## Analytic Element Method

The analytic element method (AEM) makes use of computers and the principle of superposition for linear partial differential equations. Most groundwater textbooks cover superposing a few well functions by hand to simulate a well near a linear boundary, in what is called the “method of images”. With the AEM, computers superpose large numbers of functions (thousands or more). The concept of superposition of analytic solutions is an old one, but using computers to superpose large numbers of more sophisticated solutions in groundwater flow models was an idea pioneered by Otto Strack. Readers interested in more thorough coverage of the theory of AEM will find it in textbooks by Strack (1989) and Haitjema (1995). A shorter summary of the AEM is covered in the general groundwater textbook (Fitts, 2012).

An *analytic element* is a mathematical function that is associated with a particular boundary condition within the flow domain. The functions are solutions to the governing flow equations. One kind of element represents flow to a pumping well, a second kind represents discharge to a stream segment, a third kind represents an area of recharge, and so on. The mathematical functions of all elements in a model are superposed (summed) to create a composite equation that may have as many as thousands of terms.

Unknown parameters are associated with most elements. For example, a head-specified well has one unknown parameter: the well discharge. Boundary condition equations are written for each unknown parameter, creating a system of simultaneous equations that is solved using standard solvers. In cases with non-linear equations, iteration is required to solve the system. The solved system results in composite equations for potential (head) and discharge that are functions of location and time; *h(x,y,t)* and *q(x,y,t)* . These composite equations are used to make head contour plots, trace pathlines, etc.

Many computer programs have been written, implementing the analytic element method. They differ in many ways which can be baffling to those less familiar with the AEM. The following table comparing AEM modeling programs was assembled to help sift through the options.

## AEM Software Compared

**Definitions:**

**Comprehensive transient flow:**Most things that can be modeled in steady state can also be modeled in transient simulations. Important parameters such as specified heads, discharges, and recharge rates can vary with time during the simulation.**Multi-level models:**Simulation of multiple levels with vertical leakage and resistance between levels, as opposed to single-layer (2D) simulation.**Layer scheme transitions:**Different layering schemes in different regions of the model. For example, a single layer in one area and four layers in the area of interest.**Horizontal anisotropy:**K anisotropy in the plane of the layer, and the angle and ratio can be heterogeneous within a layer.**Fresh/salt interface aquifers:**Aquifers with a sharp fresh-salt interface using the Ghyben-Herzberg equation.**Multi-level wells:**Wells that draw from multiple layers in a multi-layer model, but have a consistent head.**Uniform area sinks – circular:**Uniform rate of recharge applied within a circular area.**Uniform area sinks – irregular polygon:**Uniform rate of recharge applied within a polygon area.**Spatially-variable leakage/storage:**Smoothly-varying recharge, leakage, (or storage in transient models).**Top/bottom head-dependent leakage:**Recharge or leakage in/out the top or bottom of the model based on head difference from a specified head.**Heterogeneity (interdomain) line elements:**Line elements across which the aquifer definition changes.**Circular/elliptical heterogeneities:**Same as above but for circle and/or ellipse-shaped boundaries.**Head-specified line elements:**Elements that extract at a rate to meet specified head condition(s) on the line.**Normal flux-specified line elements:**Elements that create a specified discharge normal to the line (e.g. impermeable).**Head-dependent discharge (river) line elements:**Internal elements where the discharge per length is proportional to the difference between a specified stage and the underlying aquifer head, and proportional to a specified resistance.**Head-dependent normal discharge line elements:**Like the river elements described above, but for an external lateral boundary. Like the GHB boundary in MODFLOW.**Discharge-specified line elements:**Internal elements where the discharge/length or total discharge is specified.**Conductive drain/fracture line elements:**Internal elements where the flux carrierd by the line is proportional to a specified conductance and the head gradient along the line.**Resistant barrier line elements:**Internal elements where the flux normal to the line is proportional to a specified conductance and the head difference across the line.**High order line elements:**Each line segment may have at least 3 degrees of freedom (parameters), the number of which can be adjusted to improve accuracy.**Conjunctive surface water – groundwater:**Solutions that also solve for river or lake stage, based on leakage rates between the surface water and aquifer.**Graphic user interface:**User interface with integrated menus, data entry forms, graphics, etc. Programs without a GUI are run from text commands.**Detailed tools to analyze solution:**Solution may be examined in detail to check accuracy of solution along boundaries, discharges along elements, hydrographs, head profiles, and more.

**Software Notes:**

- Visual AEM is a user interface for several AEM computational engines, including Bluebird, Cardinal, Split, and dated versions of TimML. The current version of TimML is not supported in Visual AEM, so TimML capabilities are listed separately.
- Winflow is the AEM portion of Aquifer Win32, which is a user interface for several programs.

**Sources other than AnAqSim (accessed July 24-26, 2016):**

- SLAEM/MLAEM: http://www.strackconsulting.com/aem-products/
- GFLOW: http://www.haitjema.com/documents/OverviewofGFLOWfeatures.pdf
- TimML: https://github.com/mbakker7/timml/blob/master/docs/timml.pdf
- TTIM: https://github.com/mbakker7/ttim/blob/master/docs/ttimman.pdf
- Visual AEM: http://www.civil.uwaterloo.ca/jrcraig/visualaem/About.html
- ModAEM: http://beta.xmswiki.com/wiki/GMS:MODAEM, http://gmsdocs.aquaveo.com.s3.amazonaws.com/GMS_8.3_User_Manual.pdf
- WhAEM: https://www.epa.gov/sites/production/files/documents/User_Manual.PDF
- WinFlow: Demo download with documentation available at https://www.rockware.com/product/overview.php?id=71