Friday, March 29, 2019

Groundwater Modeling with AEM – A Look Back, and a Look Ahead to the Future

The analytic element method (AEM) approach to groundwater modeling is efficient, powerful, easy to implement, fast to run, and provides the analyst a means for gaining insight at various steps of the modeling process that can guide data collection and subsequent model development. It is also a method that is apparently underutilized by groundwater modelers today. (We say apparently because there is some hope that modelers employ the method more than they report it in their final project documentation.)

This article will briefly describe the AEM groundwater modeling approach and how it evolved to its present state.

Definition of AEM groundwater modeling method
The Analytic Element Method is often described as being based on (1) mathematical solutions for the groundwater flow equation written in terms of discharge potential, and (2) the principle of superposition.

What that means in practical terms is that the flow equation is not written in terms of hydraulic head; it is written and solved in terms of discharge potential, which for a constant thickness aquifer is essentially head multiplied by hydraulic conductivity and aquifer thickness (Brikowski 2013). This allows the AEM, by superposition (adding of individual solutions), to be applicable to both confined and unconfined flow conditions as well as to aquifers that include heterogeneities (Strack and Haitjema 1981).

A mathematical solution, in terms of strength coefficients, is generated for the space around each element in the model; with each element representing a hydraulic or hydrologic feature - - a well, infiltration basin, or river reach for example. Then all of the element solutions are added at each point in model space to develop a 2D or 3D picture of the discharge potential field, and hydraulic head field. That field can be contoured to produce a head distribution map, or used as a basis for pathline tracing, or processed to calculate the groundwater flux though a boundary or user-defined transect.

The AEM is newer than FD and FE
Several authors have pointed out that the AEM is newer than finite-difference (FD) and finite-element (FE) methods. While that is technically true - - FD methods late 1960s; FE methods early 1970s; AEM about 10 years later in the late 1970s / early 1980s - - AEM is hardly the “new kid on the block.” It has been developed, applied and tested for almost 40 years now. And AEM was originally developed to address the limitations of prior FD and FE techniques. (To read the story of how AEM’s inventor Dr. Otto Strack first applied AEM on a large US Army Corps of Engineers project see Deming 2003. For more information on the history, applications, and strengths of AEM in general see Hunt 2006.)

While computer power has, in recent years, made up for some of the inefficiencies of FD and FE groundwater flow modeling methods, the continued development of AEM over the past four decades provide it with a number of advantages that still make it the method of choice in many situations. That development, and those advantages, are discussed in the following sections.

Brief history and progress in the field of AEM method of groundwater modeling
As mentioned above, the AEM groundwater modeling method that was originally developed in the Netherlands came to be first used in the United States by Dr. Otto Strack on a large-scale USACE dewatering project. Dr. Strack was assisted on the project by Henk Haitjema. Together these two PhD hydraulicians and mathematicians would go on to expand the capabilities of the method and make it available to groundwater modelers in the form of widely used AEM software packages. They also trained dozens of students (see figure below) who rose to prominence in the AEM field and developed their own methods and software.

    (from Kraemer 2007)

Some examples of the more popular and widely used AEM groundwater modeling software include:
Table 1. AEM Groundwater Modeling Software
Software / Solver
Dr. Otto Strack
Dr. Henk Haitjema
Dr. Igor Jankovic
Dr. Mark Bakker
Dr. James Craig
Dr. Charles Fitts

Advantages of AEM
The Analytic Element Method in general, and all of the AEM codes described above (and others not described here), allow the modeler to develop groundwater flow models with the following advantages:
  1. Easy and accurate setup in representing hydrologic features and boundaries
    • As simple as drawing points and lines on the screen; works well with GIS representation of hydrologic features.
    • Accurate, fine-scale, representation of model boundaries, river features, etc. There is no need to construct million-grid-cell models to represent curved watershed boundaries or sinuous river channels; and thus there is no corresponding computational penalty.
  2. Faster more accurate simulations
    • Fewer equations in simple to moderately complex AEM models mean faster run times.
    • Continuity of flow within the model domain, and between model subdomains is assured by virtue of the underlying solution techniques.
  3. The AEM model is capable of providing a wide “range of view” with accurate results one foot from an infiltration basin or at the far reaches of the watershed. Heads and velocities can be calculated at any location within the model space.
  4. Model post-processing for pathline tracing, flux through transects, and zone budgets can be accomplished accurately and efficiently using the calculate discharge potential generated by the AEM model.
  5. The AEM groundwater modeling method provides the perfect tool for application of the Stepwise Modeling approach recommended by many investigators and organizations.
    • When starting simple with a finite difference model you often start with a coarse grid. This can cause problems in representing model features and inaccuracy in the results.
    • AEM moves smoothly from a “back of the envelope” or analytic solution in easily implemented steps (no grid to refine or rearrange) to a fairly complex model; with insight gained at each step in the process.
The Future
Significant advancements in the field of AEM modeling of groundwater flow have been in progress over the past several years, and are likely to continue into the future. New, more flexible, elements are being developed (e.g. Ranjram and Craig 2018), and new solver methods are being implemented to address difficult modeling problems such as the instabilities that can occur when severely dewatering multilayer aquifer models (e.g. Fitts 2018). There are developments underway that will expand analytic element flow modeling into the area of contaminant fate and transport simulation (e.g. Majumder and Eldho 2019), and research into rapidly and efficiently solving large complex AEM models in the cloud (Fullerton 2017).

As Steve Kraemer (2007) noted, “There is a small but active community of ground water modelers who use an innovative solution technique known as the analytic element method.” That method has evolved over the years, and its use and publications describing its use continue to grow.

For the reasons outlined in this brief summary, AEM presents an ease of use, flexibility and power that make it ideally suited for a wide variety of groundwater modeling applications; and the perfect tool for Stepwise Modeling. For more on that subject see the National Groundwater Association White Paper on Groundwater Modeling ( -  free to NGWA members). In addition, we are planning an article on AEM and Stepwise Modeling in an upcoming flexAEM blog post.

At McLane Environmental, we have been using a variety of AEM groundwater modeling tools since the early 1990s, and have applied them successfully in dozens of hydrogeologic investigations. To hopefully foster the wider use of AEM techniques in the modeling community, we have published a series of AEM modeling exercises, and model development software tools, centered around AnAqSim modeling software.

These exercises demonstrate the usefulness and accuracy of results generated by AEM models. While focused on AnAqSim, the concepts and tools are applicable to other AEM software and to groundwater modeling in general. For more information please visit the flexAEM website (

Brikowski, T. 2013. Introduction to the analytic element method (GEOS-5311 Lecture Notes). 23 p.
Deming, D. 2003. Autobiographical sketch of Otto D.L. Strack. Ground Water Vol. 41, No. 4:550 – 554.
Fitts, C.R. 2018. Modeling dewatering domains in multilayer analytic element models. Ground Water Vol. 56, No. 4:557 – 561.
Fullerton, J.B. 2017. Cloud-based analytic element groundwater modeling. Masters Thesis, Brigham Young University, 54 p.
Hunt, R.J. 2006. Ground water modeling applications using the Analytic Element Method. Ground Water Vol. 44, No. 1:5 – 15.
Kraemer, S.R. 2007. Analytic element ground water modeling as a research program (1980 to 2006). Ground Water Vol. 45, No. 4:402 – 408.
Majumder, P. and T.I. Eldho 2019. Reactive contaminant transport simulation using analytic element method, random walk particle tracking and kernel density estimator. Jour. Contam. Hydrol. Vol 222, p. 76-88.
Ranjram, M. and J.R. Craig 2018. Closed analytic elements with flexible geometry. Ground Water Vol. 56, No. 5:816 – 822.    
Strack, O.D.L and H.M. Haitjema 1981. Modeling double aquifer flow using a comprehensive potential and distributed singularities. 2. Solution for inhomogeneous permeabilities. Water Resources REsearch, Vol. 17, No. 5:1551-1560.

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