiments. Of the pore-scale models, two used a pore-network (PN) method, two others are based on a lattice Boltzmann (LB) approach, and one used a computational fluid dynamics (CFD) technique. The learning experiments were used by the PN models to modify the standard perfect mixing approach in pore bodies into approaches to simulate the observed incomplete mixing. The LB and CFD models used the learning experiments to appropriately discretize the spatial grid representations. For the continuum modeling, the required dispersivity input values were estimated based on published nonlinear relations between transverse dispersion coefficients and Peclet number. Comparisons between experimental and numerical results for the four challenge experiments show that all pore-scale models were all able to satisfactorily simulate the experiments. The continuum model underestimated the required dispersivity values, resulting in reduced dispersion. The PN models were able to complete the simulations in a few minutes, whereas the direct models, which account for the micromodel geometry and underlying flow and transport physics, needed up to several days on supercomputers to resolve the more complex problems.

The water levels in wells are known to fluctuate in response to earth tides and changes in atmospheric pressure. These water level fluctuations can be analyzed to estimate transmissivity (T). A new method to estimate transmissivity, which assumes that the atmospheric pressure varies in a sinusoidal fashion, is presented. Data analysis for this simplified method involves using a set of type curves and estimating the ratio of the amplitudes of the well response over the atmospheric pressure. Type curves for this new method were generated based on a model for ground water flow between the well and aquifer developed by Cooper et al. (1965). Data analysis with this method confirmed these published results: (1) the amplitude ratio is a function of transmissivity, the well radius, and the frequency of the sinusoidal oscillation; and (2) the amplitude ratio is a weak function of storativity. Compared to other methods, the developed method involves simpler, more intuitive data analysis and allows shorter data sets to be analyzed. The effect of noise on estimating the amplitude ratio was evaluated and found to be more significant at lower T. For aquifers with low T, noise was shown to mask the water level fluctuations induced by atmospheric pressure changes. In addition, reducing the length of the data series did not affect the estimate of T, but the variance of the estimate was higher for the shorter series of noisy data.

We present a novel computational framework for diffusive-reactive systems that satisfies the non-negative constraint and maximum principles on general computational grids. The governing equations for the concentration of reactants and product are written in terms of tensorial diffusion-reaction equations. % We restrict our studies to fast irreversible bimolecular reactions. If one assumes that the reaction is diffusion-limited and all chemical species have the same diffusion coefficient, one can employ a linear transformation to rewrite the governing equations in terms of invariants, which are unaffected by the reaction. This results in two uncoupled tensorial diffusion equations in terms of these invariants, which are solved using a novel non-negative solver for tensorial diffusion-type equations. The concentrations of the reactants and the product are then calculated from invariants using algebraic manipulations. The novel aspect of the proposed computational framework is that it will always produce physically meaningful non-negative values for the concentrations of all chemical species. Several representative numerical examples are presented to illustrate the robustness, convergence, and the numerical performance of the proposed computational framework. We will also compare the proposed framework with other popular formulations. In particular, we will show that the Galerkin formulation (which is the standard single-field formulation) does not produce reliable solutions, and the reason can be attributed to the fact that the single-field formulation does not guarantee non-negative solutions. We will also show that the clipping procedure (which produces non-negative solutions but is considered as a variational crime) does not give accurate results when compared with the proposed computational framework.

Due to the considerable computational demands of modeling solute transport in heterogeneous porous media, there is a need for upscaled models that do not require explicit resolution of the small-scale heterogeneity. This study investigates the development of upscaled solute transport models using genetic programming (GP), a domain-independent modeling tool that searches the space of mathematical equations for one or more equations that describe a set of training data. An upscaling methodology is developed that facilitates both the GP search and the implementation of the resulting models. A case study is performed that demonstrates this methodology by developing vertically averaged equations of solute transport in perfectly stratified aquifers. The solute flux models developed for the case study were analyzed for parsimony and physical meaning, resulting in an upscaled model of the enhanced spreading of the solute plume, due to aquifer heterogeneity, as a process that changes from predominantly advective to Fickian. This case study not only demonstrates the use and efficacy of GP as a tool for developing upscaled solute transport models, but it also provides insight into how to approach more realistic multi-dimensional problems with this methodology.

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.

The aerobic biodegradation of quinoline, a two-ring nitrogen heterocycle, offers an outstanding example of when structured modeling including cosubstrates is required for a biofilm system. In this case, the cosubstrate is oxygen, which is used as a direct cosubstrate in oxygenase reactions and as a primary electron acceptor in respiration. Quinoline biodegradation is numerically simulated as occurring in five key steps, two of which involve oxygen as a direct cosubstrate. Modeling evaluation of experimental results from a laboratory-scale biofilm column shows that the oxygenation steps are much more sensitive to low oxygen concentrations than are steps in which oxygen only participates through respiration. The result of this differential oxygen sensitivity is that the fast intermediate product, 2-hydroxyquinoline, builds up, because its degradation through an oxygenase reaction is slowed preferentially by oxygen depletion.

In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions when applied to one-dimensional problems with constant coefficients and forcing function. We first present a variational principle for a multi-dimensional advective-diffusive system, and then derive a new stable weak formulation. When applied to one-dimensional problems with constant coefficients and forcing function, this resulting weak formulation will be equivalent to the optimal artificial diffusion method. We present representative numerical results to corroborate our theoretical findings.

Quinoline biodegradation in a biofilm reactor was numerically simulated by a structured model in which each step was controlled by the dissolved oxygen (DO) concentration. It was essential to identify and characterize each reaction step according to how DO was utilized and how each step consumed oxygen. Quinoline biodegradation involved 5 steps, 2 of which used oxygen as a direct cosubstrate. The hydroxylation rate of quinoline was less sensitive to low DO levels than the dioxygenase-catalysed degradation rates of the intermediate 2-hydroxyquinoline. This differential oxygen sensitivity was incorporated by using dual-limitation kinetics and separate oxygen half-maximal rate concentrations for dioxygenase-controlled reactions compared with respiration-controlled reactions. The former were around 2 and the latter 0.02 mg per litre. The oxygen steps were more sensitive to low DO than those steps in which oxygen was simply required for respiration. In practice, this meant an accumulation of 2-hydroxyquinoline because low DO slowed the oxygenase reaction. There are 32 references.