8. Analytic element modeling of steady two-dimensional flow

Pumping wells form an important part of the two-dimensional solutions that have been discussed so far. The principle of superposition allows for the solution of problems with an arbitrary number of wells and the method of images allows for the simulation of straight specified-head and no-flow boundaries. The organization of solutions with multiple wells becomes cumbersome when the number of wells grows. The analytic element method is a modeling technique that facilitates the superposition of many solutions, including wells.
     The analytic element method is based on the superposition of analytic solutions (see, e.g., Strack, 2003). Each analytic solution represents a feature in the aquifer and has at least one free parameter. For example, the solution for a pumping well is an analytic element and the free parameter is the discharge of the well. The free parameter of an analytic element may either be specified or computed from specified conditions. For example, the discharge of a well may be computed from the head specified at a point, referred to as the collocation point. All unknown free parameters in a model must be computed simultaneously as, e.g., the discharge of one well may affect the head at another well and vice versa.
     In this chapter, the principles of the analytic element method are discussed. Python code for the analytic element method is developed using an object-oriented design based on Bakker and Kelson (2009). The analytic element program includes uniform flow, wells, line-sinks for the modeling of river segments, and area-sinks for the modeling of areal recharge. The formulation presented here is entirely based on real variables. A comparable formulation in terms of complex variables and complex potentials can be found in Bakker and Post (2022).
     Many advanced analytic elements exist for the simulation of steady single-layer flow, e.g., for the simulation of impermeable boundaries or inhomogeneity boundaries (e.g., Haitjema, 1995; Strack, 2017). In addition, formulations exist for both steady and transient multi-layer flow (Bakker and Strack, 2003; Fitts, 2010; Bakker, 2013). Computer programs for analytic element modeling of groundwater flow include Gflow (Haitjema, 1995) for steady single layer flow, and Timflow (Bakker and Brakenhoff, 2026) and AnAqSim (Fitts, 2021) for steady and transient multi-aquifer flow.