Daniel Fernàndez-Garcia

Optimal reconstruction of concentrations, gradients and reaction rates from particle distributions

Random walk particle tracking methodologies to simulate solute transport of conservative species constitute an attractive alternative for their computational efficiency and absence of numerical dispersion. Yet, problems stemming from the reconstruction of concentrations from particle distributions have typically prevented its use in reactive transport problems. The numerical problem mainly arises from the need to first reconstruct the concentrations of species/components from a discrete number of particles, which is an error prone process, and then computing a spatial functional of the concentrations and/or its derivatives (either spatial or temporal). Errors are then propagated, so that common strategies to reconstruct this functional require an unfeasible amount of particles when dealing with nonlinear reactive transport problems. In this context, this article presents a methodology to directly reconstruct this functional based on kernel density estimators. The methodology mitigates the error propagation in the evaluation of the functional by avoiding the prior estimation of the actual concentrations of species. The multivariate kernel associated with the corresponding functional depends on the size of the support volume, which defines the area over which a given particle can influence the functional. The shape of the kernel functions and the size of the support volume determines the degree of smoothing, which is optimized to obtain the best unbiased predictor of the functional using an iterative plug-in support volume selector. We applied the methodology to directly reconstruct the reaction rates of a precipitation/dissolution problem involving the mixing of two different waters carrying two aqueous species in chemical equilibrium and moving through a randomly heterogeneous porous medium.

Toward efficiency in heterogeneous multispecies reactive transport modeling: A particle‐tracking solution for first‐order network reactions

Modeling multispecies reactive transport in natural systems with strong heterogeneities and complex biochemical reactions is a major challenge for assessing groundwater polluted sites with organic and inorganic contaminants. A large variety of these contaminants react according to serial-parallel reaction networks commonly simplified by a combination of first-order kinetic reactions. In this context, a random-walk particle tracking method is presented. This method is capable of efficiently simulating the motion of particles affected by first-order network reactions in three-dimensional systems, which are represented by spatially variable physical and biochemical coefficients described at high resolution. The approach is based on the development of transition probabilities that describe the likelihood that particles belonging to a given species and location at a given time will be transformed into and moved to another species and location afterward. These probabilities are derived from the solution matrix of the spatial moments governing equations. The method is fully coupled with reactions, free of numerical dispersion and overcomes the inherent numerical problems stemming from the incorporation of heterogeneities to reactive transport codes. In doing this, we demonstrate that the motion of particles follows a standard random walk with time-dependent effective retardation and dispersion parameters that depend on the initial and final chemical state of the particle. The behavior of effective parameters develops as a result of differential retardation effects among species. Moreover, explicit analytic solutions of the transition probability matrix and related particle motions are provided for serial reactions. An example of the effect of heterogeneity on the dechlorination of organic solvents in a threedimensional random porous media shows that the power-law behavior typically observed in conservative tracers breakthrough curves can be largely compromised by the effect of biochemical reactions.

Apparent directional mass-transfer capacity coefficients in three-dimensional anisotropic heterogeneous aquifers under radial convergent transport

Aquifer hydraulic properties such as hydraulic conductivity (K) are ubiquitously heterogeneous and typically only a statistical characterization can be sought. Additionally, statistical anisotropy at typical characterization scales is the rule. Thus, regardless of the processes governing solute transport at the local (pore) scale, transport becomes non-Fickian. Mass-transfer models provide an efficient tool that reproduces observed anomalous transport; in some cases though, these models lack predictability as model parameters cannot readily be connected to the physical properties of aquifers. In this study, we focus on a multirate mass-transfer model (MRMT), and in particular the apparent capacity coefficient (β), which is a strong indicator of the potential of immobile zones to capture moving solute. We aim to find if the choice of an apparent β can be phenomenologically related to measures of statistical anisotropy. We analyzed an ensemble of random simulations of three-dimensional log-transformed multi-Gaussian permeability fields with stationary anisotropic correlation under convergent flow conditions. It was found that apparent β also displays an anisotropic behavior, physically controlled by the aquifer directional connectivity, which in turn is controlled by the anisotropic correlation model. A high hydraulic connectivity results in large β values. These results provide new insights into the practical use of mass-transfer models for predictive purposes.

Debates—Stochastic subsurface hydrology from theory to practice: Why stochastic modeling has not yet permeated into practitioners?

This is the peer reviewed version of the following article: [Sanchez-Vila, X., and D. Fernandez-Garcia (2016), Debates—Stochastic subsurface hydrology from theory to practice: Why stochastic modeling has not yet permeated into practitioners?, Water Resour. Res., 52, 9246–9258, doi:10.1002/2016WR019302], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/2016WR019302/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-ArchivingWe address modern topics of Stochastic Hydrogeology from their potential relevance to real modeling efforts at the field scale. While the topics of stochastic hydrogeology and numerical modelling have become routine in hydrogeological studies, non-deterministic models have not yet permeated into practitioners. We point out a number of limitations of stochastic modelling when applied to real applications and comment on the reasons why stochastic models fail to become an attractive alternative for practitioners. We specifically separate issues corresponding to flow, conservative transport and reactive transport. The different topics addressed are: emphasis on process modeling, need for upscaling parameters and governing equations, relevance of properly accounting for detailed geological architecture in hydrogeological modeling, and specific challenges of reactive transport. We end up by concluding that the main responsible for non-deterministic models having not yet permeated in industry can be fully attributed to researchers in stochastic hydrogeology. This article is protected by copyright. All rights reserved.

Probabilistic human health risk assessment of degradation‐related chemical mixtures in heterogeneous aquifers: Risk statistics, hot spots, and preferential channels

An edited version of this paper was published by AGU. Copyright (2015) American Geophysical Union.

Visualization of Mixing Processes in a Heterogeneous Sand Box Aquifer

Mixing is increasingly recognized as a critical process for understanding and modeling reactive transport. Yet, mixing is hard to characterize because it depends nonlinearly on concentrations. Visualization of optical tracers in the laboratory at high spatial and temporal resolution can help advance the study of mixing processes. The solute distribution is obtained by analyzing the relationship between pixel intensity and tracer concentration. The problem with such techniques is that grain borders, light fluctuations, and nonuniform brightness contribute to produce noisy images of concentrations that cannot be directly used to estimate mixing at the local scale. We present a nonparametric regression methodology to visualize local values of mixing from noisy images of optical tracers that minimizes smoothing in the direction of concentration gradients. This is achieved by weighting pixel data along concentration isolines. The methodology is used to provide a full visualization of mixing dynamics in a tracer experiment performed in a reconstructed aquifer consisting of two materials with contrasting hydraulic properties. The experiment reveals that mixing is largest at the contact area of regions with different permeability. Also, the temporal evolutions of mixing and dilution rates are significantly different. The mixing rate is more persistent than the dilution rate during tracer invasion, and the opposite is true during flushing, which helps in understanding the complementary nature of these two measures.

Do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods? A kernel density estimation approach

Random walk particle tracking methods are a computationally efficient family of methods to solve reactive transport problems. While the number of particles in most realistic applications is in the order of 10 6 - 10 9 , the number of reactive molecules even in diluted systems might be in the order of fractions of the Avogadro number. Thus, each particle actually represents a group of potentially reactive molecules. The use of a low number of particles may result not only in loss of accuracy, but also may lead to an improper reproduction of the mixing process, limited by diffusion. Recent works have used this effect as a proxy to model incomplete mixing in porous media. In this work, we propose using a Kernel Density Estimation (KDE) of the concentrations that allows getting the expected results for a well-mixed solution with a limited number of particles. The idea consists of treating each particle as a sample drawn from the pool of molecules that it represents; this way, the actual location of a tracked particle is seen as a sample drawn from the density function of the location of molecules represented by that given particle, rigorously represented by a kernel density function. The probability of reaction can be obtained by combining the kernels associated to two potentially reactive particles. We demonstrate that the observed deviation in the reaction vs time curves in numerical experiments reported in the literature could be attributed to the statistical method used to reconstruct concentrations (fixed particle support) from discrete particle distributions, and not to the occurrence of true incomplete mixing. We further explore the evolution of the kernel size with time, linking it to the diffusion process. Our results show that KDEs are powerful tools to improve computational efficiency and robustness in reactive transport simulations, and indicates that incomplete mixing in diluted systems should be modeled based on alternative mechanistic models and not on a limited number of particles.

The effects of sediment depth and oxygen concentration on the use of organic matter: An experimental study using an infiltration sediment tank

Water flowing through hyporheic river sediments or artificial recharge facilities promotes the development of microbial communities with sediment depth. We performed an 83-day mesocosm infiltration experiment, to study how microbial functions (e.g., extracellular enzyme activities and carbon substrate utilization) are affected by sediment depth (up to 50 cm) and different oxygen concentrations. Results indicated that surface sediment layers were mainly colonized by microorganisms capable of using a wide range of substrates (although they preferred to degrade carbon polymeric compounds, as indicated by the higher β-glucosidase activity). In contrast, at a depth of 50 cm, the microbial community became specialized in using fewer carbon substrates, showing decreased functional richness and diversity. At this depth, microorganisms picked nitrogenous compounds, including amino acids and carboxyl acids. After the 83-day experiment, the sediment at the bottom of the tank became anoxic, inhibiting phosphatase activity. Coexistence of aerobic and anaerobic communities, promoted by greater physicochemical heterogeneity, was also observed in deeper sediments. The presence of specific metabolic fingerprints under oxic and anoxic conditions indicated that the microbial community was adapted to use organic matter under different oxygen conditions. Overall the heterogeneity of oxygen concentrations with depth and in time would influence organic matter metabolism in the sediment tank.

Mathematical equivalence between time‐dependent single‐rate and multirate mass transfer models

The often observed tailing of tracer breakthrough curves is caused by a multitude of mass transfer processes taking place over multiple scales. Yet, in some cases, it is convenient to fit a transport model with a single-rate mass transfer coefficient that lumps all the non-Fickian observed behavior. Since mass transfer processes take place at all characteristic times, the single-rate mass transfer coefficient derived from measurements in the laboratory or in the field vary with time ω(t). The literature review and tracer experiments compiled by Haggerty et al. (2004) from a number of sites worldwide suggest that the characteristic mass transfer time, which is proportional to ω(t)−1, scales as a power law of the advective and experiment duration. This paper studies the mathematical equivalence between the multirate mass transfer model (MRMT) and a time-dependent single-rate mass transfer model (t-SRMT). In doing this, we provide new insights into the previously observed scale-dependence of mass transfer coefficients. The memory function, g(t), which is the most salient feature of the MRMT model, determines the influence of the past values of concentrations on its present state. We found that the t-SRMT model can also be expressed by means of a memory function φ(t,τ). In this case, though the memory function is nonstationary, meaning that in general it cannot be written as φ(t−τ). Nevertheless, the full behavior of the concentrations using a single time-dependent rate ω(t) is approximately analogous to that of the MRMT model provided that the equality ω(t)=−dln⁡g(t)/dt holds and the field capacity is properly chosen. This relationship suggests that when the memory function is a power law, g(t)∼t1−k, the equivalent mass transfer coefficient scales as ω(t)∼t−1, nicely fitting without calibration the estimated mass transfer coefficients compiled by Haggerty et al. (2004).The often observed tailing of tracer breakthrough curves is caused by a multitude of mass transfer processes taking place over multiple scales. Yet, in some cases, it is convenient to fit a transport model with a single-rate mass transfer coefficient that lumps all the non-Fickian observed behavior. Since mass transfer processes take place at all characteristic times, the single-rate mass transfer coefficient derived from measurements in the laboratory or in the field vary with time w(t). The literature review and tracer experiments compiled by Haggerty et al. (2004) from a number of sites worldwide suggest that the characteristic mass transfer time, which is proportional to w(t)^-1, scales as a power law of the advective and experiment duration. This paper studies the mathematical equivalence between the multirate mass transfer model (MRMT) and a time-dependent single-rate mass transfer model (t-SRMT). In doing this, we provide new insights into the previously observed scale-dependence of mass transfer coefficients. The memory function, g(t), which is the most salient feature of the MRMT model, determines the influence of the past values of concentrations on its present state. We found that the t-SRMT model can also be expressed by means of a memory function \phi(t,\tau). In this case, though the memory function is nonstationary, meaning that in general it cannot be written as \phi(t-\tau). Nevertheless, the full behavior of the concentrations using a single time-dependent rate w(t) is approximately analogous to that of the MRMT model provided that the equality w(t) = -dlng(t)/dt holds and the field capacity is properly chosen. This relationship suggests that when the memory function is a power law, g(t) \approx t^{1-k}, the equivalent mass transfer coefficient scales as w(t) \approx t^-1, nicely fitting without calibration the estimated mass transfer coefficients compiled by Haggerty et al. (2004).

Improving the accuracy of risk prediction from particle-based breakthrough curves reconstructed with kernel density estimators

While particle tracking techniques are often used in risk frameworks, the number of particles needed to properly derive risk metrics such as average concentration for a given exposure duration is often unknown. If too few particles are used, error may propagate into the risk estimate. In this work, we provide a less error-prone methodology for the direct reconstruction of exposure duration averaged concentration versus time breakthrough curves from particle arrival times at a compliance surface. The approach is based on obtaining a suboptimal kernel density estimator that is applied to the sampled particle arrival times. The corresponding estimates of risk metrics obtained with this method largely outperform those by means of traditional methods (reconstruction of the breakthrough curve followed by the integration of concentration in time over the exposure duration). This is particularly true when the number of particles used in the numerical simulation is small ( <105), and for small exposure times. Percent error in the peak of averaged breakthrough curves is approximately zero for all scenarios and all methods tested when the number of particles is ≥105. Our results illustrate that obtaining a representative average exposure concentration is reliant on the information contained in each individual tracked particle, more so when the number of particles is small. They further illustrate the usefulness of defining problem-specific kernel density estimators to properly reconstruct the observables of interest in a particle tracking framework without relying on the use of an extremely large number of particles.An edited version of this paper was published by AGU. Copyright (2015) American Geophysical Union.