Paper Abstracts

Series Solutions for Flow in Stratified Aquifers with Natural Geometry
(Advances in Water Resources, 2010)

Sanders Wong and James R. Craig

An analytical series solution method is presented for modeling regional steady-state groundwater flow in a 2-dimensional stratified aquifer cross-section where the water table is well-characterized. The aquifer system may have any number of contiguous or non-contiguous layers and the geometry of each layer is restricted only by the requirement that the elevation of the stratigraphic unconformities between layers is a function of the x-coordinate alone. Various techniques may be used to handle pinching layers, faults, and other discontinuities. The solutions are obtained by minimizing head and flow continuity errors between layers and errors in the Dirichlet surface at a set of control points along these unconformities; the governing equation is met exactly. The solutions are derived and demonstrated on multiple test cases. The errors for some specific, geometrically challenging cases are assessed and discussed.

Analytic elements for flow in harmonically heterogeneous aquifers
(Water Resources Research, 2009)

James R. Craig

Existing implementations of the analytic element method for simulating groundwater flow rely upon a zonal representation of hydraulic conductivity, where the conductivity is piecewise-constant within the modeled domain. Using a dependent variable transformation, the analytic element method is extended to simulate two-dimensional groundwater flow through smoothly heterogeneous media represented using pilot points, i.e., the conductivity field is interpolated between a set of locations with specified conductivity. This new implementation maintains the existing attributes of AEM solutions: the governing equation is met exactly within the domain and boundary/continuity conditions are met with extremely high precision, all without volumetric discretization. A number of test cases are used to demonstrate and evaluate the efficacy of the new method.

Coordinate mapping of analytical contaminant transport solutions to non-uniform flow fields
(Advances in Water Resources, 2009)

James R. Craig and Thomas Heidlauf

Existing analytical solutions to 2D and 3D contaminant transport problems are limited by the mathematically convenient assumption of uniform flow. An approximate method is developed herein for coordinate mapping of 2D (vertically-averaged) transport solutions to non-uniform steady-state irrotational and divergence-free flow fields in single-layer aquifers. The method enables existing analytical transport solutions to be applied to aquifer systems with wells, non-uniform saturated thickness, surface water features, and (to a limited degree) heterogeneous hydraulic conductivity and recharge. This mass-conservative coordinate mapping approach is inexact in its approximation of the dispersion process but is still sufficiently accurate for many simple flow systems. The degree of model error is directly proportional to the variation of velocity magnitude within the domain. These mapped analytical solutions are compared to numerical simulation results and the coordinate mapping errors are investigated. The methods described herein may be used in the traditional capacity of analytical transport models, i.e., screening and preliminary site assessment, without sacrificing accuracy by assuming locally uniform flow conditions or applying an ad-hoc coordinate transformation. The solutions benefit from the traditional advantages of analytical methods, particularly the removal of artifacts due to spatial and temporal discretization: no time-stepping or numerical discretization is required.

Topography-driven flow in stratified sloping and syncline aquifers
(Advances in Water Resources, 2008)

James R. Craig

Two analytical solution methods are presented for regional steady-state groundwater flow in a two-
dimensional stratified aquifer cross section where the water table is approximated by the topographic surface. For the first solution, the surficial aquifer is represented as a set of dipping parallel layers with different, but piecewise constant, anisotropic hydraulic conductivities, where the anisotropy is aligned with the dip of the layered formation. The model may be viewed as a generalization of the solutions developed by [Tóth JA. A theoretical analysis of groundwater flows in small drainage basins. J Geophys Res 1963;68(16):4795–812; Freeze R, Witherspoon P. Theoretical analysis of regional groundwater flow 1) analytical and numerical solution to the mathematical model. Water Resour Res 1966;2(4):641–56; Selim HM. Water flow through multilayered stratified hillside. Water Resour Res 1975;11:949–57] to an multi-layer aquifer with general anisotropy, layer orientation, and a topographic surface that may intersect multiple layers. The second solution presumes curved (syncline) layer stratification with layer-dependent anisotropy aligned with the polar coordinate system. Both solutions are exact everywhere in the domain except at the topographic surface, where a Dirichlet condition is met in a least-squared sense at a set of control points; the governing equation and no-flow/continuity conditions are met exactly. The solutions are derived and demonstrated on multiple test cases. The error incurred at the location where the layer boundaries intersect the surface is assessed.


Finite element modeling of contaminant transport using analytic element flow solutions
(Water Resources Research, 2006)

James R. Craig and Alan J. Rabideau

Finite element methods for solute transport simulation typically use a discrete representation of the flow domain obtained from a finite element solution of the associated groundwater flow problem. Velocity, saturated thickness, and components of the dispersion coefficient tensor are represented as a set of nodal and/or element-averaged values. In contrast, the analytic element method (AEM) provides continuous mesh-independent solutions for these variables. In this paper, a set of techniques for using two-dimensional AEM flow solutions as the basis of finite element solute transport models is introduced. First, a general AEM-based discretization approach is presented that addresses the existence of curved boundaries, singularities, and discontinuities in vertically-averaged concentration. Second, residual integration methods that handle continuous parameters with internal and boundary singularities are developed and evaluated. Third, an approach is introduced for handling internal discontinuities in concentration across certain analyticelements. This new approach uses a non-standard mesh topology and a new formulation for internal coupled boundary conditions. The AEM-based transport simulation methods introduced in this paper are demonstrated to be robust and accurate for a variety of test problems.

Analytic element modeling of Supra-regional Groundwater Flow: Concepts and Tools for Automated Model Configuration
(Journal of Hydrologic Engineering, 2007)

Alan J. Rabideau, James R. Craig, Warit Silaviserith, Douglas M. Flewelling, Kyle Frederick, Matthew W. Becker, L. Shawn Matott, Igor Janković, and Karl Bandilla

The Analytic Element method (AEM) is an appealing technique for modeling steady-state groundwater flow at the supra-regional scale (> 10,000 sq. km.) because the computational demand is determined primarily by the number of modeled hydrologic features and not constrained by the size of the domain.  In this paper, we present modeling concepts and tools designed to facilitate the automated processing of AEM models containing thousands of hydrologic features.  Topics include assignment of element types for surface water features, automated simplification of lines and polygons, conversion of polygonal elements to less computationally demanding circles and ellipses, iterative solution algorithms for models that include nonlinear resistance elements, software implementation, and integration of AEM simulators with calibration utilities and GIS.  Software implementation of the concepts and tools is discussed and demonstrated for a case study of Wisconsin’s Northern Highland Lakes Region. 
 
Finite Difference Modeling of Contaminant Transport using Analytic Element Flow Solutions
(Advances in Water Resources, 2006)

James R. Craig and Alan J. Rabideau
 

The two-dimensional implementation of the analytic element method (AEM) is commonly used to simulate a variety of steady-state saturated groundwater flow phenomena at regional and local scales. However, unlike alternative groundwater flow simulation methods, AEM results have never been used as the basis for robust simulation of reactive solute transport. The use of AEM-simulated flow fields is inhibited by the discrepancy between a continuous representation of flow and a typically discrete representation of transport. Improper translation of AEM flow fields to a discrete analog for use in finite difference-based transport simulation can lead to significant errors in solute mass balance or solute mass distribution. This paper presents a variety of methods for analytically calculating conservative discrete water fluxes and integrated dispersion coefficients across cell interfaces, thus enabling the use of AEM flow solutions for accurate reactive transport simulation. Eulerian and Eulerian-Lagrangian finite difference methods are implemented for use in 2D (vertically-averaged) simulation of solute transport. These approaches are benchmarked against existing methods that use finite-difference derived flow parameters for simulation of advection and dispersion.
  Pump-and-treat optimization using analytic element method flow models
(Advances in Water Resources, 2006)

L.Shawn Matott, Alan J. Rabideau, and James R. Craig

Plume containment using pump-and-treat (PAT) technology continues to be a popular remediation technique for sites with extensive groundwater contamination. As such, optimization of PAT systems, where cost is minimized subject to various remediation constraints, is the focus of an important and growing body of research. While previous pump-and-treat optimization (PATO) studies have used discretized (finite element or finite difference) flow models, the present study examines the use of analytic element method (AEM) flow models. In a series of numerical experiments, two PATO problems adapted from the literature are optimized using a multi-algorithmic optimization software package coupled with an AEM flow model. The experiments apply several different optimization algorithms and explore the use of various pump-and-treat cost and constraint formulations. The results demonstrate that AEM models can be used to successfully optimize the number, location and rates of wells in a pump-and-treat containment system. Furthermore, the results illustrate that constraints placed on the zone budget output of AEM models can be used to effectively enforce plume capture. Such constraints are shown to be efficient and reliable alternatives to conventional particle capture and gradient control techniques. Finally, the particle swarm optimization (PSO) technique is identified as an effective algorithm for solving pump-and-treat optimization problems. A parallel version of the PSO algorithm is shown to have linear speedup, suggesting that the algorithm is suitable for application to computationally demanding problems involving large numbers of wells.

 Analytical Expressions for the Hydraulic Design of Continuous Permeable Reactive Barriers
(Advances in Water Resources, 2006)

James R. Craig, Alan J. Rabideau, Raghavendra Suribhatla

Various analytical expressions describing the hydraulic behavior of a continuous permeable reactive barrier (PRB) are developed based upon a two-dimensional approximation of the local groundwater flow system. The fully-penetrating PRB is represented as an arbitrarily-oriented elliptical "analytic element" with a hydraulic conductivity different from that of the aquifer. The validity of this elliptical geometry approximation as a surrogate for rectangular PRB performance is evaluated and put into context. Closed-form expressions for solute travel time distributions along the extent of the barrier and PRB capture zone geometry are evaluated for general barrier dimension (length and width), hydraulic conductivity, and orientation with respect to regional flow. These expressions are used as the foundation of a simple PRB design process, and provide some interesting insights into the hydraulic behavior of continuous permeable reactive barriers.

Analytical Models for the Design of Iron-based Permeable Reactive Barriers
(Journal of Environmental Engineering, 2005)

Alan J. Rabideau, Raghu Suribhatla, and James R. Craig

The preliminary design of iron-based permeable reactive barriers is often accomplished using analytic expressions for one-dimensional groundwater flow and contaminant transport. Typically, one or more of the governing processes is simplified or neglected to facilitate development of a tractable solution. This paper presents a set of improved design equations that include the effects of dispersion, finite domain boundary, sequential decay and production processes, and increased flow through high conductivity barriers. When applied to realistic example problems, application of the expanded design equations typically results in the specification of a larger PRB thickness than obtained using conventional approaches.

The Nested Superblock Approach for Regional Scale Analytic Element Models
(Ground Water, 2006)

James R. Craig, Igor Jankovic, and Randal Barnes

A new approach is presented for improving the computational efficiency of regional-scale ground water models based on the analytic element method (AEM). The algorithm is an extension of the existing "superblock" algorithm, which combines the effects of multiple analytic elements into Laurent series and Taylor series (superblock expansions). With the new "nested superblock" formulation, Laurent series are nested in a hierarchical (quad-tree) data structure with direct mathematical relationships between parent and child superblock coefficients. Nested superblocks significantly accelerate the evaluation of the complex potential and discharge function in models that contain a large number of analytic elements at multiple scales. This evaluation process, the primary computational cost of AEM models, is required to determine the element coefficients, generate contour plots, and trace pathlines. The performance of the nested superblocks is demonstrated with a simplified model based on the Lake Ontario watershed geometry comprised of thousands of hydrogeologic features at multiple geographic scales. 

Influence of numerical precision on the calibration of AEM-based groundwater flow models
(Environmental Geology, 2005)

Alan J. Rabideau, L. Shawn Matott, Igor Jankovic, James R. Craig, Matthew Becker

Ground-water modelers have embraced the use of automated calibration tools based on classical nonlinear regression techniques. While clearly an improvement over trial-and-error calibration, it is not clear to what extent these popular inverse modeling tools yield accurate parameter sets for ground-water flow models. The impact of model configuration and precision upon automated parameter estimation is also unclear. An extensive set of numerical experiments was performed to explore the influence of model configuration on the calibration of a regional ground-water flow model developed using the Analytic Element Method. The results provided insight into the manner in which the specified level of model precision and the location of observation points influence the results of inverse modeling based on nonlinear regression. While the importance of these issues is application specific, obtaining an accurate model calibration for the case study required both a careful placement of test observations and a greater-than-anticipated level of model precision. The required level of model precision for calibration was more than necessary to produce an acceptable flow solution.

Contaminant Transport Modeling using Analytic Element Flow Solutions
(Ph.D. Dissertation, 2005)

James R. Craig

A new approach to modeling reactive contaminant transport in groundwater is developed and evaluated. The approach is unique in that it uses a grid- or mesh-independent representation of model input parameters, including continuous velocities, dispersion coefficients, and saturated thickness values obtained directly from analytic element groundwater flow solutions.

The approach is realized within a suite of revised finite element, finite difference, and characteristic methods that are designed to improve the accuracy and reduce the computational costs of complex reactive vertically-averaged transport simulations in surficial aquifers. These methods are implemented in a fully object-oriented parallel-friendly software framework, benchmarked against existing analytic and numerical solutions, tested against traditional discrete methods, and applied to a set of difficult field-scale test problems.

It was found that the majority of the methods benefited from continuous representation, and that the use of the analytic element method can facilitate the development of computationally efficient multi-scale reactive transport models.

Importantly, this work represents the first thorough implementation of a linkage between reactive contaminant transport models and the analytic element method for modeling groundwater flow, and the first detailed analysis of such a linkage. 

 

Conference Paper Abstracts

Combining the strengths of analytic element and finite element methods for mixed-scale simulation modeling
  (Modflow & More 2006)

James R. Craig

The analytic element method (AEM) is a very effective technique for simulation of regional-scale groundwater flow phenomena. Unlike traditional discrete simulation models, AEM models are not limited by geographic extent, but rather by the number of hydrogeologic features included. However, AEM is constrained by its current inability to efficiently simulate flow in transient, fully 3D, and/or highly heterogeneous systems. Standard discrete methods, including finite element methods, can efficiently simulate these more complicated systems, but are often limited by system scale, i.e., the number of nodes required to resolve both regional- and local-scale detail can lead to a computationally intractable problem. Here, some initial approaches are proposed for overcoming these obstacles through the integration of finite element and analytic element methods.

First, the AEM has been used to provide simple fixed-head or flux boundary conditions to a nested 2D FEM groundwater flow model. The 2D vertically-averaged transient FEM sub-model has been implemented and tested in an existing AEM code. The sub-model is based upon the three-node iso- and superparametric finite elements developed by the author and colleagues for reactive contaminant transport simulation (Craig and Rabideau, submitted). The nested model has been coupled through an iterative computational scheme, allowing for feedback from the local system to the regional.

In conjunction with work on the nested model approach, a novel method for superimposing FEM solutions and analytic element solutions is being developed. It is expected this mixed-scale hybrid method will benefit from both the spatial resolution of AEM and the local robustness of FEMs. It is known that superposition of independent solutions to the linear governing equation for groundwater flow is feasible; it is, in fact, the basis of the analytic element method. However, multiple difficulties lie in properly posing the finite element system of equations, accurately evaluating the new continuous integrals introduced by the analytic element solution, and evaluating or constraining the discharge from the superimposed finite element model at mesh boundaries. Hence, only a discussion and preliminary results of the hybrid superimposed AEM-FEM simulation method will be presented.

An overview of using analytic element flow solutions for contaminant transport simulation
(ICAEM 2006)

James R. Craig, Alan J. Rabideau, and Karl Bandilla

Newly developed techniques have enabled the robust use of analytic element flow solutions for reactive contaminant transport modeling. Vertically-averaged finite difference, finite element, streamline, and characteristic methods have recently been revised to directly utilize AEM flow solutions with minimal loss of accuracy due to discretization of the transport problem. An overview of these methods is presented and suggestions are made regarding the future of AEM-based transport simulation, including the extension to multi-layer, transient, and 3D analytic flow models.  

The area vortex for modeling flow through smoothly heterogeneous aquifers
(ICAEM 2006)

James R. Craig

The 2-dimensional single layer analytic element method is typically applied to aquifers characterized by uniform or piecewise-constant aquifer properties such as hydraulic conductivity. In systems characterized by significant gradients in aquifer base, thickness, or conductivity, the standard library of analytic elements is computationally inefficient—to accurately simulate flow through such systems, fine discretization of system properties is often necessary. The discretization process also neutralizes one of the primary strengths of the analytic element method, i.e., that only boundaries of the system require discretization. The area vortex, proposed by Strack [Strack, O.D.L. (1999). Principles of the analytic element method. Journal of Hydrology 226 (3-4), 128–138] may be used to circumvent this need for discretization by explicitly adding curl to the vector flow field, just as the area-sink introduces divergence. The solution for the area-vortex may be superimposed upon standard analytic element solutions. Unfortunately, the development and application of the area vortex is not as clear-cut as the implementation of a standard area-sink. Multiple factors complicate the application of such an element: (1) the potential function is undefined in regions with non-zero curl (2) the curl itself is a function of the discharge in the system, and (3) the spatial distribution of the curl function is neither uniform nor easily represented with unsmoothed radial basis functions, as used for the multi-quadric area sink of Strack and Janković [Strack, O.D.L. and I. Jankovi´c (1999). A multi-quadric area-sink for analytic element modeling of groundwater flow. Journal of Hydrology 226 (3-4), 188– 196]. In this paper, multiple methods are presented for circumventing these difficulties, including the use of an artificial potential, means of numerically integrating the discharge function, the use of an iterative solution method, and the development of a smoothed radial basis function solution to the Poisson equation.

Quantifying the Efficacy of Multicriteria Generalization (MCG) of Geospatial Data for AEM Groundwater Modeling
(TIES 2004)

Gaurav Sinha, Warit Silavisesrith, James R. Craig, and Douglas M. Flewelling

Regional scale environmental simulation models need generalization of high resolution data to multiple lower resolutions to obtain an acceptable numerical solution within a reasonable interval of time. Methods for geospatial data generalization have been developed in the past but mainly for addressing cartographic concerns. We therefore use a new framework called Multicriteria Generalization (MCG) that generalizes geospatial data under constraints determined both by cartographic and geophysical considerations. These constraints are derived from experts’ domain knowledge and realized within the generalization system as multiple interactive criteria. This paper evaluates the efficacy of MCG for a groundwater model (SPLIT) that relies on the vector-based analytic element method (AEM) to conceptualize and implement the groundwater system. The tradeoff between computation time and the errors introduced in model predictions is analyzed at several generalization levels for different weighted combinations of input criteria. To minimize the uncontrolled perturbations introduced within the groundwater model due to modifications of elemental interaction patterns, only polylinear analytic elements are generalized in this study. The results are used to advocate the future use of this framework for applications other than analytic element models of groundwater flow.

Discretization of Analytic Element Flow Solutions for Transport Modeling
(CMWR XIV, 2004)

James Craig and Alan Rabideau

The 2D analytic element method (AEM) is an alternative to finite difference methods for simulating steady-state unconfined groundwater systems. AEM is commonly used for capture zone delineation, but has not yet been used as a basis for contaminant transport simulation. However, the analytic element method is a promising source of flow information in that it is independent of any grid, and thus suitable for developing less constrictive reactive transport discretizations. One of the reasons AEM has never been used in this capacity is that most conventional contaminant transport simulation algorithms are based upon discrete finite difference flow solutions, whereas AEM represents the flow variables needed for transport as continuous functions of space. While it is adequate to numerically integrate cell-averaged saturated thickness values for use in transport simulations, discretizing system fluxes is a more demanding process. Eulerian transport algorithms require local (cell-by-cell) mass balance of water for both accuracy and stability. While the analytic element method provides exceptionally precise globally and locally conservative results, simply interpolating the analytic functions that describe flux in AEM is not adequate for obtaining mass-conservative interfacial fluxes. Likewise, the vector geometry of the analytic element method makes it challenging to accurately distribute the various sink and source terms to grid cells. This includes terms for the influence of leakage and recharge, which may also be represented as continuous functions of space. The following paper presents a suite of algorithms used to properly translate the continuous aquifer fluxes and vector-based source and sink fluxes into a format amenable to finite-difference-based transport simulation.

Iterative Solutions for the Analytic Element Method:
Recent Computational Advances and Object Oriented Design
(ICAEM 2004)

James R. Craig, Karl Banidlla, and Igor Jankovic

The iterative solution procedure for analytic element-based models is an alternative to the traditional explicit method. Instead of solving a single system of equations simultaneously, the iterative method cycles through the system of elements. Each element solves for its own coefficients based upon the most recent solution. Recent developments in the representational and computational aspects of the iterative procedure have allowed for faster performance, resolution of convergence issues, and more elegant code organization. Four primary advances in the iterative method have contributed to improvements in speed, stability, and modularity. Parallelization has resulted in near-linear speedup for domains with millions of degrees of freedom. The nested superblock approach results in significantly improved speedup of the solution for models containing elements at disparate geographic scales. Improved understanding of convergence behavior has allowed for implementation of more intelligent relaxation schemes and continuous correction schemes, providing stability of solution for nearly all model configurations. Lastly, implementation of the iterative algorithm inside the fully object-oriented code BLUEBIRD enables modifications and additions to be performed with minimal intrusion. The combined influence of all of these advances in the iterative method has allowed for faster, more stable solution of massive domains which were previously impossible to model.

Linking the Analytic Element Method to Reactive Contaminant Transport Models
(Modflow & More 2003)

James Craig and Alan Rabideau

The analytic element method (AEM) is an alternative solution method for steady state saturated groundwater flow problems, one which is particularly effective for modeling two-dimensional confined/unconfined systems which operate under the Dupuit-Forcheimer assumption. The analytic element method is independent of the grid or mesh associated with many numerical techniques and provides a continuous solution for piezometric head and discharge at any location in the model domain. It is expected that these solution attributes may prove useful as the basis for the advective portion of contaminant transport models and may reduce the grid dependence of a wide array of transport simulation methods. However, the use of AEM flow solutions as the basis for transport models has, to this point, been limited to basic random walk simulations or identification of purely advective travel time distributions.

A combined AEM flow and characteristic/streamline-based transport simulation software library, Bluebird/Cardinal has been developed to test multiple hypotheses regarding the efficacy of using AEM as a basis for simulating advective transport of reactive contaminants. The program has been used to examine 1) the effects of velocity interpolation on mass balance, 2) the decoupling of reaction and hydrodynamic transport, and 3) the possibilities for grid-independent transport simulation. In addition, an object-oriented framework for linking generic transport methods to generic flow simulation methods has been developed. Expected benefits of this object-oriented merger between AEM and transport simulators include improved solution of transport problems that require highly accurate pathlines or problems that benefit from discretization based solely upon chemical considerations.


Conference Presentation Abstracts

Coordinate mapping of analytical transport solutions to non-uniform flow fields
(AGU  Fall Meeting 2007)

James R. Craig

While analytical solutions for solute transport continue to be used as screening and predictive tools for contaminated site analysis, there remain many limitations to their applicability. Despite the recent advances in their ability to simulate complex chemistry (e.g., parent-daughter decay and mixing-driven chemical equilibrium), most existing analytical transport models depend upon the critical assumptions of uniform flow and uniform saturated thickness. A time-of-flight-based mapping procedure has been developed to take existing analytical solutions for transport in uniform flow and map them to simple non-uniform flow fields. This presentation will discuss some of the inherent assumptions in the mapping method, as well as its strengths and its weaknesses. It was found that the mapping approach, exact for purely advective transport problems, can be reliably used in steady-state flow systems with wells, surface water features, and (to a limited degree) mild recharge and heterogeneity. The mapping errors are on the same magnitude as those due to discretization error or upstream weighting in numerical simulation methods. 
Back to Papers

Incorporating Search History into the Dynamically Dimensioned Search (DDS) Optimization Algorithm
(AGU  Fall Meeting 2007)

Bryan A. Tolson, James R. Craig, M. A. Esfahani

The Dynamically Dimensioned Search (DDS) algorithm (Tolson and Shoemaker, 2007) was recently introduced as a parsimonious, efficient and robust optimization algorithm for automatic calibration of environmental models. DDS was designed to find practical or high quality solutions to a model calibration problem within a reasonable computational timeframe rather than the globally optimal solution. The simple structure of the original DDS algorithm only stores and utilizes the best current solution to guide the search. Population-based global optimization algorithms maintain a population of typically good quality solutions to influence the search. In this research, we examine how to utilize the search history of DDS to improve algorithm performance while maintaining the parsimonious and algorithmically simple nature of the original DDS algorithm. The modification to the original DDS algorithm involves storing a subset of relatively high quality solutions previously identified in the search and selecting one solution from which to make the next perturbation in order to sample a new candidate solution. Both the function value and their proximity to one another in multi-dimensional parameter space influences the likelihood of selecting a particular solution to perturb. This approach is motivated by initial results showing that for the same total computational budget, DDS with multiple restarts can sometimes be more effective than one longer DDS optimization trial. The history-based revisions discussed above allow the algorithm to search more of the parameter subspace, thus exploiting the strength of the less- refined restart approach, but with a higher likelihood of success. Results will be presented for a relatively simple problem as well as a more complex, high-dimensional automatic calibration problem. Results will also be assessed for various computational budgets.
 
Back to Papers

An Open-source Community Web Site To Support Ground-Water Model Testing
(AGU  Fall Meeting 2007)

Stephen R. Kraemer, Mark Bakker, James R. Craig

A community wiki wiki web site has been created as a resource to support ground-water model development and testing. The Groundwater Gourmet wiki is a repository for user supplied analytical and numerical recipes, howtos, and examples. Members are encouraged to submit analytical solutions, including source code and documentation. A diversity of code snippets are sought in a variety of languages, including Fortran, C, C++, Matlab, Python. In the spirit of a wiki, all contributions may be edited and altered by other users, and open source licensing is promoted. Community accepted contributions are graduated into the library of analytic solutions and organized into either a Strack (Groundwater Mechanics, 1989) or Bruggeman (Analytical Solutions of Geohydrological Problems, 1999) classification. The examples section of the wiki are meant to include laboratory experiments (e.g., Hele Shaw), classical benchmark problems (e.g., Henry Problem), and controlled field experiments (e.g., Borden landfill and Cape Cod tracer tests). Although this work was reviewed by EPA and approved for publication, it may not necessarily reflect official Agency policy. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.
 
Back to Papers

Handling continuous and singular parameter fields in mixed finite element-analytic element models of flow and transport
(ModelCARE 2007)

James R. Craig

Most finite element methods (FEMs) for simulating groundwater flow and solute transport rely upon a discrete representation of the specified independent parameters. In flow models, hydraulic conductivity and storage coefficients are specified as uniform within each element. Likewise, in solute transport models, velocity, dispersion tensor components, and porosity are typically treated as piecewise-constant. If these parameters (and their respective contributions to finite element residual integral expressions) are singular, alternative integration and discretization approaches are required.

In most cases, a discrete approach is more than sufficient. However, new approaches for reducing discretization requirements near sinks and sources in finite element flow and transport models have recently been developed. These approaches require that the finite element model be able to handle smoothly continuous and singular parameter distributions. For transport models, analytic element method (AEM) flow solutions are used to parameterize the transport model, thereby removing the need for high discretization near wells and surface water features. For flow modeling, a novel mixed AEM-FEM has been developed based upon superposition: the analytic element method is used to simulate regional steady-state flow, sinks, and sources and the FEM is used to resolve local detail, transience, and heterogeneity. Both methods require the ability to evaluate both continuous and singular finite element residual integrals. Some intermediate results of investigations into the handling and behavior of the resultant system integrals are addressed and discussed within the context of integrated or hybrid AEM / FEM solution methods. 

 

Extending the applicability of analytical contaminant transport models
(EGU  General Assembly 2007) [Invited]

James R. Craig

Both numerical and analytical subsurface transport models are used to support important policy decisions regarding the management of polluted aquifers. Numerical models are able to simulate complex reactive transport phenomena, but can be time-consuming to construct, difficult to validate, and subject to numerical and user error. These models typically include tens to hundreds of input parameters. In contrast, simpler analytical models often capture the key features of reactive transport using a smaller set of "lumped" input parameters. They are free from computational error and are typically easier to apply, understand, and calibrate. However, due to necessary simplifying assumptions, particularly that of uniform flow, they are currently limited in their applicability.

The research project described herein addresses two key limitations of existing 2D and 3D analytical models through the development of novel time-of-flight mapping techniques and more flexible treatment of source terms and geometry in analytical solutions. The time-of-flight coordinate mapping approach allows the existing library of 2D and 3D solute transport solutions to be mapped to non-uniform flow fields with wells, surface water features, and recharge. Preliminary results from the development of these new methods are introduced for discussion and error analysis is applied to interpret the error incurred from the mapping procedures. These initial improvements of analytical transport models are hoped to contribute to a powerful new suite of hybrid analytical-numerical tools for science-based environmental management.

Topography-driven flow in a stratified sloping aquifer: A general semi-analytical solution
(AGU Fall 2006)

James R. Craig

A general closed-form solution for steady-state cross-sectional flow in a stratified aquifer is developed, and may be viewed as a robust generalization of the topography-driven solutions presented by Toth (1963), Freeze and Witherspoon (1966), and Selim (1975) to the M-layer case where the topographic surface may intersect an arbitrary number of sloping anisotropic layers. Both continuity conditions at the layer interfaces and no-flow conditions along the bottom and sides of the aquifer system are met exactly. The head-specified condition at the known topographic surface is met in an approximate (but precise) manner using a least-squares formulation at a set of control points derived from digital elevation models. The highly accurate solution may be used to test numerical solvers and rapidly determine subsurface flow patterns in topography-driven subsurface flow systems. The models are quickly generated from a single DEM and a set of borehole data along a cross-section. Similar approaches are currently being developed for two-dimensional flow in a syncline aquifer and fully three-dimensional systems.

Development of a Regional-Scale Groundwater Modeling System for Research, Education, and Outreach
(AEESP 2005)

James R. Craig, Alan J. Rabideau, Matthew Becker, Karl Bandilla, Douglas Flewelling, Kyle Fredrick, Igor Jankovic, L. Shawn Matott, Warit Silvasreith

Regional-scale models of groundwater flow and contaminant transport can be used to simulate large-scale environmental processes occurring in multiple adjacent watersheds. Understanding the behavior of regional aquifer systems to external pressures (e.g., climate change, urbanization strategies, large-scale groundwater withdrawal, non-point source pollution) is essential to developing sound regional water resources, environmental, and ecological management policies. Until recently, regional groundwater models were limited in extent by computational constraints. However, researchers at the University at Buffalo have recently developed a variety of algorithms that enable the development of super-regional analytic element models on the order of hundreds of square kilometers in spatial extent.

While these models are now tractable from a computational standpoint, there are a variety of limitations that impede the construction of viable models and the understanding of their results. Construction and evaluation of a model with thousands of hydrogeologic features requires a significant user effort. Pertinent aquifer parameters must be specified across the modeled domain, the geometry of hydrologic features (e.g., rivers) must be specified, and the results of models and model calibrations must be both physically and conceptually accessible to the modeler.

To address these needs, the University at Buffalo Groundwater Research Group has developed a variety of simulation codes and user interfaces to enable the construction and understanding of both local- and regional-scale models. High-performance groundwater flow models (SPLIT and BLUEBIRD), contaminant transport models (CARDINAL), and parameter estimation/calibration tools (OSTRICH) have been incorporated into two GIS-integrated user interfaces, ARCAEM and VISUAL BLUEBIRD. These interfaces use a variety of methods to simplify the model creation and data visualization process, including feature simplification, robust data management, extensive user guides, intelligent interface design, and robust model quality checking features. These interfaces are currently being used in multiple undergraduate and graduate courses at UB, are the basis of an annual short course for professional engineers, and are widely distributed on the web, having a user base in more than 20 countries worldwide.

Optimal Mesh Generation for AEM-based Eulerian Transport Simulators
(AGU Fall 2004)

James R. Craig, Alan J. Rabideau, and L. Shawn Matott

The analytic element method (AEM) is a grid -independent approach for simulating groundwater flow in shallow aquifer systems. Recent advances have enabled AEM flow solutions to be used as the basis for Eulerian (finite difference or finite element) contaminant transport simulators. One of the benefits of such a merger is the removal of the constraints imposed by the flow grid or mesh. The resultant model discretization may be specified to accommodate only the relevant transport constraints (i.e., the Peclet and Courant limitations). Design of the mesh is typically limited by discretization requirements of the flow problem. With the use of AEM flow solutions, grid and mesh geometry may be optimized for a specific transport system without regard for flow system discretization.

A two-dimensional mesh generation algorithm is presented that maximizes the required node spacing (governed by Peclet limitations) and therefore the time step (governed by Courant limitations) required for transport models using the spatially continuous velocities and dispersion coefficients obtained from AEM flow solutions. The optimized meshes reduce the computational cost of contaminant transport models by reducing the total number of degrees of freedom. Results from the optimized mesh algorithm illuminate some artifacts of flow discretization that are commonly neglected. The increased computational efficiency of models simulated without these discretization artifacts is quantified, and some non-intuitive results concerning the optimal mesh design for transport simulation are presented.

Automated Geographic Simplification Tools for Development of Regional Scale Groundwater Flow Models
(AGU Fall 2004)

James R. Craig, Gaurav Singha, Douglas M. Flewelling, Warit Silavisesrith, Alan J. Rabideau

The analytic element method is well suited for modeling regional scale saturated groundwater flow. Recent advances enable the solution of models with tens of thousands of hydrogeologic features over scales of hundreds of kilometers. In order to implement such models, automated techniques are desired to translate regional scale conceptual models and/or readily available hydrologic base maps into model features. A suite of tools derived from standard cartographic generalization operators have been developed to perform these simplification tasks. Highly detailed digitized surface features (e.g. river and lake boundaries) are simplified into representative elements and strings of elements using algorithms designed to capture important geometric and physical properties. These simplified models are more computationally efficient and achieve similar (often nearly identical) results. In addition, a general framework for application of simplification operators to vector-based numerical models has been developed.

Role of the analytic element method in regional-scale GIS -based modeling of groundwater flow and transport
(AGU Fall 2003)

Igor Jankovic, Karl Bandilla, James Craig, and Alan Rabideau

The basic principle of the Analytic Element Method (AEM) is the superposition: complex regional-scale flows are simulated by adding the influences of individual analytic elements. Each analytic element contains geographic information and mathematical functions that describe its influence on regional groundwater flow. The main AEM advantage, relevant to GIS-based modeling, is a direct correspondence of analytic elements and hydrologic aquifer features (such as lakes, rivers, water-supply wells). The flow solution (water table elevation and fluxes) is created without discretization by including all relevant elements (or features). Several regional-scale groundwater models have been developed using the AEM include NAGROM (the National Groundwater Model of The Netherlands) and the Twin Cities Metropolitan Area Groundwater Model (Metro Model). The one-to-one correspondence of elements to hydrologic features allows for elegant integration of groundwater modeling and GIS software. This simplifies both model development and management. In contrast to finite-difference and finite-element based methods, the AEM does not require any grid-based data; all input data are vector-based. This representational style coincides well with the strengths of Geographic Information Systems.

This presentation will focus on both the practical aspects of the AEM relevant to GIS-based modeling and the development of a new GIS based user interface for AEM modeling. Recent computational advances related to regional-scale modeling using AEM will be discussed and demonstrated. These advances include (1) development of solution algorithms designed for massively parallel supercomputers where the same advantage (one-to-one correspondence of analytic elements and hydrologic features) is responsible for efficient implementation in parallel environments, and (2) algorithm enhancements designed for single-processor machines. Groundwater models with thousands of elements can hence be solved. A public domain GIS graphical user interface has been developed to aid in model construction and output visualization within the ArcGIS

Visual Bluebird: Software for Teaching Groundwater Modeling and Potential Flow to Undergraduate Students
(FAME 2003)

James Craig, Igor Jankovic, and Alan Rabideau

A new groundwater modeling program, Visual Bluebird, has been developed as an easy-to-use educational software tool to help familiarize students with groundwater modeling and basic fluid mechanics. The public domain software, based upon the two-dimensional analytic element method, has been used effectively for undergraduate and graduate projects, including remediation assessment and capture zone determination. It has also been used, in conjunction with laboratory apparatus, as a visualization tool to help students gain an understanding of simple potential flow problems. The simple design, comprehensive help files, and extensive error-checking modules allow the student to learn groundwater modeling without the high learning curve associated with commercial software.

Vertically Averaged Contaminant Transport with the Streamline method in Near-Surface aquifers
(EGU-AGU-EGS 2003)

James Craig, Alan Rabideau

Contaminant transport modeling is an important tool for assessing the impact of pollution in the subsurface. Transport modeling in unconfined shallow aquifers is particularly important because of aquifer susceptibility to contamination and direct linkage to surface water bodies. Such “near-surface” aquifers are often characterized by transience, sensitivity to recharge, and sensitivity to surface water behavior. Such aquifers are also often affected by non-point pollution. Models can be developed that accurately depict the behavior of contaminant transport in near-surface aquifers on watershed scales in order to effectively characterize the risk associated with multi-source contamination. These models may provide information that may be able to drive policy decisions for water use or environmental regulation on a watershed-by-watershed basis.

The streamline method, which has been effective in modeling miscible and immiscible displacement in petroleum engineering, has been adapted for use atop a flow solution provided by the analytic element method (AEM). The AEM effectively models regional scale unconfined flow with recharge by using the superposition of analytic functions associated with hydrologic features (i.e. rivers, wells). Recent attempts have been made by the authors to link contaminant transport models with AEM. The linkage takes advantage of the continuous velocities provided by analytic flow fields. In order to have a similar conceptual model for both transport and flow (which relies upon the Dupuit Forcheimer assumption), models have initially been constrained to 2D vertically averaged transport.

While the streamline method has been successfully revised for contaminant transport simulation in confined aquifers, it has not yet been applied to simulation of vertically averaged transport with recharge. Techniques are proposed to handle the vertical averaging of contaminant concentration and the influence of recharge upon streamline representation. These techniques allow for rapid simulation of complex contaminant transport systems on a geographic scale greater than was previously possible. The revisions also indirectly allow for inclusion of regional scale contaminant transport of non-point source pollution as input into local scale models.

Reducing Dependence upon "the Grid":
A Framework for Contaminant Transport Modeling using Analytic Flow Solutions
(AGU Fall 2002)

James Craig and Alan Rabideau

<>The development of localized reactive remediation strategies (i.e. reactive barriers, bioremediation, chemical oxidation, etc.) has progressed rapidly in recent years. With such progress, there is a distinct need for contaminant transport models that simulate complicated reaction phenomena at much finer scales than the extent of hydrodynamic transport. These systems are currently difficult to model at the desired local resolution due to the computational burden of cell size constraints. One of the tactics for adequately representing such systems is to reduce the dependence on grid-based flow models. Operator splitting allows transport models to be integrated into analytic, grid-free flow solutions such as those provided by the analytic element method (AEM). Modeling contaminant advection via AEM combined with Eulerian-Lagrangian methods or streamline techniques removes many of the constraints placed upon transport models by grid cell size. In addition, AEM provides the benefit of continuous, analytic velocities, thus freeing the model from inaccuracies attributable to velocity interpolation techniques.

The analytic element method models groundwater flow via the superposition of analytic functions ("elements") that represent hydrogeologic features. Each of these functions (and therefore the final velocity field) is continuous. The computational burden associated with a model is dependent upon the number of features, rather than the scale, of the system. Therefore, while the analytic element method represents the flow field in an entirely grid-free manner, it also is quite efficient in modeling the mixed-scale hydraulic systems associated with many remediation techniques.

A framework is developed for incorporating advection-dominated contaminant transport into AEM. The random walk method, the method of characteristics, and streamline methods are revised to operate within a continuous flow domain and exploit the analytic nature of the velocity field. The benefits of these revised methods are described in the context of systems with complex chemical behavior at short length scales and significant hydrodynamic transport behavior at long length scales (i.e. remediation systems). The development of this new framework for mechanical contaminant transport will allow for the computational burden of complex reactive transport models to lie almost entirely within the reactive portion of the transport simulation.

An Overview of the Object-Oriented Iterative Model for the Analytic Element Method
(AGU Spring 2002)

James Craig and Igor Jankovic

The Analytic Element Method (AEM) models groundwater flow via the superposition of analytic functions ("elements") which represent hydrologic features. Conventionally, the method has utilized explicit methods for solving for unknown coefficients. Purely explicit methods are traditionally characterized by fully populated matrices, strict adherence to linear boundary conditions, and highly coupled relationships between all of the elements in the system.

The development of iterative solution methods for AEM in the last few years, demonstrated by the public domain code SPLIT, has resolved some of the dependence upon fully linear systems of equations. At the same time, it has been able to introduce implementations of higher-order representations and high-speed solution algorithms for extremely large-scale systems.

AEM as a whole suggests fruitful application of object-orientation constructs such as inheritance and data encapsulation. In an iterative method, elements may individually solve for themselves and have minimal coupling with other elements, further lending credence to an object-oriented representation. The tradeoff is that the implementation of iterative methods requires more sophisticated comprehension of the convergence behavior of systems.

The development of a fully object-oriented program, BlueBird, has exploited the high modularity of the iterative model. The developers have had to consider how to generalize rules for systems with complex convergence behavior. Likewise, new techniques have been developed which take advantage of both the iterative framework and object organization. Such techniques are based upon extensive discussion within the AEM community of object orientation and organizational opportunity associated with a purely iterative model.

Modeling Groundwater/Surface Water Interaction with the Analytic Element Method
(Great Lakes Research Consortium Conference, 2002)

James Craig

Many environmental and hydrologic models of surface water systems treat groundwater reservoirs as a constant source or sink of water. Likewise, groundwater models often use surface water features as a basis for constant head conditions. However, groundwater flow and surface water flow are much more interconnected than these assumptions imply. Ignoring the close relationship between the two significantly affects the mass balance of the system of interest, whether it is an aquifer or a surface body.

The Analytic Element Method (AEM) is a groundwater modeling technique well suited for steady-state regional-scale models, because the influence of each hydrologic feature is represented across the entire domain, without the need for discretization. These features, ("elements"), are superimposed to form a complete flow solution. By crafting elements with realistic boundary conditions that directly depend upon surface water behavior, regional scale coupled groundwater-surface water models may be developed. Such models are required for a more complete representation of the Great Lakes and their associated watersheds.

New elements have been enhanced for use in an iterative AEM model- the high-order resistance linesink and the resistance area sink. Both model the flux of water to and from lakes and streams based upon pressure differentials across the bottom of the water body. These elements are not only more accurate depictions of actual groundwater/surface water interaction, but they can represent space-variable boundary conditions, produce more realistic flux terms, and are more effective for model calibration. In the future, they may be used as a basis for comprehensive models that include the entire water cycle over large geographic domains.

Accommodating Multi-scale Analytic Element Models with the Nested Superblock Approach
(AGU Fall 2001)

James Craig, Igor Jankovic, and Randal Barnes

The Analytic Element Method (AEM) models groundwater flow via the superposition of analytic functions ("elements"). A large library of functions exists to account for linear and non-linear conditions prescribed along internal features and external boundaries with analytic precision. AEM is a grid-independent method that eliminates the need for constrained domain extents.

<> <>AEM has been traditionally used either to represent detailed local-scale groundwater flow or to provide boundary conditions for a local-scale model via regional-scale simulations (e.g. AEM model of the Yucca Mountain). The exception to this tradition is the National Groundwater Model of The Netherlands (NAGROM) which covers a complete country with a sufficient amount of detail to represent both local and regional flow. However, the required computational effort is prohibitive. A new, more efficient AEM formulation is required to shorten simulation runs and to allow for inclusion of all relevant local hydrologic features in regional-scale models.

The superblock method offers a possibility to achieve this goal. This approach has already been applied in groundwater dispersion research, and is capable of improving efficiency of an analytic element method by as much as two orders of magnitude. However, the previously existing superblock technique had never been applied to regional groundwater models. The target application for the new approach is a regional model composed of hundreds of thousands of elements, each corresponding to a relevant hydrologic feature (such as a section of a river or a lake).

The power of the superblock approach comes from representing a large group of geographically proximate elements with a single equivalent series expansion valid outside of the boundary of the "superblock". The nested superblock method allows for further enhancement of this technique by using a quad-tree structure to represent the elements at varying levels of conglomeration. The nesting technique reduces the amount of functional evaluations required to simulate the flow system by several orders of magnitude. This advantage allows for inclusion of a very large number (e.g. 10,000) of elements, thus enabling simultaneous detailed modeling of local and regional groundwater flow without the use of artificial model boundaries or grid refinement techniques.