Numerical approach to Cryptosporidium risk assessment using reliability method

A previously developed Cryptosporidium transport model is solved numerically to investigate the transport and interactions between Cryptosporidium, water and surface sediment and to estimate the risk of surface water contamination by Cryptosporidium. The primary objective of this study is to expand the work of Yeghiazarian (Ph.D. dissertation, Cornell University 2001)where the analytical solution of the Cryptosporidium transport model was obtained for a simple case of specific attachment of Cryptosporidium oocysts to fine soil particles wherein some parameters have zero values. However, some studies have shown several cases where these parameters are not zero. This necessitated further study to generate a solution to the complete Cryptosporidium transport model. Utilizing the finite difference method, the Cryptosporidium transport model is solved numerically for the general case of a system with any parameter values. Previously, first- and second-order reliability methods (FORM and SORM) were employed for risk assessment using analytical transport results (Yeghiazarian, Ph.D. dissertation, Cornell University, 2001), but in this work, FORM and SORM are applied to the numerical solution of the Cryptosporidium transport model to estimate the risk of Cryptosporidium contamination in surface water. The risk of surface water contamination is estimated by the probability that the Cryptosporidium concentration in surface water at a given time and location exceeds a safety threshold. The numerical solution is interfaced with the general-purpose reliability code, CALREL, to estimate the probability of failure on one hillslope. The sensitivity of system reliability to process parameters is reported.     

A finite element solution for the fractional advection–dispersion equation

The fractional advection–dispersion equation (FADE) known as its non-local dispersion, has been proven to be a promising tool to simulate anomalous solute transport in groundwater. We present an unconditionally stable finite element (FEM) approach to solve the one-dimensional FADE based on the Caputo definition of the fractional derivative with considering its singularity at the boundaries. The stability and accuracy of the FEM solution is verified against the analytical solution, and the sensitivity of the FEM solution to the fractional order α and the skewness parameter β is analyzed. We find that the proposed numerical approach converge to the numerical solution of the advection–dispersion equation (ADE) as the fractional order α equals 2. The problem caused by using the first- or third-kind boundary with an integral-order derivative at the inlet is remedied by using the third-kind boundary with a fractional-order derivative there. The problems for concentration estimation at boundaries caused by the singularity of the fractional derivative can be solved by using the concept of transition probability conservation. The FEM solution of this study has smaller numerical dispersion than that of the FD solution by Meerschaert and Tadjeran (J Comput Appl Math 2004). For a given α, the spatial distribution of concentration exhibits a symmetric non-Fickian behavior when β = 0. The spatial distribution of concentration shows a Fickian behavior on the left-hand side of the spatial domain and a notable non-Fickian behavior on the right-hand side of the spatial domain when β = 1, whereas when β = −1 the spatial distribution of concentration is the opposite of that of β = 1. Finally, the numerical approach is applied to simulate the atrazine transport in a saturated soil column and the results indicat that the FEM solution of the FADE could better simulate the atrazine transport process than that of the ADE, especially at the tail of the breakthrough curves.     

Analytical power series solutions to the two-dimensional advection–dispersion equation with distance-dependent dispersivities

As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection–dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection–dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio e_L exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large e_L values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd.     

Prediction of Remediation of a Heterogeneous Aquifer: A Case Study

Contaminant plumes whose characteristic length is smaller than the horizontal integral scale of the hydraulic conductivity, K, are abundant in shallow, phreatic aquifers. In such cases, the aquifer can be regarded as layered, with K being only a function of the vertical coordinate. The heterogeneity of K has a critical role upon the efficiency of remediation of such sites, for example, by Pump and Treat schemes. The expected efficiency is a random variable, with uncertainty. Quantifying this uncertainty can be of great importance to decision making. In this study, we focus on a case study in the coastal aquifer of Israel and compare two different approaches for constructing realizations of K: continuous and indicator. We observe a significant difference between the constructed realizations, which results in a considerable difference in the predicted remediation efficiency and its uncertainty. Furthermore, we study the effect of conditioning the realizations by a rather limited number of K data points. We find that the conditioning results in a major reduction of the uncertainty. In addition, we compare the results of the transport model to a simplified semi‐analytical solution that is based on assuming radial flow. We find a good agreement with the three‐dimensional numerical model. This result illustrates that the simplified solution can be used for prediction of the remediation efficiency when the flow at the plume vicinity can be regarded as radial.     

An Electron-Balance Based Approach to Predict the Decreasing Denitrification Potential of an Aquifer

Numerical models for reactive transport can be used to estimate the breakthrough of a contaminant in a pumping well or at other receptors. However, as natural aquifers are highly heterogeneous with unknown spatial details, reactive transport predictions on the aquifer scale require a stochastic framework for uncertainty analysis. The high computational demand of spatially explicit reactive-transport models hampers such analysis, thus motivating the search for simplified estimation tools. We suggest performing an electron balance between the reactants in the infiltrating solution and in the aquifer matrix to obtain the hypothetical time of dissolved-reactant breakthrough at a receptor if the reaction with the matrix was instantaneous. This time we denote as the advective breakthrough time for instantaneous reaction (τ_inst ). It depends on the amount of the reaction partner present in the matrix, the mass flux of the dissolved reactant, and the stoichiometry. While the shape of the reactive-species breakthrough curve depends on various kinetic parameters, the overall timing scales with τ_inst . We calculate the latter by particle tracking. The effort of computing τ_inst is so low that stochastic calculations become feasible. We apply the concept to a two-dimensional test case of aerobic respiration and denitrification. A detailed spatially explicit reactive-transport model includes microbial dynamics. Scaling the time of local breakthrough curves observed at individual points by τ_inst decreased the variability of electron-donor breakthrough curves significantly. We conclude that the advective breakthrough time for instantaneous reaction is efficient in estimating the time over which an aquifer retains its degradation potential.     

Lagrangian particle tracking of a toxic dinoflagellate bloom within the Tampa Bay estuary

A coastal risk assessment system simulates the basic physical mechanisms underlying contaminant transport in Tampa Bay. This risk assessment system, comprised of a three-dimensional numerical circulation model coupled to a Lagrangian particle tracking model, simulates the transport and dispersion of a toxic dinoflagellate bloom. Instantaneous velocity output from the circulation model drives the movement of particles, each representing a fraction of a K. brevis bloom, within the model grid cells. Hindcast simulations of the spatial distribution of the K. brevis bloom are presented and compared with water sample concentrations collected during the peak of the bloom. Probability calculations, herein called transport quotients, allow for rapid analysis of bay-wide K. brevis transport showing locations most likely to be impacted by the contaminant. Maps constructed from the transport quotients provide managers with a bay-wide snapshot of areas in Tampa Bay most at risk during a hazardous bloom event.     

Analytical solutions for sequentially coupled one-dimensional reactive transport problems – Part I: Mathematical derivations

Multi-species reactive transport equations coupled through sorption and sequential first-order reactions are commonly used to model sites contaminated with radioactive wastes, chlorinated solvents and nitrogenous species. Although researchers have been attempting to solve various forms of these reactive transport equations for over 50 years, a general closed-form analytical solution to this problem is not available in the published literature. In Part I of this two-part article, we derive a closed-form analytical solution to this problem for spatially-varying initial conditions. The proposed solution procedure employs a combination of Laplace and linear transform methods to uncouple and solve the system of partial differential equations. Two distinct solutions are derived for Dirichlet and Cauchy boundary conditions each with Bateman-type source terms. We organize and present the final solutions in a common format that represents the solutions to both boundary conditions. In addition, we provide the mathematical concepts for deriving the solution within a generic framework that can be used for solving similar transport problems.     

Solutions to the stochastic transport problem of O(σ V N ) for conservative solutes

A recursive perturbation solution to the eulerian transport problem for a conservative solute in a random conductivity field is reported. The stochastic concentration is given to arbitrary order inσ _ν, the variance of fluctuating velocity. The result gives the stochastic concentration as a perturbation to the deterministic concentration for constant mean flow. The closed form solution is easy to implement numerically via FFT.     

Axisymmetric annulus convection at unit Prandtl number

Abstract

We examine the role played in annulus flows by mechanisms dependent upon the Prandtl number, σ. Solutions are obtained at σ = 1 for both the real annulus system and for the hypothetical “free annulus” system (free slip lateral boundaries). These solutions are compared with previously obtained solutions at σ = 7.

In the free annulus, the solution at σ = 1 differs radically from that at σ = 7. The σ = 1 solution appears to be essentially a finite amplitude mode due to Solberg instability whereas the solution at σ = 7 manifests a flow caused by the diffusive overturning mechanism.

The variation with σ of the real annulus flow is not so fundamental but some differences in the dynamical structures are noted.     

Numerical experiments on the long-term morphodynamics of the Colorado River Delta

The interaction among tidal currents, sediment transport, and long-term changes of the sea bottom in the Colorado River Delta have been investigated applying a two dimensional nonlinear hydrodynamic-numerical finite differences model. The system was forced by the dominant M ₂ tidal component at the open boundary. We carried out calculations to study the morphodynamics of the actual bathymetry caused by the bedload sediment transport. To investigate the origin of actual morphological features, we performed experiments using a smoothed bathymetry, in which the islands Montague, Gore, and Pelícano were eliminated. Under the imposed tidal hydrodynamics, the results indicate that the bedload transport contributes significantly in the genesis of sandbanks and in the formation and maintaining of the Montague and Gore Islands.     

Finite-difference solution of the transport equation: First results

The ray-theoretical transport equation for inhomogeneous isotropic media (2D-SH case) is solved by the method of finite differences on a rectangular grid, both for an incident plane wave (explicit scheme) and a line source (implicit scheme). Results for homogeneous models and for heterogeneous models with structural discontinuities are discussed. First-arrival travel times calculated by various techniques serve as input for the solution of the transport equation and the computation of amplitudes of first arrivals. To obtain correct amplitudes the travel times must be highly accurate and the discontinuities must be smoothed out; the reason is that the spatial second derivatives of the travel time field enter the transport equation. In the simple cases studied, finite differences provide a fast and efficient tool for the computation of first-arrival amplitudes.     

Escherichia coli strains harvested from springs in Kampala,Uganda: cell characterization and transport in saturated porous media

We hypothesized that the transport of Escherichia coli strains harvested from springs could be characterized by a similar set of cell characteristics and transport parameters. The hypothesis was tested by sampling springs throughout the Lubigi catchment in Kampala, Uganda. Chemo‐physical parameters in addition to total coliform concentrations were determined. Furthermore, E. coli strains were harvested, and cell properties determined. Column experiments in saturated quartz columns of 7 cm height were conducted to determine transport parameters of selected E. coli strains. Using a two‐site non‐equilibrium sorption model, transport was modelled by fitting breakthrough data in HYDRUS 1D. Results indicated faecal contamination of the springs with high concentrations of total coliforms, chloride and nitrate. Furthermore, the maximum relative E. coli concentrations (C/C₀)_max in the column experiments were high. Compared with our previous work on E. coli strains, collected from a pasture and from zoo animals, attachment was low. Modelling revealed that both equilibrium and kinetic sorption were not important under conditions employed in the experiments. These observations are explained by the way in which the strains were harvested: from termination points of flow lines (springs). Such strains may possess characteristics that might have influenced their transport in the subsurface leading to their low attachment efficiency and possibly contributing to the lack of influence of equilibrium and kinetic sorption characteristics. There was no significant correlation between cell properties and transport parameters. Furthermore, 58% of the tested strains were of the O21:H7 serotype, and all definable serotypes identified were associated with diseases. We speculate that this serotype may possess characteristics that allow preferential transport through the aquifers of the area. We demonstrated that bacteria harvested from termination points of flow lines compared with those obtained from pollution sources, which have not undergone transport yet, present a good option for the assessment of bacteria transport characteristics in aquifers. Copyright © 2013 John Wiley & Sons, Ltd.     

NON-NORMAL INCIDENCE INVERSION: EXISTENCE OF SOLUTION*

An inverse problem is one in which the parameters of a model are determined from measured seismic data. Important to the solution of inverse problems is the issue of whether or not a solution exists. In this paper we show, in a constructive manner, that a solution does exist to the specific inverse problem of determining the parameters of a horizontally stratified, lossless, isotropic and homogeneous layered system that is excited by a non-normal incidence (NNI) plane wave. Mode conversion between P- and S-waves is included. We develop a seven-step layer-recursive procedure for determining all of the parameters for layer j. These parameters are P-wave and S-wave velocities and angles of incidence, density, thickness, traveltimes, and reflection- and transmission-coefficient matrices. Downward continuation of data from the top of one layer to the top of the next lower layer is an important step in our procedure, just as it is in normal incidence (NI) inversion. We show that, in order to compute all parameters of layer j, we need to (and can) compute some parameters for layer j+ 1. This is a non-causal phenomenon that seems to be necessary in NNI inversion but is not present in NI inversion.     

Dynamics of Wood Transport in Streams: A Flume Experiment

The influence of woody debris on channel morphology and aquatic habitat has been recognized for many years. Unlike sediment, however, little is known about how wood moves through river systems. We examined some dynamics of wood transport in streams through a series of flume experiments and observed three distinct wood transport regimes: uncongested, congested and semi-congested. During uncongested transport, logs move without piece-to-piece interactions and generally occupy less than 10 per cent of the channel area. In congested transport, the logs move together as a single mass and occupy more than 33 per cent of the channel area. Semi-congested transport is intermediate between these two transport regimes. The type of transport regime was most sensitive to changes in a dimensionless input rate, defined as the ratio of log volume delivered to the channel per second (Q_log) to discharge (Q_W); this ratio varied between 0·015 for uncongested transport and 0·20 for congested transport. Depositional fabrics within stable log jams varied by transport type, with deposits derived from uncongested and semi-congested transport regimes having a higher proportion of pieces orientated normal to flow than those from congested transport. Because wood input rates are higher and channel dimensions decrease relative to piece size in low-order channels, we expect congested transport will be more common in low-order streams while uncongested transport will dominate higher-order streams. Single flotation models can be used to model the stability of individual pieces, especially in higher-order channels, but are insufficient for modelling the more complex intractions that occur in lower-order streams. © 1997 John Wiley & Sons, Ltd.     

Scale‐dependent Poiseuille flow alternatively explains enhanced dispersion in geothermal environments

Fluid flow exerts a critical impact on the convection of thermal energy in geological media, whereas heat transport in turn affects fluid properties, including fluid dynamic viscosity and density. The interplay of flow and heat transport also affects solute transport. To unravel these complex coupled flow, heat, and solute transport processes, here, we present a theory for the idealized scale‐dependent Poiseuille flow model considering a constant temperature gradient (?T) along a single fracture, where fluid dynamic viscosity connects with temperature via an exponential function. The idealized scale‐dependent model is validated based on the solutions from direct numerical simulations. We find that the hydraulic conductivity (K) of the Poiseuille flow either increases or decreases with scales depending on ?T > 0°C/m or ?T < 0°C/m, respectively. Indeed, the degree of changes in K depends on the magnitude of ?T and fracture length. The scale‐dependent model provides an alternative explanation for the well‐known scale‐dependent transport problem, for example, the dispersion coefficient increases with travel distance when ?T > 0°C/m according to the Taylor dispersion theory, because K (or equivalently flux through fractures) scales with fracture length. The proposed theory unravels intertwined interactions between flow and transport processes, which might shed light on understanding many practical geophysical problems, for example, geothermal energy exploration.     

A Phenomenological Model for Particle Retention in Single,Saturated Fractures

Fractured aquifers are some of the most poorly characterized subsurface environments despite posing one of the highest risks to the protection of potable groundwater. This research was designed to improve the understanding of the factors affecting particle transport through fractures by developing a phenomenological model based on laboratory‐scale transport data. The model presented in this research employed data from over 70 particle tracer tests conducted in single, saturated, variable‐aperture fractures that were obtained from the natural environment and fractured in the laboratory or cast from epoxy in the laboratory. The particles employed were Escherichia coli RS2‐GFP and microspheres. The tracer experiments were conducted in natural (dolomitic limestone and granite) as well as epoxy replicas of the natural fractures. The multiple linear regression analysis revealed that the most important factors influencing particle retention in fractures are the ratio of the ionic strength of solution to collector charge, the ratio of particle to collector charge, and the ratio of advective to diffusive forces as described by the Peclet number. The model was able to reasonably (R² = 0.64) predict the fraction of particles retained; however, it is evident that some factors not accounted for in the model also contributed to retention. This research presents a novel approach to understanding particle transport in fractures, and illustrates the relative importance of various factors affecting the transport mechanisms. The utility of this model lies in the increased understanding of particle transport in fractures, which is extremely useful for directing future research.     

Solution of the nonlinear transport equation using modified Picard iteration

The transport and fate of reactive chemicals in groundwater is governed by equations which are often difficult to solve due to the nonlinear relationship between the solute concentrations for the liquid and solid phases. The nonlinearity may cause mass balance errors during the numerical simulation in addition to numerical errors for linear transport system. We have generalized the modified Picard iteration algorithm of Celia et al.⁵ for unsaturated flow to solve the nonlinear transport equation. Written in a ‘mixed-form’ formulation, the total solute concentration is expanded in a Taylor series with respect to the solution concentration to linearize the transport equation, which is then solved with a conventional finite element method. Numerical results of this mixed-form algorithm are compared with those obtained with the concentration-based scheme using conventional Picard iteration. In general, the new solver resulted in negligible mass balance errors (< ∥10⁻⁸∥%) and required less computational time than the conventional iteration scheme for the test examples, including transport involving highly nonlinear adsorption under steady-state as well as transient flow conditions. In contrast, mass balance errors resulting from the conventional Picard iteration method were higher than 10% for some highly nonlinear problems. Application of the modified Picard iteration scheme to solve the nonlinear transport equation may greatly reduce the mass balance errors and increase computational efficiency.     

Mechanisms of the meridional heat transport in the Southern Ocean

The Southern Ocean (SO) transports heat towards Antarctica and plays an important role in determining the heat budget of the Antarctic climate system. A global ocean data synthesis product at eddy-permitting resolution from the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) project is used to estimate the meridional heat transport (MHT) in the SO and to analyze its mechanisms. Despite the intense eddy activity, we demonstrate that most of the poleward MHT in the SO is due to the time-mean fields of the meridional velocity, V, and potential temperature, θ. This is because the mean circulation in the SO is not strictly zonal. The Antarctic Circumpolar Current carries warm waters from the region south of the Agulhas Retroflection to the lower latitudes of the Drake Passage and the Malvinas Current carries cold waters northward along the Argentinian shelf. Correlations between the time-varying fields of V and θ (defined as transient processes) significantly contribute to the horizontal-gyre heat transport, but not the overturning heat transport. In the highly energetic regions of the Agulhas Retroflection and the Brazil-Malvinas Confluence the contribution of the horizontal transient processes to the total MHT exceeds the contribution of the mean horizontal flow. We show that the southward total MHT is mainly maintained by the meridional excursion of the mean geostrophic horizontal shear flow (i.e., deviation from the zonal average) associated with the Antarctic Circumpolar Current that balances the equatorward MHT due to the Ekman transport and provides a net poleward MHT in the SO. The Indian sector of the SO serves as the main pathway for the poleward MHT.