Monday, March 2, 2020

Interview with Dr. Charles Fitts Regarding the Analytic Element Method and AnAqSim Groundwater Modeling Software

You can probably tell from our flexAEM website and blog that we like AnAqSim. So we wanted to find out a little more about its author, Dr. Charles Fitts, how he became involved in Analytic Element Method (AEM) groundwater modeling, and how AnAqSim was developed to have the features it has. We sent Dr. Fitts some questions and he has graciously responded. You can read the “interview” below.

We were fascinated to learn about how Dr. Fitts became interested in learning, coding, and applying AEM models. But we are especially grateful for the in-depth responses that he provided on some of AnAqSim’s features, complete with references for those who want to explore this topic further. We want to offer our thanks to Dr. Fitts, and we hope you enjoy his responses below.

Q: How did you first become interested in groundwater flow, mathematical modeling, and the AEM?

A: I always liked mathematics, maps, water, and geology. As a teen, my parents called me “Chuck Atlas” because I spent so much time perusing the Rand McNally atlas. As an undergrad I majored in Geology/Biology at Colby and then I studied engineering geology and soil mechanics at Cornell for an MS. Then in the early 1980s I worked for four years doing geotechnical consulting in the Boston area. During those years I worked on many projects with seepage and groundwater issues. I took a semester off work to take engineering classes at U. Minnesota (my home state) to prepare for PE exams, and one of the courses I took was Groundwater Mechanics with Dr. Otto Strack, the founder of the analytic element method (AEM). He is a gifted teacher and I really enjoyed learning analytic techniques and applying them to groundwater flow problems. I decided to pursue more graduate study under Otto, and eventually got a PhD in civil engineering. My research there focused on AEM methods for vertical plane flow, 2D heterogeneities, and 3D heterogeneities. I was drawn to the AEM because of the beauty of the mathematics and the promise of the method with its high accuracy, minimal inputs, and geometric flexibility.

Q: What led you to develop AnAqSim with an approach that is quite different from previous AEM programs?

A: For my research at U. Minnesota, I wrote Fortran programs which I kept improving and using in my consulting work after leaving school. Ultimately this code became TWODAN (two-dimensional analytic groundwater model), which I distributed in the 1990s and 2000s. TWODAN was a good tool with a decent interface, but like most AEM programs at that time, it was limited to a single layer (2D) and had almost no transient capabilities (merely adding Theis solutions to the steady solution). In the 2000s, the AEM declined in popularity while numerical methods like FDM (MODFLOW) and FEM were on the rise. This trend was probably due to the robust multilayer (3D) and transient capabilities of numerical models, whereas AEM programs were generally limited to 2D steady flow. Rather than keep up with multiple modeling techniques and programs, many modelers just used one numerical modeling program. I felt that the key to growing AEM use was to improve its 3D and transient capabilities.

I saw a significant break when Strack and Jankovic (1999) published their paper on solutions for spatially variable area sinks (SVAS). These are 2D functions with a sink term whose strength (rate) varies with position in the plane using radial basis functions. SVAS functions create a smooth, irregular surface of sink strength that passes through any number of basis points where the strength is set. The radial basis functions used by Strack and Jankovic (1999) were first employed by Hardy (1971) to interpolate smooth topographic surfaces from point elevation data. Strack and Jankovic demonstrated the SVAS functions in a model simulating leakage from a single-layer aquifer to an overlying surface water using relatively few basis points. I saw that these SVAS functions could be used in a different way with larger numbers of basis points to dramatically improve multi-layer and transient modeling with the AEM. The distributions of vertical leakage between layers in multi-layer models and the storage fluxes in transient models may be combined and viewed as spatially variable distributed sinks. The functions presented by Strack and Jankovic could be used to approximate the required sink distribution.  

On another front, I worked on methods for modeling anisotropy in the plane of the domain, using some old (Muskat, 1937) and new coordinate transformations (Fitts, 2006). At this time, I also worked on a subdomain approach in AEM where you break up a model layer into separate finite subdomains instead of one infinite domain (Fitts, 2004; Fitts, 2010). Each subdomain has its own mathematical model and can have its own direction and ratio of anisotropy, making for very robust anisotropy capabilities. A subdomain approach has been used with boundary element methods, which are like AEM but use numerical integrations along boundary elements instead of analytic solutions (Liggett and Liu, 1983; Bruch and Lejeune, 1989). The subdomain approach has other nice side-effects: (1) it makes the equations for potential or discharge shorter, (2) model layering can vary from one region of a model to another, and (3) heterogeneity boundaries can be a mix of different boundary types such as specified head, specified normal flux, and interdomain.

It was exciting to see how all these pieces could come together. I began writing code that used subdomains, SVAS, and coordinate transformations to make AEM models that had multi-layer and general anisotropy capabilities. I also incorporated high-order line elements which make it easier to tune accuracy and use longer line segments (Jankovic and Barnes, 1999). I later added storage terms and transience using the same SVAS functions as are used for vertical leakage between layers. The result is a program that has the accuracy, minimal inputs and other benefits of AEM, but with comprehensive capabilities for anisotropy, multi-layer systems, and transient flow.

The vertical leakages are computed using finite difference approximations of dh/dz and storage fluxes are computed using finite difference approximations of dh/dt. The flow models are based on the AEM while these derivatives are approximated with finite differences. Such approximations are employed by MODFLOW and other numerical modeling programs as well and afford a wide range of application to both AnAqSim and numerical models.

Q: What are the main differences between AnAqSim’s methods and methods used in other AEM modeling codes?

A:  This topic has many facets, and for a quick overview you may want to see this web page comparing AEM program capabilities:  I will discuss the major differences below.

Most AEM programs have elements that are all internal to a single infinite domain in each layer. In contrast, AnAqSim uses finite subdomains that extend laterally only as far as their polygonal external line boundaries. Each subdomain in AnAqSim has its own mathematical model, whereas the infinite domain approach has a single model per layer. This means the equations in AnAqSim are shorter (only elements within or on the boundary of a subdomain contribute to the potential or discharge functions of that subdomain), which leads to computational savings while solving the system of equations and while generating graphic outputs like contours and pathlines. Subdomains also allow complex layering schemes where, for example, three subdomains (layers) on one side of an interdomain boundary abut a single subdomain (layer) on the other. This lets you easily have more layers in the areas of interest and fewer layers farther out. No other AEM program has this feature.

Other AEM programs typically specify a reference head at a point outside the area of interest, which determines the flow to or from infinity. Since AnAqSim uses finite subdomains, this is not needed. The external boundary conditions in AnAqSim’s bounded models are more clearly defined than the infinity boundary condition generated by the reference head in infinite-domain AEM models. I know from my years of supporting TWODAN that the reference head was a source of confusion.

Most AEM programs are single layer (2D), but AnAqSim, MLAEM, and TimML all offer multi-level aquifer simulation (3D). TimML (and MLU, which is pumping test analysis software based on the same approach) is a multi-level simulator that provides exact distributions of the sink terms simulating vertical leakage and storage, whereas AnAqSim approximates these distributions between basis points. TimML has some limitations at this point in its development. For example all domains must be fixed transmissivity (confined), layers must be infinite, fewer kinds of line and area element boundary conditions are possible, and it lacks a graphic user interface. MLAEM appears to allow simulation of leakage in multi-level aquifer systems but after considerable online searching, I could not find documentation of how that is accomplished with area sinks, and the models are only for steady-state flow (not transient).

AnAqSim has comprehensive transient capabilities meaning that just about all key inputs (discharges, heads, recharge rates, normal fluxes, river stages) can be transient, varying from one time period to the next. Head-specified wells and line boundaries can have transient heads and they can turn off during specified periods. This on/off feature is handy for dewatering simulations. TTim is a transient AEM program like TimML that can also do transient simulations with transient discharges and heads, but TTim is subject to the same kinds of limitations listed above for TimML.

AnAqSim has general anisotropy capabilities, allowing unique directions and ratios in each subdomain. There are no other AEM programs that allow anisotropy in the plane of domains.

Q: Describe the process of creating and updating AnAqSim.  What were (are) some of the big hurdles?

A: AnAqSim is written in C#, an object-oriented language with many nice built-in components supplied by Microsoft. In the 2000s I switched from FORTRAN to C# because AnAqSim’s subdomains and transient time-stepping are complex and require a lot of bookkeeping that C#’s classes and objects are good at. I chose C# because it had the right language features, Microsoft is likely to be around for a while, it comes with a great development environment (Visual Studio), and there were good libraries for math and graphics. AnAqSim has tens of thousands of lines of code, so I had to build it incrementally and slowly from the ground up. I started with fundamental classes to represent spots (x, y, level, subdomain, potential and discharge functions), subdomains (elevations, properties, coordinate transformations, conversions between head and potential), and elements (wells, line elements, area sinks). I tested the daylights out of these fundamental classes before building higher-level classes (e.g. boundary conditions, equation-building) that use these lower-level classes.

It’s important to test the code thoroughly every step of the way and whenever anything is modified or added. Bugs are a fact of life with complex software and it takes time to root them out. Fortunately, AnAqSim has now been in use for almost a decade with many hundreds of users, so it is mature in the sense that serious bugs have been found and cured. I built lots of analysis features into AnAqSim, in part because the outputs are useful to users, but also because it makes thorough testing of the code much easier for me.

The biggest hurdle was the sheer magnitude of the project. It is a lot of work to write the computational parts of the code, the user interface, the documentation, the training tools, and the web site. I like the challenge of big projects and at this point I have invested probably a decade of work on this. I use bought components for the plotting, graphics, and math (complex variables, solvers), and these have saved time and made it possible. A difficult, but unseen part is creating an installation package that a user runs to install the software. This is a whole field unto itself, with specialized software that creates the install file and numerous arcane considerations such as .net frameworks, C++ libraries, dynamic link libraries, digital signatures, target platforms, etc.

Q: How has your use of groundwater modeling software evolved over time? 

A: In the 1990s I used TWODAN for 2-D steady flow analyses and MODFLOW for anything that was 3D and/or transient. I also used Aqtesolv, which is good software for analyzing pumping tests using various radial single-layer analytic solutions. I still use Aqtesolv for some pumping test analyses, but it has been a few years since I’ve used MODFLOW. I find that I can reasonably model most any situation with AnAqSim. I have been using AnAqSim often to analyze pumping tests in a more sophisticated way when one or more of these conditions are present: partial penetration of pumping or observation wells, heterogeneity, anisotropy, or irregular boundary conditions. With AnAqSim’s hydrograph features, it is easy to plot simulated vs. observed heads at all observation wells and tweak aquifer parameters to get a good match of those hydrographs.


Q: What types of groundwater projects have you or others applied AnAqSim to?

A: AnAqSim has been used on many projects for construction or mining dewatering simulations, which appears to be the most common application. Dewatering simulations have involved wells, sumps, trenches, sheet-pile barriers, and recharge trenches. The ability to turn specified head wells and line boundaries off is handy for dewatering projects that proceed in stages (e.g. an expanding and deepening excavation). I have also done transient analysis of the impact of constructed ponds on local groundwater flows and heads.

Another common application is capture zone analysis for remediation design and for water supply zone of contribution studies. These often involve multilayer models in the area of interest to simulate partially penetrating wells, drains, and surface waters; and they can be performed as part of groundwater protection zone delineation or groundwater remediation projects. AnAqSim’s area pathlines and capture constrain features give it unique capabilities for capture zone analysis. 

Several of my projects have involved simulating vertical plane flow. One was analyzing discharge to a ship channel with bulkheads and heterogeneous subsurface conditions, and another was looking at patterns of discharge from a sand and gravel aquifer to a lake. The anisotropy features are helpful with vertical plane models, where stratification typically makes subdomains anisotropic.

I often use AnAqSim to analyze pumping tests with partially penetrating wells, heterogeneous and layered systems, and surface water connections near the tested well. An analysis done this way can be accomplished in just a few hours. You get a much more sophisticated and realistic analysis compared to the typical pump test solution for a single layer, no boundaries, and radial symmetry. 

I’ve analyzed fresh/salt interface aquifers a couple of times and I know of other consultants who have done many projects analyzing interface positions in sandy coastal aquifers.

Q: What materials are available to help users learn to use AnAqSim?

A: The more capable a modeling software is, the more there is to learn. Since AnAqSim can do so many things, there is a lot to learn. So, I have worked hard to provide detailed documentation and training for users.  There is a 120+ page user guide that is available online and with the software, which explains all inputs and outputs. There are also instructional videos and tutorials at this web page, which are a good way to get up and running: McLane Environmental’s flexAEM training materials are well-designed and are a great way to learn AEM modeling and AnAqSim in particular.

The website has some other useful learning resources, including a detailed explanation of a transient construction dewatering model, a page explaining how to use AnAqSim with PEST (parameter estimation software), 19 different checks of AnAqSim complete with model files, and a reference list of helpful books and journal articles. 

Recently, we have been offering a six-session webinar series called AnAqSim in Depth at least once a year. The series aims to help a user learn how AnAqSim works and how to apply it. Each session is one hour and there is one session per week. This gives time for optional home exercises each week that walk you through building or modifying AnAqSim models that use the content of that week’s session. The next AnAqSim in Depth series starts March 10, 2020 and it is described at this web page:

Q: What are some useful, but possibly obscure, modeling techniques that you could share with AnAqSim users?

A: My first tips are about analyzing pumping tests. I often do this using multi-level models with an initial condition of head = zero everywhere, and just simulate the deviation from zero with the transient model, like is done in typical pumping test analysis. To do this, set the average head in all domains to zero, and have your elevation datum be the static water level if you have unconfined domains at the top. I select the constant head option for Starting Heads in the solve settings, which uses a domain’s average head as the starting head. I typically use Head-Specified External with Gradient line boundaries for the far-field boundary condition in all levels of the model, and I set the head and gradient at these to zero. Place these external boundaries far enough out so the drawdown doesn’t reach there by the end of the test. If there are constant head boundaries such as a surface water nearby, simulate them with a specified head = 0, so there is no drawdown there. These sorts of models set up quickly in an hour or so. I calibrate them manually using the Analysis/Hydrograph features in AnAqSim, but for PEST enthusiasts calibration can be automated that way.

Another feature that few people seem to use are the SVAS top/bottom condition surfaces. Most people apply SVAS top conditions of a constant recharge rate or sometimes of a constant head for leakage to an overlying surface water. But it is possible to use these surfaces to model recharge that varies with location, or leakage to an overlying surface water with a sloping surface (e.g. a wetland or stream). Likewise, at the bottom of a model, it is most common to specify zero flux (impermeable) everywhere. If there is leakage to/from a deeper layer that is not in your model, you can use a surface to represent the heads in that deeper layer below your lowest model layer, and simulate leakage to/from that deeper layer using a head-dependent flux for the bottom condition.

Q: Any closing remarks?

A: Thank you for the opportunity to do this Q&A. I want to encourage feedback from users, who help me track down bugs but also come up with suggestions for capabilities they’d like to see added. I keep a database “wish list” of these and chip away at them over time.


Bruch, E., and A. Lejeune (1989), An effective solution of the numerical problems at multi‐domain points for anisotropic Laplace problems, in
Advances in Boundary Elements, vol. 2, Field and Flow Solutions, edited by C. A. Brebbia and J. J. Connor, pp. 3–14, Comput. Mech., Boston.

Fitts, C.R. (2010), Modeling Aquifer Systems with Analytic Elements and Subdomains, Water Resources Research, 46, W07521, doi:10.1029/2009WR008331.

Fitts, C.R. (2006), Exact Solution for Two-Dimensional Flow to a Well in an Anisotropic Domain, Groundwater, 44(1), 99-101.

Fitts, C.R. (2004) Discrete Analytic Domains: A New Technique for Groundwater Flow Modeling in Layered, Anisotropic, and Heterogeneous Aquifer Systems, American Geophysical Union Fall Meeting, San Francisco, CA.

Hardy, R.L. (1971), Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research 76, 1905–

Jankovic, I., and R. Barnes (1999), High‐order line elements in modeling two‐dimensional groundwater flow, Journal of Hydrology, 226, 211–223.

Liggett, J. A., and P. L‐F. Liu (1983), The Boundary Integral Equation Method for Porous Media Flow, Allen and Unwin, London.

Muskat, M. (1937) The Flow of Homogeneous Fluids through Porous Media. New York: McGraw-Hill.