An ELLAM approximation for advective–dispersive transport with nonlinear sorption

We consider an Eulerian–Lagrangian localized adjoint method (ELLAM) applied to nonlinear model equations governing solute transport and sorption in porous media. Solute transport in the aqueous phase is modeled by standard advection and hydrodynamic dispersion processes, while sorption is modeled with a nonlinear local-equilibrium model. We present our implementation of finite volume ELLAM (FV-ELLAM) and finite element (FE-ELLAM) discretizations to the reactive transport model and evaluate their performance for several test problems containing self-sharpening fronts.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

Two-dimensional modeling of contaminant transport in porous media in the presence of colloids

It has long been known that colloids can facilitate the transport of contaminants in groundwater systems by reducing the effective retardation factor. A significant effort has been devoted to study colloid-facilitated contaminant transport during the past decade. Many of the previous studies were restricted to one-dimensional analyses and comparisons with finite-column experiments. In this work, a two-dimensional numerical model is developed and used to study the different interactions between colloids, contaminants, and porous media under homogeneous conditions. The numerical formulation of the model is based on discretizing mass balance equations and reaction equations using finite differences having a third-order, total variance-diminishing scheme for the advection terms. This scheme significantly reduces numerical dispersion and leads to greater accuracy compared to the standard central-differencing scheme. The model is tested against analytical solutions under simplified conditions as well as against experimental data, and the results are favorable. The model is used to investigate the impact of the various reaction rates and parameter values on the movement of contaminant plumes in two dimensions. The model is also used to investigate the hypothesis that colloids may increase the effective retardation factor of contaminant plumes. The analysis shows that assuming kinetic mass exchange between contaminant and colloids with constant reaction rate coefficients that are not related to the concentrations may lead to inaccurate results. These inaccurate results are exemplified in the finding that under the kinetic assumption the ratio of the initial concentration of colloids to the initial concentration of contaminant does not affect the amount of facilitation or retardation that occurs in the system. It is also found that colloids can increase the effective retardation factor for the contaminant under certain combinations of reaction rates and distribution coefficients. A quantitative empirical expression to identify whether colloids retard or facilitate the contaminant movement is presented.     

Groundwater contaminant transport including adsorption and first order decay

This study deals with the transport of a contaminant in groundwater. The contaminant is subject to first order decay or linear adsorption. Its displacement can be modeled by a random walk process in which particles are killed at exponentially distributed times. Dirichlet problems are derived for the rate and mean time at which contaminated particles reach a particular part of the boundary of a certain domain. These Dirichlet problems are solved asymptotically for two types of 2D-flow patterns: flow parallel to the boundary of a domain and arbitrary flow towards a well in an aquifer.     

Groundwater contaminant transport including adsorption and first order decay

This study deals with the transport of a contaminant in groundwater. The contaminant is subject to first order decay or linear adsorption. Its displacement can be modeled by a random walk process in which particles are killed at exponentially distributed times. Dirichlet problems are derived for the rate and mean time at which contaminated particles reach a particular part of the boundary of a certain domain. These Dirichlet problems are solved asymptotically for two types of 2D-flow patterns: flow parallel to the boundary of a domain and arbitrary flow towards a well in an aquifer.     

Non-uniqueness of inverse transmissivity field calibration and predictive transport modeling

Recent work with stochastic inverse modeling techniques has led to the development of efficient algorithms for the construction of transmissivity (T) fields conditioned to measurements of T and head. Small numbers of calibration targets and correlation between model parameters in these inverse solutions can lead to a relatively large region in parameter space that will produce a near optimal calibration of the T field to measured heads. Most applications of these inverse techniques have not considered the effects of non-unique calibration on subsequent predictions made with the T fields. Use of these T fields in predictive contaminant transport modeling must take into account the non-uniqueness of the T field calibration. A recently developed ‘predictive estimation’ technique is presented and employed to create T fields that are conditioned to observed heads and measured T values while maximizing the conservatism of the associated predicted advective travel time. Predictive estimation employs confidence and prediction intervals calculated simultaneously on the flow and transport models, respectively. In an example problem, the distribution of advective transport results created with the predictive estimation technique is compared to the distribution of results created under traditional T field optimization where model non-uniqueness is not considered. The predictive estimation technique produces results with significantly shorter travel times relative to traditional techniques while maintaining near optimal calibration. Additionally, predictive estimation produces more accurate estimates of the fastest travel times.     

A parallel mesh-free contaminant transport model based on the Analytic Element and Streamline Methods

This paper introduces a new method for simulating large-scale subsurface contaminant transport that combines an Analytic Element Method (AEM) groundwater flow solution with a split-operator Streamline Method for modeling reactive transport. The key feature of the method is the manner in which the vertically integrated AEM flow solution is used to construct three-dimensional particle tracks that define the geometry of the Streamline Method. The inherently parallel nature of the algorithm supports the development of reactive transport models for spatial domains much larger than current grid-based methods. The applicability of the new approach is verified for cases with negligible transverse dispersion through comparisons to analytic solutions and existing numerical solutions, and parallel performance is demonstrated through a realistic test problem based on the regional-scale transport of agricultural contaminants from spatially distributed sources.     

Quantifying geological uncertainty for flow and transport modeling in multi-modal heterogeneous formations

In this work, we address the problem of characterizing the heterogeneity and uncertainty of hydraulic properties for complex geological settings. Hereby, we distinguish between two scales of heterogeneity, namely the hydrofacies structure and the intrafacies variability of the hydraulic properties. We employ multiple-point geostatistics to characterize the hydrofacies architecture. The multiple-point statistics are borrowed from a training image that is designed to reflect the prior geological conceptualization. The intrafacies variability of the hydraulic properties is represented using conventional two-point correlation methods, more precisely, spatial covariance models under a multi-Gaussian spatial law. We address the different levels and sources of uncertainty in characterizing the subsurface heterogeneity, and explore their effect on groundwater flow and transport predictions. Typically, uncertainty is assessed by way of many images, termed realizations, of a fixed statistical model. However, in many cases, sampling from a fixed stochastic model does not adequately represent the space of uncertainty. It neglects the uncertainty related to the selection of the stochastic model and the estimation of its input parameters. We acknowledge the uncertainty inherent in the definition of the prior conceptual model of aquifer architecture and in the estimation of global statistics, anisotropy, and correlation scales. Spatial bootstrap is used to assess the uncertainty of the unknown statistical parameters. As an illustrative example, we employ a synthetic field that represents a fluvial setting consisting of an interconnected network of channel sands embedded within finer-grained floodplain material. For this highly non-stationary setting we quantify the groundwater flow and transport model prediction uncertainty for various levels of hydrogeological uncertainty. Results indicate the importance of accurately describing the facies geometry, especially for transport predictions.     

On a monotonicity preserving Eulerian–Lagrangian localized adjoint method for advection–diffusion equations

Eulerian–Lagrangian localized adjoint methods (ELLAMs) provide a general approach to the solution of advection-dominated advection–diffusion equations allowing large time steps while maintaining good accuracy. Moreover, the methods can treat systematically any type of boundary condition and are mass conservative. However, all ELLAMs developed so far suffer from non-physical oscillations and are usually implemented on structured grids. In this paper, we propose a finite volume ELLAM which incorporates a novel correction step rendering the method monotone while maintaining conservation of mass. The method has been implemented on fully unstructured meshes in two space dimensions. Numerical results demonstrate the applicability of the method for problems with highly non-uniform flow fields arising from heterogeneous porous media.     

3D finite volume groundwater and heat transport modeling with non-orthogonal grids,using a coordinate transformation method

Many popular groundwater modeling codes are based on the finite differences or finite volume method for orthogonal grids. In cases of complex subsurface geometries this type of grid either leads to coarse geometric representations or to extremely fine meshes. We use a coordinate transformation method (CTM) to circumvent this shortcoming. In computational fluid dynamics (CFD), this method has been applied successfully to the general Navier–Stokes equation. The method is based on tensor analysis and performs a transformation of a curvilinear into a rectangular unit grid, on which a modified formulation of the differential equations is applied. Therefore, it is not necessary to reformulate the code in total. We applied the CTM to an existing three-dimensional code (SHEMAT), a simulator for heat conduction and advection in porous media. The finite volume discretization scheme for the non-orthogonal, structured, hexahedral grid leads to a 19-point stencil and a correspondingly banded system matrix. The implementation is straightforward and it is possible to use some existing routines without modification. The accuracy of the modified code is demonstrated for single phase flow on a two-dimensional analytical solution for flow and heat transport. Additionally, a simple case of potential flow is shown for a two-dimensional grid which is increasingly deformed. The result reveals that the corresponding error increases only slightly. Finally, a thermal free-convection benchmark is discussed. The result shows, that the solution obtained with the new code is in good agreement with the ones obtained by other codes.     

Heavy metal content in relation to particle size and organic content of surficial sediments in Miami River and transport potential

Transport of contaminants in Miami River (Florida, USA) sediments with river currents is a concern due to their potential impact in areas that are away from the potential sources. Accumulation profiles of five metals (As, Cd, Hg, Zn, Pb) in the surficial sediments of the Miami River were evaluated in relation to grain size (from less than 0.075 mm to 6.3 mm) and organic content. Surficial sediment samples were collected along the river basin as well as in bay waters. Fine sediments ( o 0.106 mm) contained more than 10 times the levels of Cd and Hg and more than 6 times the levels of arsenic in comparison to the sediments that are larger than 0.850 mm. Zn and Pb levels were more than 10 times in the fine sediments ( o 0.106 mm) in comparison to those that were larger than 4.750 mm. Cd, Hg, and Zn levels had sig-nificant correlation with the total organic carbon content of the sediments. This indicates that Cd, Hg, and Zn in fine sediments have the potential to be metabolized in addition to potential to be mobilized with river currents. Analysis of the Gibbs settling velocities of particles showed that particles smaller than 0.5 mm can be transported with the river currents. The levels of heavy metal in fine sediments ( o 0.425 mm) along the river bed showed that fine particles had tendency to be mobilized and accu-mulate at locations where the rivers currents are low and carried out to the bay.     

First-principles derivation of reactive transport modeling parameters for particle tracking and PDE approaches

Both Eulerian and Lagrangian reactive transport simulations in natural media require selection of a parameter that controls the “promiscuity” of the reacting particles. In Eulerian models, measurement of this parameter may be difficult because its value will generally differ between natural (diffusion-limited) systems and batch experiments, even though both are modeled by reaction terms of the same form. And in Lagrangian models, there previously has been no a priori way to compute this parameter. In both cases, then, selection is typically done by calibration, or ad hoc. This paper addresses the parameter selection problem for Fickian transport by deriving, from first principles and D (the diffusion constant) the reaction-rate-controlling parameters for particle tracking (PT) codes and for the diffusion–reaction equation (DRE). Using continuous time random walk analysis, exact reaction probabilities are derived for pairs of potentially reactive particles based on D and their probability of reaction provided that they collocate. Simultaneously, a second PT scheme directly employing collocation probabilities is derived. One-to-one correspondence between each of D, the reaction radius specified for a PT scheme, and the DRE decay constant are then developed. These results serve to ground reactive transport simulations in their underlying thermodynamics, and are confirmed by simulations.     

Modeling the effects of nonlinear equilibrium sorption on the transport of solute plumes in saturated heterogeneous porous media

Transport of sorbing solutes in 2D steady and heterogeneous flow fields is modeled using a particle tracking random walk technique. The solute is injected as an instantaneous pulse over a finite area. Cases of linear and Freundlich sorption isotherms are considered. Local pore velocity and mechanical dispersion are used to describe the solute transport mechanisms at the local scale. This paper addresses the impact of the degree of heterogeneity and correlation lengths of the log-hydraulic conductivity field as well as negative correlation between the log-hydraulic conductivity field and the log-sorption affinity field on the behavior of the plume of a sorbing chemical. Behavior of the plume is quantified in terms of longitudinal spatial moments: center-of-mass displacement, variance, 95% range, and skewness. The range appears to be a better measure of the spread in the plumes with Freundlich sorption because of plume asymmetry. It has been found that the range varied linearly with the travelled distance, regardless of the sorption isotherm. This linear relationship is important for extrapolation of results to predict behavior beyond simulated times and distances. It was observed that the flow domain heterogeneity slightly enhanced the spreading of nonlinearly sorbing solutes in comparison to that which occurred for the homogeneous flow domain, whereas the spreading enhancement in the case of linear sorption was much more pronounced. In the case of Freundlich sorption, this enhancement led to further deceleration of the solute plume movement as a result of increased retardation coefficients produced by smaller concentrations. It was also observed that, except for plumes with linear sorption, correlation between the hydraulic conductivity and the sorption affinity fields had minimal effect on the spatial moments of solute plumes with nonlinear sorption.     

Multi-resolution adaptive modeling of groundwater flow and transport problems

Many groundwater flow and transport problems, especially those with sharp fronts, narrow transition zones, layers and fingers, require extensive computational resources. In this paper, we present a novel multi-resolution adaptive Fup approach to solve the above mentioned problems. Our numerical procedure is the Adaptive Fup Collocation Method (AFCM), based on Fup basis functions and designed through a method of lines (MOL). Fup basis functions are localized and infinitely differentiable functions with compact support and are related to more standard choices such as splines or wavelets. This method enables the adaptive multi-resolution approach to solve problems with different spatial and temporal scales with a desired level of accuracy using the entire family of Fup basis functions. In addition, the utilized collocation algorithm enables the mesh free approach with consistent velocity approximation and flux continuity due to properties of the Fup basis functions. The introduced numerical procedure was tested and verified by a few characteristic groundwater flow and transport problems, the Buckley–Leverett multiphase flow problem, the 1-D vertical density driven problem and the standard 2-D seawater intrusion benchmark–Henry problem. The results demonstrate that the method is robust and efficient particularly when describing sharp fronts and narrow transition zones changing in space and time.     

Stochastic modelling of groundwater flow and solute transport in aquifers

This paper presents an introductory overview of recently developed stochastic theories for tackling spatial variability problems in predicting groundwater flow and solute transport. Advantages and limitations of the theories are discussed. Lastly, strategies based on the stochastic approaches to predict solute transport in aquifers are recommended.     

Flow and transport through two-dimensional isotropic media of binary conductivity distribution. Part 2: NUMERICAL simulations and comparison with theoretical results

In the present part the results of numerical simulations of flow and transport in media made up from circular inclusions of conductivity K that are submerged in a matrix of conductivity K ₀, subjected to uniform mean velocity, are presented. This is achieved for a few values of =K/K ₀ (0.01, 0.1 and 10) and of the volume fraction n (0.05, 0.1 and 0.2). The numerical simulations (NS) are compared with the analytical approximate models presented in Part 1: the composite elements (CEA), the effective medium (EMA), the dilute system (DSA) and the first-order in the logconductivity variance (FOA). The comparison is made for the longitudinal velocity variance and for the longitudinal macrodispersivity. This is carried out for n<0.2, for which the theoretical and simulation models represent the same structure of random and independent inclusions distribution. The main result is that transport is quite accurately modeled by the EMA and CEA for low , for which _L is large, whereas in the case of =10, the EMA matches the NS for n<0.1. The first-order approximation is quite far apart from the NS for the values of examined . This material is based upon work supported by the National Science Foundation under Grant No. 0218914. Authors also wish to thank the Center of Computational Research, University at Buffalo for assistance in running numerical simulations.