No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
The Nested Superblock Approach for Regional‐Scale Analytic Element Models
Abstract
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 comprising thousands of hydrogeologic features at multiple geographic scales.
... The number of samples taken is greater than the number of unknowns (known as the overspecification principle [9]) and the resulting system is solved in a least-squares sense. For simplicity QR decomposition is used, although in practice methods like the nested super-block method [5] would give better performance by using a hierarchical datastructure to compute long-range interactions. ...
... Furthermore, based on methods developed in the literature it has the potential to be fast as well. For example, in Craig et al. [5] the problem domain is subdivided hierarchically to allow far-field contributions of elements to be combined as much as possible. Having the potential to give both good and fast solutions makes the method developed in this thesis a good addition to the arsenal of solvers for diffusion curves. ...
... The solver as developed in this thesis could potentially be made faster by using the nested superblock approach developed in Craig et al. [5]. It would be interesting to see if this could lead to a version of the AEM for diffusion curves that is fast enough for interactive use. ...
Diffusion curves provide an exciting new vector primitive that allows for easily editable, freeform, color gradients. They allow an artist to focus on the color boundaries in an image. The rest of the image is filled in a smooth manner by solving a linear system of equations based on the Laplace equation. Current solvers for diffusion curves struggle to be both fast and precise though, making it unattractive to standardize diffusion curves or use them in a vector graphics editor. This thesis gives some insight into why it is hard to obtain a good result and shows how a different kind of solver can make it fundamentally easier to get high quality results.
... The convergence was typically achieved in 50 or fewer iterations for models lacking head-specified elements. However, models containing many head-specified elements converge much slower, requiring thousands instead of tens of iterations [6], and tend to diverge when parallel processing is used. ...
... The total discharge potential at a specific location in the domain is obtained through the superposition of the discharge potentials of all the analytic elements. As head-specified elements require many more iterations to converge [6] than other element types (inhomogeneities, leaky walls, drains, . . .) [11] the total discharge potential is here divided into the following parts: ...
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.
... Modeling in canal command systems are regional scale problems. Groundwater flow in regional scale aquifers (Bakker et al. 1999;Craig et al. 2006) can be modeled using AEM. The challenge of modeling regional scale problems with limited data sets can be addressed with the Analytic Element Method (AEM) based model. ...
Regional scale modeling of coupled groundwater-surface water interaction in canal command areas is difficult due to high computational requirements and data insufficiency. Groundwater plays an essential role in the interaction process to fulfil the irrigation requirement in tail reaches of canal command areas. A comprehensive coupled model is required to simulate the canal command systems by incorporating the processes: (a) saturated groundwater flow, (b) unsaturated flow and (c) overland flow. In the present work, a fully-coupled model is developed that simulates saturated groundwater flow using Analytic Element Method (AEM), unsaturated flow using analytical solution and overland flow using Finite Volume Method (FVM) based Zero-inertia model. The Capability of the developed coupled model is demonstrated for Damodar Left Bank Main Canal (LBMC) under two canal regulation scenarios for “Boro Rice” cultivation season (Jan-Apr). Major canal water shortage is observed in LBMC during this season. It can be observed from the results that hydraulic heads in the upper reach are quite high whereas it is significantly lowering down as we move away from the main canal or in the lower reach where the groundwater is the main source of Boro rice irrigation. The considerable decline in hydraulic head values can be observed in LBMC which can be justified with a decrease in water supply and an increase in the area under Boro rice cultivation.
... Here, the mathematical functions for individual features are gathered into families of elements, each with a more computationally efficient series expansions for evaluation at larger distances from a group of features. The computational advantage of superblocks has been demonstrated for regional modeling with Cauchy integrals [13], for separation of variables with circles and spheroids [23,24], and for functions associated with groundwater uptake by fields of phreatophytes [34]. However, the superblock approach does not explicitly allow for jump in aquifer properties between different domains. ...
Groundwater studies face computational limitations when providing local
detail (such as well drawdown) within regional models. We adapt the
Analytic Element Method (AEM) to extend separation of variable solutions
for a rectangle to domains composed of multiple interconnected
rectangular elements. Each rectangle contains a series solution that
satisfies the governing equations and coefficients are adjusted to match
boundary conditions at the edge of the domain and continuity conditions
across adjacent rectangles. A complete mathematical implementation is
presented including matrices to solve boundary and continuity
conditions. This approach gathers the mathematical functions associated
with head and velocity within a small set of functions for each
rectangle, enabling fast computation of these variables. Benchmark
studies verify that conservation of mass and energy conditions are
accurately satisfied using a method of images solution, and also develop
a solution for heterogeneous hydraulic conductivity with log normal
distribution. A case study illustrates that the methods are capable of
modeling local detail within a large-scale regional model of the High
Plains Aquifer in the central USA and reports the numerical costs
associated with increasing resolution, where use is made of GIS datasets
for thousands of rectangular elements each with unique geologic and
hydrologic properties, Methods are applicable to interconnected
rectangular domains in other fields of study such as heat conduction,
electrical conduction, and unsaturated groundwater flow.
... Modern implementation of the AEM is capable of handling thousands of straight elements with strengths of order MZ100 (Craig et al. 2006;Bandilla et al. 2007). These models enable accurate computation of local detail (singularities in the vicinity of corners and tips) within the context of large regional models. ...
Subject collections (60 articles) mathematical modelling (27 articles) analysis Articles on similar topics can be found in the following collections A general framework for analytic evaluation of singular integral equations with a Cauchy kernel is developed for higher order line elements of curvilinear geometry. This extends existing theory which relies on numerical integration of Cauchy integrals since analytic evaluation is currently published only for straight lines, and circular and hyperbolic arcs. Analytic evaluation of Cauchy integrals along straight elements is presented to establish a context coalescing new developments within the existing body of knowledge. Curvilinear boundaries are partitioned into sectionally holomorphic elements that are conformally mapped from a local curvilinear Z-plane to a straight line in the Z-plane. Cauchy integrals are evaluated in these planes to achieve a simple representation of the complex potential using Chebyshev polynomials and a Taylor series expansion of the conformal mapping. Bell polynomials and the Faà di Bruno formula provide this Taylor series for mappings expressed as inverse mappings and/or compositions. Examples illustrate application of the general framework to boundary-value problems with boundaries of natural coordinates, Bezier curves and B-splines. Strings formed by the union of adjacent curvilinear elements form a large class of geometries along which Dirichlet and/or Neumann conditions may be applied. This provides a framework applicable to a wide range of fields of study including groundwater flow, electricity and magnetism, acoustic radiation, elasticity, fluid flow, air flow and heat flow.
... Initial numerical experiments using the analytic element method on 186 inhomogeneities (Strack 1992) suggested the addition of an inertia term in addition to classic diffusive theory. Subsequent work with a higher number of inhomogeneities took advantage of the analytic element method's use of the principle of superposition, where it was found that computational performance could be significantly increased by selective solving of equations (superblocks, Strack et al. 1999; Craig et al. 2006), iterative solvers (Barnes and Jankovic 1999), faster direct solvers (Haitjema 2006), overspecification of inhomogeneity boundaries (Jankovic and Barnes 1999a), and parallel processing. Using combinations of these techniques, analytic element models solved three-dimensional flow through tens of thousands of spherical inhomogeneities (Jankovic and Barnes 1999b ) and now use massively parallel supercomputer clusters to analyze ground water flow and transport through hundreds of thousands of circular (Figure 3) and spherical inhomogeneities (Jankovic et al. 2003a). ...
Though powerful and easy to use, applications of the analytic element method are not as widespread as finite-difference or finite-element models due in part to their relative youth. Although reviews that focus primarily on the mathematical development of the method have appeared in the literature, a systematic review of applications of the method is not available. An overview of the general types of applications of analytic elements in ground water modeling is provided in this paper. While not fully encompassing, the applications described here cover areas where the method has been historically applied (regional, two-dimensional steady-state models, analyses of ground water-surface water interaction, quick analyses and screening models, wellhead protection studies) as well as more recent applications (grid sensitivity analyses, estimating effective conductivity and dispersion in highly heterogeneous systems). The review of applications also illustrates areas where more method development is needed (three-dimensional and transient simulations).
Groundwater is the most significant source of fresh water for a variety of uses, including industrial, irrigation, drinking and domestic purposes. Nowadays, excessive usage of groundwater resource is taking place, to meet water demand for industrial, agricultural and domestic uses. Nevertheless, excessive use of groundwater has resulted in the depletion of this natural resource. A gradual decline in groundwater quality is also taking place, as industrial, farming and domestic effluents entering into the hydrological cycle. To counteract groundwater resource depletion and deterioration, it is pertinent to understand the physics of groundwater flow and contaminant transport processes and to develop strategies for groundwater resources management and groundwater remediation.
Numerical techniques such as finite difference method (FDM) and finite element method (FEM) are commonly used for groundwater flow and transport simulation. However, the analytic element method (AEM) has certain capabilities which overcome the difficulties associated with grid-based algorithms. In AEM, only the hydrogeologic features in the domain are broken up into sections and entered into the model as input data. AEM eliminates the compromise between model resolution and size of the model area. Also, AEM generates very accurate hydraulic head at pumping well location, which in turn improves the quality of the groundwater management model. On the contrary, in FDM/FEM, the hydraulic head at the pumping well location is the averaged hydraulic head over the grid. In the particle tracking method to track particles at each time step, it is necessary to know the position of a particle as well as its velocity. AEM-based flow models compute continuous velocities over the entire aquifer domain, and hence for the reverse particle tracking (RPT) and random walk particle tracking (RWPT) simulation, there is no need to use any velocity interpolation schemes as generally required in FDM or FEM based models. Further, the Eulerian transport models, such as FDM/FEM based models are often plagued by numerical dispersion and artificial oscillations if spatial and temporal discretization criteria do not meet properly. As an alternative, the random walk particle tracking (RWPT) simulates the advection-dispersion equation in a different manner and it is completely free from the numerical dispersion. The analytic element method is amenable for reverse particle tracking or random walk particle tracking and they have various advantages as mentioned above.
In this context, the main scope of the present study is to develop groundwater flow and contaminant transport simulation models using analytic element method, reverse particle tracking, and random walk particle tracking and to couple the simulation models with efficient optimization algorithm such as cat swarm optimization to get the effective simulation-optimization model for groundwater management and remediation. In this study, an AEM-RPT model is developed by combining analytic element method with reverse particle tracking. The AEM-RPT model is used to delineate the time-related capture zone of well-field. Further, the AEM-RWPT model is developed by combining analytic element method with random walk particle tracking. The AEM-RWPT model is applied to simulate groundwater flow and contaminant transport processes (advection and hydrodynamic dispersion) of heterogeneous hypothetical and field aquifer. Furthermore, the accuracy and computational efficiency of the AEM-RWPT model is enhanced by combining it with kernel density estimator (KDE). Additional features are included in the AEM-RWPT-KDE model to simulate radioactive decay and linear adsorption isotherm. The AEM-RWPT-KDE model is effectively used to solve the advection-dispersion-reaction equation (ADRE). The effectiveness of the developed model is verified with MODFLOW-MT3DMS and found to be satisfactory.
Heuristic optimization technique, such as Genetic Algorithm (GA), Simulated Annealing (SA), Particle Swarm Optimization (PSO), Harmony Search (HS), Tabu Search (TS) and Differential Evolution (DE) are most commonly used for various groundwater management and groundwater remediation studies. All these optimization algorithms have advantages as well as limitations. Among these optimization algorithms, the particle swarm optimization is very popular and relatively easy to implement. However, the particle swarm optimization (PSO) can be influenced by stagnation point problem and parameter convergence error. Recently, a relatively new swarm optimization technique, namely Cat Swarm Optimization (CSO) is gaining considerable attention in various engineering fields. In Cat swarm optimization (CSO), search operation takes place via two modes (Seeking and Tracking mode). So, in case the solution is trapped in stagnation point, then there are greater chances of escaping from stagnation point, via inertia term and seeking mode process. Considering, the recent popularity of CSO, in the present study the cat swarm optimization is considered to develop the optimization model for groundwater resources management and groundwater remediation design.
In this study, two new simulation-optimization (S-O) models for groundwater management (AEM-CSO and AEM-RPT-CSO) are developed by coupling analytic element method with reverse particle tracking and cat swarm optimization. The AEM-CSO model is applied for groundwater management in a hypothetical unconfined aquifer considering two objectives separately: maximization of the total pumping and minimization of the total pumping cost. Also, an attempt is made to minimize groundwater contamination risk through capture zone management of pumping wells by AEM-RPT-CSO model. Further, a coupled AEM-MOCSO model is also developed by coupling analytic element method with multiobjective cat swarm optimization (MOSCO). The AEM-MOCSO model is applied to a hypothetical unconfined aquifer by considering two objectives together: maximization of the total pumping and minimization of the total pumping cost.
There are significant challenges, to directly incorporate analytic element method and random walk particle tracking in an optimization model for groundwater remediation, as both of them are computationally expensive. To deal this issue, in this study, an artificial neural network (ANN) and cat swarm based surrogate simulation-optimization model are developed for groundwater remediation. The ANN mimics the behavior of AEM-RWPT-KDE model. The ANN-CSO model is applied to remediate a hypothetical and field case study. Further, a simulation-optimization model is developed for multiobjective groundwater remediation by coupling artificial neural network with multiobjective cat swarm optimization (MOCSO). Here also, the ANN model acts as a proxy simulator for AEM-RWPT-KDE model. The ANN-MOCSO model is applied to a hypothetical and field case study for multiobjective groundwater remediation showing the effectiveness of the developed model.
The present study shows that AEM, RPT, and RWPT based models are very effective in groundwater flow and transport simulation. When these models are coupled with an efficient optimization tool such as CSO, we get robust simulation-optimization models, which can be effectively used in groundwater management and remediation designs.
The analytic-element method (AEM) is an appealing technique for modeling steady-state groundwater flow at the supraregional scale (defined here as > 10,000 km(2)) 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 introduce AEM to practitioners unfamiliar with the approach and present modeling concepts and new 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.
This work introduces a new iterative Analytic Element Method (AEM) algorithm for solving 2D steady-state groundwater flow models containing large numbers of head-specified elements (e.g., rivers and lakes). The new algorithm improves convergence of models containing head-specified elements by explicitly computing fluxes of all such features at the start of each iteration. The new algorithm also enables the use of efficient parallel processing on distributed-memory super-computers. The combination of parallel processing and reduced number of iterations significantly extends the size and complexity of problems that can be modeled using AEM.
1] A new approach for analytic element (AE) modeling of groundwater flow is presented. The approach divides the modeled region into polygonal subdomains, each with its own analytic flow model and its own local isotropic or anisotropic aquifer parameters. This allows analytic modeling of systems where the anisotropy ratio and direction vary spatially, an AE capability not possible without subdomains. It also allows for flexible layering in a model, with more layers in the area of interest abutting fewer layers in the far field. The approach is demonstrated in a model with seven subdomains and a mix of single‐layer and triple‐layer areas. Checks of the model indicate that the inter‐subdomain boundary conditions can be approximated well, and where the differential equation is approximated (multilayer areas and transient flow), that approximation can be quite accurate.
The study aims at deriving the effective conductivity K ef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale I Y . We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere Ω, of radius R 0 , and they occupy a volume fraction n. Uniform flow of flux U ∞ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere Ω, under the limits n→1 and R 0 /I Y →∞. Following a qualitative argument, we derive an exact expression of K ef by computing it at the dilute limit n→0. It turns out that K ef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulations for σ Y 2 ≤10 and for spheres of uniform radii. By using a faced-centered cubic lattice arrangement, the values of the volume fraction are in the interval 0<n<0·7. The simulations are carried out by the means of an analytic element procedure. To exchange space and ensemble averages, a large number N=10000 of inclusions is used for most simulations. We surmise that the self-consistent model is an exact one for this type of medium that is different from the multi-Gaussian one.
1] In parts 1 [Dagan et al., 2003] and 2 [Fiori et al., 2003] a multi-indicator model of heterogeneous formations is devised in order to solve flow and transport in highly heterogeneous formations. The isotropic medium is made up from circular (2-D) or spherical (3-D) inclusions of different conductivities K, submerged in a matrix of effective conductivity. This structure is different from the multi-Gaussian one, even for equal log conductivity distribution and integral scale. A snapshot of a two-dimensional plume in a highly heterogeneous medium of lognormal conductivity distribution shows that the model leads to a complex transport picture. The present study was limited, however, to investigating the statistical moments of ergodic plumes. Two approximate semianalytical solutions, based on a self-consistent model (SC) and on a first-order perturbation in the log conductivity variance (FO), are used in parts 1 and 2 in order to compute the statistical moments of flow and transport variables for a lognormal conductivity pdf. In this paper an efficient and accurate numerical procedure, based on the analytic-element method [Strack, 1989], is used in order to validate the approximate results. The solution satisfies exactly the continuity equation and at high-accuracy the continuity of heads at inclusion boundaries. The dimensionless dependent variables depend on two parameters: the volume fraction n of inclusions in the medium and the log conductivity variance s Y 2 . For inclusions of uniform radius, the largest n was 0.9 (2-D) and 0.7 (3-D), whereas the largest s Y 2 was equal to 10. The SC approximation underestimates the longitudinal Eulerian velocity variance for increasing n and increasing s Y 2 in 2-D and, to a lesser extent, in 3-D, as compared to numerical results. The FO approximation overestimates these variances, and these effects are larger in the transverse direction. The longitudinal velocity pdf is highly skewed and negative velocities are present at high s Y 2 , especially in 2-D. The main results are in the comparison of the macrodispersivities, computed with the aid of the Lagrangian velocity covariances, as functions of travel time. For the longitudinal macrodispersivity, the SC approximation yields results close to the numerical ones in 2-D for n = 0.4 but underestimates them for n = 0.9. The asymptotic, large travel time values of macrodispersivities in the SC and FO approximations are close for low to moderate s Y 2 , as shown and explained in part 1. However, while the slow tendency to Fickian behavior is well reproduced by SC, it is much quicker in the FO approximation. In 3-D the SC approximation is closer to numerical one for the highest n = 0.7 and the different s Y 2 = 2, 4, 8, and the comparison improves if molecular diffusion is taken into account. Transverse macrodispersivity for small travel times is underestimated by SC in 2-D and is closer to numerical results in 3-D, whereas FO overestimates them. Transverse macrodispersivity asymptotically tends to zero in 2-D for large travel times. In 3-D the numerical simulations lead to a small but persistent transverse macrodispersivity for large travel times, whereas it tends to zero in the approximate solutions. The results suggest that the self-consistent semianalytical approximation provides a valuable tool to model transport in highly heterogeneous isotropic formations of a 3-D structure in terms of trajectories statistical moments. It captures effects like slow transition to Fickian behavior and to Gaussian trajectory distribution, which are neglected by the first-order approximation. Citation: Jankovi, I., A. Fiori, and G. Dagan, Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results, Water Resour. Res., 39(9), 1270, doi:10.1029/2002WR001721, 2003.
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 chapter draws heavily on building the theoretical framework for a mathematical description of groundwater flow. In the process numerous practical problems are solved. A recharge area or aquifer in homogeneity, for instance, is taken to be circular, and the problem of a well near a stream is approximated by modeling a well opposite an infinitely long equipotential line. Although many of the solutions serve useful educational purposes, they are not very convincing representations of groundwater flow in regional aquifers with, among others, complex stream networks, wetlands, and varying areal recharge rates and transmissivities. It is true that there exist many more analytic solutions to groundwater flow than are presented in this text, and some of them are very ingenious. By themselves, however, they all share the limitation of a relatively simple hydrogeological setting.
Analytic Element Modeling of Groundwater Flow offers much more than one might anticipate from its title. It is an excellent tutorial on the subjects of groundwater flow and modeling groundwater with the analytic element method. I greatly enjoyed reading it, and I recommend it to any student of groundwater hydrology, as well as to researchers and professionals concerned with modeling regional groundwater flow in productive aquifers. Chapters 3 and 6 are particularly interesting.
An approach, the superblock approach, is presented for increasing computational efficiency of analytic element models. The approach is based on computing the combined effect of functions using both asymptotic expansions and Taylor Series expansions. The superblocks are used to reduce both the computational effort required to determine the coefficients in the analytic element model and to reduce the effort expended in generating contour plots and streamlines. An application of flow is presented in an aquifer with one hundred thousand circular impermeable objects. The errors in the simulation are within the machine accuracy.
An implicit analytic solution is presented for two-dimensional groundwater flow through a large number of non-intersecting circular inhomogeneities in the hydraulic conductivity. The locations, sizes and conductivity of the inhomogeneities may be arbitrarily selected. The influence of each inhomogeneity is expanded in a series that satisfies the Laplace equation exactly. The unknown coefficients in this expansion are related to the coefficients in the expansion of the combined discharge potential from all other elements. Using a least squares formulation for the boundary conditions and an iterative algorithm, solutions can be obtained for a very large number of inhomogeneities (e.g. 10,000) on a personal computer to any desired precision, up to the machine's limit. Such precision and speed allows the development of a numerical laboratory for investigating two-dimensional flow and convective transport.
This paper presents a new Analytic Element formulation for high-order line elements in modeling two-dimensional groundwater flow. These elements are line-doublets, line-dipoles and line-sinks. The jump functions for line elements are expressed as Chebyshev series. The unknown coefficients are computed by applying the principle of overspecification to the boundary conditions. The use of the high-order elements and the principle of overspecification have resulted in high precision and significant improvements in computational efficiency compared to the existing collocation-based formulation. The new formulation is currently being used in the development of the Metropolitan Area Groundwater Model for Twin Cities, Minnesota, USA and for enhancements of the next release of National Groundwater Model for The Netherlands.
Bluebird Developer's Manual
- J R Craig
Craig, J.R. 2002. Bluebird Developer's Manual. http://www.
groundwater.buffalo.edu/ (accessed July 2003).
- G H Golub
- C F Van-Loan
Golub, G.H., and C.F. Van-Loan. 1996. Matrix Computations.
Baltimore, Maryland: The Johns Hopkins University Press.
Analytic Element Modeling of Groundwater Flow
- H M Haitjema
Haitjema, H.M. 1995. Analytic Element Modeling of Groundwater Flow. San Diego, California: Academic Press.