Theory and applications of the Analytic Element Method
Abstract
[1] We present a review of the theory and applications of the Analytic Element Method as it exists today. The paper begins with a presentation of analytic elements used for modeling of divergence-free and irrotational flow in both two and three dimensions, including a description of the superblock approach, which makes it possible to deal effectively with very large models both in terms of accuracy and speed. We briefly discuss a particular application of the Analytic Element Method to multiaquifer problems, present the theory of the Analytic Element Method for general vector fields, and discuss the inclusion of source terms (irrotational vector fields with nonzero divergence). We finish the paper with a brief discussion of three major applications on a regional scale.
1. INTRODUCTION
[2] The objectives of this paper are fivefold: to introduce the reader to a computational method known as the Analytic Element Method, to give an overview of the method as it currently exists, to present the principles on which this method is based, to discuss several applications to regional modeling, and to present one new application in detail. The Analytic Element Method is a computational method based upon the superposition of analytic expressions; it is approximate and can be applied, in principle, to represent any three-dimensional or two-dimensional vector field.
[3] The Analytic Element Method is applicable to both finite and infinite domains and typically is applied to problems that contain internal boundaries. The method was first developed in the early 1970s in order to solve a groundwater flow problem: the modeling of the effect of the Tennessee-Tombigbee Waterway on the surrounding aquifers. The project demanded the groundwater model to reproduce flow both on a regional scale (∼80 by 130 km initially) and on a local scale; the effect of some 60 wells in a 915 m long test section of the excavation needed to be simulated accurately [see Strack and Haitjema, 1981a, 1981b]. The most comprehensive application of the Analytic Element Method in the field at present is the Dutch National Groundwater Model (NAGROM). The nature of the method, superposition of analytic expressions, makes it possible to deal with very large models, while still maintaining a high degree of accuracy on a small scale. This property is particularly useful for models on a state or national scale, such as NAGROM [de Lange, 1996b] and the Twin Cities Metropolitan Groundwater Model in Minnesota [Seaberg et al., 1997].
[4] The Analytic Element Method utilizes Helmholz's decomposition theorem, which states that any vector field can be partitioned into a solenoidal field and an irrotational field plus a vector field that contributes neither to the divergence nor to the rotation. The various attributes of the vector field are then represented by analytic expressions, each suitable for modeling a particular feature or aspect of the vector field. Boundary conditions, often internal, are met in an approximate sense: The sum of the squares of the errors at control points is minimized for each analytic element.
[5] The Analytic Element Method makes use of complex variable methods for two-dimensional problems. This complex variable approach, known as Wirtinger calculus (see Wirtinger [1927] or, e.g., Remmert [1991]), makes it possible to apply complex variable methods to any two-dimensional problem that requires the determination of a vector field. Boundary conditions, often internal, are applied along lines or curves and are formulated in terms of discontinuities or jumps. The analytic elements that control these jumps are often, but not always, formulated by means of Cauchy integrals or Legendre functions. Each analytic element is chosen such that it represents a jump in either the normal component or the tangential component of the vector. The solution to the problem is formulated in such a way that continuity of flow is satisfied exactly. The jumps are computed from the boundary conditions and are approximated, usually by a polynomial. The sum of the squares of the errors at the control points is minimized for each analytic element.
[6] Applications of boundary elements and singular Cauchy integrals to problems in geotechnical engineering are common, e.g., in groundwater flow and the theory of linear elasticity. Applications of such integrals are given by Polubarinova-Kochina [1962, 1977] and Pilatovski [1966]. Vortex distributions are applied to a variety of problems by Kalinin [1941], Risenkampf [1940], Kozlov [1941], and De Josselin de Jong [1960]. Muskhelishvili's [1958] work contains a comprehensive discussion of singular line elements, and many applications of such elements in the theory of linear elasticity are given by Muskhelishvili [1952, 1977].
[7] A difference between the Analytic Element Method and existing applications of singular line elements (such as in the boundary integral equation method) is that the harmonic functions are only part of the solution; they are combined with functions that create divergence or curl. A second difference is that the elements are considered as independent functions with degrees of freedom to be determined such that the final solution meets specified conditions. A third difference is that the elements may not be, and are often not, constructed as line integrals.
[8] Applications to date have been concerned with Laplace's equation, the Poisson equation, the biharmonic equation, the modified Helmholz equation, and the heat equation. Although the method is, in principle, applicable to general vector fields, it has been almost exclusively applied to the field of groundwater flow and will be discussed in this paper with this application in mind. As a new application and illustration the Analytic Element Method will be applied to the problem of infiltration or leakage over an area bounded by an ellipse.
2. DUPUIT-FORCHHEIMER FLOW IN SINGLE AQUIFERS WITHOUT INFILTRATION
[9] Single aquifers are modeled with the Analytic Element Method by superposition of functions that are used to represent the features of the aquifer. The resistance to flow in vertical direction is usually neglected; the Dupuit-Forchheimer approximation is adopted [e.g., Strack, 1989]. It may be noted that the Dupuit-Forchheimer approximation produces solutions which are accurate in terms of discharge but are approximate in terms of piezometric heads in areas where vertical components of flow are relatively large, e.g., in the immediate vicinity of a partially penetrating well. This approximation does not limit the flow to two dimensions; it is possible to determine streamlines in three dimensions for such models [Strack, 1984].
[10] The analytic elements that represent the features in the aquifer that affect the flow, such as streams, may be straight, curved, or closed and curved. Infiltration into the aquifer is modeled either as uniform or as being constant in areas bounded by polygons.
[11] The analytic elements are represented by complex potentials, except the functions that represent infiltration, which is assumed to be given in magnitude. These complex potentials contain parameters that are determined so as to meet the boundary conditions, usually jump conditions, i.e., conditions that involve jumps in either the tangential or the normal component of the discharge vector along the analytic elements. These boundary conditions are not met exactly but as accurately as possible, given the number of degrees of freedom and the nature of the elements. It will be shown in what follows that the accuracy of the model for single aquifers is limited only by machine accuracy and memory.







2.1. Line Elements: Linear Jump Functions
[15] The cornerstone of the Analytic Element Method is the complex potential for a line element, obtained from a singular Cauchy integral [Strack, 1989]. The complex potential for a line element is characterized by the jump it exhibits. We distinguish between two kinds of line elements: the line doublet and the line dipole. The complex potential for a line doublet is also known as the complex potential for a double layer; it may be viewed as resulting from a double layer of sources and sinks distributed along a line. The complex potential for a line doublet exhibits a jump in the tangential component of flow, whereas the normal component of flow is continuous. Note that a jump in the tangential component of flow corresponds to a jump in the potential, except for the special case that this jump is constant. The complex potential for a line dipole is a special case of the complex potential for a single layer; it corresponds to a jump in the normal component of flow, whereas the tangential component of flow is continuous. The imaginary part of the complex potential, the stream function, jumps across the line dipole and the line sink, whereas the real part, the potential, is continuous.

[17] Line elements with jumps approximated by polynomials are discussed in detail by Strack [1989]. Applications of higher-order line elements, i.e., line elements with jumps represented by high-order polynomials, became attractive after Janković [1997] introduced the principle of overspecification (discussed in section 2.1.1).
2.1.1. Line Doublet



















[20] The line doublets are used in a variety of applications, e.g., for modeling the effect of an impermeable wall in a flow field. The jump λ then corresponds to a jump in head across the wall, and the polynomial representation of λ is an approximation. The coefficients aj can be determined such that the jump condition is met at control points distributed along the element; elsewhere the jump deviates from the exact solution. The accuracy of the approximation can be increased significantly by distributing the control points in a well-defined way and/or by using the principle of overspecification, introduced by Janković [1997], as will be shown in section 2.1.3.
[21] The benefit of the functions Ij can now be clarified; separation of the functions Ij from the coefficients aj makes it possible to carry out the analysis to determine the coefficients of the far-field correction polynomial (the second sum in equation (17)) as well as those of the far-field expansion equations (23) and (24) without knowledge of the values of the coefficients aj.
2.1.2. Control Point Distribution
[22] Solutions obtained using analytic elements can be improved by distributing collocation points as Chebyshev sample points. For further information, see, for example, Hamming [1973].
2.1.3. Overspecification



[24] The flow nets shown in Figures 2a and 2b correspond to the case of an inhomogeneity in the hydraulic conductivity in a confined aquifer with a uniform far field. Again, overspecification is used. The jump in hydraulic conductivity is ki/k = 0.01 for Figure 2a and ki/k = 100 for Figure 2b. Note that the piezometric contours are shown in the plots; the condition of continuity of flow is met exactly across the boundary of the inhomogeneity, but the condition of continuity of head is met approximately. However, the piezometric contours appear to be continuous; the solution is very accurate. The order of the polynomials that represent the jump in potential is 30, the overspecification fold is 1.5, and the control points are distributed as Chebychev sample points. Again, the flow is confined, and the shape of the flow net is not affected by the values of the aquifer parameters.

[25] Note that uniqueness of solutions to Laplace's equation is guaranteed; the largest error in the solution occurs along the boundary, in this case the leaky wall. The error can be determined empirically by computing values of the stream function along the wall; this function should be equal to a constant along the entire wall.
2.1.4. Line Dipole


[27] The difference between the complex potential for a line dipole and a line doublet is a factor i. Line dipoles create a jump in the normal component of flow but do not extract a net amount of water. Line dipoles are useful for modeling features of zero net discharge, such as drains that are not being pumped and narrow fissures filled with highly permeable material. The picture for such a fissure is identical to that for a slurry wall shown in Figure 1, provided that the equipotentials shown in Figure 1 are interpreted as streamlines and the streamlines are interpreted as equipotentials.
2.1.5. Line Sink
[28] Elements that extract a net amount of water from the aquifer are modeled by line elements called line sinks. The difference between a line sink and a line dipole is that the line sink can extract a net amount of water from the aquifer, whereas the line dipole cannot. Line sinks thus are more generally applicable and can be used for modeling streams and boundaries of rivers and lakes, for example. Line dipoles can be used only for modeling features that cannot extract a net amount of water yet generate a discontinuity in the normal component of flow. Examples of such features are fissures and drains that are not being pumped.
[29] Strack [1989] shows that the complex potential for a line sink is obtained from that for a line dipole by adding complex potentials for wells at the end points. These wells have discharges equal to minus the jumps in the stream function at the end points.
2.1.6. Implementation
[30] Modeling groundwater flow in a single aquifer using the elementary analytic elements discussed in sections 2.1 is the most common and most direct application of the Analytic Element Method. It lends itself to elegant implementations in computer programs. The reader interested in either using or expanding upon these implementations is referred to two web sites. The first one is http://www.groundwater.buffalo.edu; it contains information regarding the public domain code SPLIT. The computer program SPLIT was used to generate Figures 1 and 2.
[31] A second web site is http://www.engr.uga.edu/∼mbakker/tim.html, which contains the source code of a computer program called Tim, which is an implementation of analytic elements for the modeling of groundwater flow. The program has an object-oriented design that was developed with the input from a large group of analytic element developers. The object-oriented design of Tim is basic but flexible, the structure of the program is transparent, and the design is suitable for changes and additions. The source of Tim is open and written in Python, an open-source, interpreted, interactive, object-oriented programming language. The source code of Tim and a manual may be downloaded from the latter web site. Both web sites are updated as the code is enhanced and new features are added.
2.2. Curvilinear Elements: Flow Nets and Boundary Values
[32] A major advantage of the Analytic Element Method is that computer programs based on the Analytic Element Method can be validated without the need for access to the source code. Since the method is based upon the superposition of analytic functions, it is sufficient to demonstrate (1) that each analytic element satisfies its governing equation and (2) that the analytic elements are added properly by the computer program. To demonstrate that a single element satisfies the governing differential equation is particularly straightforward for the case of harmonic elements (elements that are represented by a harmonic potential); it suffices to show that flow nets are produced by the program on any scale. That an element satisfies its boundary conditions can be verified visually in case the boundary condition consists of constant potential or stream function. In other cases it may be necessary to let the program produce, either graphically or numerically, proof that the boundary condition is met with adequate precision.
[33] We demonstrate this important property by presenting the flow net for a curvilinear element. The mathematical description of this element has not been published other than in a report; nonetheless, the curvilinear elements have been used in regional modeling using the computer program Multi-Layer Analytic Element Model (MLAEM). To illustrate how a flow net can be used to validate the accuracy of the solution, a flow net is shown in Figure 3 for a lake bounded by curvilinear elements in a field of uniform flow. The head along the lake boundary is unknown but constant, the lake has no net extraction, the uniform flow field is given, and the head at a point downstream of the lake is given as well. It is clear from Figure 3 that the flow net is accurate, and it can be seen also that the head along the lake boundary is nearly constant.

[34] This method of validation was followed when the computer program MLAEM was applied to the modeling of the aquifers near Yucca Mountain [see Bakker et al., 1999]. The application of the program to this project required that the computer program MLAEM be approved for quality assurance. This was done by demonstrating, for each analytic element, that the differential equation was satisfied and that the boundary conditions were met as claimed.
2.3. Closed Two-Sided Elements: Circular Inhomogeneities
[35] Jump functions need not be associated with straight lines. As seen from the example in section 2.2 and as explained by Strack [1989], jumps can be created along curves, which may be either closed or open. Closed two-sided elements are usually constructed using conformal mapping techniques. Examples of such elements are curvilinear elements shaped as Bézier curves [see Le Grand, 1999], circular inhomogeneities, and elliptical inhomogeneities. The advantage of creating analytic elements based upon the Bézier curves is that these curves are often used in geographical information systems (GIS), which facilitates the transferal of shapes from the GIS to the groundwater model. Analytic elements for modeling inhomogeneities bounded by curved closed elements were developed by Strack [1989] for circular and elliptical inhomogeneities in terms of Laurent expansions, chosen in such a way that the condition of continuity of flow is satisfied exactly. The formulation for the case of circles is straightforward, and the generalization to the case of ellipses was obtained using conformal mapping.
[36] The condition of continuity of head across the boundary of the inhomogeneity was met in an approximate fashion by requiring that the boundary conditions were met exactly at collocation points that were distributed uniformly along the boundary. Barnes and Janković [1999] improved the formulation significantly by representing the jump in potential required to enforce continuity of head in terms of a Fourier series. Their formulation allows for modeling with large numbers of circles or ellipses with an accuracy that is limited only by the machine. Results obtained with this formulation are presented in the four plots shown in Figure 4. The case concerns 100,000 cylindrical inhomogeneities in a confined aquifer. The flow in the aquifer is horizontal. The hydraulic conductivity inside the inhomogeneities is zero; outside the inhomogeneities the hydraulic conductivity is uniform. The far field is uniform flow. The top left plot shows 100,000 cylindrical objects. A square defines the boundaries of the area shown in the top right plot, where equipotentials and streamlines are shown. Again, a square is shown, and the plot in the bottom left-hand corner shows the equipotentials and streamlines inside the square. This process is repeated once more to obtain the plot in the bottom right-hand corner. It is apparent from the plots that the computer program produces accurate flow nets that meet the boundary conditions precisely.

[37] The modeling of such large numbers of analytic elements is possible by using an approach known as the superblock approach, which relies on the property of each analytic element that it is fully defined by a harmonic function outside some predefined boundary, such as a polygon, a circle, or an ellipse, of finite dimensions. The interested reader is referred to Strack et al. [1999] for further information on the superblock approach, which is applicable in both two and in three dimensions.
[38] The Analytic Element Method has recently been used in the development of stochastic dispersion models. Numerical simulations, based on the Analytic Element Method, were the basis for validation of two new dispersion theories. Both theories are based on a multi-indicator permeability structure. Multi-indicator permeability structure is created using inclusions (zones of constant hydraulic conductivity) that are represented using analytic elements. Examples of such structures and movies showing the dispersion can be found at http://www.groundwater.buffalo.edu/movies/Movies.html.
[39] First, dispersion theory was developed to model flow and transport in porous formations of lognormal conductivity distribution. The second theory applies to porous formations of binary conductivity distribution. An additional example is the use of the Analytic Element Method in testing (and rejection) of Matheron-Landau conjecture that deals with effective conductivity of three-dimensional flow through porous formations of lognormal conductivity distribution [Janković et al., 2003].
3. PRINCIPLES OF THE ANALYTIC ELEMENT METHOD
[40] As stated above, the principle of the Analytic Element Method is to superimpose solutions that can be used to model features in the aquifer system. The method, however, need not be restricted to problems of groundwater flow but can be stated in general terms. The purpose, then, of the Analytic Element Method is to describe a general vector field by developing analytic expressions that can be superimposed. Each element is chosen or developed to simulate a certain characteristic of the vector field. Perhaps the most important characteristic of the Analytic Element Method is the freedom it offers in choosing the elements; they can be developed using conformal mapping techniques, Cauchy integrals, Fourier analysis, Laplace transforms, and separation of variables. The method makes possible, even encourages us, to find and use new ways to develop and combine suitable functions. This observation sounds obvious and is, in fact, a very simple principle. Nonetheless, the reader may observe from the material presented in this paper that it does, in fact, lead to analytic descriptions and solutions that were heretofore considered either impossible or impractical.
[41] We will formulate the Analytic Element Method for the case of a two-dimensional vector field for the sake of simplicity, following Strack [1999]. The principle extends to three-dimensional problems with the usual increase in level of difficulty. According to Helmholz's decomposition theorem each vector field may be represented by a combination of an irrotational field with nonzero divergence, a divergence-free rotational field, and a vector field that is both irrotational and divergence-free. The three fields that in combination meet the boundary conditions constitute the solution to the problem.







[43] For the case of groundwater flow the divergence is either associated with infiltration (given divergence) or with leakage (divergence unknown a priori) or with change in storage in transient models. The curl is nonzero for cases of variable hydraulic conductivity, aquifer base elevation, or aquifer thickness. The curl is also nonzero for cases of anisotropic hydraulic conductivity. In the theory of linear elasticity the curl and divergence may be expressed in such a way that biharmonic equations are obtained for the potential and stream functions. In this case the vector field is the displacement field, the divergence represents the volume strain, and the curl represents the infinitesimal rotation.
4. IRROTATIONAL FLOW WITH NONZERO DIVERGENCE (POTENTIAL FLOW)
4.1. Piecewise Uniform Infiltration
[44] Infiltration can be added to the solutions for regional groundwater flow problems by the use of area sinks as introduced by Strack [1989]. These area sinks simulate a uniform rate of extraction (or infiltration) in an area bounded by a polygon. The potential is determined such that it satisfies the Poisson equation inside the polygon, so that the divergence of the discharge vector equals the extraction rate inside the polygon and vanishes outside. The potential for an area sink bounded by an ellipse, useful for modeling infiltration through the bottom of elliptical ponds, will be presented at the end of this paper as an example.
4.2. Conjunctive Surface Water and Groundwater Flow
[45] Regional models based on the Analytic Element Method are particularly suitable to handle the interaction between groundwater and surface water. Infiltration (in this context the entry of water through the phreatic surface) is modeled using area sinks, and the streams are represented by line elements and thus can be easily constructed to follow the natural streambeds. These line elements are characterized by a jump in the stream function, which is linked directly with the amount of water captured by the stream. If a string of these line elements represents a stream, then the jump in the stream function at one end of this stream can be set to zero. The value of the jump in the stream function at any point of the stream then represents the amount of water collected by the stream from the end point to the point of consideration. It can be determined at run time whether the stream is in direct contact with the groundwater; if the piezometric head falls below the level of the streambed, then the stream is not in direct contact, otherwise it is. It is perhaps of interest to note that the jump in the stream function is meaningful even if the stream function itself is not, i.e., in cases of nonzero divergence of the discharge vector. This is true because the jump occurs over a line element of zero width, so that the infiltration does not affect the flow in between the two sides of the element over which the stream function jumps.
[46] Regional groundwater models are constructed by including all or most streams and lakes in the model area. These surface waters are boundaries of the groundwater flow domain; they either supply or withdraw water from the aquifer. Whether these surface waters are modeled as being in direct contact with the aquifer, or separated from it by a leaky bottom, a losing stream in the groundwater flow model may well supply more water than it has available as streamflow. This is particularly true for small tributaries and head waters of streams.
[47] Mitchell-Bruker and Haitjema [1996] proposed a conjunctive surface water and groundwater flow modeling approach that is designed to prevent overinfiltration of streams or lakes. The line sinks that represent streams and lakes are organized into networks in which base flow is calculated by accumulating all groundwater inflows and outflows. An overland inflow rate can be specified to arrive at a complete streamflow rate everywhere in the network under steady state conditions. During an iterative groundwater flow solution process the infiltration rates of line sinks that represent losing stream sections are limited to the available streamflow. The conjunctive surface water and groundwater flow solutions serve primarily to improve the reality of the hydrological boundaries to the groundwater flow regime. These solutions have also proven to be valuable in calibrating the groundwater flow model, using as targets both piezometric head data and streamflow data. Calibration of a model to observed heads alone provides insight in the ratio of aquifer recharge to the hydraulic conductivity; additional calibration to observed streamflow provides a second equation that makes it possible to determine these two parameters individually [Haitjema, 1995]. The formation of stream networks in an analytic element model (GFLOW) appears rather simple and intuitive, making it a practical tool for routine groundwater flow modeling [Haitjema, 1995].
4.3. Approximate Modeling of Leakage Using Area Elements of Variable Strength
[48] Leakage in multiaquifer systems may be modeled effectively using the exact solution for Dupuit-Forchheimer flow in multiaquifer systems, but this approach is restricted to constant properties of the leaky layer and aquifer. An alternative to this approach is to develop special analytic elements capable of simulating leakage through leaky layers and with the flexibility that they can be added to any other analytic elements and reduce to harmonic functions at some distance from the elements. The first attempt at developing such elements was using area elements of constant strength; that is, the divergence of the vector field was assumed to be piecewise constant. We will briefly discuss the use of area elements of variable strength to simulate leakage between aquifers in a multiaquifer setting.
[49] The leakage between aquifers through leaky separating layers is equal to the difference in head across the leaky layer, divided by the resistivity. Leakage is thus generated by a difference in head, which, in turn, can be caused by infiltration into the uppermost aquifer, and by features in the system that withdraw water, such as wells, streams, lakes, and rivers. The distribution of the leakage is highly dependent on the distribution of the aquifer properties and, in particular, on discontinuities in these properties.
[50] A logical manner of modeling the leakage in an aquifer system therefore is to create analytic elements that make it possible to model the effect of the features that influence leakage.
[51] To date, the approximate modeling of leakage in analytic element models was done by the use of area sinks. The area sink of constant strength, i.e., constant extraction rate, was already mentioned in this paper. Strack and Janković [1999] present the mathematical description of an area sink with variable strength; the strength of this area sink varies as a multiquadric interpolator.
[52] Both the area sink of constant strength and the area sink of variable strength have been used to simulate leakage in multiaquifer systems. Good results have been obtained with the area sinks of constant strength [de Lange, 1996a, 1996b]. The area sinks of variable strength are useful primarily for cases where the leakage is not concentrated along features but smoothly distributed over the area. These area elements have been shown to converge to the proper solution, given enough degrees of freedom [see Hansen, 2002], but they require a large number of degrees of freedom along with considerable computational effort. The primary reason for this drawback is that the elements do not focus on the causes of the leakage, i.e., the streams, wells, or inhomogeneity boundaries that affect the leakage directly. A new formulation of modeling leakage that focuses on the causes of leakage has been developed and implemented but has not been published.
5. LEAKAGE IN MULTIAQUIFER SYSTEMS: EXACT SOLUTION FOR DUPUIT-FORCHHEIMER FLOW
[53] Bakker and Strack [2003] developed an analytic element formulation for steady state flow in a multiaquifer system with leaky layers separating the aquifers. The Dupuit-Forchheimer approximation is adopted, and flow in the leaky layers is approximated as vertical. Groundwater flow in a system consisting of an arbitrary number (M) of aquifers is governed by a system of M differential equations. Polubarinova-Kochina [1962] presents a solution for the case of wells in a system of aquifers separated by leaky layers; the original work is attributed to Mjatiev [1947]. Hemker [1984] presented a general approach based on an eigenvalue analysis to decouple these M linked differential equations into M unlinked differential equations. For steady state flow in a confined aquifer system the decoupling results in one Laplace equation and M-1 modified Helmholtz equations. Maas [1986] demonstrated that, alternatively, a solution to the system of equations may be obtained with matrix calculus when the aquifer system is semiconfined, i.e., when the upper aquifer is bounded above by a leaky layer with a fixed water level above it. Bruggeman [1999] used matrix functions to derive exact solutions to over 50 problems of multiaquifer flow.
[54] Hemker [1984] presented a solution for the head distribution and flow to a pumping well in a multiaquifer system. This solution consists of a logarithm plus a sum of modified Bessel functions of the second kind and order zero. This solution may be integrated along a line to obtain an expression for flow to a line sink. The integral of the modified Bessel function is unknown, however. Nienhuis [1997] implemented a numerical integration technique. Heitzman [1977] and Keil [1982] used a polynomial approximation for the Bessel function and performed the integration analytically for flow in a three-aquifer system, but they used a polynomial approximation that is valid in a limited area only. Bakker and Strack [2003] improved this procedure by combining it with the theory of Hemker [1984] and by using a polynomial approximation that is accurate for an infinite domain; they also presented solutions for a circular area sink. Bakker [2003] developed an approach for the simulation of flow through many cylindrical inhomogeneities in a multiaquifer system using a separation of variables approach.
[55] An example is presented for flow in a confined system of two aquifers separated by a leaky layer. Two river segments are present in the upper aquifer, and there are two pumping wells. The well on the right-hand side is screened in the upper aquifer, and the well on the left-hand side is screened in the lower aquifer. The river segments are represented by line sinks with a constant strength. Infiltration through the upper boundary of the upper aquifer is represented by one large circular area sink. Contour plots of the head in the upper and lower aquifers are shown in Figure 5. A contour plot of the leakage between the two aquifers is also shown in Figure 5; the shaded area represents upward leakage.

6. TRANSIENT FLOW
[56] The applications of the Analytic Element Method to date have been primarily to problems of steady flow. This by no means implies, however, that transient modeling of groundwater flow is not feasible or advantageous. Transient modeling of flow in regional aquifers may be carried out for a confined system, where the storage is elastic, or for unconfined systems, where the storage is due primarily to vertical movement of the phreatic surface. The differential equation for the latter case is linearized [e.g., Strack, 1989].
[57] There are two kinds of transient analytic element models. Zaadnoordijk [1988] used a formulation based on the superposition of transient line doublets and line sinks that satisfy the heat equation. Zaadnoordijk and Strack [1993] extended this approach to include transient area sinks, useful for the modeling of transient infiltration, due to rainfall events, for example. An alternative is to solve the governing differential equation numerically in time, discretizing time using finite differences. This approach has the advantage that it can be combined with all existing analytic elements valid for steady flow; abrupt changes in the system, such as those caused by streams that become dry at some time (i.e., stop having an effect on the flow), can be taken into account. The latter approach was applied by Haitjema and Strack [1985].
7. THREE-DIMENSIONAL FLOW
[58] The Analytic Element Method has been applied to a variety of three-dimensional problems. As for the two-dimensional applications of analytic elements the basic functions may be obtained by integration, in this case integration of singularities along lines and over planes [Haitjema, 1982, 1985]. The elementary analytic elements, such as three-dimensional line sinks, were used in a variety of engineering applications prior to the development of the Analytic Element Method. The application of these elements, in particular their combination with two-dimensional elements in regional models, differs from that in other techniques such as boundary element methods and panel methods. As the Analytic Element Method matured, however, new functions were developed, for example, functions for modeling partially penetrating wells, high-order line elements for modeling horizontal drains, and functions for modeling inhomogeneities bounded by ellipsoids.
7.1. Line Elements and Panels
[59] Suitable three-dimensional analytic elements can be superimposed to approximate boundary conditions for three-dimensional potential flow, as is done, among other fields, in the aerospace industry for modeling flow around airfoils [Hess and Smith, 1967]. Three-dimensional analytic elements are useful for modeling a variety of features in groundwater flow, such as partially penetrating wells, horizontal wells, aquifer stratification, shallow rivers and lakes, and a phreatic surface. Haitjema and Kraemer [1988] developed an analytic element representation for a partially penetrating well using a line sink at the center of the well with a singular sink density distribution near the extremities. The square root singularity in the sink density yields a good approximation to a cylindrical equipotential around a line sink in three dimensions. Doublet discs (double layers) with radially varying doublet density have been used to model the transition in aquifer hydraulic conductivity between different geological layers penetrated by a well [Haitjema, 1987a].
[60] The three-dimensional analytic elements may be placed in a confined aquifer with parallel upper and lower impermeable boundaries. The no-flow conditions along these boundaries are maintained by the use of images, with the remote images replaced by an infinite continuous sink or dipole distribution [Bischoff, 1981; Haitjema, 1982, 1985]. The three-dimensional effects are local; their measurable effect extends to ∼2 times the aquifer thickness from a three-dimensional feature [Haitjema 1987b, 1995]. Three-dimensional analytic elements can be superimposed to two-dimensional ones for confined flow in a single model, combining the efficiency of two-dimensional flow on a regional scale, while providing fully three-dimensional flow where needed, such as near a partially penetrating well [Haitjema, 1982, 1985].
7.2. Closed Analytic Elements
[61] As in two dimensions, two-sided closed analytic elements may be constructed in three dimensions. One such three-dimensional analytic element applies to spheroidal inhomogeneities, i.e., inhomogeneities shaped as rotational ellipsoids. The first to construct an analytic element model with such three-dimensional elements was Fitts [1990]. Fitts used collocation to obtain approximate solutions for flow with a limited number of such inhomogeneities. This approach was generalized to a formulation with explicit expressions for the coefficients in the expansions and later was simplified using overspecification by Janković [1997] and Janković and Barnes [1999b]. The specific discharge potential due to each inhomogeneity is based on the general solution for Laplace's equation in terms of ellipsoidal coordinates, which was developed using the theory of ellipsoidal harmonics [e.g., Byerly, 1893; Hobson, 1931]. The coefficients in the solution are obtained analytically from the condition that the head and normal component of specific discharge are continuous across the boundary of each inclusion. The example in Figure 6 shows intersections of constant head surfaces and a vertical plane for a flow problem containing a group of infinitely conductive thin prolate inhomogeneities. These inhomogeneities are created by rotation of ellipses about their long axis and may be used to model three-dimensional flow and transport around an arbitrary number of linear cracks.

[62] Computer models capable of dealing with large numbers of inhomogeneities in two and three dimensions are extremely valuable as numerical laboratories for studying macroscopic dispersion in highly heterogenous porous media [see Janković, 2001]. The numerical laboratory used to generate the plots in Figure 4 is presently used by G. Dagan and I. Janković to estimate the effective hydraulic conductivity and the dispersion coefficients using the simplified dispersion model of Dagan [1989] (the self-consistent model).
7.3. Unconfined Three-Dimensional Flow
[63] Simply superimposing two- and three-dimensional analytic elements is prohibited under unconfined flow conditions by the presence of the phreatic surface. One approach to modeling a phreatic surface is to represent it by sink or dipole panels that are placed at the estimated position of the phreatic surface. The sink or dipole distributions and the location of the panels are adjusted iteratively until convergence of the solution is obtained; that is, all conditions are met [Lennon et al., 1980]. Luther and Haitjema [1999, 2000] used a function similar to that introduced for two-dimensional flow by Zhukovski [1949] and placed singularities outside the flow domain to satisfy the conditions along the phreatic surface.
[64] An extension of this approach for modeling three-dimensional groundwater flow with a free surface was introduced by Janković and Barnes [2000]. They used the Analytic Element Method to model local three-dimensional unconfined transient flow in the vicinity of partially penetrating wells. The flow domain is bounded by an impermeable horizontal base, a phreatic surface with recharge, and a cylindrical lateral boundary. The analytic element solution for this problem contains sources outside the flow domain to satisfy the boundary conditions along the phreatic surface: atmospheric pressure and a flux condition to match release from storage. Other sources are placed outside the domain in order to approximate the given head along the cylindrical boundary. The sources are imaged with respect to the lower boundary to satisfy the no-flow condition across the impermeable base. The solution contains, as the driving elements, the potential for a line sink of constant strength and spheroidal harmonics to account for inhomogeneities in hydraulic conductivity. The temporal effect is accounted for by centered finite differences in time; the finite difference formulation is used to compute the change in storage during each time step. Note that the aquifer is considered incompressible.
[65] The solution provides a precise description of local groundwater flow with an arbitrary number of wells of any orientation and an arbitrary number of ellipsoidal inhomogeneities of any size and conductivity. These inhomogeneities may be used to model local hydrogeological features, such as gravel packs and clay lenses, which affect the flow in the vicinity of partially penetrating wells. The example shown in Figure 7 includes two oblate inclusions 10 times less conductive than the background and a well in a steady state flow without recharge. Intersections of piezometric surfaces and a vertical plane and the phreatic surface are shown.

7.4. Stream Surfaces and Vector Potentials
[66] The Analytic Element Method makes it possible to generate computer models of very high accuracy. Such accurate models offer the reader an opportunity to investigate features of flow that were not well understood. Precise computation of fluxes over areas of nonuniform leakage and fluxes through three-dimensional surfaces requires special tools; existing tools using numerical integration do not do justice to the ability of the analytic element models to describe accurately highly variable flow fields as nearly all illustrations in this paper demonstrate. The development of such tools and their application may be considered as a specialized field inside the Analytic Element Method; a brief overview of this work will be given in what follows.
[67] The shapes of streamlines and stream surfaces in three dimensions was studied and described mathematically by Steward [1998] and Steward and Janković [2001]. Steward [1998] proved that a two-dimensional model cannot reproduce the rearrangement of neighboring streamlines that are generated by a three-dimensional flow. This provided the motivation for the development of three-dimensional models to understand and quantify contaminant transport. Steward and Janković [2001] examined the deformation of streamlines in axisymmetric flow where flow is symmetric about an axis of symmetry and streamlines lie in half planes containing this axis. This study proved that there may occur in axisymmetric flow a permanent rearrangement of the relative position between two neighboring streamlines in the direction transverse to the prevailing regional flow but only if a nonzero discharge is removed from the aquifer by a feature.
[68] The study, and accurate description with analytic elements, of the flow near horizontal wells is the main focus of Steward [1999] and Steward and Jin [2001]. The capture zones geometry was examined by Steward [1999] for fully penetrating, partially penetrating, and horizontal wells. Nomographs were developed to quantify the minimum pumping rate required to capture a contaminated leachate over typical ranges of aquifer parameters. It was shown that a horizontal well oriented perpendicularly to the direction of regional flow can capture a leachate with the minimum pumping rate. Steward and Jin [2001] examined the distribution of flux along a horizontal well. Wells with uniform head and low pumping rates may have a gaining section, along which water enters the well, and a losing section, along which water leaves the well. Such a well may provide a conduit where contaminated water could enter the well, travel a large distance within the well (wells can be placed with screen intervals up to 1.6 km long), and then be injected into an uncontaminated region of the aquifer. The minimum pumping rate required to eliminate a losing section was quantified using dimensionless ratios of aquifer parameters. These ratios were used to quantify the geometry of capture zones.
[69] Steward [2001, 2002] developed an analytic framework for the determination and use of vector potentials in three-dimensional flow. Steward [2001] introduced the vector potential in groundwater flow when developing the vector potential for a partially penetrating well in a horizontal aquifer. The net flux obtained via the vector potential was used to quantify and interpret approximations associated with the method of images. Steward [2002] developed a framework to determine a vector potential for any three-dimensional divergence-free flow that can be decomposed into two-dimensional and axisymmetric components. This vector potential is directly linked to the Lagrange and Stokes stream functions. Integrals were developed to provide the exact flux through a three-dimensional surface, and closed form expressions were obtained for flow to a partially penetrating well in a uniform flow field.
[70] The reader interested in using public domain analytic element code for three-dimensional flow can download the computer program 3DFlow from the web site http://groundwater.ce.ksu.edu, which models three-dimensional groundwater flow generated by fully penetrating, partially penetrating, and horizontal wells in a uniform flow field.
8. APPLICATIONS
[71] The Analytic Element Method has been applied to many problems in the field since 1976, from small-scale studies, often concerning wellhead protection, to large-scale models on a national or statewide scale. The inherent scale independence of the method makes it possible to deal effortlessly with local issues such as wellhead protection using a large-scale groundwater model. A variety of computer programs are available for modeling with analytic elements; besides the two public domain versions already mentioned, there exist commercial programs such as QUICKFLOW, Groundwater Flow (GFLOW), the Two-Dimensional Analytic Model (TWODAN), the Single-Layer Analytic Element Model (SLAEM), and MLAEM.
[72] The description of the large-scale regional models is beyond the scope of this paper; brief descriptions of three major applications will be presented along with the mention of capture zone analysis. The package WHAEM [Haitjema et al., 1995] was developed specifically for the United States Environmental Protection Agency for the purpose of wellhead protection and is used throughout the nation for wellhead protection analysis.
[73] An application that illustrates the ability of the Analytic Element Method to deal with very large domains is the modeling of the Yucca Mountain area. The reader is referred to Bakker et al. [1999] for a complete description of that modeling project. The modeling was carried out with the Multi-Layer Analytic Element Model (MLAEM).
[74] The largest and most comprehensive application of the Analytic Element Method is the Dutch National Groundwater Model (NAGROM) [de Lange, 1996b]. NAGROM has been created, used, and updated over more than 15 years. As an illustration of the use of NAGROM, a plot of fluxes and heads generated by NAGROM is shown in Figure 8. It is of interest to note that the plot of fluxes represents the inflow into the saturated zone to match an unsaturated flow model used in conjunction with NAGROM. The scale independence of NAGROM made it possible to match the far higher detail of the unsaturated flow model. NAGROM is based on MLAEM.

[75] The Twin Cities Metropolitan Groundwater Model [Seaberg et al., 1997] was constructed with MLAEM over many years and is currently used in the Twin Cities metropolitan area as a starting point for local studies (detailed information regarding this model can be found on the web site http://www.pca.state.mn.us/water/groundwater/metromodel.html. Again, a detailed description of the model is beyond the scope of this paper, but a single illustration will serve to show the flexibility of analytic elements. The plot in Figure 9 shows the extent of aquifers 5 and 4 in the Twin Cities Metropolitan Groundwater Model. It is interesting to note the very large scale of these aquifers. The reason is that aquifer 5, the Hinckley aquifer, which is the lowest aquifer in the Twin Cities metropolitan area basin, extends all the way to the town of Hinckley, which is roughly 110 km north of the Twin Cities. The analytic element model allowed such remote boundaries to be included, making it possible to generate realistic flow patterns in the lower aquifer, capable of reacting appropriately to a man-made change in conditions.

9. FUTURE DEVELOPMENTS
[76] Presently, there are two major thrusts in the development of the Analytic Element Method. One thrust is toward the development of relatively simple yet powerful analytic element models suitable for small-scale groundwater modeling. The work by Bakker and Strack [2003] is an example of such a model. Developments of transient models based on a similar premise are currently underway and are scheduled for implementation in the computer program Tim, referenced in this paper.
[77] A second thrust is toward developing powerful and efficient analytic elements suitable for large-scale regional modeling. Line elements have recently been developed for simulating leakage near linear elements that are a primary influence on the leakage distribution (e.g., streams and boundaries of inhomogeneities). These new analytic elements produce their own leakage distribution, and the leakage field is generated by the superposition of the effect of these line elements. At present, the leakage field is approximated as being harmonic in between linear and elliptical elements that control the leakage distribution. The same elements will be used for the modeling of transient flow; the transient effect will be simulated using a finite difference discretization in time. These new elements have been implemented in MLAEM.
10. SUMMARY AND CONCLUSIONS
[78] As stated above, the principle of the Analytic Element Method is to superimpose analytic elements that can be used to model features in a vector field, such as the discharge vector field in an aquifer system. Each element is chosen or developed to simulate a certain property of the vector field. Perhaps the most important characteristic of the Analytic Element Method is the freedom it offers in choosing the elements; they can be developed using conformal mapping techniques, Cauchy integrals, Fourier analysis, and Laplace transforms, for example. The method makes possible, even encourages us, to find and use new ways to develop and combine suitable functions. This observation sounds obvious and is, in fact, a very simple principle. Nonetheless, it is clear from the material presented in this paper that it does, in fact, lead to analytic descriptions and solutions that were heretofore considered either impossible or impractical.
[79] A major drawback of the Analytic Element Method is the high degree of effort required to develop elements and to implement these in computer codes. The movement toward object-oriented code development is aimed at decreasing effort in development and is designed to take advantage of the portability of computer libraries of mathematical functions.
[80] There is no fundamental limitation to the applicability of the Analytic Element Method; the approach may be more advantageous in some cases than in others. The Analytic Element Method offers clear advantages over other numerical methods; the analytic formulation allows for scale independence, flexibility, and a high degree of accuracy. Using analytic elements constructed such that they reduce to harmonic functions outside a certain chosen boundary around the element makes it possible to apply the superblock approach, which makes it possible to deal with very large problems at unprecedented efficiency. The combination of the superblock approach with accurate and efficient elements makes it possible to create groundwater flow models of very large scale that may function both as an intelligent database and as a starting point for a variety of investigations.
[81] Although there is no fundamental limitation to the applicability of the Analytic Element Method, further development is necessary to realize the potential of the method in practice. Examples of such developments are the efficient modeling of leakage in multiaquifer systems in general settings and the efficient modeling of transient flow. In these respects the method lags behind in popular implementations of discrete numerical methods such as the finite difference and finite element methods.
Acknowledgments
[82] I thank R. Barnes, M. Bakker, H. M. Haitjema, D. R. Steward, and I. Jankovi for providing information regarding their work. I am grateful to R. Barnes, M. Bakker, and I. Janković for carefully reading the manuscript; their constructive critical comments and useful suggestions were indispensable to me in preparing this manuscript. I also acknowledge that I. Janković created Figures 1, 2, 4, 5, and 6, M. Bakker created Figure 7, W. de Lange created Figure 8, and J. Seaberg created Figure 9. I am indebted to A. Kacimov for providing me with valuable references in the Russian literature.
[83] Daniel Tartakovsky is the Editor responsible for this paper. He thanks technical reviewer A. R. Kacimov and an anonymous technical reviewer and one anonymous cross-disciplinary reviewer.