Olaf A. Cirpka

Numerical simulation of biodegradation controlled by transverse mixing

Abstract Microbial activity in aquifers is controlled by the mixing between the reacting substrates. Conventional modelling methods that are commonly used to analyze reactive transport of organics in heterogeneous systems may give erroneous results because mixing is often over-represented in the model. This effect will be strongest when the reaction is controlled by transverse dispersion as in the case of aerobic degradation of waste-water introduced into an aquifer by an injection well. We show that fictitious transverse mixing can be created by a numerical model based on rectangular grids, and that this problem can be controlled by formulating the problem in streamline-oriented coordinates. In both model formulations, nonlinear high-resolution techniques minimizing the amount of artificial diffusion were applied, so that fictitious mixing is exclusively due to grid-orientation effects. Additionally it is shown that applying dispersivity values based on the second spatial moment transverse to the direction of flow leads to an overestimation of mixing. The fictitious degradation produced by model-dependent transverse dispersion caused the modelled plume to degrade much faster, and therefore appear much shorter, than the actual plume. Thus, the choice of appropriate dispersivity values as well as the control of artificial transverse diffusion is crucial when modelling mixing-controlled reactive transport.

Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments

Breakthrough curves of a conservative tracer in a heterogeneous two-dimensional aquifer are analyzed by means of their temporal moments. The average velocity and the longitudinal macrodispersion coefficient of the equivalent one-dimensional aquifer obtained through cross-sectional averaging of concentration can be defined from the first and second central moments of a breakthrough curve integrated over the outflow boundary of the domain. On the basis of an integrated breakthrough curve, one cannot distinguish between actual solute dilution, which involves concentration reduction, and variability of arrival times among parts of the plume at different cross-sectional positions. Analyzing the temporal moments of breakthrough curves at a “point” within the domain gives additional information about the dilution of the tracer. From these local first and second central moments an apparent seepage velocity υa and an apparent dispersivity of mixing αa can be derived. For short travel distances, αa equals the local-scale longitudinal dispersivity. It increases with the travel distance but much more slowly than the macrodispersivity. At the large-distance limit, αa may eventually reach the level of macrodispersivity. Lenses of high conductivity where groundwater flow converges are identified as regions of preferential enhanced mixing. The spatial distribution of these regions causes a high degree of variability of αa within a domain, indicating a high degree of uncertainty in the quantification of dilution at early stages. In an accompanying paper [Cirpka and Kitanidis, this issue] the results of conservative tracer transport are utilized for the study of mixing-controlled reactive transport.

Enhancement of dilution and transverse reactive mixing in porous media: experiments and model-based interpretation.

Transport and natural attenuation of contaminant plumes in groundwater are often controlled by transverse dispersion. The extent of mixing between dissolved reaction partners at the fringe of a plume determines its length and depends strongly on the groundwater flow field. Transient flow conditions as well as the focusing of the flow in high-permeability zones may enhance transverse mixing of dissolved species and, therefore, create favorable conditions for the natural attenuation of contaminant plumes. The aim of the present study is to experimentally test the influence of these processes on solute mixing and to directly compare the results with those under analogous homogeneous and steady-state conditions. We have performed conservative and reactive tracer experiments in a quasi two-dimensional tank filled with glass beads of different sizes. The experiments have been carried out in both homogeneous and heterogeneous porous media under steady-state and transient (i.e. oscillating) flow fields. We used fluorescein as conservative tracer; whereas an alkaline solution (NaOH) was injected into ambient acidic water (HCl) in the reactive experiments. A pH indicator was added to the reacting solutions in order to visualize the emerging plume. We simulated the laboratory experiments with a numerical model and compared the outcomes of the model with the measured concentrations at the outlet of the tank and with the observed tracer plumes. Spatial moments, a newly defined flux-related dilution index, the product mass fluxes and the reaction enhancement factors were calculated to quantify the differences in mixing and reaction extent under various experimental conditions. The results show that flow focusing in heterogeneous porous media significantly enhances transverse mixing and mixing-controlled reactions, whereas temporally changing flow fields appear to be of minor importance.

An advective‐dispersive stream tube approach for the transfer of conservative‐tracer data to reactive transport

Conservative-tracer data are used for the parameterization of mixing-controlled reactive transport. Temporal moments of the tracer breakthrough curve integrated over the outflow boundary of the domain yield the average velocity and the path-averaged macrodispersion coefficient. On the basis of this information alone, no distinction is possible between spreading and mixing of the tracer. Analyzing the temporal moments of breakthrough curves locally obtained at single points in the domain gives additional information about the dilution of the tracer. In an accompanying paper (Cirpka and Kitanidis, this issue) we derive an apparent Peclet number of mixing Pea from local temporal moments. Assuming that Pea is constant over the cross section of the outflow boundary, a corrected probability density function of arrival times is determined. By interpreting the spatially integrated breakthrough curve as the result of advective- dispersive transport in independent stream tubes with identical Peclet number but differing seepage velocity, it is possible to transfer results of conservative transport to the transport of interacting compounds for cases in which mixing of the compounds is influenced significantly by local-scale dispersion. This is an improvement to the stochastic- convective model of Simmons et al. (1995) for the transfer of integrated tracer data to reactive transport. The approach is applied to the hypothetical case of a bimolecular reaction in a heterogeneous two-dimensional aquifer.

Sensitivity of temporal moments calculated by the adjoint-state method and joint inversing of head and tracer data

Abstract Including tracer data into geostatistically based methods of inverse modeling is computationally very costly when all concentration measurements are used and the sensitivities of many observations are calculated by the direct differentiation approach. Harvey and Gorelick (Water Resour Res 1995;31(7):1615–26) have suggested the use of the first temporal moment instead of the complete concentration record at a point. We derive a computationally efficient adjoint-state method for the sensitivities of the temporal moments that require the solution of the steady-state flow equation and two steady-state transport equations for the forward problem and the same number of equations for each first-moment measurement. The efficiency of the method makes it feasible to evaluate the sensitivity matrix many times in large domains. We incorporate our approach for the calculation of sensitivities in the quasi-linear geostatistical method of inversing (“iterative cokriging”). The application to an artificial example of a tracer introduced into an injection well shows good convergence behavior when both head and first-moment data are used for inversing, whereas inversing of arrival times alone is less stable.

Evidence of Compound-Dependent Hydrodynamic and Mechanical Transverse Dispersion by Multitracer Laboratory Experiments

Mass transfer, mixing, and therefore reaction rates during transport of solutes in porous media strongly depend on dispersion and diffusion. In particular, transverse mixing is a significant mechanism controlling natural attenuation of contaminant plumes in groundwater. The aim of the present study is to gain a deeper understanding of vertical transverse dispersive mixing of reaction partners in saturated porous media. Multitracer laboratory experiments in a quasi two-dimensional tank filled with glass beads were conducted and transverse dispersion coefficients were determined from high-resolution vertical concentration profiles. We investigated the behavior of conservative tracers (i.e., fluorescein, dissolved oxygen, and bromide), with different aqueous diffusion coefficients, in a range of grain-related Peclet numbers between 1 and 562. The experimental results do not agree with the classical linear parametric model of hydrodynamic dispersion, in which the transverse component is approximated as the sum of pore diffusion and a compound-independent mechanical dispersion term. The outcome of the multitracer experiments clearly indicates a nonlinear relation between the dispersion coefficient and the average linear velocity. More importantly, we show that transverse mechanical dispersion depends on the diffusion coefficient of the compound, at least at the experimental bench-scale. This result has to be considered in reactive-transport models, because the typical assumption that two reactants with different aqueous diffusive properties are characterized by the same dispersive behavior does not hold anymore.

Efficient Computation of Linearized Cross-Covariance and Auto-Covariance Matrices of Interdependent Quantities

In many geostatistical applications, spatially discretized unknowns are conditioned on observations that depend on the unknowns in a form that can be linearized. Conditioning takes several matrix–matrix multiplications to compute the cross-covariance matrix of the unknowns and the observations and the auto-covariance matrix of the observations. For large numbers n of discrete values of the unknown, the storage and computational costs for evaluating these matrices, proportional to n2, become strictly inhibiting. In this paper, we summarize and extend a collection of highly efficient spectral methods to compute these matrices, based on circulant embedding and the fast Fourier transform (FFT). These methods are applicable whenever the unknowns are a stationary random variable discretized on a regular equispaced grid, imposing an exploitable structure onto the auto-covariance matrix of the unknowns. Computational costs are reduced from O(n2) to O(nlog2n) and storage requirements are reduced from O(n2) to O(n).

Three-Dimensional Geostatistical Inversion of Flowmeter and Pumping Test Data

We jointly invert field data of flowmeter and multiple pumping tests in fully screened wells to estimate hydraulic conductivity using a geostatistical method. We use the steady-state drawdowns of pumping tests and the discharge profiles of flowmeter tests as our data in the inference. The discharge profiles need not be converted to absolute hydraulic conductivities. Consequently, we do not need measurements of depth-averaged hydraulic conductivity at well locations. The flowmeter profiles contain information about relative vertical distributions of hydraulic conductivity, while drawdown measurements of pumping tests provide information about horizontal fluctuation of the depth-averaged hydraulic conductivity. We apply the method to data obtained at the Krauthausen test site of the Forschungszentrum Jülich, Germany. The resulting estimate of our joint three-dimensional (3D) geostatistical inversion shows an improved 3D structure in comparison to the inversion of pumping test data only.

Streamline-oriented grid generation for transport modelling in two-dimensional domains including wells

Flownets are useful tools for the visualization of groundwater flow fields. Using orthogonal flownets as grids for transport modeling is an effective way to control numerical dispersion, especially transverse to the direction of flow. Therefore tools for automatic generation of flownets may be seen both as postprocessors for groundwater flow simulations and preprocessors for contaminant transport models. Existing methods to generate streamline-oriented grids suffer from drawbacks such as the inability to include sources in the interior of the grid. In this paper, we introduce a new method for the generation of streamline-oriented grids which handles wells in the grid interior, and which produces orthogonal grids for anisotropic systems. Streamlines are generated from an accurate velocity field obtained from the solution of the mixed-hybrid finite element method for flow, while pseudopotentials, which are orthogonal to the streamlines, are obtained by a standard finite element solution of the pseudopotential equation. A comprehensive methodology for the generation of orthogonal grids, including the location of stagnation points and dividing streamlines, is introduced. The effectiveness of the method is illustrated by means of examples. A related paper presents a compatible formulation of the solution for reactive transport, while a second related paper gives a detailed quantitative assessment of the various forms of modelled mixing and their effect on the accuracy of simulations of the biodegradation of groundwater contaminants.

Concentration statistics for mixing-controlled reactive transport in random heterogeneous media

Uncertainty in the distribution of hydraulic parameters leads to uncertainty in flow and reactive transport. Traditional stochastic analysis of solute transport in heterogeneous media has focused on the ensemble mean of conservative-tracer concentration. Studies in the past years have shown that the mean concentration often is associated with a high variance. Because the range of possible concentration values is bounded, a high variance implies high probability weights on the extreme values. In certain cases of mixing-controlled reactive transport, concentrations of conservative tracers, denoted mixing ratios, can be mapped to those of constituents that react with each other upon mixing. This facilitates mapping entire statistical distributions from mixing ratios to reactive-constituent concentrations. In perturbative approximations, only the mean and variance of the mixing-ratio distribution are used. We demonstrate that the second-order perturbative approximation leads to erroneous or even physically impossible estimates of mean reactive-constituent concentrations when the variance of the mixing ratio is high and the relationship between the mixing ratio and the reactive-constituent concentrations strongly deviates from a quadratic function. The latter might be the case in biokinetic reactions or in equilibrium reactions with small equilibrium constant in comparison to the range of reactive-constituent concentrations. When only the mean and variance of the mixing ratio is known, we recommend assuming a distribution that meets the known bounds of the mixing ratio, such as the beta distribution, and mapping the assumed distribution of the mixing ratio to the distributions of the reactive constituents.