Bayesian evaluation of groundwater age distribution using radioactive tracers and anthropogenic chemicals

pre-printThe development of a Bayesian modeling approach for estimation of the age distribution of groundwater using radioactive isotopes and anthropogenic chemicals is described. The model considers the uncertainties associated with the measured tracer concentrations as well as the parameters affecting the concentration of tracers in the groundwater, and it provides the posterior probability densities of the parameters defining the groundwater age distribution using a Markov chain Monte Carlo method. The model also incorporates the effect of dissolution of aquifer minerals on diluting the 14C signature and the uncertainties associated with this process on the inferred age distribution parameters. Two demonstration modeling cases have been performed. First, the method was applied to simulated tracer concentrations at a discharge point of a hypothetical 2-D vertical aquifer with two recharge zones, leading to a mixed groundwater age distribution under different presumed uncertainties. When the error variance of the observed tracer concentrations is considered unknown, the method can estimate the parameters of the fitted exponential-lognormal distribution with a relatively narrow credible interval when five hypothetical samples are assumed to be collected at the discharge point. However, when a single sample is assumed, the credible intervals become wider, and credible estimations of the parameters are not obtained. Second, the method was applied to the data collected at La Selva Biological Station in Costa Rica. In this demonstration application, nine different forms of presumed groundwater age distributions have been considered, including four single forms and five mixed forms, assuming the groundwater consists of distinct young and old fractions. For the medium geometrical standard deviation dc,i = 1.41, the model estimates a young groundwater age of between 0 and 350 years, with the largest odds being given to a mean age of approximately 100 years, and a fraction of young groundwater of between 15% to roughly 60%, with the largest odds for 30%. However, the method cannot definitively rule out larger fractions of young groundwater. The model provides a much more uncertain estimation of the age of old groundwater, with a credible interval of between 20,000 to 200,000 years

Stochastic And Deterministic Parameter Estimation Of Coupled Bacteria-Sediment Fate And Transport In Streams

E. Coli is widely used as an indicator organism to assess the risk of pathogenic bacteria in water bodies. Due to their strong association with suspended and bed sediments, the fate and transport of micro-organisms in water bodies is strongly controlled by sediment dynamics. It has been shown that bed sediments can contain orders of magnitude larger pathogen concentration than the water column and these sediment-associated bacteria can be released into the water column as a result of high flow velocities that cause sediment resuspension. In this presentation parameter estimation of a mechanistic model of bacteria-sediment interaction using a deterministic method through a hybrid genetic algorithm and also stochastically through Makov-Chain Monte Carlo (MCMC) approach will be presented. The physically-based model considers the advective-dispersive transport of sediments as well as both free-floating and sediment-associated bacteria in the water column and also the fate and transport of bacteria in the bed sediments. The bed sediments are treated as a distributed system which allows modeling the evolution of the vertical distribution of bacteria as a result of sedimentation, resuspension, diffusion, and bioturbation in the sediments. The model is applied to sediment and E. coli concentration data collected during a high flow event in a small stream historically receiving agricultural runoff. The genetic algorithm and MCMC method are used to estimate the likeliest values as well as the joint probability density functions of model parameters including sediment deposition and erosion rates, critical shear stress for deposition and erosion, attachment and detachment rate constants of E. coli to/from sediments and also the effective diffusion coefficients of E. coli in the bed sediments. The uncertainties associated with the estimated parameters are quantified via the MCMC approach and the correlation between the posterior distribution of parameters have been used to assess the model adequacy and parsimony

Residence time distributions for hydrologic systems: Mechanistic foundations and steady-state analytical solutions

International audienceThis review presents the physical mechanisms generating residence time distributions (RTDs) in hydrologic systems with a focus on steady-state analytical solutions. Steady-state approximations of the RTD in hydrologic systems have seen widespread use over the last half-century because they provide a convenient, simplified modeling framework for a wide range of problems. The concept of an RTD is useful anytime that characterization of the timescales of flow and transport in hydrologic systems is important, which includes topics like water quality, water resource management, contaminant transport, and ecosystem preservation. Analytical solutions are often adopted as a model of the RTD and a broad spectrum of models from many disciplines has been applied. Although these solutions are typically reduced in dimensionality and limited in complexity, their ease of use makes them preferred tools, specifically for the interpretation of tracer data. Our review begins with the mechanistic basis for the governing equations, highlighting the physics for generating a RTD, and a catalog of analytical solutions follows. This catalog explains the geometry, boundary conditions and physical aspects of the hydrologic systems, as well as the sampling conditions, that altogether give rise to specific RTDs. The similarities between models are noted, as are the appropriate conditions for their applicability. The presentation of simple solutions is followed by a presentation of more complicated analytical models for RTDs, including serial and parallel combinations, lagged systems, and non-Fickian models. The conditions for the appropriate use of analytical solutions are discussed, and we close with some thoughts on potential applications, alternative approaches, and future directions for modeling hydrologic residence time

A Stochastic Simulation Procedure for Selecting Herbicides with Minimum Environmental Impact

A mathematical environmental transport model of roadside applied herbicides at the site scale (∼100 m) was stochastically applied using a Monte-Carlo technique to simulate the concentrations of 33 herbicides in stormwater runoff. Field surveys, laboratory sorption data, and literature data were used to generate probability distribution functions for model input parameters to allow extrapolation of the model to the regional scale. Predicted concentrations were compared to EPA acute toxicity end points for aquatic organisms to determine the frequency of potentially toxic outcomes. Results are presented for three geographical regions in California and two highway geometries. For a given herbicide, frequencies of potential toxicity (FPTs) varied by as much as 36% between region and highway type. Of 33 herbicides modeled, 16 exhibit average FPTs greater than 50% at the maximum herbicide application rate, while 20 exhibit average FPTs less than 50% at the minimum herbicide application rate. Based on these FPTs and current usage statistics, selected herbicides were determined to be more environmentally acceptable than others in terms of acute toxicity and other documented environmental effects. This analysis creates a decision support system that can be used to evaluate the relative water quality impacts of varied herbicide application practices

[Dataset] Upscaling non-linear reactive transport in correlated velocity fields

Dataset from Upscaling non-linear reactive transport in correlated velocity fieldsPeer reviewe

Upscaling non-linear reactive transport in correlated velocity fields

While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous aquifers successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. Time-domain random walk approaches based on velocity Markov models provide a framework to resolve these issues by incorporating the spatial correlation of velocity of a solute particle in consecutive locations along its trajectory. These approaches often rely on particle tracking approaches, which, however, can be computationally burdensome especially when non-linear reactions are sought to be modeled. In this paper, an integro-differential equation is proposed for upscaling multi-component reactive transport in heterogeneous media that relies on copulas for representing the velocity correlation structure. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized copula function which expresses the correlation between velocity-ranks of a solute particle along its trajectory. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields to the results of the upscaled model.Marco Dentz acknowledges the support of the European Research Council (ERC) through the project MHetScale (617511).Peer reviewe

The Effect of Velocity Correlation on the Spatial Evolution of Breakthrough Curves in Heterogeneous Media

International audienceIn heterogeneous media, the velocity distribution and the spatial correlation structure of velocity for solute particles determine the breakthrough curves and how they evolve as one moves away from the solute source. The ability to predict such evolution can help relating the spatio-statistical hydraulic properties of the media to the transport behavior and travel time distributions. While commonly used non-local transport models such as anomalous dispersion and classical continuous time random walk (CTRW) can reproduce breakthrough curve successfully by adjusting the model parameter values, they lack the ability to relate model parameters to the spatio-statistical properties of the media. This in turns limits the transferability of these models. In the research to be presented, we express concentration or flux of solutes as a distribution over their velocity. We then derive an integrodifferential equation that governs the evolution of the particle distribution over velocity at given times and locations for a particle ensemble, based on a presumed velocity correlation structure and an ergodic cross-sectional velocity distribution. This way, the spatial evolution of breakthrough curves away from the source is predicted based on cross-sectional velocity distribution and the connectivity, which is expressed by the velocity transition probability density. The transition probability is specified via a copula function that can help construct a joint distribution with a given correlation and given marginal velocities. Using this approach, we analyze the breakthrough curves depending on the velocity distribution and correlation properties. The model shows how the solute transport behavior evolves from ballistic transport at small spatial scales to Fickian dispersion at large length scales relative to the velocity correlation length.https://doi.org/10.1002/2017WR02057

An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models

Abstract A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system's stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters